Tangent and Normal Lines - Basic Exercises
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Tangent and Normal Lines – Exercises with a Constant
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Tangent and Normal Lines of Implicit Functions
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Tangent and Normal lines - Parametric Functions
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The Angle Between Two Curves
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Vertical Tangents and Cusps
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Linear Approximation
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[{"Name":"Tangent and Normal Lines - Basic Exercises","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tangent","Duration":"5m 30s","ChapterTopicVideoID":8258,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this clip, we\u0027ll be talking about"},{"Start":"00:02.040 ","End":"00:05.279","Text":"the equation of the tangent to the graph of a function."},{"Start":"00:05.279 ","End":"00:07.970","Text":"Now, let\u0027s take a coordinate system for the plane,"},{"Start":"00:07.970 ","End":"00:10.950","Text":"we have an x-axis and a y-axis."},{"Start":"00:10.950 ","End":"00:12.330","Text":"In this coordinate system,"},{"Start":"00:12.330 ","End":"00:13.964","Text":"I\u0027d like to draw a curve,"},{"Start":"00:13.964 ","End":"00:16.665","Text":"customary to call the curve f of x,"},{"Start":"00:16.665 ","End":"00:21.765","Text":"or even y equals f of x. I want to draw a tangent to this curve."},{"Start":"00:21.765 ","End":"00:26.715","Text":"The place where the tangent touches the curve is called the point of contact,"},{"Start":"00:26.715 ","End":"00:30.240","Text":"it\u0027s denoted by x_1, y_1,"},{"Start":"00:30.240 ","End":"00:34.885","Text":"where x_1 is the x coordinate and y_1 is the y-coordinate of this point."},{"Start":"00:34.885 ","End":"00:37.640","Text":"Sometimes we denote it as x_1,"},{"Start":"00:37.640 ","End":"00:41.150","Text":"F of x_1 instead of y_1, but remember,"},{"Start":"00:41.150 ","End":"00:46.200","Text":"we can always find y_1 from x_1 because y_1 is F of x_1."},{"Start":"00:46.200 ","End":"00:50.375","Text":"The question arises, what is the equation of this line?"},{"Start":"00:50.375 ","End":"00:53.390","Text":"Other words, what\u0027s the equation of the tangent line to"},{"Start":"00:53.390 ","End":"00:56.720","Text":"the curve f at the point x_1, y_1."},{"Start":"00:56.720 ","End":"01:02.225","Text":"Well, I\u0027m going to write down an equation, a formula and then I\u0027ll go into it in detail."},{"Start":"01:02.225 ","End":"01:03.760","Text":"Here\u0027s the formula."},{"Start":"01:03.760 ","End":"01:10.845","Text":"Y minus y_1 equals f prime of x 1 times x minus x_1."},{"Start":"01:10.845 ","End":"01:14.825","Text":"I\u0027ll go through all the terms in this equation and explain."},{"Start":"01:14.825 ","End":"01:17.175","Text":"First of all, let\u0027s go to x_1."},{"Start":"01:17.175 ","End":"01:22.070","Text":"X_1 is just the x coordinate of the point of contact."},{"Start":"01:22.070 ","End":"01:23.510","Text":"It\u0027s this x_1 here."},{"Start":"01:23.510 ","End":"01:25.775","Text":"Next is y_1."},{"Start":"01:25.775 ","End":"01:31.025","Text":"Y_1 is just the y-coordinate of the point of contact."},{"Start":"01:31.025 ","End":"01:35.345","Text":"Next, we have to explain what is f prime of x_1."},{"Start":"01:35.345 ","End":"01:41.750","Text":"This term, f prime of x_1 is the result of what I get when I"},{"Start":"01:41.750 ","End":"01:48.870","Text":"substitute the value x_1 in the derivative function that comes from f. Again,"},{"Start":"01:48.870 ","End":"01:52.450","Text":"f prime of x 1 is what I get if I differentiate"},{"Start":"01:52.450 ","End":"01:56.510","Text":"f and get f prime and then substitute x_1 in it."},{"Start":"01:56.510 ","End":"02:00.290","Text":"There is an interpretation of this f prime of x_1,"},{"Start":"02:00.290 ","End":"02:04.475","Text":"it\u0027s simply the slope of the tangent, and I\u0027ll write that."},{"Start":"02:04.475 ","End":"02:08.660","Text":"To summarize, all I need in order to get the equation of"},{"Start":"02:08.660 ","End":"02:14.150","Text":"the tangent to the function at this point of contact is the x_1,"},{"Start":"02:14.150 ","End":"02:15.845","Text":"the x coordinate of the point."},{"Start":"02:15.845 ","End":"02:16.910","Text":"I need y_1,"},{"Start":"02:16.910 ","End":"02:18.590","Text":"which is the y coordinate of the point,"},{"Start":"02:18.590 ","End":"02:23.905","Text":"and I also need the derivative of the function at that point, x_1."},{"Start":"02:23.905 ","End":"02:27.110","Text":"Then I just plug these in here and I have my equation."},{"Start":"02:27.110 ","End":"02:29.630","Text":"Of course, the x and the y remain variables."},{"Start":"02:29.630 ","End":"02:33.125","Text":"Let\u0027s get to an example and hope that that clarifies things."},{"Start":"02:33.125 ","End":"02:37.310","Text":"Find the equation of the tangent to the graph of the function f"},{"Start":"02:37.310 ","End":"02:41.945","Text":"of x equals x squared at the point where x equals 4."},{"Start":"02:41.945 ","End":"02:45.290","Text":"First of all, I\u0027m going to write down the formula,"},{"Start":"02:45.290 ","End":"02:50.240","Text":"and that was y minus y_1 equals"},{"Start":"02:50.240 ","End":"02:56.545","Text":"f prime of x_1 times x minus x_1."},{"Start":"02:56.545 ","End":"02:59.420","Text":"The first thing we\u0027ll be needing will be the points of"},{"Start":"02:59.420 ","End":"03:03.124","Text":"contact between the tangent and the function."},{"Start":"03:03.124 ","End":"03:07.255","Text":"In other words, I need x_1 and y_1."},{"Start":"03:07.255 ","End":"03:09.870","Text":"But look x_1, I\u0027m already given."},{"Start":"03:09.870 ","End":"03:11.640","Text":"I\u0027m given the x_1 of the point,"},{"Start":"03:11.640 ","End":"03:13.930","Text":"this is exactly this 4 here."},{"Start":"03:13.930 ","End":"03:16.035","Text":"That\u0027s the x of the point."},{"Start":"03:16.035 ","End":"03:17.610","Text":"Now I need the y_1."},{"Start":"03:17.610 ","End":"03:18.885","Text":"As I mentioned before,"},{"Start":"03:18.885 ","End":"03:20.670","Text":"y_1 is f of x_1."},{"Start":"03:20.670 ","End":"03:23.205","Text":"Just have to plug in 4 into the function."},{"Start":"03:23.205 ","End":"03:27.050","Text":"If x here is for f of x is 16,"},{"Start":"03:27.050 ","End":"03:28.760","Text":"4 squared is 16."},{"Start":"03:28.760 ","End":"03:32.600","Text":"Now, if you look, we actually have 2 of the 3 quantities we need."},{"Start":"03:32.600 ","End":"03:35.120","Text":"We have x_1, which is 4,"},{"Start":"03:35.120 ","End":"03:37.975","Text":"and that\u0027s the x_1 here,"},{"Start":"03:37.975 ","End":"03:40.590","Text":"and we have y_1,"},{"Start":"03:40.590 ","End":"03:44.385","Text":"and that\u0027s 16, and that\u0027s the y_1 that\u0027s here."},{"Start":"03:44.385 ","End":"03:47.885","Text":"We\u0027re missing the third 1 is this quantity,"},{"Start":"03:47.885 ","End":"03:50.225","Text":"this f prime of x_1."},{"Start":"03:50.225 ","End":"03:51.725","Text":"How do we get that?"},{"Start":"03:51.725 ","End":"03:54.650","Text":"What I have to do is to get f prime,"},{"Start":"03:54.650 ","End":"03:59.290","Text":"which I can do by differentiating f and then substitute x_1,"},{"Start":"03:59.290 ","End":"04:02.205","Text":"which is 4 into that and then I\u0027ll get the answer."},{"Start":"04:02.205 ","End":"04:05.205","Text":"Here it goes, f of x is x squared."},{"Start":"04:05.205 ","End":"04:09.435","Text":"F prime of x in general is 2x."},{"Start":"04:09.435 ","End":"04:15.225","Text":"Now f prime of x_1 is just f prime of 4."},{"Start":"04:15.225 ","End":"04:20.485","Text":"F prime of 4 means I substitute instead of x 4 in this equation,"},{"Start":"04:20.485 ","End":"04:24.785","Text":"so it\u0027s 2 times 4, which is 8."},{"Start":"04:24.785 ","End":"04:27.860","Text":"Now, I have that third quantity that I needed."},{"Start":"04:27.860 ","End":"04:29.315","Text":"I have x_1,"},{"Start":"04:29.315 ","End":"04:31.295","Text":"y_1, and f prime of x_1."},{"Start":"04:31.295 ","End":"04:36.095","Text":"All that remains now is to substitute the known quantities into this formula."},{"Start":"04:36.095 ","End":"04:41.600","Text":"We\u0027ll get that y minus y_1 is 16,"},{"Start":"04:41.600 ","End":"04:49.870","Text":"is equal to f prime of x_1 is 8 times x is just x and x_1 is 4."},{"Start":"04:49.870 ","End":"04:52.685","Text":"I\u0027ll highlight it just so you can see,"},{"Start":"04:52.685 ","End":"04:54.920","Text":"the x_1 is this 4,"},{"Start":"04:54.920 ","End":"05:01.925","Text":"the 16 is our y_1 and the 8 is f prime of x_1."},{"Start":"05:01.925 ","End":"05:08.270","Text":"Now, it\u0027s customary to simplify and extract just y so we can get from"},{"Start":"05:08.270 ","End":"05:17.300","Text":"here that y is equal to 8x minus 8 times 4 is minus 32,"},{"Start":"05:17.300 ","End":"05:21.800","Text":"but plus the 16 leaves us with minus 16."},{"Start":"05:21.800 ","End":"05:25.400","Text":"This here is the equation of"},{"Start":"05:25.400 ","End":"05:31.160","Text":"the tangent to the graph of the function at the given point. We are done."}],"ID":8418},{"Watched":false,"Name":"Normal Lines","Duration":"5m 9s","ChapterTopicVideoID":8257,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.485","Text":"In the previous clip,"},{"Start":"00:01.485 ","End":"00:05.100","Text":"we did the equation of a tangent to the graph of a function."},{"Start":"00:05.100 ","End":"00:07.485","Text":"Now, I want to talk about the normal,"},{"Start":"00:07.485 ","End":"00:11.100","Text":"but I\u0027m going to use what I did about the tangent to help me."},{"Start":"00:11.100 ","End":"00:17.595","Text":"I copy pasted the stuff from tangent where we had this equation and this diagram."},{"Start":"00:17.595 ","End":"00:20.340","Text":"I\u0027ll just label this tangent."},{"Start":"00:20.340 ","End":"00:25.410","Text":"Now, I want to modify this to get the equation of the normal."},{"Start":"00:25.410 ","End":"00:28.005","Text":"First, I\u0027ll show you what the normal is on the diagram."},{"Start":"00:28.005 ","End":"00:29.100","Text":"Let me just label them."},{"Start":"00:29.100 ","End":"00:34.980","Text":"This line here was the tangent which just grazes the function,"},{"Start":"00:34.980 ","End":"00:36.690","Text":"touches it at 1 point,"},{"Start":"00:36.690 ","End":"00:40.565","Text":"and the 1 perpendicular to it like 90 degrees,"},{"Start":"00:40.565 ","End":"00:43.550","Text":"this 1 would be the normal."},{"Start":"00:43.550 ","End":"00:46.340","Text":"That\u0027s the key thing that the normal is"},{"Start":"00:46.340 ","End":"00:50.660","Text":"just the perpendicular line to the tangent through the given point."},{"Start":"00:50.660 ","End":"00:52.985","Text":"We know the slope of the normal,"},{"Start":"00:52.985 ","End":"00:59.000","Text":"let me remind you that if we have something which has a line with slope m,"},{"Start":"00:59.000 ","End":"01:04.340","Text":"then the perpendicular line or normal line to it,"},{"Start":"01:04.340 ","End":"01:07.625","Text":"or orthogonal line to it that we just say the perpendicular,"},{"Start":"01:07.625 ","End":"01:17.000","Text":"has the property that the slope is minus 1 over m. So if I know that this slope,"},{"Start":"01:17.000 ","End":"01:19.235","Text":"which I wrote here is f prime of x,"},{"Start":"01:19.235 ","End":"01:24.840","Text":"all I have to do is replace f prime of x by minus 1 over f prime of x."},{"Start":"01:24.840 ","End":"01:28.994","Text":"Here I\u0027ll put minus 1 over f prime."},{"Start":"01:28.994 ","End":"01:31.075","Text":"In this case it\u0027s x_1."},{"Start":"01:31.075 ","End":"01:34.159","Text":"That is the slope of the normal."},{"Start":"01:34.159 ","End":"01:39.860","Text":"So this is the equation of the normal to the curve through the given point."},{"Start":"01:39.860 ","End":"01:41.845","Text":"That\u0027s all there is to it,"},{"Start":"01:41.845 ","End":"01:43.610","Text":"except that I\u0027ll give you an example."},{"Start":"01:43.610 ","End":"01:49.200","Text":"I copy pasted the previous exercise in the tutorial on the tangent,"},{"Start":"01:49.200 ","End":"01:53.840","Text":"and I\u0027m just going to change the word tangent to the word normal."},{"Start":"01:53.840 ","End":"01:58.265","Text":"You might want to go back and look and see how I solve the sample exercise there."},{"Start":"01:58.265 ","End":"02:00.335","Text":"But I\u0027ll just start a fresh."},{"Start":"02:00.335 ","End":"02:06.330","Text":"We have that x_1 is equal to 4, that\u0027s given here."},{"Start":"02:06.330 ","End":"02:09.225","Text":"We can also figure out what y_1 is."},{"Start":"02:09.225 ","End":"02:11.820","Text":"Y_1 is just f of x_1,"},{"Start":"02:11.820 ","End":"02:15.915","Text":"which is x_1 squared, which is 16."},{"Start":"02:15.915 ","End":"02:18.310","Text":"So now we have y_1."},{"Start":"02:18.310 ","End":"02:20.675","Text":"Next we need f prime of x."},{"Start":"02:20.675 ","End":"02:25.700","Text":"In general, f prime of x is equal to 2x,"},{"Start":"02:25.700 ","End":"02:27.755","Text":"because that\u0027s the derivative of this."},{"Start":"02:27.755 ","End":"02:32.710","Text":"So f prime of x_1 is equal to twice,"},{"Start":"02:32.710 ","End":"02:35.935","Text":"x_1 is 4, which is 8."},{"Start":"02:35.935 ","End":"02:38.780","Text":"So all I have to do now is substitute here."},{"Start":"02:38.780 ","End":"02:44.105","Text":"I\u0027ve got that the equation of the normal at the point x_1,"},{"Start":"02:44.105 ","End":"02:47.765","Text":"y_1, which is 4, 16,"},{"Start":"02:47.765 ","End":"02:52.650","Text":"is given by y minus y_1,"},{"Start":"02:52.650 ","End":"02:57.240","Text":"which is 16 equals minus 1 over,"},{"Start":"02:57.240 ","End":"03:04.255","Text":"this was 8 times x minus and x_1 was 4."},{"Start":"03:04.255 ","End":"03:05.630","Text":"That\u0027s the answer."},{"Start":"03:05.630 ","End":"03:10.775","Text":"But sometimes you might want to bring the 16 to the other side and open brackets."},{"Start":"03:10.775 ","End":"03:15.455","Text":"So although this is y equals minus 1/8x,"},{"Start":"03:15.455 ","End":"03:20.430","Text":"and then we have plus 4 over 8 is a 1/2 plus 16,"},{"Start":"03:20.430 ","End":"03:22.915","Text":"so it\u0027s plus 16 and 1/2."},{"Start":"03:22.915 ","End":"03:25.580","Text":"But this would be perfectly acceptable."},{"Start":"03:25.580 ","End":"03:28.210","Text":"That\u0027s an example with the normal."},{"Start":"03:28.210 ","End":"03:31.955","Text":"This is 1 special case and we haven\u0027t considered it."},{"Start":"03:31.955 ","End":"03:36.035","Text":"What happens if this denominator is 0?"},{"Start":"03:36.035 ","End":"03:40.220","Text":"If f prime and x_1 is 0, if the slope is 0,"},{"Start":"03:40.220 ","End":"03:42.550","Text":"then the line is horizontal,"},{"Start":"03:42.550 ","End":"03:45.495","Text":"and so the normal would be a vertical line."},{"Start":"03:45.495 ","End":"03:47.410","Text":"We wouldn\u0027t use this formula."},{"Start":"03:47.410 ","End":"03:49.570","Text":"Let me illustrate with an example."},{"Start":"03:49.570 ","End":"03:53.655","Text":"Let\u0027s make this 2x minus x squared."},{"Start":"03:53.655 ","End":"03:57.805","Text":"Let\u0027s take it at x equals 1 and see what happens."},{"Start":"03:57.805 ","End":"04:01.240","Text":"Now we have that x_1 is equal to 1."},{"Start":"04:01.240 ","End":"04:03.955","Text":"Actually, I don\u0027t even need y_1."},{"Start":"04:03.955 ","End":"04:05.575","Text":"I could substitute here."},{"Start":"04:05.575 ","End":"04:07.690","Text":"You\u0027ll see why I don\u0027t need it in a moment."},{"Start":"04:07.690 ","End":"04:11.710","Text":"F prime of x in general is equal to the derivative of this,"},{"Start":"04:11.710 ","End":"04:13.865","Text":"which is 2 minus 2x."},{"Start":"04:13.865 ","End":"04:18.520","Text":"So f prime of x_1 is what we get when we put 1 in here,"},{"Start":"04:18.520 ","End":"04:22.555","Text":"which is equal to 2 minus 2 times 1, which is 0."},{"Start":"04:22.555 ","End":"04:25.095","Text":"When it\u0027s 0, we can\u0027t use the formula,"},{"Start":"04:25.095 ","End":"04:27.839","Text":"because minus 1 over 0 is not defined."},{"Start":"04:27.839 ","End":"04:30.920","Text":"But if the derivative is 0,"},{"Start":"04:30.920 ","End":"04:34.655","Text":"it means that the tangent is horizontal,"},{"Start":"04:34.655 ","End":"04:38.150","Text":"which means that the normal is vertical line,"},{"Start":"04:38.150 ","End":"04:43.595","Text":"and we know that it goes through x equals 1, y equals something."},{"Start":"04:43.595 ","End":"04:45.680","Text":"Because of the x equals 1,"},{"Start":"04:45.680 ","End":"04:49.450","Text":"the equation of the normal is just x equals 1."},{"Start":"04:49.450 ","End":"04:51.380","Text":"Because for a vertical line,"},{"Start":"04:51.380 ","End":"04:52.700","Text":"the x is constant."},{"Start":"04:52.700 ","End":"04:54.080","Text":"So this gives us this,"},{"Start":"04:54.080 ","End":"04:55.970","Text":"so that\u0027s the equation of the normal."},{"Start":"04:55.970 ","End":"05:00.005","Text":"So it\u0027s pretty easy if you get 0 for the derivative,"},{"Start":"05:00.005 ","End":"05:03.955","Text":"then you just take x equals whatever the x that was given."},{"Start":"05:03.955 ","End":"05:09.720","Text":"So that\u0027s the exception for the previous rule. Now we\u0027re done."}],"ID":8417},{"Watched":false,"Name":"Exercise 1","Duration":"4m 12s","ChapterTopicVideoID":3340,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to find the equation of the line that is tangent to the curve,"},{"Start":"00:05.280 ","End":"00:09.405","Text":"this 1, at the point on the curve where x equals 2."},{"Start":"00:09.405 ","End":"00:12.975","Text":"Now this might be a little bit difficult to comprehend."},{"Start":"00:12.975 ","End":"00:14.535","Text":"Let me explain further."},{"Start":"00:14.535 ","End":"00:16.310","Text":"The line that is tangent,"},{"Start":"00:16.310 ","End":"00:18.095","Text":"also known as a tangent line,"},{"Start":"00:18.095 ","End":"00:21.995","Text":"is a line that touches the curve at only 1 point."},{"Start":"00:21.995 ","End":"00:24.844","Text":"In this case, the curve is given by a function."},{"Start":"00:24.844 ","End":"00:27.925","Text":"When you sketch the graph of the function, that\u0027s a curve."},{"Start":"00:27.925 ","End":"00:29.250","Text":"Now, on the curve,"},{"Start":"00:29.250 ","End":"00:33.935","Text":"there is a place where x equals 2 and y equals something and that point,"},{"Start":"00:33.935 ","End":"00:35.795","Text":"we can draw a tangent line."},{"Start":"00:35.795 ","End":"00:38.830","Text":"We have to find the equation of that line."},{"Start":"00:38.830 ","End":"00:44.630","Text":"To help you understand what\u0027s going on and for those who are more visually inclined,"},{"Start":"00:44.630 ","End":"00:47.014","Text":"I\u0027ve prepared in advance a sketch."},{"Start":"00:47.014 ","End":"00:48.365","Text":"Here\u0027s the sketch."},{"Start":"00:48.365 ","End":"00:51.190","Text":"This is the function,"},{"Start":"00:51.190 ","End":"00:56.375","Text":"y equals x squared minus 2x plus 3 or something. That\u0027s the curve."},{"Start":"00:56.375 ","End":"01:05.555","Text":"This is the point on the curve where x equals 2 and this is the tangent line."},{"Start":"01:05.555 ","End":"01:07.975","Text":"This is just to give you an idea."},{"Start":"01:07.975 ","End":"01:15.275","Text":"In general, what we\u0027re going to do is I mean we\u0027ll find the coordinates of this point,"},{"Start":"01:15.275 ","End":"01:19.280","Text":"what y is, and we\u0027ll use the calculus to find the derivative,"},{"Start":"01:19.280 ","End":"01:22.230","Text":"which will give us the slope of the tangent."},{"Start":"01:22.230 ","End":"01:28.045","Text":"Well, let\u0027s just go up and do it but a picture often helps."},{"Start":"01:28.045 ","End":"01:33.995","Text":"The first thing we need is to find the y of the point."},{"Start":"01:33.995 ","End":"01:37.880","Text":"When x equals 2, the y, this is y."},{"Start":"01:37.880 ","End":"01:39.890","Text":"When x equals 2,"},{"Start":"01:39.890 ","End":"01:43.475","Text":"then y equals f of 2,"},{"Start":"01:43.475 ","End":"01:47.115","Text":"which is equal to 2 squared minus twice 2 plus 3."},{"Start":"01:47.115 ","End":"01:48.570","Text":"This is equal to 3,"},{"Start":"01:48.570 ","End":"01:51.139","Text":"so we have our point,"},{"Start":"01:51.139 ","End":"01:55.700","Text":"and sometimes we call this point x_1,"},{"Start":"01:55.700 ","End":"02:00.735","Text":"y_1, which in our case is 2, 3."},{"Start":"02:00.735 ","End":"02:05.380","Text":"We\u0027ll also need the derivative and you\u0027ll soon see what for."},{"Start":"02:05.380 ","End":"02:09.140","Text":"F prime of x or y prime is equal to,"},{"Start":"02:09.140 ","End":"02:11.120","Text":"I\u0027m looking at x squared minus 2x plus 3,"},{"Start":"02:11.120 ","End":"02:14.090","Text":"simple differentiation, 2x minus"},{"Start":"02:14.090 ","End":"02:18.980","Text":"2 and now I\u0027ll come to the formula that we\u0027re going to use."},{"Start":"02:18.980 ","End":"02:22.790","Text":"The formula for a tangent that is,"},{"Start":"02:22.790 ","End":"02:28.655","Text":"let\u0027s write it as formula for tangent is,"},{"Start":"02:28.655 ","End":"02:31.040","Text":"I\u0027m going to use a different style of x and y for"},{"Start":"02:31.040 ","End":"02:34.220","Text":"the line not to mix it up with the curve,"},{"Start":"02:34.220 ","End":"02:39.330","Text":"is Y minus y_1 is equal to"},{"Start":"02:39.330 ","End":"02:47.205","Text":"f prime of x_1 times X minus x_1."},{"Start":"02:47.205 ","End":"02:52.275","Text":"Now, we already have y_1 which is this,"},{"Start":"02:52.275 ","End":"02:54.150","Text":"and x_1, which is this."},{"Start":"02:54.150 ","End":"02:57.285","Text":"All we\u0027re missing is f prime of x_1."},{"Start":"02:57.285 ","End":"03:02.240","Text":"Now x_1 is 2 so all we\u0027re missing now is f prime of"},{"Start":"03:02.240 ","End":"03:08.130","Text":"2 and if I put 2 in here twice 2 minus 2 is equal to 2."},{"Start":"03:08.130 ","End":"03:10.190","Text":"Now I have everything I need."},{"Start":"03:10.190 ","End":"03:12.650","Text":"I just have to plug it in the formula."},{"Start":"03:12.650 ","End":"03:21.660","Text":"The tangent line is actually Y minus the y_1 is equal to f prime of x_1,"},{"Start":"03:21.660 ","End":"03:28.405","Text":"which is f prime of 2, which is 2 times X minus the x_1,"},{"Start":"03:28.405 ","End":"03:30.660","Text":"which is also 2."},{"Start":"03:30.660 ","End":"03:32.385","Text":"There are 2 different 2s here."},{"Start":"03:32.385 ","End":"03:34.425","Text":"I\u0027ll just say that this 2,"},{"Start":"03:34.425 ","End":"03:35.805","Text":"or maybe I\u0027ll highlight it."},{"Start":"03:35.805 ","End":"03:40.710","Text":"This 2, is this 2 and"},{"Start":"03:40.710 ","End":"03:46.650","Text":"use a different color now and this 2 is actually this 2."},{"Start":"03:46.650 ","End":"03:50.525","Text":"It\u0027s confusing when there was twice the digit, the number 2."},{"Start":"03:50.525 ","End":"03:58.740","Text":"That is basically it but it is customary to leave just Y on 1 side and simplify it."},{"Start":"03:58.740 ","End":"04:00.650","Text":"If I just manipulate a bit,"},{"Start":"04:00.650 ","End":"04:05.690","Text":"I\u0027ll get y equals 2x minus 4 plus 3,"},{"Start":"04:05.690 ","End":"04:13.020","Text":"which is 2x minus 1 and that is the answer and we are done."}],"ID":3351},{"Watched":false,"Name":"Exercise 2","Duration":"5m 8s","ChapterTopicVideoID":4334,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to find the equation of the line that"},{"Start":"00:03.270 ","End":"00:06.915","Text":"is normal to the curve at the point on the curve where x equals 2."},{"Start":"00:06.915 ","End":"00:08.400","Text":"What does all this mean?"},{"Start":"00:08.400 ","End":"00:11.010","Text":"I have below a picture, a sketch,"},{"Start":"00:11.010 ","End":"00:13.950","Text":"that might help, but let me just say a few things now."},{"Start":"00:13.950 ","End":"00:20.305","Text":"This is a function f of x is equal to or if you like, y equals."},{"Start":"00:20.305 ","End":"00:24.590","Text":"If you plot this function, you get a curve."},{"Start":"00:24.590 ","End":"00:26.780","Text":"It\u0027s a graph which is a curve."},{"Start":"00:26.780 ","End":"00:28.650","Text":"The concept of a normal,"},{"Start":"00:28.650 ","End":"00:32.190","Text":"I\u0027ll explain below when we get to the picture that you should know."},{"Start":"00:32.190 ","End":"00:36.680","Text":"On the curve, there is a place where x equals 2 and y equals something else."},{"Start":"00:36.680 ","End":"00:39.950","Text":"Let\u0027s go down to the picture I prepared."},{"Start":"00:39.950 ","End":"00:41.270","Text":"It\u0027s all in green,"},{"Start":"00:41.270 ","End":"00:44.750","Text":"but this part is the graph which is the curve."},{"Start":"00:44.750 ","End":"00:48.260","Text":"I\u0027ll just explain, make a little glossary of terms here."},{"Start":"00:48.260 ","End":"00:51.020","Text":"This is the curve which is given by y"},{"Start":"00:51.020 ","End":"00:54.710","Text":"equals x squared minus 2x plus 3 or whatever it was."},{"Start":"00:54.710 ","End":"00:57.860","Text":"Now this is the point where x equals 2."},{"Start":"00:57.860 ","End":"01:02.780","Text":"The point where x equals 2 means that if I"},{"Start":"01:02.780 ","End":"01:07.925","Text":"take here on the x-axis the point 2 and I go above it,"},{"Start":"01:07.925 ","End":"01:10.520","Text":"y equals something based on the formula,"},{"Start":"01:10.520 ","End":"01:11.825","Text":"it doesn\u0027t matter for now,"},{"Start":"01:11.825 ","End":"01:14.275","Text":"this is the point where x equals 2."},{"Start":"01:14.275 ","End":"01:16.125","Text":"That\u0027s our point."},{"Start":"01:16.125 ","End":"01:17.630","Text":"Now there\u0027s 2 other things here."},{"Start":"01:17.630 ","End":"01:18.980","Text":"There\u0027s a couple of lines,"},{"Start":"01:18.980 ","End":"01:21.965","Text":"one\u0027s dotted because that\u0027s not the 1 we have to find,"},{"Start":"01:21.965 ","End":"01:24.515","Text":"but this is actually more basic,"},{"Start":"01:24.515 ","End":"01:26.570","Text":"is the tangent line."},{"Start":"01:26.570 ","End":"01:34.130","Text":"This line here, which is perpendicular at right angles to the tangent,"},{"Start":"01:34.130 ","End":"01:38.019","Text":"is called the normal line or simply the normal."},{"Start":"01:38.019 ","End":"01:40.560","Text":"What we have to do is,"},{"Start":"01:40.560 ","End":"01:42.995","Text":"find the equation of the normal,"},{"Start":"01:42.995 ","End":"01:45.755","Text":"y equals something x plus something."},{"Start":"01:45.755 ","End":"01:48.440","Text":"This, I hope, gives you an idea,"},{"Start":"01:48.440 ","End":"01:52.410","Text":"but you certainly don\u0027t have to have a sketch in order to solve the equation."},{"Start":"01:52.410 ","End":"01:56.525","Text":"It can be done purely algebraically with calculus."},{"Start":"01:56.525 ","End":"01:59.885","Text":"Let\u0027s go back now that you have maybe a better idea."},{"Start":"01:59.885 ","End":"02:02.990","Text":"What we need to do is obviously, first of all,"},{"Start":"02:02.990 ","End":"02:06.830","Text":"we might need the actual coordinates of the point."},{"Start":"02:06.830 ","End":"02:10.485","Text":"When x equals 2,"},{"Start":"02:10.485 ","End":"02:16.470","Text":"that gives us that y or f of x is equal to 2 squared minus twice 2 plus 3."},{"Start":"02:16.470 ","End":"02:18.660","Text":"We can do it in our heads, it\u0027s equal to 3."},{"Start":"02:18.660 ","End":"02:20.780","Text":"This gives us our point,"},{"Start":"02:20.780 ","End":"02:23.255","Text":"which is in general,"},{"Start":"02:23.255 ","End":"02:26.955","Text":"we use the terms x_1 and y_1."},{"Start":"02:26.955 ","End":"02:30.295","Text":"In our case, this comes out to be 2, 3."},{"Start":"02:30.295 ","End":"02:33.170","Text":"At this stage, I\u0027ll introduce you to the formula that we\u0027re going to"},{"Start":"02:33.170 ","End":"02:35.900","Text":"be using for the normal and"},{"Start":"02:35.900 ","End":"02:43.590","Text":"the formula for the normal line at that point etc,"},{"Start":"02:43.590 ","End":"02:48.660","Text":"is given by Y minus y_1,"},{"Start":"02:48.660 ","End":"02:50.075","Text":"this is in general,"},{"Start":"02:50.075 ","End":"02:56.135","Text":"is equal to minus 1 over f prime"},{"Start":"02:56.135 ","End":"03:03.015","Text":"of x_1 times X minus x_1."},{"Start":"03:03.015 ","End":"03:06.650","Text":"I deliberately used a different style of x and y so as not to"},{"Start":"03:06.650 ","End":"03:10.730","Text":"confuse the line with the x and y in the curve."},{"Start":"03:10.730 ","End":"03:13.070","Text":"Now, let\u0027s see what we have and what we don\u0027t have."},{"Start":"03:13.070 ","End":"03:14.435","Text":"We have x_1,"},{"Start":"03:14.435 ","End":"03:17.720","Text":"we have y_1 because this is x_1, y_1."},{"Start":"03:17.720 ","End":"03:19.310","Text":"What we\u0027re missing is this."},{"Start":"03:19.310 ","End":"03:22.545","Text":"We need this bit here that\u0027s missing."},{"Start":"03:22.545 ","End":"03:28.160","Text":"Let\u0027s differentiate and get f prime and then substitute x_1."},{"Start":"03:28.160 ","End":"03:34.310","Text":"In general, f prime of x is equal to just the derivative of this,"},{"Start":"03:34.310 ","End":"03:36.980","Text":"which is 2x minus 2 clearly."},{"Start":"03:36.980 ","End":"03:43.385","Text":"Then what we want is f prime of x_1 and x_1 in our case is 2,"},{"Start":"03:43.385 ","End":"03:49.290","Text":"so f prime of 2 is equal twice 2 minus 2, which is 2."},{"Start":"03:49.290 ","End":"03:52.980","Text":"What we need now is we can just write the formula."},{"Start":"03:52.980 ","End":"03:57.630","Text":"The formula for the normal is Y minus,"},{"Start":"03:57.630 ","End":"04:00.040","Text":"now y_1 is this 3,"},{"Start":"04:00.040 ","End":"04:06.155","Text":"which is equal to minus 1 over f prime of x_1 is"},{"Start":"04:06.155 ","End":"04:13.590","Text":"2 times X minus and x_1 is also 2."},{"Start":"04:13.590 ","End":"04:15.510","Text":"The number 2 appears twice,"},{"Start":"04:15.510 ","End":"04:19.910","Text":"or maybe I think I\u0027d like to just emphasize which 2 is which."},{"Start":"04:19.910 ","End":"04:23.915","Text":"This 2 is our x_1 is this 2,"},{"Start":"04:23.915 ","End":"04:28.805","Text":"but this 2 is the 1 that comes from here."},{"Start":"04:28.805 ","End":"04:34.140","Text":"In general, I often remember it Y minus the y of the point and"},{"Start":"04:34.140 ","End":"04:40.265","Text":"X minus the x of the point and minus 1 here over the derivative."},{"Start":"04:40.265 ","End":"04:42.950","Text":"That\u0027s basically the answer,"},{"Start":"04:42.950 ","End":"04:48.515","Text":"except that it\u0027s customary to leave just Y on 1 side and tidy up a bit."},{"Start":"04:48.515 ","End":"04:55.650","Text":"Let\u0027s just do that and write it as Y equals minus 1/2x."},{"Start":"04:55.650 ","End":"04:59.700","Text":"Then minus 1/2 times minus 2 is 1,"},{"Start":"04:59.700 ","End":"05:01.605","Text":"and if I add the 3 there,"},{"Start":"05:01.605 ","End":"05:05.115","Text":"so it\u0027s 1 plus the 3 plus 4."},{"Start":"05:05.115 ","End":"05:08.890","Text":"I\u0027ll underline it, and that\u0027s the answer."}],"ID":4343},{"Watched":false,"Name":"Exercise 3","Duration":"5m 45s","ChapterTopicVideoID":31843,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":34098},{"Watched":false,"Name":"Exercise 4","Duration":"8m 20s","ChapterTopicVideoID":4337,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.785","Text":"This exercise is for the more advanced students and you know who you are."},{"Start":"00:04.785 ","End":"00:09.450","Text":"Find the equations of the lines tangent to the curve, dah, dah,"},{"Start":"00:09.450 ","End":"00:11.730","Text":"dah, and they\u0027re parallel to the line dah,"},{"Start":"00:11.730 ","End":"00:14.399","Text":"dah, dah, that\u0027s an unusual question."},{"Start":"00:14.399 ","End":"00:18.225","Text":"Lines tangent to a curve and parallel to a given line."},{"Start":"00:18.225 ","End":"00:19.695","Text":"Well, to help you,"},{"Start":"00:19.695 ","End":"00:23.700","Text":"what I\u0027ve done is I\u0027ve prepared a sketch that helps you understand the question."},{"Start":"00:23.700 ","End":"00:25.620","Text":"This is our curve,"},{"Start":"00:25.620 ","End":"00:28.150","Text":"y equals x cubed and all the rest of it."},{"Start":"00:28.150 ","End":"00:32.300","Text":"This is the line that we have to find tangents parallel to."},{"Start":"00:32.300 ","End":"00:34.160","Text":"This is our reference line,"},{"Start":"00:34.160 ","End":"00:36.350","Text":"I\u0027ll call it the word reference,"},{"Start":"00:36.350 ","End":"00:39.065","Text":"but this is the line that was mentioned."},{"Start":"00:39.065 ","End":"00:44.030","Text":"What we have to do is go along the curve and see at each point there\u0027s a tangent."},{"Start":"00:44.030 ","End":"00:50.270","Text":"But find those places where the tangent is actually parallel to the reference line."},{"Start":"00:50.270 ","End":"00:54.870","Text":"It turns out that at this point the tangent is parallel,"},{"Start":"00:54.870 ","End":"00:57.170","Text":"and at this point the tangent is parallel."},{"Start":"00:57.170 ","End":"01:02.000","Text":"We have to find the equation of each of these tangents."},{"Start":"01:02.000 ","End":"01:06.630","Text":"Here\u0027s 1 tangent and here\u0027s another tangent."},{"Start":"01:07.010 ","End":"01:10.520","Text":"I hope this makes it a little bit clearer."},{"Start":"01:10.520 ","End":"01:13.610","Text":"Now we\u0027ll go back up and start doing it."},{"Start":"01:13.610 ","End":"01:16.760","Text":"There is 1 other important thing is that I should"},{"Start":"01:16.760 ","End":"01:19.460","Text":"explain what we talk about parallel here."},{"Start":"01:19.460 ","End":"01:23.825","Text":"I want to mention that how we know that things are parallel,"},{"Start":"01:23.825 ","End":"01:25.475","Text":"that 2 lines are parallel."},{"Start":"01:25.475 ","End":"01:28.375","Text":"Parallel means have the same slope."},{"Start":"01:28.375 ","End":"01:30.965","Text":"That\u0027s something that\u0027s essential."},{"Start":"01:30.965 ","End":"01:36.920","Text":"We also write down the standard formula for the tangent."},{"Start":"01:36.920 ","End":"01:43.320","Text":"The formula for the tangent at a given point on a curve is,"},{"Start":"01:43.320 ","End":"01:44.685","Text":"if we know a point,"},{"Start":"01:44.685 ","End":"01:46.200","Text":"x_1, y_1,"},{"Start":"01:46.200 ","End":"01:49.655","Text":"and the slope, it\u0027s y. I\u0027ll just write it and explain it."},{"Start":"01:49.655 ","End":"01:59.260","Text":"Y minus y_1 is equal to f prime of x_1 times x minus x_1."},{"Start":"01:59.260 ","End":"02:03.710","Text":"This is a formula for a tangent to the curve at the place where x is"},{"Start":"02:03.710 ","End":"02:08.570","Text":"equal to x_1 and y_1 is the corresponding y for this x."},{"Start":"02:08.570 ","End":"02:11.165","Text":"Now I\u0027ve written the y here and the x here in large,"},{"Start":"02:11.165 ","End":"02:14.660","Text":"different style, not to confuse with the x and the y of the curve."},{"Start":"02:14.660 ","End":"02:23.240","Text":"What we need to do is to find places x_1 where this is parallel to this given line."},{"Start":"02:23.240 ","End":"02:27.170","Text":"Now, this coefficient here is actually"},{"Start":"02:27.170 ","End":"02:32.810","Text":"the slope and the same thing as in the line y equals 6x minus 2."},{"Start":"02:32.810 ","End":"02:37.590","Text":"This thing is the slope of the line."},{"Start":"02:37.590 ","End":"02:40.340","Text":"What we have to do is compare"},{"Start":"02:40.340 ","End":"02:44.704","Text":"the tangent slope to the line slope and then they\u0027ll be parallel."},{"Start":"02:44.704 ","End":"02:50.555","Text":"Basically, our equation is going to be f prime of x_1."},{"Start":"02:50.555 ","End":"02:52.910","Text":"Now, slowly, first of all,"},{"Start":"02:52.910 ","End":"02:54.830","Text":"f prime of x in general,"},{"Start":"02:54.830 ","End":"02:56.720","Text":"if this is y,"},{"Start":"02:56.720 ","End":"03:00.290","Text":"it\u0027s written as y, but it could have been written as f of x."},{"Start":"03:00.290 ","End":"03:02.135","Text":"On the 1 hand,"},{"Start":"03:02.135 ","End":"03:04.115","Text":"f prime of x in general,"},{"Start":"03:04.115 ","End":"03:13.365","Text":"the derivative of this function is equal to 3x squared minus 6, and that\u0027s it."},{"Start":"03:13.365 ","End":"03:17.450","Text":"When we compare that is the slope here,"},{"Start":"03:17.450 ","End":"03:19.370","Text":"and this is the slope here."},{"Start":"03:19.370 ","End":"03:22.135","Text":"Basically if we compare these 2,"},{"Start":"03:22.135 ","End":"03:31.170","Text":"then what we will get will be 3x_1 squared minus 6 is going to be equal,"},{"Start":"03:31.170 ","End":"03:34.324","Text":"yeah, when we put x_1 is equal to 6."},{"Start":"03:34.324 ","End":"03:41.285","Text":"What that will give us is we\u0027ll get that 3x_1 squared is equal to 12,"},{"Start":"03:41.285 ","End":"03:45.570","Text":"x_1 squared is equal to 4,"},{"Start":"03:45.570 ","End":"03:50.015","Text":"x_1 is equal to plus or minus 2."},{"Start":"03:50.015 ","End":"03:52.070","Text":"Now this is 2 possibilities."},{"Start":"03:52.070 ","End":"03:55.470","Text":"What we\u0027ll need is for each of these,"},{"Start":"03:55.470 ","End":"03:57.765","Text":"we\u0027ll need to find y_1,"},{"Start":"03:57.765 ","End":"04:00.720","Text":"and we\u0027ll need to find f prime of x_1."},{"Start":"04:00.720 ","End":"04:03.905","Text":"Let\u0027s take the 2 cases. We have case a."},{"Start":"04:03.905 ","End":"04:05.750","Text":"This will give us tangent a,"},{"Start":"04:05.750 ","End":"04:09.540","Text":"which will be that x_1 is equal to,"},{"Start":"04:09.540 ","End":"04:10.580","Text":"let\u0027s say the plus 2,"},{"Start":"04:10.580 ","End":"04:11.930","Text":"I\u0027m just emphasizing it."},{"Start":"04:11.930 ","End":"04:13.625","Text":"I don\u0027t need to write the plus."},{"Start":"04:13.625 ","End":"04:22.065","Text":"If x is 2, then y_1 will equal f of x_1 is equal to f of x_1,"},{"Start":"04:22.065 ","End":"04:23.625","Text":"which is equal 2."},{"Start":"04:23.625 ","End":"04:28.125","Text":"How do I get that? I just have to substitute 2 in here,"},{"Start":"04:28.125 ","End":"04:30.710","Text":"and if I substitute 2 in there,"},{"Start":"04:30.710 ","End":"04:32.010","Text":"what do I get?"},{"Start":"04:32.010 ","End":"04:33.920","Text":"Is minus 2."},{"Start":"04:33.920 ","End":"04:37.880","Text":"We\u0027ll also get that f prime of,"},{"Start":"04:37.880 ","End":"04:39.905","Text":"well, the slope has to be 6."},{"Start":"04:39.905 ","End":"04:43.970","Text":"If I substitute it then it isn\u0027t 6 then there\u0027s something very wrong."},{"Start":"04:43.970 ","End":"04:47.465","Text":"For example, if I substitute x is equal to 2 here,"},{"Start":"04:47.465 ","End":"04:51.320","Text":"I\u0027ll get 2 squared is 4 times 3 is 12 minus 6,"},{"Start":"04:51.320 ","End":"04:53.975","Text":"it has to be 6 otherwise we made a mistake."},{"Start":"04:53.975 ","End":"04:56.630","Text":"The slope is,"},{"Start":"04:56.630 ","End":"05:01.775","Text":"we call f prime of x_1 is equal to 6."},{"Start":"05:01.775 ","End":"05:03.830","Text":"In part b,"},{"Start":"05:03.830 ","End":"05:06.935","Text":"x_1 will equal minus 2,"},{"Start":"05:06.935 ","End":"05:10.745","Text":"y_1 will equal f of minus 2,"},{"Start":"05:10.745 ","End":"05:15.934","Text":"which is, let\u0027s see if I do it with minus 2,"},{"Start":"05:15.934 ","End":"05:18.995","Text":"that should equal 6."},{"Start":"05:18.995 ","End":"05:20.660","Text":"Let it be 6."},{"Start":"05:20.660 ","End":"05:27.560","Text":"But an f prime of x_1 will also be equal to 6."},{"Start":"05:27.560 ","End":"05:28.865","Text":"Is it the same?"},{"Start":"05:28.865 ","End":"05:31.535","Text":"It has to come out 6 otherwise it\u0027s not going to be parallel."},{"Start":"05:31.535 ","End":"05:34.925","Text":"There\u0027s 2 possibilities, and for each of them,"},{"Start":"05:34.925 ","End":"05:37.360","Text":"I\u0027ll write this formula."},{"Start":"05:37.360 ","End":"05:40.070","Text":"Continuing to scroll down,"},{"Start":"05:40.070 ","End":"05:42.500","Text":"I\u0027ll write tangent a."},{"Start":"05:42.500 ","End":"05:48.410","Text":"Let\u0027s just call them tangent a and tangent b. Tangent a will be given"},{"Start":"05:48.410 ","End":"05:56.100","Text":"by the formula which is y minus y_1, which is 2."},{"Start":"05:56.100 ","End":"05:57.850","Text":"Minus minus 2,"},{"Start":"05:57.850 ","End":"05:59.650","Text":"I\u0027ll write it straight away as plus 2,"},{"Start":"05:59.650 ","End":"06:01.205","Text":"your are the advanced students,"},{"Start":"06:01.205 ","End":"06:07.370","Text":"y minus minus 2 is equal to the slope which is 6."},{"Start":"06:07.370 ","End":"06:13.955","Text":"Take it from here or from here or from the line times x minus x_1,"},{"Start":"06:13.955 ","End":"06:15.800","Text":"which is minus 2."},{"Start":"06:15.800 ","End":"06:18.230","Text":"As I said, we usually simplify it,"},{"Start":"06:18.230 ","End":"06:23.930","Text":"so we\u0027ll write it as y equals the words to start to simplification,"},{"Start":"06:23.930 ","End":"06:32.490","Text":"y equals 6x minus 12 minus 2 so 6x minus 14."},{"Start":"06:32.490 ","End":"06:34.260","Text":"That\u0027s 1 answer."},{"Start":"06:34.260 ","End":"06:35.785","Text":"Then tangent b."},{"Start":"06:35.785 ","End":"06:38.630","Text":"Let\u0027s see afterwards in the picture which is a and which is b."},{"Start":"06:38.630 ","End":"06:40.955","Text":"We get that again,"},{"Start":"06:40.955 ","End":"06:44.405","Text":"y minus the y of the point, the y_1."},{"Start":"06:44.405 ","End":"06:47.335","Text":"In the other case it\u0027s y minus 6."},{"Start":"06:47.335 ","End":"06:50.870","Text":"Y minus y_1 is y minus 6 is equal to,"},{"Start":"06:50.870 ","End":"06:52.770","Text":"again, this has to be the same 6,"},{"Start":"06:52.770 ","End":"06:54.920","Text":"that\u0027s the slope of the reference line."},{"Start":"06:54.920 ","End":"06:57.240","Text":"X minus x_1,"},{"Start":"06:57.240 ","End":"06:59.670","Text":"which in this case is minus 2."},{"Start":"06:59.670 ","End":"07:02.220","Text":"Here we have plus 2."},{"Start":"07:02.220 ","End":"07:07.580","Text":"After simplification that gives us y is equal"},{"Start":"07:07.580 ","End":"07:14.810","Text":"to 6x plus 12 plus 6 is plus 18."},{"Start":"07:14.810 ","End":"07:17.945","Text":"I\u0027ll just highlight them."},{"Start":"07:17.945 ","End":"07:24.170","Text":"Tangent a will be this and tangent b will be this."},{"Start":"07:24.170 ","End":"07:26.210","Text":"Those are the 2 answers, that\u0027s it."},{"Start":"07:26.210 ","End":"07:31.020","Text":"I just asked for those tangents and we got 2 solutions as expected."},{"Start":"07:31.020 ","End":"07:34.775","Text":"We\u0027re done, but those who would like to stay can just"},{"Start":"07:34.775 ","End":"07:39.050","Text":"go with me and see how these makes sense,"},{"Start":"07:39.050 ","End":"07:41.934","Text":"6x minus 14 and 6x plus 18."},{"Start":"07:41.934 ","End":"07:47.360","Text":"Now, they\u0027re all 6x plus something just like the line is."},{"Start":"07:47.360 ","End":"07:50.210","Text":"It has to be 6x because then it wouldn\u0027t be parallel."},{"Start":"07:50.210 ","End":"07:54.040","Text":"Now 1 of them was plus 18."},{"Start":"07:54.040 ","End":"07:59.070","Text":"That has to be this one because it\u0027s off the charts,"},{"Start":"07:59.070 ","End":"08:01.965","Text":"but it\u0027s going to cross the plus 18,"},{"Start":"08:01.965 ","End":"08:04.860","Text":"and this one over here looks like it\u0027s going to hit,"},{"Start":"08:04.860 ","End":"08:06.695","Text":"again it may be off the charts,"},{"Start":"08:06.695 ","End":"08:09.480","Text":"but it\u0027s going to hit the y-axis."},{"Start":"08:09.590 ","End":"08:12.260","Text":"This is the x-axis of course,"},{"Start":"08:12.260 ","End":"08:13.460","Text":"and this is the y axis."},{"Start":"08:13.460 ","End":"08:15.650","Text":"Here y is going to be minus 14."},{"Start":"08:15.650 ","End":"08:20.460","Text":"It makes sense. It looks reasonable. Done."}],"ID":8422},{"Watched":false,"Name":"Exercise 5","Duration":"4m 11s","ChapterTopicVideoID":4339,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to find the equation of the line that is tangent to"},{"Start":"00:04.590 ","End":"00:09.420","Text":"the curve that\u0027s written here but at the point on the curve where x equals 3."},{"Start":"00:09.420 ","End":"00:11.865","Text":"The curve is a function,"},{"Start":"00:11.865 ","End":"00:13.320","Text":"and the function, if you sketch it,"},{"Start":"00:13.320 ","End":"00:16.195","Text":"is a curve, and somewhere along the curve,"},{"Start":"00:16.195 ","End":"00:17.960","Text":"there is a place where x equals 3,"},{"Start":"00:17.960 ","End":"00:19.475","Text":"and I don\u0027t know what y is there."},{"Start":"00:19.475 ","End":"00:21.290","Text":"At that point, there\u0027s a tangent,"},{"Start":"00:21.290 ","End":"00:23.335","Text":"and we have to find the equation of that."},{"Start":"00:23.335 ","End":"00:25.515","Text":"To help you understand it,"},{"Start":"00:25.515 ","End":"00:28.325","Text":"I\u0027ve actually prepared a sketch. Here it is."},{"Start":"00:28.325 ","End":"00:30.125","Text":"This is the curve,"},{"Start":"00:30.125 ","End":"00:32.840","Text":"the y equals the square root of something,"},{"Start":"00:32.840 ","End":"00:37.800","Text":"and this is the point where x equals 3."},{"Start":"00:37.800 ","End":"00:40.835","Text":"This is the tangent line,"},{"Start":"00:40.835 ","End":"00:42.800","Text":"and we have to find its equation."},{"Start":"00:42.800 ","End":"00:44.915","Text":"I hope this picture helps a bit."},{"Start":"00:44.915 ","End":"00:49.985","Text":"But these exercises can be done without any visual aids."},{"Start":"00:49.985 ","End":"00:53.150","Text":"So let\u0027s get on with it."},{"Start":"00:53.150 ","End":"00:55.145","Text":"I\u0027ll write the formula, and you\u0027ll see."},{"Start":"00:55.145 ","End":"00:58.635","Text":"The formula for the tangent line,"},{"Start":"00:58.635 ","End":"01:03.600","Text":"y minus y_1 equals f prime"},{"Start":"01:03.600 ","End":"01:09.000","Text":"of x_1 times x minus x_1."},{"Start":"01:09.000 ","End":"01:12.200","Text":"First of all, I like to write the x and y in a different style"},{"Start":"01:12.200 ","End":"01:15.475","Text":"so as not to confuse with the x and the y from the curve."},{"Start":"01:15.475 ","End":"01:20.070","Text":"X_1 and y_1 are our particular x and y."},{"Start":"01:20.070 ","End":"01:22.070","Text":"So when I say x equals 3,"},{"Start":"01:22.070 ","End":"01:26.990","Text":"then I know right away that I mean that x_1 is equal to 3."},{"Start":"01:26.990 ","End":"01:28.775","Text":"It just means our x."},{"Start":"01:28.775 ","End":"01:33.070","Text":"Next thing we do is we find y_1,"},{"Start":"01:33.070 ","End":"01:35.320","Text":"which is the y that corresponds to this x,"},{"Start":"01:35.320 ","End":"01:39.055","Text":"which means that when x is 3, what is y?"},{"Start":"01:39.055 ","End":"01:44.790","Text":"Y_1 is f of 3, which is,"},{"Start":"01:44.790 ","End":"01:46.210","Text":"if I put 3 in here twice,"},{"Start":"01:46.210 ","End":"01:47.950","Text":"3 is 6 plus 3 is 9,"},{"Start":"01:47.950 ","End":"01:50.590","Text":"square root of 9 is just 3."},{"Start":"01:50.590 ","End":"01:54.355","Text":"So we have x_1 and then we have y_1."},{"Start":"01:54.355 ","End":"01:56.755","Text":"Next we need is f prime of x_1."},{"Start":"01:56.755 ","End":"01:58.930","Text":"Let\u0027s do f prime in general."},{"Start":"01:58.930 ","End":"02:02.605","Text":"F prime of x is equal to the derivative of this."},{"Start":"02:02.605 ","End":"02:05.110","Text":"Now, the derivative of the square root."},{"Start":"02:05.110 ","End":"02:09.955","Text":"Again, there is a formula for when you have square root of something."},{"Start":"02:09.955 ","End":"02:11.410","Text":"Let\u0027s call it, I don\u0027t know, box,"},{"Start":"02:11.410 ","End":"02:14.200","Text":"some function of x, and I want to know what the derivative of that is,"},{"Start":"02:14.200 ","End":"02:18.590","Text":"the formula is 1 over twice the square root"},{"Start":"02:18.590 ","End":"02:23.630","Text":"of whatever was in the box times the derivative of the box."},{"Start":"02:23.630 ","End":"02:25.670","Text":"Instead of writing it at the side,"},{"Start":"02:25.670 ","End":"02:28.700","Text":"I prefer to write it up on the top of the fraction."},{"Start":"02:28.700 ","End":"02:32.470","Text":"In our case, the box is 2x plus 3."},{"Start":"02:32.470 ","End":"02:35.955","Text":"So I have here 1 over,"},{"Start":"02:35.955 ","End":"02:37.620","Text":"in fact, it\u0027s not 1."},{"Start":"02:37.620 ","End":"02:40.970","Text":"It\u0027s the derivative of its box prime,"},{"Start":"02:40.970 ","End":"02:45.410","Text":"and box prime is 2x plus 3 derivative,"},{"Start":"02:45.410 ","End":"02:47.125","Text":"which is just 2."},{"Start":"02:47.125 ","End":"02:50.645","Text":"That\u0027s the 2. Then on the denominator,"},{"Start":"02:50.645 ","End":"02:54.110","Text":"twice the square root of,"},{"Start":"02:54.110 ","End":"02:56.700","Text":"again, the box was 2x plus 3."},{"Start":"02:56.700 ","End":"02:59.340","Text":"This 2 and this 2 cancels,"},{"Start":"02:59.340 ","End":"03:05.355","Text":"I can say it\u0027s just 1 over the square root of 2x plus 3."},{"Start":"03:05.355 ","End":"03:08.520","Text":"What is f prime of x_1?"},{"Start":"03:08.520 ","End":"03:09.780","Text":"X_1 is 3,"},{"Start":"03:09.780 ","End":"03:12.930","Text":"f prime of 3 is equal to,"},{"Start":"03:12.930 ","End":"03:15.825","Text":"we\u0027ve already done the square root when x is 3,"},{"Start":"03:15.825 ","End":"03:17.830","Text":"and we got the answer 3."},{"Start":"03:17.830 ","End":"03:20.345","Text":"We did that above,"},{"Start":"03:20.345 ","End":"03:24.075","Text":"so that\u0027s equal to 1 over 3, 1/3."},{"Start":"03:24.075 ","End":"03:26.400","Text":"Now, we have all the ingredients,"},{"Start":"03:26.400 ","End":"03:27.900","Text":"the x_1, y_1,"},{"Start":"03:27.900 ","End":"03:29.010","Text":"and the f prime of x_1."},{"Start":"03:29.010 ","End":"03:31.215","Text":"Those are the 3 things that we need to get."},{"Start":"03:31.215 ","End":"03:36.210","Text":"So we just go ahead and substitute in this formula."},{"Start":"03:36.210 ","End":"03:43.860","Text":"So we get the tangent that Y minus y_1 is 3 is equal to f prime of x_1,"},{"Start":"03:43.860 ","End":"03:45.705","Text":"we found as 1/3,"},{"Start":"03:45.705 ","End":"03:51.435","Text":"times x minus and our x_1 is equal to also 3."},{"Start":"03:51.435 ","End":"03:54.620","Text":"3 is in here. That\u0027s actually the answer,"},{"Start":"03:54.620 ","End":"03:59.015","Text":"but it\u0027s customary to simplify and isolate y,"},{"Start":"03:59.015 ","End":"04:05.210","Text":"so y equals 1/3 of x from here minus 1/3 times 3 is 1,"},{"Start":"04:05.210 ","End":"04:08.745","Text":"minus 1 plus 3 is plus 4,"},{"Start":"04:08.745 ","End":"04:11.650","Text":"and this is the answer."}],"ID":4348},{"Watched":false,"Name":"Exercise 6","Duration":"6m 53s","ChapterTopicVideoID":4340,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have to find the equation of"},{"Start":"00:03.480 ","End":"00:07.635","Text":"the tangent line to this curve at the origin."},{"Start":"00:07.635 ","End":"00:11.250","Text":"That\u0027s one part of the question and then we\u0027re also asked,"},{"Start":"00:11.250 ","End":"00:13.275","Text":"does this line, this tangent line,"},{"Start":"00:13.275 ","End":"00:15.270","Text":"intersect the curve at any other point?"},{"Start":"00:15.270 ","End":"00:17.595","Text":"Let me just make a couple of remarks."},{"Start":"00:17.595 ","End":"00:21.435","Text":"First of all, we don\u0027t know that it is at the origin that has to be checked."},{"Start":"00:21.435 ","End":"00:24.210","Text":"But it\u0027s easy to see that if you put x equals 0,"},{"Start":"00:24.210 ","End":"00:27.810","Text":"then f of x is also equal to 0."},{"Start":"00:27.810 ","End":"00:30.795","Text":"I\u0027ll call this y."},{"Start":"00:30.795 ","End":"00:35.900","Text":"Second thing is that I\u0027ve made a sketch which I\u0027d like to show you."},{"Start":"00:35.900 ","End":"00:38.690","Text":"It might help you imagine what\u0027s going on here."},{"Start":"00:38.690 ","End":"00:42.350","Text":"Here\u0027s the x and y axis there we have the function,"},{"Start":"00:42.350 ","End":"00:45.380","Text":"the curve xe to the x squared."},{"Start":"00:45.380 ","End":"00:48.330","Text":"I just want to label it to see what\u0027s going on."},{"Start":"00:48.330 ","End":"00:49.800","Text":"If this is the curve,"},{"Start":"00:49.800 ","End":"00:52.490","Text":"this is the origin obviously,"},{"Start":"00:52.490 ","End":"00:56.210","Text":"but I\u0027m mentioning it because that\u0027s the point that the curve goes"},{"Start":"00:56.210 ","End":"01:01.600","Text":"through and that\u0027s where we have the tangent and this is the tangent line."},{"Start":"01:01.600 ","End":"01:03.765","Text":"We can see that it say,"},{"Start":"01:03.765 ","End":"01:06.300","Text":"the tangent cuts the curve at"},{"Start":"01:06.300 ","End":"01:10.870","Text":"the origin and then we\u0027re asked whether it intersects to anywhere else."},{"Start":"01:10.870 ","End":"01:12.350","Text":"According to this picture,"},{"Start":"01:12.350 ","End":"01:13.635","Text":"the answer would be no."},{"Start":"01:13.635 ","End":"01:18.665","Text":"Let\u0027s get back there and do everything algebraically."},{"Start":"01:18.665 ","End":"01:24.110","Text":"Now, we\u0027re going to use our standard formula for the tangent."},{"Start":"01:24.110 ","End":"01:31.265","Text":"The formula for the tangent line is that Y minus"},{"Start":"01:31.265 ","End":"01:41.355","Text":"y1 is equal to f prime of x1 times X minus x1."},{"Start":"01:41.355 ","End":"01:43.365","Text":"Now what does this mean?"},{"Start":"01:43.365 ","End":"01:46.700","Text":"x1 and y1 are the point in question."},{"Start":"01:46.700 ","End":"01:50.210","Text":"Well, in our case, we\u0027ve already said that x1,"},{"Start":"01:50.210 ","End":"01:54.860","Text":"y1 is 0,0 it\u0027s the origin and we even checked that"},{"Start":"01:54.860 ","End":"01:59.420","Text":"when you substitute x1 equals 0 in here,"},{"Start":"01:59.420 ","End":"02:02.900","Text":"that we really do get y1 equals 0."},{"Start":"02:02.900 ","End":"02:06.120","Text":"This we have the X and the Y,"},{"Start":"02:06.120 ","End":"02:10.040","Text":"I\u0027ve chosen a different style so as not to confuse with the curve."},{"Start":"02:10.040 ","End":"02:12.650","Text":"All we\u0027re missing is f prime of x1,"},{"Start":"02:12.650 ","End":"02:20.240","Text":"so let\u0027s do the general differentiation and I\u0027m going to use the product rule on this."},{"Start":"02:20.240 ","End":"02:22.010","Text":"Product rule is that,"},{"Start":"02:22.010 ","End":"02:23.690","Text":"if you have 2 functions of x,"},{"Start":"02:23.690 ","End":"02:28.400","Text":"let\u0027s say u and v, I don\u0027t want to use f and g because f is already used there."},{"Start":"02:28.400 ","End":"02:34.175","Text":"So u and v derivative is the first one derived"},{"Start":"02:34.175 ","End":"02:40.290","Text":"and the second one not and then the first one as is and the second one derived."},{"Start":"02:40.290 ","End":"02:48.810","Text":"In our case, the u is the x part and the V is the e to the x squared part."},{"Start":"02:49.090 ","End":"02:58.775","Text":"Using this formula, what we\u0027re going to get is that f prime of x is equal to u prime,"},{"Start":"02:58.775 ","End":"02:59.900","Text":"which is x prime,"},{"Start":"02:59.900 ","End":"03:02.705","Text":"which is 1 times the other 1,"},{"Start":"03:02.705 ","End":"03:05.015","Text":"e to the x squared as is,"},{"Start":"03:05.015 ","End":"03:10.115","Text":"and then the x as is and e to the x squared derived."},{"Start":"03:10.115 ","End":"03:12.215","Text":"Well, it\u0027s a simple chain rule."},{"Start":"03:12.215 ","End":"03:16.760","Text":"Normally e to the something because derivative is just e to the something."},{"Start":"03:16.760 ","End":"03:19.580","Text":"If it\u0027s x, but if it\u0027s a function of x,"},{"Start":"03:19.580 ","End":"03:21.889","Text":"we need to multiply by the internal derivative,"},{"Start":"03:21.889 ","End":"03:29.195","Text":"which is 2x and I suggest that we collect the e to the x squared,"},{"Start":"03:29.195 ","End":"03:30.560","Text":"take it outside the brackets."},{"Start":"03:30.560 ","End":"03:35.645","Text":"What we\u0027re left with is x times 2x is 2x squared plus the 1 from here."},{"Start":"03:35.645 ","End":"03:39.740","Text":"All this times e to the x squared."},{"Start":"03:39.740 ","End":"03:42.245","Text":"Now what we want is f prime at x1."},{"Start":"03:42.245 ","End":"03:44.045","Text":"Now our x1 is 0."},{"Start":"03:44.045 ","End":"03:47.600","Text":"So f prime of 0,"},{"Start":"03:47.600 ","End":"03:49.205","Text":"you put 0 here,"},{"Start":"03:49.205 ","End":"03:53.090","Text":"That\u0027s just 1 and 0 here, so we get e to the 0."},{"Start":"03:53.090 ","End":"03:55.350","Text":"The answer is 1."},{"Start":"03:55.640 ","End":"03:59.480","Text":"Now we can really get the formula of the tangent line."},{"Start":"03:59.480 ","End":"04:03.860","Text":"This is the tangent, this is the curve."},{"Start":"04:03.860 ","End":"04:10.010","Text":"This is the tangent is that y minus 0 is equal to"},{"Start":"04:10.010 ","End":"04:17.825","Text":"1 times x minus 0 and if we simplify this,"},{"Start":"04:17.825 ","End":"04:22.290","Text":"i.e it just comes down to y equals x,"},{"Start":"04:22.290 ","End":"04:28.165","Text":"which is the straight line at 45 degrees through the origin."},{"Start":"04:28.165 ","End":"04:33.835","Text":"Now we have a curve and a straight line and I summarize it again."},{"Start":"04:33.835 ","End":"04:35.845","Text":"So we have our curve."},{"Start":"04:35.845 ","End":"04:41.185","Text":"The curve is y equals xe to the x squared and we have"},{"Start":"04:41.185 ","End":"04:45.460","Text":"the line which is the tangent and we\u0027ll go back to"},{"Start":"04:45.460 ","End":"04:49.780","Text":"the regular y and x and the question is,"},{"Start":"04:49.780 ","End":"04:51.085","Text":"where do they intersect?"},{"Start":"04:51.085 ","End":"04:53.320","Text":"Now, we\u0027re looking for the intersection."},{"Start":"04:53.320 ","End":"04:54.805","Text":"When we look for intersection,"},{"Start":"04:54.805 ","End":"04:56.470","Text":"we just have to compare them because at"},{"Start":"04:56.470 ","End":"04:59.410","Text":"the intersection the x is the same and the y is the same."},{"Start":"04:59.410 ","End":"05:01.355","Text":"So we have to find out which x?"},{"Start":"05:01.355 ","End":"05:07.045","Text":"Basically, we have to compare this part with this part and get an equation."},{"Start":"05:07.045 ","End":"05:12.755","Text":"Let\u0027s try that, xe to the x squared is equal to x."},{"Start":"05:12.755 ","End":"05:19.930","Text":"From here I can bring x to this side and then take it outside the brackets,"},{"Start":"05:19.930 ","End":"05:26.015","Text":"xe to the x squared minus 1 is equal to 0."},{"Start":"05:26.015 ","End":"05:31.685","Text":"Now, let\u0027s suppose there was a solution other than 0,"},{"Start":"05:31.685 ","End":"05:33.380","Text":"0 we know fits,"},{"Start":"05:33.380 ","End":"05:34.955","Text":"but if it isn\u0027t 0,"},{"Start":"05:34.955 ","End":"05:38.985","Text":"then I can cancel by x because it\u0027s not 0, but x."},{"Start":"05:38.985 ","End":"05:41.450","Text":"Let\u0027s look for x not equal to 0."},{"Start":"05:41.450 ","End":"05:46.820","Text":"So we get e to the x squared minus 1 equals"},{"Start":"05:46.820 ","End":"05:52.940","Text":"0 and let me skip a step and then take that minus 1 and bring it as 1 on the other side."},{"Start":"05:52.940 ","End":"05:57.245","Text":"Now the only number that e to the power of it is 1 is 0."},{"Start":"05:57.245 ","End":"05:58.850","Text":"E to the 0 is 1."},{"Start":"05:58.850 ","End":"06:02.465","Text":"The other way is to take the natural logarithm of this."},{"Start":"06:02.465 ","End":"06:04.475","Text":"In any event you get 0."},{"Start":"06:04.475 ","End":"06:07.370","Text":"So we get that x squared equals 0."},{"Start":"06:07.370 ","End":"06:09.729","Text":"So x is 0."},{"Start":"06:09.729 ","End":"06:12.515","Text":"If x is not 0 then x is 0."},{"Start":"06:12.515 ","End":"06:14.780","Text":"But in any case, x is 0,"},{"Start":"06:14.780 ","End":"06:17.495","Text":"so there is no other point."},{"Start":"06:17.495 ","End":"06:22.105","Text":"I just want to write that. How did they phrase the question? Does the line?"},{"Start":"06:22.105 ","End":"06:30.295","Text":"Well, I\u0027ll just say no answer to question and to say, no other intersection."},{"Start":"06:30.295 ","End":"06:37.700","Text":"Let\u0027s see, what did they ask, what the equation of the line and the question."},{"Start":"06:37.700 ","End":"06:42.810","Text":"The equation of the tangent is Y equals X."},{"Start":"06:42.810 ","End":"06:44.245","Text":"Maybe I\u0027ll highlight that,"},{"Start":"06:44.245 ","End":"06:49.085","Text":"this is the tangent line and this is the answer to the question."},{"Start":"06:49.085 ","End":"06:53.250","Text":"The question is no and we\u0027re done."}],"ID":4350},{"Watched":false,"Name":"Exercise 7","Duration":"5m 8s","ChapterTopicVideoID":4341,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this exercise, we have to find the equation of the tangent to"},{"Start":"00:03.960 ","End":"00:09.269","Text":"the curve given by this formula at the point on the curve where x equals Pi."},{"Start":"00:09.269 ","End":"00:10.650","Text":"Just to help you,"},{"Start":"00:10.650 ","End":"00:14.980","Text":"I\u0027ve prepared a sketch and we\u0027ll take a look at it."},{"Start":"00:15.800 ","End":"00:19.680","Text":"This part here is the curve."},{"Start":"00:19.680 ","End":"00:21.840","Text":"Notice there was a sine,"},{"Start":"00:21.840 ","End":"00:29.550","Text":"sine of 4x and that\u0027s what makes it periodic and it was e to the power of something."},{"Start":"00:29.550 ","End":"00:32.910","Text":"Then the event, there is a point here where"},{"Start":"00:32.910 ","End":"00:37.035","Text":"x equals Pi that gives us a point on the curve,"},{"Start":"00:37.035 ","End":"00:39.525","Text":"and at that point there\u0027s a tangent."},{"Start":"00:39.525 ","End":"00:41.655","Text":"This is the tangent,"},{"Start":"00:41.655 ","End":"00:44.845","Text":"and we have to find the equation of this tangent,"},{"Start":"00:44.845 ","End":"00:46.500","Text":"draw the standard formula."},{"Start":"00:46.500 ","End":"00:48.765","Text":"Let\u0027s get back to the algebra."},{"Start":"00:48.765 ","End":"00:51.185","Text":"The picture just helps to give you an idea,"},{"Start":"00:51.185 ","End":"00:52.715","Text":"but for those who don\u0027t need it,"},{"Start":"00:52.715 ","End":"00:54.380","Text":"then forget about it."},{"Start":"00:54.380 ","End":"00:57.490","Text":"We\u0027re back at the exercise."},{"Start":"00:57.490 ","End":"00:59.775","Text":"This is the equation."},{"Start":"00:59.775 ","End":"01:01.830","Text":"I also like to call f of x,"},{"Start":"01:01.830 ","End":"01:05.535","Text":"sometimes which I also denote it as y."},{"Start":"01:05.535 ","End":"01:08.270","Text":"Let\u0027s just put that in the case when you need that form."},{"Start":"01:08.270 ","End":"01:12.245","Text":"I\u0027ll start out by writing the formula for the tangent."},{"Start":"01:12.245 ","End":"01:19.560","Text":"The formula is Y minus y_1 is"},{"Start":"01:19.560 ","End":"01:27.795","Text":"equal to f prime of x_1 times X minus x_1."},{"Start":"01:27.795 ","End":"01:33.500","Text":"Now, x_1 and y_1 are the coordinates of our particular point."},{"Start":"01:33.500 ","End":"01:35.750","Text":"We know that x_1 is Pi,"},{"Start":"01:35.750 ","End":"01:39.020","Text":"and what we\u0027re going to do is find the other things that we need."},{"Start":"01:39.020 ","End":"01:40.525","Text":"We\u0027re going to find y_1,"},{"Start":"01:40.525 ","End":"01:42.370","Text":"we\u0027re going to find f prime of x_1,"},{"Start":"01:42.370 ","End":"01:43.685","Text":"and then we\u0027re going to plug it in."},{"Start":"01:43.685 ","End":"01:49.280","Text":"Of course, Y and X remain as Y and X because formula for equation of a line."},{"Start":"01:49.280 ","End":"01:52.310","Text":"y_1 is just f of x_1."},{"Start":"01:52.310 ","End":"01:53.935","Text":"y_1 is like,"},{"Start":"01:53.935 ","End":"01:56.600","Text":"on the graph Pi was below on the curve,"},{"Start":"01:56.600 ","End":"01:57.965","Text":"we need f of Pi."},{"Start":"01:57.965 ","End":"01:59.420","Text":"Now, what is f of Pi?"},{"Start":"01:59.420 ","End":"02:02.440","Text":"That means that plugging Pi into here."},{"Start":"02:02.440 ","End":"02:07.335","Text":"It\u0027s e to the power of sine of 4 Pi."},{"Start":"02:07.335 ","End":"02:09.435","Text":"Now, what is the sine of 4 Pi?"},{"Start":"02:09.435 ","End":"02:12.530","Text":"Well, the sine is periodic of 2 Pi."},{"Start":"02:12.530 ","End":"02:14.180","Text":"Sine of 0, 2 Pi,"},{"Start":"02:14.180 ","End":"02:16.905","Text":"4 Pi, 6 Pi, they\u0027re all 0."},{"Start":"02:16.905 ","End":"02:19.980","Text":"A 2 Pi shift is like sine of 0."},{"Start":"02:19.980 ","End":"02:23.760","Text":"If this is 0, e^0 is 1."},{"Start":"02:23.760 ","End":"02:25.400","Text":"Now we also have y_1,"},{"Start":"02:25.400 ","End":"02:29.320","Text":"but I just like to write the point in coordinate,"},{"Start":"02:29.320 ","End":"02:31.935","Text":"which is x_1, y_1,"},{"Start":"02:31.935 ","End":"02:35.680","Text":"and this is Pi, 1."},{"Start":"02:35.680 ","End":"02:40.835","Text":"The only thing missing now is the f prime of x_1."},{"Start":"02:40.835 ","End":"02:47.655","Text":"Let\u0027s find f prime in general and then substitute the particular x_1 into it."},{"Start":"02:47.655 ","End":"02:53.270","Text":"This is a straightforward formula for e to the power of something."},{"Start":"02:53.270 ","End":"02:58.474","Text":"I think we can handle it without writing formulae at this stage."},{"Start":"02:58.474 ","End":"03:00.785","Text":"If it\u0027s e to the something,"},{"Start":"03:00.785 ","End":"03:08.140","Text":"then the derivative is also equal to e to that something for a start."},{"Start":"03:08.140 ","End":"03:10.430","Text":"But because this is not x,"},{"Start":"03:10.430 ","End":"03:11.570","Text":"but it\u0027s a function of x,"},{"Start":"03:11.570 ","End":"03:13.460","Text":"we need the internal derivative."},{"Start":"03:13.460 ","End":"03:15.920","Text":"We need the derivative of sine 4x,"},{"Start":"03:15.920 ","End":"03:18.620","Text":"now what\u0027s the derivative of sine 4x?"},{"Start":"03:18.620 ","End":"03:21.655","Text":"Well, derivative of sine is cosine."},{"Start":"03:21.655 ","End":"03:23.670","Text":"It\u0027s cosine 4x,"},{"Start":"03:23.670 ","End":"03:25.800","Text":"but that\u0027s not all,"},{"Start":"03:25.800 ","End":"03:31.020","Text":"because it\u0027s 4x and it\u0027s not x so there but there is still another inner derivative,"},{"Start":"03:31.020 ","End":"03:34.250","Text":"and you have to multiply all this by 4."},{"Start":"03:34.250 ","End":"03:36.415","Text":"F prime of Pi,"},{"Start":"03:36.415 ","End":"03:38.340","Text":"which is what we want here,"},{"Start":"03:38.340 ","End":"03:41.450","Text":"f prime of Pi, let\u0027s see if x is Pi."},{"Start":"03:41.450 ","End":"03:42.935","Text":"We already did this bit."},{"Start":"03:42.935 ","End":"03:45.965","Text":"This e to this, is e^0 is 1."},{"Start":"03:45.965 ","End":"03:48.575","Text":"This part was 1,"},{"Start":"03:48.575 ","End":"03:53.170","Text":"cosine of 4x is cosine of 4Pi,"},{"Start":"03:53.170 ","End":"03:55.895","Text":"is cosine of 0, which is also 1."},{"Start":"03:55.895 ","End":"03:57.410","Text":"Here we have a 4."},{"Start":"03:57.410 ","End":"04:00.800","Text":"Let\u0027s see what is 1 times 1 times 4?"},{"Start":"04:00.800 ","End":"04:03.135","Text":"That equals 4."},{"Start":"04:03.135 ","End":"04:08.055","Text":"F prime of Pi is equal to 4."},{"Start":"04:08.055 ","End":"04:10.070","Text":"Now we have all the ingredients."},{"Start":"04:10.070 ","End":"04:13.730","Text":"We have y_1 and x_1 here,"},{"Start":"04:13.730 ","End":"04:17.330","Text":"and we have the f prime of x_1 here."},{"Start":"04:17.330 ","End":"04:25.505","Text":"All we have to do now is substitute that Y minus y_1 is 1."},{"Start":"04:25.505 ","End":"04:30.485","Text":"Y minus y_1 is equal to f prime of x_1,"},{"Start":"04:30.485 ","End":"04:35.860","Text":"which is 4 times X minus the x_1,"},{"Start":"04:35.860 ","End":"04:39.625","Text":"which is X minus Pi."},{"Start":"04:39.625 ","End":"04:41.480","Text":"That is the answer,"},{"Start":"04:41.480 ","End":"04:43.565","Text":"but we\u0027ll tidy it up a little bit,"},{"Start":"04:43.565 ","End":"04:49.010","Text":"just customary to give Y in terms of X. Y is equal to"},{"Start":"04:49.010 ","End":"04:55.295","Text":"4X minus 4Pi plus 1."},{"Start":"04:55.295 ","End":"04:59.130","Text":"This is the equation of the tangent line there."},{"Start":"04:59.440 ","End":"05:02.980","Text":"I think that\u0027s all they asked for."},{"Start":"05:02.980 ","End":"05:06.350","Text":"They asked for the tangent of the curve at this point."},{"Start":"05:06.350 ","End":"05:09.090","Text":"Well, this is it, the end."}],"ID":4351},{"Watched":false,"Name":"Exercise 8","Duration":"11m 34s","ChapterTopicVideoID":4342,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"This exercise is for the more advanced students,"},{"Start":"00:03.420 ","End":"00:04.840","Text":"and you know who you are."},{"Start":"00:04.840 ","End":"00:13.035","Text":"In it, we have to find the equation of the lines that are normal to the curve so and so,"},{"Start":"00:13.035 ","End":"00:18.930","Text":"but parallel to the line so and so, normal and parallel."},{"Start":"00:18.930 ","End":"00:20.370","Text":"Now, first of all,"},{"Start":"00:20.370 ","End":"00:22.770","Text":"let me show you a sketch just to help because I\u0027ve"},{"Start":"00:22.770 ","End":"00:26.070","Text":"prepared it in advance and I\u0027d like to show it to you."},{"Start":"00:26.070 ","End":"00:28.620","Text":"For those of you who don\u0027t like visual stuff,"},{"Start":"00:28.620 ","End":"00:30.345","Text":"you don\u0027t have to look."},{"Start":"00:30.345 ","End":"00:33.030","Text":"Now, the curve happens to be what is called a"},{"Start":"00:33.030 ","End":"00:36.120","Text":"hyperbola which actually comes in 2 pieces,"},{"Start":"00:36.120 ","End":"00:38.620","Text":"2 branches, if you like."},{"Start":"00:38.620 ","End":"00:40.775","Text":"It\u0027s in brown,"},{"Start":"00:40.775 ","End":"00:42.440","Text":"this is the curve,"},{"Start":"00:42.440 ","End":"00:45.605","Text":"and this is also part of the curve."},{"Start":"00:45.605 ","End":"00:47.270","Text":"This is the line,"},{"Start":"00:47.270 ","End":"00:48.980","Text":"I\u0027ll call it a reference line."},{"Start":"00:48.980 ","End":"00:51.155","Text":"We want the normals to be parallel to it."},{"Start":"00:51.155 ","End":"00:57.380","Text":"Reference for the concept of parallel."},{"Start":"00:57.380 ","End":"01:02.270","Text":"Now, a normal to a curve is something"},{"Start":"01:02.270 ","End":"01:07.925","Text":"through a point is the line that is perpendicular to the tangent."},{"Start":"01:07.925 ","End":"01:11.180","Text":"In this case, we\u0027ve sketched the answer."},{"Start":"01:11.180 ","End":"01:16.545","Text":"This is a tangent at this point,"},{"Start":"01:16.545 ","End":"01:20.295","Text":"and here, we have another tangent."},{"Start":"01:20.295 ","End":"01:23.510","Text":"This is not what we\u0027re looking for but this just"},{"Start":"01:23.510 ","End":"01:26.885","Text":"simply helps to explain what I mean by normal."},{"Start":"01:26.885 ","End":"01:29.060","Text":"These are the 2 things we\u0027re looking for."},{"Start":"01:29.060 ","End":"01:32.875","Text":"This is the normal corresponding to this tangent,"},{"Start":"01:32.875 ","End":"01:35.380","Text":"meaning, it\u0027s perpendicular to it."},{"Start":"01:35.380 ","End":"01:37.735","Text":"This is the normal."},{"Start":"01:37.735 ","End":"01:40.490","Text":"These are the 2 things we\u0027ll be looking for."},{"Start":"01:40.490 ","End":"01:44.525","Text":"We don\u0027t know there\u0027s 2 of them when we go to do the algebraic part."},{"Start":"01:44.525 ","End":"01:46.685","Text":"I hope this picture helps."},{"Start":"01:46.685 ","End":"01:49.820","Text":"This point and at this point, we get the tangent,"},{"Start":"01:49.820 ","End":"01:52.785","Text":"we get the normal, and this other normal,"},{"Start":"01:52.785 ","End":"01:56.300","Text":"and each case, it\u0027s parallel to this reference line."},{"Start":"01:56.300 ","End":"02:02.880","Text":"Let\u0027s go back, parallel lines are the ones that have the same slope."},{"Start":"02:02.880 ","End":"02:09.935","Text":"What I suggest is to start with the formula for a normal to the curve."},{"Start":"02:09.935 ","End":"02:15.960","Text":"The standard formula is the y minus y_1."},{"Start":"02:15.960 ","End":"02:18.225","Text":"I\u0027ll write it first and we\u0027ll explain it."},{"Start":"02:18.225 ","End":"02:23.580","Text":"y minus y_1 equals minus 1 over"},{"Start":"02:23.580 ","End":"02:30.945","Text":"f prime of x_1 times x minus x_1."},{"Start":"02:30.945 ","End":"02:35.250","Text":"Now, x_1, y_1 are usually, given points."},{"Start":"02:35.250 ","End":"02:36.590","Text":"Only here, they\u0027re not given,"},{"Start":"02:36.590 ","End":"02:38.045","Text":"we have to find them."},{"Start":"02:38.045 ","End":"02:40.700","Text":"Those would be those particular points in the sketch that"},{"Start":"02:40.700 ","End":"02:43.325","Text":"you saw where the normals went through."},{"Start":"02:43.325 ","End":"02:46.550","Text":"Now, if this is the equation of the normal,"},{"Start":"02:46.550 ","End":"02:50.015","Text":"this thing here is the slope."},{"Start":"02:50.015 ","End":"02:52.955","Text":"This would be the slope."},{"Start":"02:52.955 ","End":"02:55.460","Text":"As for the line, well,"},{"Start":"02:55.460 ","End":"03:01.670","Text":"in elementary analytical, the coefficient of x is the slope."},{"Start":"03:01.670 ","End":"03:04.550","Text":"Over here, this is the slope of the line,"},{"Start":"03:04.550 ","End":"03:07.115","Text":"I\u0027ll just write the word slope again."},{"Start":"03:07.115 ","End":"03:11.170","Text":"What we have to do is compare the slopes,"},{"Start":"03:11.170 ","End":"03:13.910","Text":"because for this line to be parallel to this line,"},{"Start":"03:13.910 ","End":"03:17.030","Text":"all it has to be is that this slope and this slope are the same."},{"Start":"03:17.030 ","End":"03:22.595","Text":"We get the equation that minus 1 over"},{"Start":"03:22.595 ","End":"03:29.760","Text":"f prime of x_1 is equal to minus 2."},{"Start":"03:29.760 ","End":"03:34.550","Text":"We don\u0027t have f prime yet, let\u0027s compute it."},{"Start":"03:34.550 ","End":"03:37.080","Text":"Let\u0027s also rewrite this in the form."},{"Start":"03:37.080 ","End":"03:41.600","Text":"If I bring this to the right side and bring this over here,"},{"Start":"03:41.600 ","End":"03:48.490","Text":"we get that f prime of x_1 is equal to 1 over 2."},{"Start":"03:48.490 ","End":"03:51.500","Text":"Now, what is the derivative?"},{"Start":"03:51.500 ","End":"03:57.440","Text":"If I do the derivative of this function whose domain is got to be,"},{"Start":"03:57.440 ","End":"04:00.860","Text":"of course, where x is not equal to 1,"},{"Start":"04:00.860 ","End":"04:03.530","Text":"otherwise, that would be a denominator 0."},{"Start":"04:03.530 ","End":"04:05.210","Text":"Also, in case I need it,"},{"Start":"04:05.210 ","End":"04:07.190","Text":"I like to write also that f of x,"},{"Start":"04:07.190 ","End":"04:10.080","Text":"I sometimes use it as y."},{"Start":"04:10.330 ","End":"04:13.190","Text":"This is the equation we have to solve,"},{"Start":"04:13.190 ","End":"04:15.635","Text":"and we have to differentiate this."},{"Start":"04:15.635 ","End":"04:18.095","Text":"We\u0027ll use the quotient rule."},{"Start":"04:18.095 ","End":"04:26.565","Text":"The quotient rule I want to remind you is this is the quotient rule for differentiation."},{"Start":"04:26.565 ","End":"04:30.530","Text":"It says that if you have 2 functions of x,"},{"Start":"04:30.530 ","End":"04:32.825","Text":"say, u over v,"},{"Start":"04:32.825 ","End":"04:34.870","Text":"and you want to differentiate that,"},{"Start":"04:34.870 ","End":"04:39.220","Text":"then the answer is that you take the first 1 derived,"},{"Start":"04:39.220 ","End":"04:40.735","Text":"the second as is,"},{"Start":"04:40.735 ","End":"04:42.970","Text":"minus the first 1 as is,"},{"Start":"04:42.970 ","End":"04:44.850","Text":"the second 1 derived,"},{"Start":"04:44.850 ","End":"04:47.805","Text":"over the denominator squared."},{"Start":"04:47.805 ","End":"04:56.625","Text":"In our case, this will be u and this will be v. Let\u0027s get to it."},{"Start":"04:56.625 ","End":"05:02.280","Text":"f prime of x is equal to."},{"Start":"05:02.280 ","End":"05:06.605","Text":"I\u0027ll even write it again at the side here that will help us,"},{"Start":"05:06.605 ","End":"05:12.840","Text":"that u is 2x and v is 1 minus x."},{"Start":"05:12.840 ","End":"05:15.930","Text":"Let\u0027s have it right here [inaudible] to keep going up there."},{"Start":"05:15.930 ","End":"05:20.280","Text":"f prime of x is u prime which is 2,"},{"Start":"05:20.280 ","End":"05:24.000","Text":"times v which is 1 minus x,"},{"Start":"05:24.000 ","End":"05:27.495","Text":"minus u which is 2x,"},{"Start":"05:27.495 ","End":"05:31.185","Text":"times v prime which is just minus 1,"},{"Start":"05:31.185 ","End":"05:36.510","Text":"all over v which is 1 minus x, squared."},{"Start":"05:36.510 ","End":"05:38.910","Text":"Let\u0027s simplify this a bit,"},{"Start":"05:38.910 ","End":"05:40.650","Text":"let\u0027s see what we can get."},{"Start":"05:40.650 ","End":"05:45.810","Text":"2 minus 2x plus 2x,"},{"Start":"05:45.810 ","End":"05:52.230","Text":"2 over 1 minus x squared."},{"Start":"05:52.230 ","End":"05:57.260","Text":"We\u0027re just about ready because this is the equation that we want solved to."},{"Start":"05:57.260 ","End":"05:59.255","Text":"It\u0027s actually an equation in x_1."},{"Start":"05:59.255 ","End":"06:04.775","Text":"I\u0027ll just put x_1 in here from this equation which I\u0027ll maybe highlight,"},{"Start":"06:04.775 ","End":"06:06.810","Text":"this is the equation,"},{"Start":"06:06.810 ","End":"06:08.995","Text":"we\u0027re looking for x_1."},{"Start":"06:08.995 ","End":"06:11.765","Text":"If I write this equation out,"},{"Start":"06:11.765 ","End":"06:22.645","Text":"I will get that 2 over 1 minus x_1 squared is equal to 1/2."},{"Start":"06:22.645 ","End":"06:27.635","Text":"It looks simple enough algebraic thing to solve."},{"Start":"06:27.635 ","End":"06:29.450","Text":"Let\u0027s try cross-multiplying."},{"Start":"06:29.450 ","End":"06:32.120","Text":"When you have 2 fractions are equal,"},{"Start":"06:32.120 ","End":"06:33.890","Text":"you can cross-multiply it."},{"Start":"06:33.890 ","End":"06:35.495","Text":"Let\u0027s do the other 1 first."},{"Start":"06:35.495 ","End":"06:39.225","Text":"1 minus x_1 squared,"},{"Start":"06:39.225 ","End":"06:44.480","Text":"this times this, has got to equal 2 times 2 which is 4."},{"Start":"06:44.480 ","End":"06:47.720","Text":"We could make a quadratic equation out of it but the easiest thing is"},{"Start":"06:47.720 ","End":"06:51.155","Text":"just to take the square root with the plus and minus."},{"Start":"06:51.155 ","End":"06:56.825","Text":"What we get is 1 minus x_1 is plus or minus 2."},{"Start":"06:56.825 ","End":"06:59.000","Text":"Looks like we\u0027re getting 2 answers and that\u0027s not"},{"Start":"06:59.000 ","End":"07:02.270","Text":"surprising since we saw the sketch and there were 2 answers."},{"Start":"07:02.270 ","End":"07:05.300","Text":"Let\u0027s take case a and case b."},{"Start":"07:05.300 ","End":"07:08.870","Text":"If we take the plus 2 and if we take the minus 2,"},{"Start":"07:08.870 ","End":"07:10.430","Text":"we\u0027ll get 2 different paths."},{"Start":"07:10.430 ","End":"07:13.295","Text":"If we take 1 minus x_1 is 2,"},{"Start":"07:13.295 ","End":"07:20.185","Text":"then we get that x_1 is 1 minus 2 is minus 1."},{"Start":"07:20.185 ","End":"07:23.820","Text":"If we take 1 minus x_1 is minus 2,"},{"Start":"07:23.820 ","End":"07:27.195","Text":"then we\u0027ll get x_1 is 3."},{"Start":"07:27.195 ","End":"07:29.290","Text":"When we have x_1,"},{"Start":"07:29.290 ","End":"07:31.640","Text":"we already have f prime of x_1,"},{"Start":"07:31.640 ","End":"07:35.400","Text":"and all we need then is y_1."},{"Start":"07:35.400 ","End":"07:37.665","Text":"y is a function of x."},{"Start":"07:37.665 ","End":"07:40.590","Text":"Here, we\u0027ll get this."},{"Start":"07:40.590 ","End":"07:44.910","Text":"y_1, here will be f of x_1,"},{"Start":"07:44.910 ","End":"07:48.015","Text":"will be f of minus 1."},{"Start":"07:48.015 ","End":"07:49.815","Text":"Just do them in parallel."},{"Start":"07:49.815 ","End":"07:54.675","Text":"Here, y_1 will equal f of 3."},{"Start":"07:54.675 ","End":"07:57.720","Text":"Now, where\u0027s our original equation?"},{"Start":"07:57.720 ","End":"08:05.240","Text":"f of x or y is equal to 2x over 1 minus x,"},{"Start":"08:05.240 ","End":"08:07.865","Text":"and that\u0027s also equal to y."},{"Start":"08:07.865 ","End":"08:11.465","Text":"In our case, y_1 is f of x_1."},{"Start":"08:11.465 ","End":"08:14.675","Text":"If I plug in minus 1 in here,"},{"Start":"08:14.675 ","End":"08:17.410","Text":"what do we get if it\u0027s minus 1?"},{"Start":"08:17.410 ","End":"08:24.630","Text":"We get minus 2 over 2 which is minus 1."},{"Start":"08:24.630 ","End":"08:29.835","Text":"If we plug in 3, it\u0027s minus 3."},{"Start":"08:29.835 ","End":"08:32.280","Text":"The last part we need,"},{"Start":"08:32.280 ","End":"08:38.480","Text":"the last ingredient is the coefficient this here,"},{"Start":"08:38.480 ","End":"08:41.030","Text":"this minus 1 over f prime of x_1."},{"Start":"08:41.030 ","End":"08:42.785","Text":"But that, we already know,"},{"Start":"08:42.785 ","End":"08:45.715","Text":"is got to be equal minus 2."},{"Start":"08:45.715 ","End":"08:53.580","Text":"Minus 1 over f prime of x_1 has got to equal minus 2."},{"Start":"08:53.580 ","End":"08:59.575","Text":"Likewise, just write same thing here, it\u0027s minus 2."},{"Start":"08:59.575 ","End":"09:04.550","Text":"That\u0027s what we started off from is the minus 1 over f prime of x_1."},{"Start":"09:04.550 ","End":"09:11.135","Text":"Now, each of these lines gives rise to a tangent."},{"Start":"09:11.135 ","End":"09:13.190","Text":"We get 2 possibilities."},{"Start":"09:13.190 ","End":"09:17.955","Text":"I mean, let\u0027s say this is a and this is possibility b."},{"Start":"09:17.955 ","End":"09:23.900","Text":"We get the normal lines,"},{"Start":"09:23.900 ","End":"09:30.875","Text":"so we have normal a which will be from this formula."},{"Start":"09:30.875 ","End":"09:35.360","Text":"Y minus the y_1 which is minus minus 1,"},{"Start":"09:35.360 ","End":"09:40.630","Text":"which I\u0027ll have myself to write as plus 1 equals"},{"Start":"09:40.630 ","End":"09:46.995","Text":"this part here minus 2 times x minus x_1."},{"Start":"09:46.995 ","End":"09:48.210","Text":"Again, it\u0027s a minus,"},{"Start":"09:48.210 ","End":"09:52.395","Text":"so I\u0027ll allow myself to write minus minus as a plus right away."},{"Start":"09:52.395 ","End":"09:54.965","Text":"That\u0027s the equation of normal a."},{"Start":"09:54.965 ","End":"09:59.990","Text":"Normal b, we knew they were going to be 2 solutions from the picture but now,"},{"Start":"09:59.990 ","End":"10:01.295","Text":"it\u0027s really confirmed,"},{"Start":"10:01.295 ","End":"10:03.890","Text":"is again, that y from the green formula,"},{"Start":"10:03.890 ","End":"10:08.480","Text":"y minus our particular y which is minus 3."},{"Start":"10:08.480 ","End":"10:10.370","Text":"Again, minus minus 3,"},{"Start":"10:10.370 ","End":"10:12.410","Text":"I\u0027ll write it as plus 3,"},{"Start":"10:12.410 ","End":"10:16.150","Text":"is equal to the same minus 2,"},{"Start":"10:16.150 ","End":"10:17.980","Text":"minus 2 is in both cases,"},{"Start":"10:17.980 ","End":"10:21.170","Text":"it\u0027s the minus 2 from here."},{"Start":"10:21.170 ","End":"10:25.535","Text":"This is the thing that appears here and also here."},{"Start":"10:25.535 ","End":"10:32.600","Text":"Maybe I\u0027ll just write equals minus 2 and this time, x minus 3."},{"Start":"10:32.600 ","End":"10:38.660","Text":"These are the 2 answers but I would like to just tidy them up a bit."},{"Start":"10:38.660 ","End":"10:42.095","Text":"It\u0027s customary to leave it as y equals."},{"Start":"10:42.095 ","End":"10:46.160","Text":"The first 1, part a is y equals, let\u0027s see,"},{"Start":"10:46.160 ","End":"10:50.190","Text":"minus 2x minus 2 plus 1,"},{"Start":"10:50.190 ","End":"10:53.580","Text":"so it\u0027s minus 2x minus 1."},{"Start":"10:53.580 ","End":"10:56.115","Text":"This is 1 possibility."},{"Start":"10:56.115 ","End":"11:04.050","Text":"Or we could have the other possibility which is that y equals minus 2x,"},{"Start":"11:04.050 ","End":"11:05.670","Text":"it has to be minus 2x,"},{"Start":"11:05.670 ","End":"11:10.905","Text":"it\u0027s going to be parallel with the reference line which was minus 2x plus something,"},{"Start":"11:10.905 ","End":"11:12.360","Text":"it has to be minus 2x."},{"Start":"11:12.360 ","End":"11:14.390","Text":"The only difference is in that constant."},{"Start":"11:14.390 ","End":"11:17.810","Text":"Here it\u0027s minus 2 times minus 3 is plus 6,"},{"Start":"11:17.810 ","End":"11:21.280","Text":"but minus 3 is plus 3."},{"Start":"11:21.280 ","End":"11:27.045","Text":"This is like a and this is b, 2 possibilities."},{"Start":"11:27.045 ","End":"11:30.705","Text":"That\u0027s the answer, 2 normal lines."},{"Start":"11:30.705 ","End":"11:35.290","Text":"I\u0027ll box it and say, we are done."}],"ID":4352},{"Watched":false,"Name":"Exercise 9","Duration":"5m 12s","ChapterTopicVideoID":4343,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this exercise, we have to find the equation of the line that is tangent to"},{"Start":"00:06.030 ","End":"00:09.450","Text":"this curve at the point where x equals"},{"Start":"00:09.450 ","End":"00:14.370","Text":"e. In case you\u0027re having difficulty visualizing this,"},{"Start":"00:14.370 ","End":"00:19.305","Text":"I prepared in advance a little sketch."},{"Start":"00:19.305 ","End":"00:21.090","Text":"This is the curve."},{"Start":"00:21.090 ","End":"00:22.680","Text":"The curve is the function."},{"Start":"00:22.680 ","End":"00:24.855","Text":"When you draw a graph, it\u0027s a curve."},{"Start":"00:24.855 ","End":"00:27.270","Text":"This is the curve f of x,"},{"Start":"00:27.270 ","End":"00:30.360","Text":"so y equals x natural log of x."},{"Start":"00:30.360 ","End":"00:32.160","Text":"Something like that doesn\u0027t matter the function,"},{"Start":"00:32.160 ","End":"00:34.200","Text":"whatever it was as the curve,"},{"Start":"00:34.200 ","End":"00:39.455","Text":"and this is the point where x equals e,"},{"Start":"00:39.455 ","End":"00:41.510","Text":"so this is the point e,"},{"Start":"00:41.510 ","End":"00:43.145","Text":"that\u0027s the x line."},{"Start":"00:43.145 ","End":"00:48.115","Text":"This green line is the tangent here,"},{"Start":"00:48.115 ","End":"00:51.650","Text":"and this is the tangent to the curve at this point."},{"Start":"00:51.650 ","End":"00:54.560","Text":"Well, we have to find out the equation of this line,"},{"Start":"00:54.560 ","End":"00:57.110","Text":"this tangent line, and that\u0027s all."},{"Start":"00:57.110 ","End":"00:59.975","Text":"If this helps then good,"},{"Start":"00:59.975 ","End":"01:03.405","Text":"if not then never mind."},{"Start":"01:03.405 ","End":"01:05.760","Text":"But everything can be done algebraically,"},{"Start":"01:05.760 ","End":"01:07.385","Text":"you don\u0027t have to have a sketch."},{"Start":"01:07.385 ","End":"01:11.000","Text":"Let me write down the formula for the tangent,"},{"Start":"01:11.000 ","End":"01:13.655","Text":"and then I\u0027ll explain it and fill it in."},{"Start":"01:13.655 ","End":"01:18.230","Text":"The formula or equation,"},{"Start":"01:18.230 ","End":"01:24.935","Text":"I call it the formula for the tangent line at our given point, x_1,"},{"Start":"01:24.935 ","End":"01:33.350","Text":"y_1 is just y minus our particular y is equal"},{"Start":"01:33.350 ","End":"01:43.230","Text":"to the derivative function at the point x_1 times x minus x_1."},{"Start":"01:43.230 ","End":"01:45.770","Text":"X_1, y_1 is our given point."},{"Start":"01:45.770 ","End":"01:48.815","Text":"Now all we\u0027re given is x_1."},{"Start":"01:48.815 ","End":"01:53.690","Text":"This by the way, f of x is y in case I need it."},{"Start":"01:53.690 ","End":"01:55.475","Text":"We have x_1,"},{"Start":"01:55.475 ","End":"01:57.400","Text":"then we need y_1,"},{"Start":"01:57.400 ","End":"01:59.490","Text":"and then we\u0027ll need f prime of x_1,"},{"Start":"01:59.490 ","End":"02:01.900","Text":"and then we can use the formula."},{"Start":"02:02.000 ","End":"02:06.340","Text":"We have f of x equals x,"},{"Start":"02:06.340 ","End":"02:08.485","Text":"natural log of x,"},{"Start":"02:08.485 ","End":"02:11.635","Text":"which is also call it y."},{"Start":"02:11.635 ","End":"02:13.525","Text":"Now our x_1,"},{"Start":"02:13.525 ","End":"02:17.370","Text":"our particular x is equal to e,"},{"Start":"02:17.370 ","End":"02:24.555","Text":"so y_1, y is just f of e. What is f of e?"},{"Start":"02:24.555 ","End":"02:26.310","Text":"X is e,"},{"Start":"02:26.310 ","End":"02:28.950","Text":"natural log of e is 1,"},{"Start":"02:28.950 ","End":"02:33.675","Text":"so e times 1 is equal to e,"},{"Start":"02:33.675 ","End":"02:39.010","Text":"so they\u0027re both equal to e. Then we need the final bit,"},{"Start":"02:39.010 ","End":"02:40.380","Text":"which is this bit here,"},{"Start":"02:40.380 ","End":"02:43.365","Text":"so let\u0027s first of all get the derivative in general."},{"Start":"02:43.365 ","End":"02:45.240","Text":"What is f prime of x,"},{"Start":"02:45.240 ","End":"02:49.565","Text":"and then we\u0027ll put in our x_1 which should be this"},{"Start":"02:49.565 ","End":"02:55.055","Text":"e. But in general we have a product maybe I"},{"Start":"02:55.055 ","End":"03:00.605","Text":"better write the formula for the product rule is the u times v"},{"Start":"03:00.605 ","End":"03:04.430","Text":"prime is equal to the derivative of the first times the"},{"Start":"03:04.430 ","End":"03:08.330","Text":"second plus the first times the derivative of the second."},{"Start":"03:08.330 ","End":"03:10.760","Text":"Sometimes you see it [inaudible] like u,v."},{"Start":"03:10.760 ","End":"03:13.520","Text":"In our case, of course,"},{"Start":"03:13.520 ","End":"03:18.680","Text":"the u is going to equal the x and the v will equal"},{"Start":"03:18.680 ","End":"03:25.850","Text":"the natural log of x. U prime is x prime,"},{"Start":"03:25.850 ","End":"03:30.560","Text":"which is 1 times the other 1 as it is,"},{"Start":"03:30.560 ","End":"03:34.535","Text":"plus vice versa, x as it is."},{"Start":"03:34.535 ","End":"03:38.750","Text":"Natural log of x derived is 1 over x,"},{"Start":"03:38.750 ","End":"03:46.370","Text":"so this actually simplifies to natural log of x plus 1."},{"Start":"03:46.370 ","End":"03:47.930","Text":"In our particular case,"},{"Start":"03:47.930 ","End":"03:51.955","Text":"what we want is f prime of x 1,"},{"Start":"03:51.955 ","End":"03:58.610","Text":"so f prime of e is equal to natural log of e,"},{"Start":"03:58.610 ","End":"04:02.450","Text":"which is 1 plus 1, makes it 2."},{"Start":"04:02.450 ","End":"04:06.440","Text":"Now we have all the ingredients, and yes,"},{"Start":"04:06.440 ","End":"04:11.865","Text":"the important ingredients are basically x_1,"},{"Start":"04:11.865 ","End":"04:14.190","Text":"which is e,"},{"Start":"04:14.190 ","End":"04:16.980","Text":"and then we have y_1,"},{"Start":"04:16.980 ","End":"04:19.020","Text":"which is also e,"},{"Start":"04:19.020 ","End":"04:21.475","Text":"and we have,"},{"Start":"04:21.475 ","End":"04:27.095","Text":"I should have colored the whole left prime of x_1 just hang on a second."},{"Start":"04:27.095 ","End":"04:29.855","Text":"That\u0027s f prime of x_1,"},{"Start":"04:29.855 ","End":"04:31.670","Text":"and that\u0027s this f prime,"},{"Start":"04:31.670 ","End":"04:33.625","Text":"that\u0027s this 2 here."},{"Start":"04:33.625 ","End":"04:37.175","Text":"These things correspond and now we can"},{"Start":"04:37.175 ","End":"04:41.845","Text":"plug all 3 parts of the formula have been computed,"},{"Start":"04:41.845 ","End":"04:48.420","Text":"and so y minus y_1 by the color is e"},{"Start":"04:48.420 ","End":"04:55.535","Text":"is equal to the green bit 2 times x minus the yellow bit,"},{"Start":"04:55.535 ","End":"04:58.880","Text":"which is e. In this customary we simplify,"},{"Start":"04:58.880 ","End":"05:02.030","Text":"we just leave y on 1 side and everything else on"},{"Start":"05:02.030 ","End":"05:07.145","Text":"the other side equals 2x minus 2e plus e,"},{"Start":"05:07.145 ","End":"05:08.855","Text":"just minus e,"},{"Start":"05:08.855 ","End":"05:12.450","Text":"and this is the answer and we are done."}],"ID":4353},{"Watched":false,"Name":"Exercise 10","Duration":"4m 55s","ChapterTopicVideoID":4344,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.680","Text":"In this exercise, we\u0027re given a function which describes a curve and we have to"},{"Start":"00:07.680 ","End":"00:11.130","Text":"find the equation of the line which is tangent to"},{"Start":"00:11.130 ","End":"00:16.410","Text":"this curve at the point on the curve where x equals 1."},{"Start":"00:16.630 ","End":"00:20.900","Text":"I\u0027d like to start by writing down the formula for"},{"Start":"00:20.900 ","End":"00:25.160","Text":"the tangent and then I\u0027ll explain it and we\u0027ll see what we have to do."},{"Start":"00:25.160 ","End":"00:30.425","Text":"The formula for the tangent line, formula or equation,"},{"Start":"00:30.425 ","End":"00:36.420","Text":"is given by Y minus y_1 equals"},{"Start":"00:36.420 ","End":"00:43.635","Text":"f\u0027 of x_1 times x minus x_1."},{"Start":"00:43.635 ","End":"00:47.890","Text":"What is x_1 and y_1?"},{"Start":"00:47.890 ","End":"00:51.130","Text":"Those are the coordinates of the point on the curve."},{"Start":"00:51.130 ","End":"00:52.760","Text":"It says here x equals 1,"},{"Start":"00:52.760 ","End":"00:55.130","Text":"so that means that\u0027s x_1."},{"Start":"00:55.130 ","End":"00:57.080","Text":"We have that x_1,"},{"Start":"00:57.080 ","End":"01:02.840","Text":"that\u0027s equal to 1, f of x is also called y and"},{"Start":"01:02.840 ","End":"01:09.585","Text":"y_1 is the corresponding y coordinate so we just substitute the x in the function."},{"Start":"01:09.585 ","End":"01:18.055","Text":"We have to get f of x_1 or f of 1, and if we plug in 1 into this function, what do we get?"},{"Start":"01:18.055 ","End":"01:22.125","Text":"Natural log of 1 is 0,"},{"Start":"01:22.125 ","End":"01:24.510","Text":"this is 0 times something,"},{"Start":"01:24.510 ","End":"01:28.140","Text":"so it\u0027s got to be equal to just 0."},{"Start":"01:28.140 ","End":"01:32.270","Text":"What we\u0027re basically doing is we\u0027re getting all the pieces here."},{"Start":"01:32.270 ","End":"01:38.070","Text":"For example, the x would be the yellow."},{"Start":"01:38.070 ","End":"01:39.980","Text":"What we have here"},{"Start":"01:39.980 ","End":"01:48.320","Text":"is what we have is this x_1 is 1 and the y_1 that we put here,"},{"Start":"01:48.320 ","End":"01:50.360","Text":"let\u0027s put that in a different color,"},{"Start":"01:50.360 ","End":"01:55.590","Text":"that\u0027s y_1, and that\u0027s going to equal 0."},{"Start":"01:55.590 ","End":"01:59.825","Text":"The bit that we\u0027re missing is this derivative."},{"Start":"01:59.825 ","End":"02:02.030","Text":"that\u0027s what we still have to find."},{"Start":"02:02.030 ","End":"02:03.380","Text":"What we do is,"},{"Start":"02:03.380 ","End":"02:06.530","Text":"we first of all find the derivative in general."},{"Start":"02:06.530 ","End":"02:11.315","Text":"In other words, we figure out what is f\u0027 of x in general,"},{"Start":"02:11.315 ","End":"02:13.655","Text":"which is just differentiating this,"},{"Start":"02:13.655 ","End":"02:15.590","Text":"but not quite so simple."},{"Start":"02:15.590 ","End":"02:19.415","Text":"You have to remember the product rule because it\u0027s something times something."},{"Start":"02:19.415 ","End":"02:21.950","Text":"Let\u0027s remember the product rule."},{"Start":"02:21.950 ","End":"02:28.780","Text":"If we have 2 functions of x product rule for differentiation is that if we have"},{"Start":"02:28.780 ","End":"02:36.430","Text":"the product uv or fg\u0027, you differentiate the first times the second as is,"},{"Start":"02:36.430 ","End":"02:37.870","Text":"and then vice versa."},{"Start":"02:37.870 ","End":"02:43.300","Text":"Take the first times the derivative of the second."},{"Start":"02:43.300 ","End":"02:51.465","Text":"In our case, it\u0027s clear that this is going to be our u and this is going to be our v."},{"Start":"02:51.465 ","End":"02:59.185","Text":"What we will get if we plug all that in is we\u0027ll get u\u0027,"},{"Start":"02:59.185 ","End":"03:04.755","Text":"which is just e^x minus 1."},{"Start":"03:04.755 ","End":"03:09.470","Text":"Theoretically, we have to multiply by the internal derivative,"},{"Start":"03:09.470 ","End":"03:12.805","Text":"which is the derivative of x minus 1, which is 1."},{"Start":"03:12.805 ","End":"03:20.510","Text":"I\u0027ll do it times v as it is, natural log of x plus, in this case,"},{"Start":"03:20.510 ","End":"03:24.965","Text":"we take e^x minus 1 as it is,"},{"Start":"03:24.965 ","End":"03:27.185","Text":"times derivative of v,"},{"Start":"03:27.185 ","End":"03:32.330","Text":"which is coming from this part, and v here is 1/x."},{"Start":"03:32.330 ","End":"03:37.780","Text":"What we have to do is to plug in x_1, which is 1."},{"Start":"03:37.780 ","End":"03:41.640","Text":"What we need is f\u0027 of 1,"},{"Start":"03:41.640 ","End":"03:44.010","Text":"which is, if we put x equals 1,"},{"Start":"03:44.010 ","End":"03:46.560","Text":"1 minus 1 is 0, e^0 is 1."},{"Start":"03:46.560 ","End":"03:49.215","Text":"The first term is 1."},{"Start":"03:49.215 ","End":"03:55.035","Text":"Natural log of 1 is 0 so the whole first bit is 0 plus,"},{"Start":"03:55.035 ","End":"03:57.695","Text":"let\u0027s see what happens here when x equals 1,"},{"Start":"03:57.695 ","End":"04:02.015","Text":"e^1 minus 1 is e^0 is 1 again,"},{"Start":"04:02.015 ","End":"04:04.310","Text":"and 1/1 is 1."},{"Start":"04:04.310 ","End":"04:07.885","Text":"This thing turns out to equal 1."},{"Start":"04:07.885 ","End":"04:11.430","Text":"I\u0027ve got all 3 colors presented here,"},{"Start":"04:11.430 ","End":"04:13.785","Text":"it\u0027s like painting by numbers."},{"Start":"04:13.785 ","End":"04:17.300","Text":"What we get if we just substitute into this formula,"},{"Start":"04:17.300 ","End":"04:22.150","Text":"is we get Y minus the green bit"},{"Start":"04:22.150 ","End":"04:28.040","Text":"is 0 is equal to the turquoise bit,"},{"Start":"04:28.040 ","End":"04:35.650","Text":"which is 1 times X minus the yellow bit, which is 1."},{"Start":"04:35.650 ","End":"04:39.905","Text":"Basically, if we just get rid of this 0,"},{"Start":"04:39.905 ","End":"04:42.140","Text":"get rid of this 1,"},{"Start":"04:42.140 ","End":"04:51.510","Text":"it basically just gives us that Y is equal to X minus 1."},{"Start":"04:51.970 ","End":"04:55.170","Text":"That\u0027s the answer."}],"ID":4354},{"Watched":false,"Name":"Exercise 11","Duration":"3m 58s","ChapterTopicVideoID":4355,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.270","Text":"In this exercise, we have to find the equation of the tangent to this curve,"},{"Start":"00:06.270 ","End":"00:07.545","Text":"this one here,"},{"Start":"00:07.545 ","End":"00:10.905","Text":"at the point where x equals Pi over 2."},{"Start":"00:10.905 ","End":"00:15.240","Text":"What I\u0027d like to do is start by giving the formula for"},{"Start":"00:15.240 ","End":"00:19.650","Text":"the equation or the equation of the tangent line,"},{"Start":"00:19.650 ","End":"00:21.570","Text":"and then I\u0027ll explain what the formula is,"},{"Start":"00:21.570 ","End":"00:24.090","Text":"and we\u0027ll see what we\u0027re missing."},{"Start":"00:24.090 ","End":"00:27.345","Text":"The formula or equation,"},{"Start":"00:27.345 ","End":"00:30.030","Text":"I like to use the word formula."},{"Start":"00:30.030 ","End":"00:38.360","Text":"Formula for the tangent line is just Y minus y_1 is"},{"Start":"00:38.360 ","End":"00:47.560","Text":"equal to f prime of x_1 times X minus x_1."},{"Start":"00:47.560 ","End":"00:49.880","Text":"I use different style of x and y,"},{"Start":"00:49.880 ","End":"00:53.975","Text":"not to confuse with the x and y from here known as the y here,"},{"Start":"00:53.975 ","End":"00:56.735","Text":"f of x is also y."},{"Start":"00:56.735 ","End":"00:59.410","Text":"Now what this means,"},{"Start":"00:59.410 ","End":"01:03.290","Text":"the x_1 and y_1 are the point on the curve."},{"Start":"01:03.290 ","End":"01:06.220","Text":"We know that this is our x_1,"},{"Start":"01:06.220 ","End":"01:08.495","Text":"that\u0027s the very first thing that we have,"},{"Start":"01:08.495 ","End":"01:10.235","Text":"here is the x_1,"},{"Start":"01:10.235 ","End":"01:11.720","Text":"which it means our x,"},{"Start":"01:11.720 ","End":"01:14.900","Text":"our particular x is Pi over 2."},{"Start":"01:14.900 ","End":"01:17.510","Text":"The next thing we know is that y_1,"},{"Start":"01:17.510 ","End":"01:20.635","Text":"the corresponding y, is just f of x_1."},{"Start":"01:20.635 ","End":"01:27.470","Text":"So it\u0027s f of Pi over 2 which means plug Pi over 2 in the formula."},{"Start":"01:27.470 ","End":"01:30.530","Text":"Pi over 2 times 2 is Pi."},{"Start":"01:30.530 ","End":"01:35.690","Text":"The sine of Pi is like the sine of 180 degrees is 0."},{"Start":"01:35.690 ","End":"01:37.490","Text":"If you look up the curve,"},{"Start":"01:37.490 ","End":"01:39.080","Text":"sine starts at 0,"},{"Start":"01:39.080 ","End":"01:40.700","Text":"goes up at 90 degrees,"},{"Start":"01:40.700 ","End":"01:44.200","Text":"and down again at 180 degrees at 0 again."},{"Start":"01:44.200 ","End":"01:46.305","Text":"This is equal to 0."},{"Start":"01:46.305 ","End":"01:48.360","Text":"We have x_1, y_1,"},{"Start":"01:48.360 ","End":"01:50.855","Text":"and we color some of that,"},{"Start":"01:50.855 ","End":"01:54.560","Text":"say, yellow for the x_1,"},{"Start":"01:54.560 ","End":"01:56.450","Text":"which is Pi over 2,"},{"Start":"01:56.450 ","End":"01:59.820","Text":"and that\u0027s what we\u0027re going to plug in here."},{"Start":"01:59.820 ","End":"02:04.515","Text":"Let\u0027s take green for the y_1,"},{"Start":"02:04.515 ","End":"02:06.600","Text":"which is 0,"},{"Start":"02:06.600 ","End":"02:09.720","Text":"that\u0027s going to get plugged in over here."},{"Start":"02:09.720 ","End":"02:13.640","Text":"The last bit is the derivative at x_1,"},{"Start":"02:13.640 ","End":"02:15.350","Text":"which we haven\u0027t yet computed,"},{"Start":"02:15.350 ","End":"02:18.110","Text":"and so let\u0027s do that next."},{"Start":"02:18.110 ","End":"02:24.470","Text":"To do that, we first of all get the f prime in general, f prime of x,"},{"Start":"02:24.470 ","End":"02:28.355","Text":"the general derivative of this function,"},{"Start":"02:28.355 ","End":"02:31.745","Text":"which is going to be the derivative of sine is cosine."},{"Start":"02:31.745 ","End":"02:35.315","Text":"So we would say cosine of 2x."},{"Start":"02:35.315 ","End":"02:37.340","Text":"But because it\u0027s 2x and not x,"},{"Start":"02:37.340 ","End":"02:39.005","Text":"we have an internal derivative,"},{"Start":"02:39.005 ","End":"02:41.720","Text":"which is the derivative of 2x times 2."},{"Start":"02:41.720 ","End":"02:44.365","Text":"Allow me to write it in front, looks better."},{"Start":"02:44.365 ","End":"02:47.055","Text":"Now, f prime of x_1,"},{"Start":"02:47.055 ","End":"02:51.185","Text":"which is f prime of Pi over 2,"},{"Start":"02:51.185 ","End":"02:57.605","Text":"means the substitute Pi over 2 in the derivative."},{"Start":"02:57.605 ","End":"03:02.135","Text":"Pi over 2 times 2 is Pi,"},{"Start":"03:02.135 ","End":"03:04.790","Text":"cosine of Pi, if you look it up,"},{"Start":"03:04.790 ","End":"03:08.195","Text":"is actually minus 1 times 2."},{"Start":"03:08.195 ","End":"03:10.805","Text":"This is minus 2,"},{"Start":"03:10.805 ","End":"03:17.055","Text":"and that would be the bit that corresponds to this."},{"Start":"03:17.055 ","End":"03:22.620","Text":"Now we\u0027re just painting by numbers and what we can say,"},{"Start":"03:22.620 ","End":"03:29.675","Text":"the tangent that we\u0027re looking for is y minus the green thing."},{"Start":"03:29.675 ","End":"03:32.960","Text":"The green bit is 0 equals"},{"Start":"03:32.960 ","End":"03:40.690","Text":"the turquoise bit times x minus the yellow bit Pi over 2,"},{"Start":"03:40.690 ","End":"03:43.340","Text":"and that is basically the answer,"},{"Start":"03:43.340 ","End":"03:46.745","Text":"except that we just simplify it a little bit."},{"Start":"03:46.745 ","End":"03:55.815","Text":"This is just y is equal to minus 2x plus Pi,"},{"Start":"03:55.815 ","End":"03:59.080","Text":"and that is the answer."}],"ID":4364},{"Watched":false,"Name":"Exercise 12","Duration":"10m 47s","ChapterTopicVideoID":4356,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"This exercise is intended for the more advanced students and you know who you are."},{"Start":"00:04.755 ","End":"00:08.250","Text":"At how many different values of x does the curve,"},{"Start":"00:08.250 ","End":"00:10.935","Text":"so and so have a tangent line,"},{"Start":"00:10.935 ","End":"00:14.715","Text":"parallel to the line so and so."},{"Start":"00:14.715 ","End":"00:22.500","Text":"First of all, let\u0027s write that again and we\u0027ll write it also in terms of a function,"},{"Start":"00:22.500 ","End":"00:25.230","Text":"y equals f of x,"},{"Start":"00:25.230 ","End":"00:30.090","Text":"which equals e x cubed minus 2x plus 1."},{"Start":"00:30.090 ","End":"00:33.525","Text":"I want to explain a bit what the question really means."},{"Start":"00:33.525 ","End":"00:34.890","Text":"We have a function,"},{"Start":"00:34.890 ","End":"00:36.795","Text":"and a function when you sketch it,"},{"Start":"00:36.795 ","End":"00:40.215","Text":"graphically describes a curve and at every x,"},{"Start":"00:40.215 ","End":"00:42.480","Text":"the curve has a tangent."},{"Start":"00:42.480 ","End":"00:46.550","Text":"We just want to know which values of x does"},{"Start":"00:46.550 ","End":"00:52.135","Text":"the tangent line to the curve become parallel to this given line."},{"Start":"00:52.135 ","End":"00:55.340","Text":"There\u0027s really 2 things I want to write down."},{"Start":"00:55.340 ","End":"00:58.310","Text":"One is how to know when 2 lines are parallel,"},{"Start":"00:58.310 ","End":"01:02.495","Text":"and the other is the formula or equation for the tangent line."},{"Start":"01:02.495 ","End":"01:06.215","Text":"Let\u0027s begin with a simple one."},{"Start":"01:06.215 ","End":"01:11.000","Text":"I\u0027ll write it here. What does it mean for 2 lines to be parallel lines."},{"Start":"01:11.000 ","End":"01:15.425","Text":"In calculus terms or in analytic geometry terms,"},{"Start":"01:15.425 ","End":"01:19.250","Text":"what it means is that they have the same slope."},{"Start":"01:19.250 ","End":"01:22.205","Text":"You can tell if you have the slope of the line,"},{"Start":"01:22.205 ","End":"01:27.060","Text":"which is the coefficient of y equals x plus b,"},{"Start":"01:27.060 ","End":"01:30.755","Text":"then it\u0027s the a but I\u0027m getting ahead of myself."},{"Start":"01:30.755 ","End":"01:32.570","Text":"Parallel lines have the same slope,"},{"Start":"01:32.570 ","End":"01:34.190","Text":"that\u0027s one thing I wanted to write."},{"Start":"01:34.190 ","End":"01:38.300","Text":"The other thing I wanted to write is the equation for a tangent line,"},{"Start":"01:38.300 ","End":"01:40.205","Text":"or let\u0027s call it the formula."},{"Start":"01:40.205 ","End":"01:47.075","Text":"Formula for tangent is that y minus"},{"Start":"01:47.075 ","End":"01:55.810","Text":"y_1 is equal to f prime of x_1 times x minus x_1."},{"Start":"01:55.810 ","End":"01:59.480","Text":"What this means is that y and x remain the variables,"},{"Start":"01:59.480 ","End":"02:01.100","Text":"but we have a specific point,"},{"Start":"02:01.100 ","End":"02:02.705","Text":"x_1, y_1,"},{"Start":"02:02.705 ","End":"02:04.265","Text":"and we have a slope."},{"Start":"02:04.265 ","End":"02:07.280","Text":"This actually is the slope,"},{"Start":"02:07.280 ","End":"02:08.930","Text":"I\u0027d like to point that out."},{"Start":"02:08.930 ","End":"02:15.200","Text":"This is the slope of the line and the slope of the tangent,"},{"Start":"02:15.200 ","End":"02:19.205","Text":"that\u0027s the tie-in between the calculus and the analytic geometry,"},{"Start":"02:19.205 ","End":"02:22.610","Text":"is that the derivative at a point x_1 is"},{"Start":"02:22.610 ","End":"02:28.580","Text":"exactly the slope of the tangent line at the point x_1 or x_1y_1."},{"Start":"02:28.580 ","End":"02:31.200","Text":"With these 2 things,"},{"Start":"02:31.200 ","End":"02:35.490","Text":"which the main things we\u0027ll start attacking this problem."},{"Start":"02:35.490 ","End":"02:39.620","Text":"The first thing I can say is that we have, okay, the other thing,"},{"Start":"02:39.620 ","End":"02:43.790","Text":"this is the curve because we also have a line and then we"},{"Start":"02:43.790 ","End":"02:47.990","Text":"also have y equals another function of x,"},{"Start":"02:47.990 ","End":"02:50.330","Text":"say g of x, we don\u0027t need the g really,"},{"Start":"02:50.330 ","End":"02:51.770","Text":"but it\u0027s just to be consistent,"},{"Start":"02:51.770 ","End":"02:56.910","Text":"which is x plus 1 and that is the line."},{"Start":"02:56.910 ","End":"02:59.045","Text":"Here we have the curve,"},{"Start":"02:59.045 ","End":"03:01.380","Text":"and here we have the line."},{"Start":"03:01.600 ","End":"03:04.205","Text":"Now, for the line,"},{"Start":"03:04.205 ","End":"03:08.445","Text":"we know the slope because a line has,"},{"Start":"03:08.445 ","End":"03:10.725","Text":"you don\u0027t need to do calculus or anything,"},{"Start":"03:10.725 ","End":"03:13.430","Text":"you just need to remember that the coefficient of x,"},{"Start":"03:13.430 ","End":"03:17.435","Text":"when y equals something x plus something that the slope is equal to 1,"},{"Start":"03:17.435 ","End":"03:19.420","Text":"I\u0027ll just write it as 1x,"},{"Start":"03:19.420 ","End":"03:21.780","Text":"just to emphasize it."},{"Start":"03:21.780 ","End":"03:25.980","Text":"1 is the slope so this,"},{"Start":"03:25.980 ","End":"03:27.765","Text":"let me see,"},{"Start":"03:27.765 ","End":"03:33.500","Text":"that the slope of the line is equal to 1 at any point on the line,"},{"Start":"03:33.500 ","End":"03:34.730","Text":"a line has the same slope,"},{"Start":"03:34.730 ","End":"03:36.665","Text":"it\u0027s not dependent on x."},{"Start":"03:36.665 ","End":"03:39.350","Text":"You could differentiate it and say, well,"},{"Start":"03:39.350 ","End":"03:42.530","Text":"a derivative is 1 and it\u0027s constant doesn\u0027t depend on x."},{"Start":"03:42.530 ","End":"03:47.750","Text":"What we want is the slope of this thing at the point x_1."},{"Start":"03:47.750 ","End":"03:52.100","Text":"Just take the derivative in general, y prime,"},{"Start":"03:52.100 ","End":"03:54.380","Text":"which is f prime of x,"},{"Start":"03:54.380 ","End":"03:55.940","Text":"is the derivative of this,"},{"Start":"03:55.940 ","End":"04:00.515","Text":"which is 3x squared minus 2, and that\u0027s it."},{"Start":"04:00.515 ","End":"04:04.895","Text":"Now, what we want is to find where this thing is"},{"Start":"04:04.895 ","End":"04:10.610","Text":"equal to the slope of the line so this is the slope."},{"Start":"04:10.610 ","End":"04:15.380","Text":"Let me just say that also that looking for a specific x,"},{"Start":"04:15.380 ","End":"04:17.630","Text":"so let\u0027s just write this again with x_1,"},{"Start":"04:17.630 ","End":"04:21.350","Text":"which is 3x_1 squared minus 2,"},{"Start":"04:21.350 ","End":"04:25.340","Text":"I\u0027ve just changed the general variable to a specific x_1."},{"Start":"04:25.340 ","End":"04:27.905","Text":"This is also the slope at the point,"},{"Start":"04:27.905 ","End":"04:31.055","Text":"this is the slope."},{"Start":"04:31.055 ","End":"04:33.815","Text":"What we have to do is compare slopes."},{"Start":"04:33.815 ","End":"04:37.490","Text":"What that gives us is if I just highlighted,"},{"Start":"04:37.490 ","End":"04:43.070","Text":"is that 3x_1 squared minus 2 has to equal"},{"Start":"04:43.070 ","End":"04:51.035","Text":"1 and then the slope of the line will equal the slope at the point where x equals x_1,"},{"Start":"04:51.035 ","End":"04:54.030","Text":"so we get an equation,"},{"Start":"04:55.170 ","End":"05:05.770","Text":"3x squared minus 2 is equal to x_1 squared rather specific."},{"Start":"05:06.380 ","End":"05:11.365","Text":"Bring the 2 over to the other side and divide by 3,"},{"Start":"05:11.365 ","End":"05:15.430","Text":"and you\u0027ll get x_1 squared is equal to 1,"},{"Start":"05:15.430 ","End":"05:23.460","Text":"see I brought the 2 over its 3 divided by 3. x_1 is equal to plus or minus 1."},{"Start":"05:23.460 ","End":"05:30.040","Text":"It looks like there are 2 possibilities for getting x_1 where the slopes are equal."},{"Start":"05:30.040 ","End":"05:36.560","Text":"The question was actually phrased as how many different values of x?"},{"Start":"05:36.560 ","End":"05:39.770","Text":"I could be smart and just say there\u0027s 2 of"},{"Start":"05:39.770 ","End":"05:43.190","Text":"them plus 1 and minus 1 and finished the exercise, and if you like,"},{"Start":"05:43.190 ","End":"05:46.520","Text":"we\u0027re finished and you can leave right now or stop this,"},{"Start":"05:46.520 ","End":"05:51.215","Text":"but I\u0027d like to actually find out what the do a bit extra."},{"Start":"05:51.215 ","End":"05:58.020","Text":"I\u0027m just to say that the answer is 2 points and it\u0027s plus 1 equals x plus 1 and minus 1,"},{"Start":"05:58.020 ","End":"06:00.200","Text":"but I\u0027d like to continue and find actually"},{"Start":"06:00.200 ","End":"06:03.720","Text":"the tangent lines so you can leave if you want."},{"Start":"06:03.740 ","End":"06:07.230","Text":"If x_1, I\u0027ll just do this quickly,"},{"Start":"06:07.230 ","End":"06:08.835","Text":"if x_1 equals 1,"},{"Start":"06:08.835 ","End":"06:10.950","Text":"then we have the y,"},{"Start":"06:10.950 ","End":"06:16.190","Text":"y_1 is equal to just by substituting as f of 1,"},{"Start":"06:16.190 ","End":"06:25.445","Text":"which is 1 plus 1 minus 2 is 0 and the derivative has to be 1."},{"Start":"06:25.445 ","End":"06:30.450","Text":"Plus or minus 1 squared is 3 minus 2."},{"Start":"06:30.450 ","End":"06:34.340","Text":"It has to be 1 because we compared it to 1 and the slope"},{"Start":"06:34.340 ","End":"06:38.405","Text":"and f prime of x_1 is also equal to 1,"},{"Start":"06:38.405 ","End":"06:47.119","Text":"and then we plug it into this formula here so we get y minus y_1 is equal to the slope,"},{"Start":"06:47.119 ","End":"06:54.475","Text":"which is 1 times x minus x_1 is 1."},{"Start":"06:54.475 ","End":"06:56.895","Text":"We don\u0027t get confused,"},{"Start":"06:56.895 ","End":"06:59.310","Text":"let\u0027s use some color."},{"Start":"06:59.310 ","End":"07:09.005","Text":"We have, I\u0027d say I used the yellow for the x_1 and x_1 is this bit,"},{"Start":"07:09.005 ","End":"07:12.645","Text":"this I\u0027ll use the,"},{"Start":"07:12.645 ","End":"07:17.465","Text":"I shouldn\u0027t have used the slope for yellow."},{"Start":"07:17.465 ","End":"07:24.450","Text":"Next thing I get is that the y_1 is 0 and that\u0027s this"},{"Start":"07:24.450 ","End":"07:33.080","Text":"0 and then this 1 is this turquoise and that gives us this value here."},{"Start":"07:33.080 ","End":"07:35.600","Text":"Now in the next one, next possibility,"},{"Start":"07:35.600 ","End":"07:43.665","Text":"this is like possibility a and possibility b is c,"},{"Start":"07:43.665 ","End":"07:51.739","Text":"x_1 equals y_1 equals f prime of x_1 equals,"},{"Start":"07:51.739 ","End":"07:56.570","Text":"we get it from the other possibility where x_1 is minus 1."},{"Start":"07:56.570 ","End":"08:02.980","Text":"If this is minus 1 then y_1 is when we plug in minus 1 into here,"},{"Start":"08:02.980 ","End":"08:08.730","Text":"so it\u0027s minus 1 plus 2 plus 1,"},{"Start":"08:08.730 ","End":"08:10.470","Text":"so that\u0027s equal to 2,"},{"Start":"08:10.470 ","End":"08:12.200","Text":"so it\u0027s f of minus 1,"},{"Start":"08:12.200 ","End":"08:14.540","Text":"which is 2, and f prime of x_1,"},{"Start":"08:14.540 ","End":"08:17.480","Text":"it has to be 1 because it has to be the same slope,"},{"Start":"08:17.480 ","End":"08:19.775","Text":"it\u0027s always the same slope."},{"Start":"08:19.775 ","End":"08:28.260","Text":"That gives me that y minus the y of ry is equal to the slope,"},{"Start":"08:28.260 ","End":"08:32.135","Text":"which is 1 times x minus our x,"},{"Start":"08:32.135 ","End":"08:34.190","Text":"which is minus minus 1,"},{"Start":"08:34.190 ","End":"08:37.310","Text":"I\u0027ll just do advanced students and what it straight for,"},{"Start":"08:37.310 ","End":"08:38.915","Text":"x plus 2,"},{"Start":"08:38.915 ","End":"08:40.325","Text":"or in this case,"},{"Start":"08:40.325 ","End":"08:47.370","Text":"the highlighting is that this one is from this one as before,"},{"Start":"08:47.370 ","End":"08:51.645","Text":"this x_1 came from,"},{"Start":"08:51.645 ","End":"08:53.915","Text":"okay, I fixed it up."},{"Start":"08:53.915 ","End":"08:56.495","Text":"Just copied something wrong somewhere."},{"Start":"08:56.495 ","End":"08:58.865","Text":"But yes, this is the x_1,"},{"Start":"08:58.865 ","End":"09:01.265","Text":"the 3 data we need at x_1y_1,"},{"Start":"09:01.265 ","End":"09:03.190","Text":"and the derivative at x_1,"},{"Start":"09:03.190 ","End":"09:06.005","Text":"and this one goes into where the slope goes."},{"Start":"09:06.005 ","End":"09:09.075","Text":"This goes y minus y,"},{"Start":"09:09.075 ","End":"09:11.925","Text":"this is the y_1 and this is the x_1 only,"},{"Start":"09:11.925 ","End":"09:13.350","Text":"it isn\u0027t really x plus 1,"},{"Start":"09:13.350 ","End":"09:14.970","Text":"it\u0027s really x minus minus 1,"},{"Start":"09:14.970 ","End":"09:17.860","Text":"so this minus1 is this minus1 here."},{"Start":"09:17.860 ","End":"09:20.690","Text":"Then we\u0027re basically done."},{"Start":"09:20.690 ","End":"09:28.715","Text":"The only other thing that one usually does is to simplify finally at the end."},{"Start":"09:28.715 ","End":"09:34.175","Text":"I\u0027ll just write it again more cleanly."},{"Start":"09:34.175 ","End":"09:36.170","Text":"I\u0027m just going to simplify this."},{"Start":"09:36.170 ","End":"09:38.525","Text":"This is 0, this is 1,"},{"Start":"09:38.525 ","End":"09:40.715","Text":"so y equals x minus 1."},{"Start":"09:40.715 ","End":"09:46.220","Text":"Y equals x minus 1 is the first tangent line and the second one,"},{"Start":"09:46.220 ","End":"09:47.630","Text":"if we simplify this,"},{"Start":"09:47.630 ","End":"09:50.445","Text":"this is just x plus 1,"},{"Start":"09:50.445 ","End":"09:51.660","Text":"and then we add another 2,"},{"Start":"09:51.660 ","End":"09:55.370","Text":"x plus 3, y equals x plus 3."},{"Start":"09:55.370 ","End":"09:59.360","Text":"But I just don\u0027t feel right by answering yes,"},{"Start":"09:59.360 ","End":"10:01.550","Text":"there are 2 points at which the tangents are"},{"Start":"10:01.550 ","End":"10:05.150","Text":"parallel and I\u0027ve actually written the tangents."},{"Start":"10:05.150 ","End":"10:10.880","Text":"Notice that our line is x plus 1 or 1x plus 1,"},{"Start":"10:10.880 ","End":"10:14.015","Text":"and these are also the coefficients of x is also 1 and1."},{"Start":"10:14.015 ","End":"10:16.895","Text":"Instead of x plus 1, we have x minus 1 and x plus 3,"},{"Start":"10:16.895 ","End":"10:19.204","Text":"and all these 3 things are parallel."},{"Start":"10:19.204 ","End":"10:25.310","Text":"If I use even a different color than x plus 1 is parallel to"},{"Start":"10:25.310 ","End":"10:27.680","Text":"both this and to this because it\u0027s"},{"Start":"10:27.680 ","End":"10:32.035","Text":"the same coefficient of x or the equation y equals x plus 1,"},{"Start":"10:32.035 ","End":"10:38.190","Text":"y equals x plus 1 and here y equals x minus 1,"},{"Start":"10:38.190 ","End":"10:40.960","Text":"and here y equals x plus 3."},{"Start":"10:41.510 ","End":"10:46.950","Text":"I\u0027ve fully done it and this was not satisfactory to end it here."}],"ID":4365},{"Watched":false,"Name":"Exercise 13","Duration":"7m 5s","ChapterTopicVideoID":6444,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we have to find the equation of the line which is"},{"Start":"00:04.560 ","End":"00:10.300","Text":"tangent to this function at the point where x equals 1."},{"Start":"00:10.300 ","End":"00:14.265","Text":"There is actually a formula for tangent lines."},{"Start":"00:14.265 ","End":"00:18.615","Text":"Let me write the formula and see how we can use it in this exercise."},{"Start":"00:18.615 ","End":"00:24.375","Text":"The formula is y minus y_1 is equal"},{"Start":"00:24.375 ","End":"00:30.795","Text":"to f prime of x_1 times x minus x_1."},{"Start":"00:30.795 ","End":"00:33.869","Text":"Now, I\u0027ll explain the meaning of this formula,"},{"Start":"00:33.869 ","End":"00:35.130","Text":"what the different things are."},{"Start":"00:35.130 ","End":"00:37.875","Text":"x_1 is just the x_1 of our point."},{"Start":"00:37.875 ","End":"00:40.950","Text":"In fact, x_1 is equal to 1 here and"},{"Start":"00:40.950 ","End":"00:45.890","Text":"y_1 is the value of the function where x is x_1."},{"Start":"00:45.890 ","End":"00:47.405","Text":"Actually, I could even write that,"},{"Start":"00:47.405 ","End":"00:50.890","Text":"y_1 is the function of x_1."},{"Start":"00:50.890 ","End":"00:52.260","Text":"For each x, you have a y,"},{"Start":"00:52.260 ","End":"00:53.555","Text":"that\u0027s what a function is."},{"Start":"00:53.555 ","End":"00:57.380","Text":"Let me emphasize that we use instead of f of x,"},{"Start":"00:57.380 ","End":"01:01.120","Text":"you can also say that y is equal to."},{"Start":"01:01.120 ","End":"01:06.735","Text":"The general strategy is to plug x_1 into f and get y_1,"},{"Start":"01:06.735 ","End":"01:11.600","Text":"then to plug x_1 into f prime and figure out what this quantity is."},{"Start":"01:11.600 ","End":"01:14.240","Text":"Then once we have this and this and this,"},{"Start":"01:14.240 ","End":"01:16.200","Text":"just to write the formula."},{"Start":"01:16.200 ","End":"01:20.160","Text":"Let\u0027s begin finding y_1."},{"Start":"01:20.160 ","End":"01:25.950","Text":"y_1 is f of x_1 but x_1 is 1,"},{"Start":"01:25.950 ","End":"01:29.070","Text":"because that\u0027s our particular x, so it\u0027s x_1."},{"Start":"01:29.070 ","End":"01:32.055","Text":"y_1 is f of x_1 which is f of 1,"},{"Start":"01:32.055 ","End":"01:36.660","Text":"which means f of 1, and f of 1 is 1^4,"},{"Start":"01:36.660 ","End":"01:42.120","Text":"1 to the power of anything is 1."},{"Start":"01:42.120 ","End":"01:43.900","Text":"To summarize these 2 things,"},{"Start":"01:43.900 ","End":"01:46.065","Text":"we have the point x_1,"},{"Start":"01:46.065 ","End":"01:48.870","Text":"y_1 where the curve goes through,"},{"Start":"01:48.870 ","End":"01:50.250","Text":"where the function goes through,"},{"Start":"01:50.250 ","End":"01:53.205","Text":"is x_1 is 1,"},{"Start":"01:53.205 ","End":"01:54.985","Text":"y_1, also 1."},{"Start":"01:54.985 ","End":"01:57.800","Text":"The next thing we need is this thing here,"},{"Start":"01:57.800 ","End":"01:59.945","Text":"f prime of 1."},{"Start":"01:59.945 ","End":"02:02.065","Text":"We need f prime first."},{"Start":"02:02.065 ","End":"02:04.140","Text":"Let me just copy, again,"},{"Start":"02:04.140 ","End":"02:09.650","Text":"the equation f of x is x^4x."},{"Start":"02:09.650 ","End":"02:12.650","Text":"Now, we need f prime of x."},{"Start":"02:12.650 ","End":"02:16.520","Text":"We have to remember how to do a function of x to the power of function of x."},{"Start":"02:16.520 ","End":"02:19.580","Text":"There is a standard trick and I\u0027ll tell you what it is."},{"Start":"02:19.580 ","End":"02:23.215","Text":"If we have a^b, in general,"},{"Start":"02:23.215 ","End":"02:27.465","Text":"we can write it [inaudible] is equal to e^b,"},{"Start":"02:27.465 ","End":"02:29.685","Text":"natural log of a."},{"Start":"02:29.685 ","End":"02:31.895","Text":"When you get a function to the power of function,"},{"Start":"02:31.895 ","End":"02:36.785","Text":"it\u0027s easier to use this formula and have it as e to the power of some function."},{"Start":"02:36.785 ","End":"02:40.835","Text":"The function might be more complex, but it\u0027s e to the power of makes it easier."},{"Start":"02:40.835 ","End":"02:42.725","Text":"If we apply that,"},{"Start":"02:42.725 ","End":"02:45.800","Text":"in our case, we have that f of x."},{"Start":"02:45.800 ","End":"02:47.525","Text":"If I write it in another way,"},{"Start":"02:47.525 ","End":"02:50.990","Text":"is e to the power of what was at the top here,"},{"Start":"02:50.990 ","End":"02:52.670","Text":"that\u0027s the b, that\u0027s the 4x,"},{"Start":"02:52.670 ","End":"02:57.185","Text":"times natural log of the base which is x."},{"Start":"02:57.185 ","End":"02:59.180","Text":"The other thing I\u0027ll have,"},{"Start":"02:59.180 ","End":"03:01.970","Text":"e to the power of something,"},{"Start":"03:01.970 ","End":"03:03.345","Text":"let\u0027s call it, say,"},{"Start":"03:03.345 ","End":"03:05.950","Text":"box, whatever, some function of x,"},{"Start":"03:05.950 ","End":"03:10.505","Text":"derived is e to the power of box,"},{"Start":"03:10.505 ","End":"03:15.575","Text":"just the same as this, but multiplied by the derivative of the box."},{"Start":"03:15.575 ","End":"03:17.360","Text":"That\u0027s another thing we\u0027ll need."},{"Start":"03:17.360 ","End":"03:19.340","Text":"Already, I see that for finding this,"},{"Start":"03:19.340 ","End":"03:21.370","Text":"we\u0027ll need the product rules,"},{"Start":"03:21.370 ","End":"03:27.605","Text":"2 functions of x multiplied u times v. I want the derivative of that."},{"Start":"03:27.605 ","End":"03:29.720","Text":"It\u0027s the derivative of the first times the"},{"Start":"03:29.720 ","End":"03:34.415","Text":"second plus the first times the derivative of the second."},{"Start":"03:34.415 ","End":"03:36.830","Text":"I think we can finally get down to doing"},{"Start":"03:36.830 ","End":"03:40.175","Text":"some actual computation and not just writing formulas."},{"Start":"03:40.175 ","End":"03:43.895","Text":"What we have is using the purple formula."},{"Start":"03:43.895 ","End":"03:49.745","Text":"We have this equals e^4x natural log of"},{"Start":"03:49.745 ","End":"03:59.000","Text":"x times the derivative times 4x natural log of x derivative."},{"Start":"03:59.000 ","End":"04:00.890","Text":"For the derivative of this,"},{"Start":"04:00.890 ","End":"04:05.905","Text":"I\u0027ll use the light blue formula product rule."},{"Start":"04:05.905 ","End":"04:12.525","Text":"This is equal to e^4x natural log of x times,"},{"Start":"04:12.525 ","End":"04:17.690","Text":"I\u0027ll just take the 4 outside the brackets and just call this part."},{"Start":"04:17.690 ","End":"04:20.509","Text":"This part here will be my u,"},{"Start":"04:20.509 ","End":"04:25.860","Text":"and this part here will be the v from this formula."},{"Start":"04:25.860 ","End":"04:29.385","Text":"It\u0027s 4 times something plus something."},{"Start":"04:29.385 ","End":"04:32.855","Text":"It\u0027s x differentiated is 1,"},{"Start":"04:32.855 ","End":"04:37.050","Text":"natural log of x as is, and the opposite,"},{"Start":"04:37.050 ","End":"04:43.195","Text":"then x as is natural log of x differentiated which is 1/x."},{"Start":"04:43.195 ","End":"04:47.650","Text":"What\u0027s inside the brackets here is natural log of x plus 1."},{"Start":"04:47.650 ","End":"04:50.390","Text":"I\u0027ll just write this bit underneath again,"},{"Start":"04:50.390 ","End":"04:53.525","Text":"this is natural log of x plus 1,"},{"Start":"04:53.525 ","End":"04:55.055","Text":"just a bit in here."},{"Start":"04:55.055 ","End":"04:58.670","Text":"Now, what I want is this thing here."},{"Start":"04:58.670 ","End":"05:04.200","Text":"F prime of x_1 is, x_1, remember,"},{"Start":"05:04.200 ","End":"05:08.715","Text":"is f prime of 1 because x_1 is 1,"},{"Start":"05:08.715 ","End":"05:12.435","Text":"which means substituting 1 in this mess."},{"Start":"05:12.435 ","End":"05:17.595","Text":"Let\u0027s see, if we put x equals 1 or x_1 is 1,"},{"Start":"05:17.595 ","End":"05:21.225","Text":"natural log of 1 is 0,"},{"Start":"05:21.225 ","End":"05:25.995","Text":"0 times whatever is 0, e^0 is 1."},{"Start":"05:25.995 ","End":"05:30.795","Text":"All this first bit is 1, it\u0027s e^0,"},{"Start":"05:30.795 ","End":"05:32.950","Text":"let\u0027s stop somewhere,"},{"Start":"05:32.950 ","End":"05:37.985","Text":"times 4 times 1 times natural log."},{"Start":"05:37.985 ","End":"05:40.910","Text":"As I said, if 1 is 0, still a 0,"},{"Start":"05:40.910 ","End":"05:44.105","Text":"hasn\u0027t changed, plus 1 from here."},{"Start":"05:44.105 ","End":"05:48.080","Text":"Altogether, this was 1, this is 4,"},{"Start":"05:48.080 ","End":"05:50.000","Text":"1 times 0 plus 1 is 1,"},{"Start":"05:50.000 ","End":"05:52.160","Text":"4 times 1 is 4,"},{"Start":"05:52.160 ","End":"05:55.360","Text":"which is just 4."},{"Start":"05:56.120 ","End":"05:59.430","Text":"That pretty much summarizes everything."},{"Start":"05:59.430 ","End":"06:02.995","Text":"I\u0027ll just emphasize this and write this again over here."},{"Start":"06:02.995 ","End":"06:05.405","Text":"So x_1, y_1 we have,"},{"Start":"06:05.405 ","End":"06:11.585","Text":"and we also have that f prime of x_1 is equal to 4."},{"Start":"06:11.585 ","End":"06:16.050","Text":"I\u0027m just saying this line here is very important,"},{"Start":"06:16.050 ","End":"06:21.790","Text":"that line, because it\u0027s got all the basic data for plugging in to this green formula."},{"Start":"06:21.790 ","End":"06:24.000","Text":"Basically, I can even tie them in."},{"Start":"06:24.000 ","End":"06:25.310","Text":"I\u0027m going to use this information,"},{"Start":"06:25.310 ","End":"06:28.595","Text":"plug it in here, and we\u0027ll write the answer down here."},{"Start":"06:28.595 ","End":"06:32.705","Text":"We\u0027ll have that y minus"},{"Start":"06:32.705 ","End":"06:39.205","Text":"y_1 is 1, x and y are the same so I don\u0027t have to get mixed up,"},{"Start":"06:39.205 ","End":"06:47.180","Text":"is equal to the slope which turned out to be 4, x minus x_1 which is also 1."},{"Start":"06:47.180 ","End":"06:48.850","Text":"That\u0027s basically the answer,"},{"Start":"06:48.850 ","End":"06:51.760","Text":"the only thing left to do is the simplification."},{"Start":"06:51.760 ","End":"06:53.885","Text":"If we simplify this,"},{"Start":"06:53.885 ","End":"07:00.935","Text":"we will get y equals 4x minus 4, but plus 1,"},{"Start":"07:00.935 ","End":"07:05.550","Text":"so it\u0027s only minus 3. We\u0027re done."}],"ID":6471},{"Watched":false,"Name":"Exercise 14","Duration":"10m 24s","ChapterTopicVideoID":6445,"CourseChapterTopicPlaylistID":1664,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, we have to find the equation of the line which is"},{"Start":"00:04.620 ","End":"00:10.545","Text":"tangent to this function or curve at the point where x equals 0."},{"Start":"00:10.545 ","End":"00:14.985","Text":"What I\u0027ve decided to do is to first of all write all the formulae."},{"Start":"00:14.985 ","End":"00:17.895","Text":"I\u0027ve been not done this exercise already."},{"Start":"00:17.895 ","End":"00:22.225","Text":"I want to just gather together all the formulae that I\u0027m going to need,"},{"Start":"00:22.225 ","End":"00:24.165","Text":"and I\u0027ll have them handy."},{"Start":"00:24.165 ","End":"00:30.570","Text":"Since we\u0027re looking for an equation of a tangent line there standard formula for tangent."},{"Start":"00:30.570 ","End":"00:32.400","Text":"First, I\u0027ll copy the exercise."},{"Start":"00:32.400 ","End":"00:34.365","Text":"That\u0027s for sure needs to be done."},{"Start":"00:34.365 ","End":"00:41.000","Text":"F of x is equal to 2x plus 1 to the x squared plus 1."},{"Start":"00:41.000 ","End":"00:43.850","Text":"What we\u0027re going to need is the equation of the tangent line."},{"Start":"00:43.850 ","End":"00:48.680","Text":"We\u0027re going to need some formula which are going to help us with this function;"},{"Start":"00:48.680 ","End":"00:51.930","Text":"function of x to the power of a function of x."},{"Start":"00:52.870 ","End":"00:57.725","Text":"Well, I\u0027ll just write it down and trust me that you\u0027ll need all of these."},{"Start":"00:57.725 ","End":"01:06.465","Text":"The first 1 is the equation of the formula for the tangent line is y"},{"Start":"01:06.465 ","End":"01:16.225","Text":"minus y_1 is equal to f prime of x_1 times x minus x_1."},{"Start":"01:16.225 ","End":"01:22.700","Text":"Now, x_1 and y_1 are just the point on the curve where we have to find the tangent line."},{"Start":"01:22.700 ","End":"01:24.440","Text":"We\u0027re not given x_1y_1,"},{"Start":"01:24.440 ","End":"01:26.105","Text":"were given just x_1."},{"Start":"01:26.105 ","End":"01:28.735","Text":"In fact, I already know that x_1 is 0,"},{"Start":"01:28.735 ","End":"01:34.120","Text":"and y_1 is just f of x_1."},{"Start":"01:34.120 ","End":"01:37.955","Text":"It\u0027s the point above or below x_1 on the curve."},{"Start":"01:37.955 ","End":"01:43.835","Text":"F prime of x_1 is just what you get when you take f prime and substitute x_1 in it."},{"Start":"01:43.835 ","End":"01:46.460","Text":"There are 3 things we need to know for this formula,"},{"Start":"01:46.460 ","End":"01:48.230","Text":"1 we already do, which is x_1."},{"Start":"01:48.230 ","End":"01:50.105","Text":"The second is we\u0027ll compute y_1,"},{"Start":"01:50.105 ","End":"01:53.060","Text":"then we\u0027ll differentiate f and substitute x_1,"},{"Start":"01:53.060 ","End":"01:56.085","Text":"which we know, and then we\u0027ll just plug all these in the formula."},{"Start":"01:56.085 ","End":"01:57.905","Text":"That\u0027s a general strategy."},{"Start":"01:57.905 ","End":"01:59.465","Text":"Now along the way,"},{"Start":"01:59.465 ","End":"02:01.655","Text":"we\u0027re going to need various things."},{"Start":"02:01.655 ","End":"02:04.879","Text":"For example, we\u0027re going to need the formula"},{"Start":"02:04.879 ","End":"02:10.370","Text":"for converting any exponent into a base e exponent,"},{"Start":"02:10.370 ","End":"02:14.330","Text":"and that formula would be a to the power of b"},{"Start":"02:14.330 ","End":"02:19.085","Text":"is e to the power of b times natural log of a."},{"Start":"02:19.085 ","End":"02:20.985","Text":"That\u0027s a formula."},{"Start":"02:20.985 ","End":"02:23.600","Text":"Then we\u0027ll convert this to e to the power of something,"},{"Start":"02:23.600 ","End":"02:29.740","Text":"and then we\u0027ll need the formula for the chain rule in the case of e to the power of."},{"Start":"02:29.740 ","End":"02:34.515","Text":"If I want to differentiate e to the power of something,"},{"Start":"02:34.515 ","End":"02:36.835","Text":"that something I\u0027ll just call it box,"},{"Start":"02:36.835 ","End":"02:38.480","Text":"I want to differentiate it,"},{"Start":"02:38.480 ","End":"02:43.160","Text":"the answer is just same as the initial thing,"},{"Start":"02:43.160 ","End":"02:47.960","Text":"except multiplied by what I call the internal derivative in the chain rule,"},{"Start":"02:47.960 ","End":"02:50.270","Text":"which is box prime."},{"Start":"02:50.270 ","End":"02:52.460","Text":"There\u0027s a similar 1 for the natural log,"},{"Start":"02:52.460 ","End":"02:54.575","Text":"and you\u0027ll see where the natural log comes in."},{"Start":"02:54.575 ","End":"02:56.970","Text":"Well, it comes in already from here, but anyway."},{"Start":"02:56.970 ","End":"03:02.495","Text":"If it\u0027s the natural log of box and I want to differentiate that again by the chain rule."},{"Start":"03:02.495 ","End":"03:03.890","Text":"If it was just x,"},{"Start":"03:03.890 ","End":"03:05.165","Text":"it would be 1 over x,"},{"Start":"03:05.165 ","End":"03:06.895","Text":"1 over box,"},{"Start":"03:06.895 ","End":"03:09.980","Text":"and then I multiply by the internal derivative and put box"},{"Start":"03:09.980 ","End":"03:12.920","Text":"prime except I prefer to write it on the top."},{"Start":"03:12.920 ","End":"03:15.035","Text":"That\u0027s what we\u0027re going to need,"},{"Start":"03:15.035 ","End":"03:18.755","Text":"and you see there\u0027s a b times the natural log of a."},{"Start":"03:18.755 ","End":"03:22.245","Text":"That is why we\u0027ll need the product rule."},{"Start":"03:22.245 ","End":"03:24.450","Text":"I\u0027ll do that and use a different color,"},{"Start":"03:24.450 ","End":"03:26.975","Text":"and say that if you have 2 functions of x,"},{"Start":"03:26.975 ","End":"03:28.985","Text":"say u and v prime,"},{"Start":"03:28.985 ","End":"03:30.880","Text":"that\u0027s equal to,"},{"Start":"03:30.880 ","End":"03:34.790","Text":"and now let\u0027s get onto solving this thing."},{"Start":"03:34.790 ","End":"03:40.505","Text":"Like I said, 1 thing we do know is that we talked about the point where x equals 0."},{"Start":"03:40.505 ","End":"03:45.570","Text":"I can straight away say that x_1 is equal to 0,"},{"Start":"03:45.570 ","End":"03:47.465","Text":"and that\u0027s 1 of my ingredients."},{"Start":"03:47.465 ","End":"03:49.640","Text":"The plan is to find all those 3 things,"},{"Start":"03:49.640 ","End":"03:51.185","Text":"x_1, y_1,"},{"Start":"03:51.185 ","End":"03:53.780","Text":"and f prime of x 1 and plug them in."},{"Start":"03:53.780 ","End":"03:56.510","Text":"Here\u0027s 1 of them, and this is important."},{"Start":"03:56.510 ","End":"03:58.750","Text":"I think I\u0027ll highlight it."},{"Start":"03:58.750 ","End":"04:03.435","Text":"Then we\u0027re going to find what y_1 is."},{"Start":"04:03.435 ","End":"04:07.315","Text":"Y_1 is simply just f of x_1,"},{"Start":"04:07.315 ","End":"04:10.670","Text":"which is f of 0,"},{"Start":"04:10.670 ","End":"04:13.850","Text":"which equals, and what happens if we put 0 in here,"},{"Start":"04:13.850 ","End":"04:16.910","Text":"twice 0 plus 1 is 1,"},{"Start":"04:16.910 ","End":"04:19.620","Text":"to the power of anything is 1."},{"Start":"04:20.150 ","End":"04:24.040","Text":"As it happened, it\u0027s 1 to the power of 1."},{"Start":"04:24.110 ","End":"04:30.395","Text":"That\u0027s y_1, and that\u0027s I think is also important and I\u0027ll highlight that also."},{"Start":"04:30.395 ","End":"04:34.135","Text":"Actually, I might as well highlight it here too."},{"Start":"04:34.135 ","End":"04:36.770","Text":"Back to real business."},{"Start":"04:36.770 ","End":"04:38.960","Text":"What we have here is y_1,"},{"Start":"04:38.960 ","End":"04:40.790","Text":"and then what else do we need?"},{"Start":"04:40.790 ","End":"04:43.160","Text":"Well, we need f prime of x_1."},{"Start":"04:43.160 ","End":"04:45.650","Text":"We need f prime of x in general."},{"Start":"04:45.650 ","End":"04:48.635","Text":"What I\u0027m going to do is to do the differentiation."},{"Start":"04:48.635 ","End":"04:51.755","Text":"F prime of x is equal to."},{"Start":"04:51.755 ","End":"04:55.610","Text":"But I can\u0027t say what it is straight away because I"},{"Start":"04:55.610 ","End":"04:58.925","Text":"want to rewrite f in more convenient terms."},{"Start":"04:58.925 ","End":"05:02.360","Text":"That\u0027s where this blue color comes in."},{"Start":"05:02.360 ","End":"05:04.560","Text":"Where I can rewrite this with"},{"Start":"05:04.560 ","End":"05:08.250","Text":"an algebraically equivalent terms as e to the power of something."},{"Start":"05:08.250 ","End":"05:10.760","Text":"It\u0027s e to the power of b,"},{"Start":"05:10.760 ","End":"05:12.980","Text":"which is the original exponent."},{"Start":"05:12.980 ","End":"05:14.630","Text":"Put it in brackets,"},{"Start":"05:14.630 ","End":"05:18.030","Text":"times natural log of a, a is the base."},{"Start":"05:18.030 ","End":"05:22.595","Text":"We have natural log of 2x plus 1."},{"Start":"05:22.595 ","End":"05:27.590","Text":"This is much more convenient to differentiate. What do you see?"},{"Start":"05:27.590 ","End":"05:29.340","Text":"I\u0027m prepared with all the tools I need,"},{"Start":"05:29.340 ","End":"05:32.210","Text":"and this is the tool I need right now"},{"Start":"05:32.210 ","End":"05:36.100","Text":"because I have e to the something and this whole thing is the box."},{"Start":"05:36.100 ","End":"05:41.915","Text":"This is e to the power of x squared plus 1,"},{"Start":"05:41.915 ","End":"05:45.890","Text":"natural log of 2x plus 1,"},{"Start":"05:45.890 ","End":"05:48.460","Text":"times the derivative,"},{"Start":"05:48.460 ","End":"05:51.950","Text":"and what I\u0027ll do is I\u0027ll just not differentiate yet,"},{"Start":"05:51.950 ","End":"05:53.090","Text":"but just say what I have to do."},{"Start":"05:53.090 ","End":"05:56.300","Text":"I have to find the derivative of x squared plus"},{"Start":"05:56.300 ","End":"06:04.145","Text":"1 times natural log of 2x plus 1 square brackets derivative."},{"Start":"06:04.145 ","End":"06:08.525","Text":"Now, what I\u0027ll do is I\u0027ll continue with this derivative."},{"Start":"06:08.525 ","End":"06:11.080","Text":"I\u0027ll do this as a side exercise."},{"Start":"06:11.080 ","End":"06:15.175","Text":"What I\u0027ll do is I\u0027ll do this thing over here,"},{"Start":"06:15.175 ","End":"06:19.715","Text":"there, and just the square bracket derivative part."},{"Start":"06:19.715 ","End":"06:22.890","Text":"That\u0027s where I have the u times v come in,"},{"Start":"06:22.890 ","End":"06:27.310","Text":"because u will be the first bit and v will be the second bit."},{"Start":"06:27.310 ","End":"06:31.695","Text":"If this is u, then u prime is 2x,"},{"Start":"06:31.695 ","End":"06:34.125","Text":"derivative of x squared plus 1,"},{"Start":"06:34.125 ","End":"06:36.300","Text":"times the other 1 as is,"},{"Start":"06:36.300 ","End":"06:39.505","Text":"natural log of 2x plus 1,"},{"Start":"06:39.505 ","End":"06:43.790","Text":"plus u, which is x squared plus 1,"},{"Start":"06:43.790 ","End":"06:46.275","Text":"times the derivative of this."},{"Start":"06:46.275 ","End":"06:48.795","Text":"Now, derivative of natural log of something."},{"Start":"06:48.795 ","End":"06:51.770","Text":"Again, I\u0027ve prepared with a formula."},{"Start":"06:51.770 ","End":"06:54.950","Text":"Natural log of something prime is,"},{"Start":"06:54.950 ","End":"06:56.465","Text":"you have 2x plus 1."},{"Start":"06:56.465 ","End":"07:01.115","Text":"The 2x plus 1 goes on the denominator and on the numerator,"},{"Start":"07:01.115 ","End":"07:03.590","Text":"it\u0027s derivative. That\u0027s 2."},{"Start":"07:03.590 ","End":"07:08.435","Text":"Basically, all this bit is what I have to multiply here."},{"Start":"07:08.435 ","End":"07:12.430","Text":"Let\u0027s see. This is equal to equals."},{"Start":"07:12.430 ","End":"07:20.390","Text":"It\u0027s going to be e to the power of x squared plus 1 times square brackets,"},{"Start":"07:20.390 ","End":"07:22.100","Text":"and then the stuff from here,"},{"Start":"07:22.100 ","End":"07:23.720","Text":"I don\u0027t want to put the original thing."},{"Start":"07:23.720 ","End":"07:28.965","Text":"2x natural log 2x plus 1 plus x squared plus 1."},{"Start":"07:28.965 ","End":"07:31.470","Text":"Time for this thing to go."},{"Start":"07:31.470 ","End":"07:37.040","Text":"We just have to remember that this bit is what was done over here,"},{"Start":"07:37.040 ","End":"07:43.600","Text":"or x squared plus 1 here times 2 over 2x plus 1,"},{"Start":"07:46.490 ","End":"07:49.470","Text":"and close the brackets."},{"Start":"07:49.470 ","End":"07:53.240","Text":"I\u0027d still like to indicate that this bits goes over there."},{"Start":"07:53.240 ","End":"07:55.205","Text":"I\u0027ll use a different color."},{"Start":"07:55.205 ","End":"08:01.295","Text":"This bit with the prime is what I\u0027ve done down here."},{"Start":"08:01.295 ","End":"08:03.020","Text":"I will set."},{"Start":"08:03.020 ","End":"08:04.520","Text":"Now, what I want to do,"},{"Start":"08:04.520 ","End":"08:06.530","Text":"this is f prime of x."},{"Start":"08:06.530 ","End":"08:08.450","Text":"The answer is all of this,"},{"Start":"08:08.450 ","End":"08:13.700","Text":"but I don\u0027t need f prime of x. I need f prime of x_1."},{"Start":"08:13.700 ","End":"08:17.945","Text":"What I need is f prime of x_1,"},{"Start":"08:17.945 ","End":"08:22.475","Text":"which is f prime of x_1 is 0,"},{"Start":"08:22.475 ","End":"08:24.140","Text":"which is equal to."},{"Start":"08:24.140 ","End":"08:29.030","Text":"Now what I have to do, is put 0 into this whole thing."},{"Start":"08:29.030 ","End":"08:32.735","Text":"Perhaps I\u0027ll use a color for the intermediate calculations,"},{"Start":"08:32.735 ","End":"08:35.300","Text":"and that\u0027s x 0 is going to make life easy."},{"Start":"08:35.300 ","End":"08:40.475","Text":"Because what I have here is twice 0 plus 1 is 1,"},{"Start":"08:40.475 ","End":"08:43.330","Text":"natural log of 1 is 0."},{"Start":"08:43.330 ","End":"08:45.070","Text":"It doesn\u0027t matter what this is,"},{"Start":"08:45.070 ","End":"08:48.210","Text":"this is 0 here."},{"Start":"08:48.210 ","End":"08:50.790","Text":"This whole thing becomes 0."},{"Start":"08:50.790 ","End":"08:52.980","Text":"What we get is e^0,"},{"Start":"08:52.980 ","End":"08:55.545","Text":"which is 1. That\u0027s the first bit."},{"Start":"08:55.545 ","End":"08:57.980","Text":"This bit, when I put x equals 0,"},{"Start":"08:57.980 ","End":"08:59.615","Text":"again natural log of this,"},{"Start":"08:59.615 ","End":"09:02.825","Text":"we already figured that out that this bit is 0,"},{"Start":"09:02.825 ","End":"09:06.130","Text":"which makes this whole thing 0."},{"Start":"09:06.130 ","End":"09:09.540","Text":"Then 0 squared plus 1 is just 1."},{"Start":"09:09.540 ","End":"09:13.005","Text":"Here we have 2 over 1, which is 2."},{"Start":"09:13.005 ","End":"09:14.630","Text":"Let\u0027s see what we get."},{"Start":"09:14.630 ","End":"09:18.575","Text":"First bit is 1 times brackets."},{"Start":"09:18.575 ","End":"09:21.370","Text":"This is also 0, but this is 0 as well."},{"Start":"09:21.370 ","End":"09:27.150","Text":"We get 0 plus 1 times 2,"},{"Start":"09:27.150 ","End":"09:28.770","Text":"and what does that give us altogether?"},{"Start":"09:28.770 ","End":"09:30.945","Text":"It looks like it\u0027s 2."},{"Start":"09:30.945 ","End":"09:38.935","Text":"What I get is that this is f prime of x_1 and it happens to equal 2."},{"Start":"09:38.935 ","End":"09:40.210","Text":"Now at this point,"},{"Start":"09:40.210 ","End":"09:43.895","Text":"everything is a piece of cake because I just do it by numbers."},{"Start":"09:43.895 ","End":"09:45.095","Text":"I\u0027ve got a green,"},{"Start":"09:45.095 ","End":"09:46.850","Text":"turquoise and a yellow here,"},{"Start":"09:46.850 ","End":"09:49.850","Text":"and I got a green, a turquoise and a yellow here."},{"Start":"09:49.850 ","End":"09:53.135","Text":"All I have to do is paint by numbers."},{"Start":"09:53.135 ","End":"09:55.340","Text":"I\u0027ll just write that this is what I\u0027m getting."},{"Start":"09:55.340 ","End":"10:01.425","Text":"The tangent is y minus the green 1, which is 1,"},{"Start":"10:01.425 ","End":"10:04.005","Text":"equals the turquoise 1,"},{"Start":"10:04.005 ","End":"10:10.710","Text":"which is 2 times x minus the yellow 1, which is 0."},{"Start":"10:10.710 ","End":"10:12.350","Text":"That\u0027s it, basically."},{"Start":"10:12.350 ","End":"10:14.855","Text":"Except that is customly to simplify."},{"Start":"10:14.855 ","End":"10:20.580","Text":"If you simplify it becomes y equals 2x plus 1,"},{"Start":"10:20.580 ","End":"10:24.850","Text":"and that\u0027s the equation of the tangent, and we\u0027re done."}],"ID":6472}],"Thumbnail":null,"ID":1664},{"Name":"Tangent and Normal Lines – Exercises with a Constant","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercises with a Constant","Duration":"4m 13s","ChapterTopicVideoID":10246,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this clip, I\u0027m going to talk about a certain class of exercises which"},{"Start":"00:03.840 ","End":"00:07.500","Text":"relate to tangent and normal lines and have a constant in them."},{"Start":"00:07.500 ","End":"00:09.240","Text":"Now this is not very clear,"},{"Start":"00:09.240 ","End":"00:12.150","Text":"so I\u0027m going to bring in an example right away."},{"Start":"00:12.150 ","End":"00:16.320","Text":"This exercise asks us to determine the constant k such that"},{"Start":"00:16.320 ","End":"00:21.225","Text":"a line y equals 2x minus 2 is tangent to the curve y equals kx squared."},{"Start":"00:21.225 ","End":"00:24.900","Text":"Notice that we have a constant that we have to find,"},{"Start":"00:24.900 ","End":"00:27.240","Text":"constant k, and here it is."},{"Start":"00:27.240 ","End":"00:30.504","Text":"We\u0027re looking for which curve has this property."},{"Start":"00:30.504 ","End":"00:36.529","Text":"Now the way we tackle this exercise is to notice that 3 conditions must hold."},{"Start":"00:36.529 ","End":"00:40.025","Text":"First of all, the point of contact,"},{"Start":"00:40.025 ","End":"00:43.910","Text":"let\u0027s call the point of contact x_1, y_1."},{"Start":"00:43.910 ","End":"00:48.350","Text":"That\u0027s the point at which the tangent makes contact with the curve."},{"Start":"00:48.350 ","End":"00:50.225","Text":"It has 3 properties."},{"Start":"00:50.225 ","End":"00:52.850","Text":"Let\u0027s call that point p for the moment."},{"Start":"00:52.850 ","End":"00:54.530","Text":"I know that p,"},{"Start":"00:54.530 ","End":"00:57.140","Text":"first of all, it\u0027s on the curve,"},{"Start":"00:57.140 ","End":"01:02.110","Text":"and also it must be on the tangent line."},{"Start":"01:02.110 ","End":"01:06.245","Text":"The third condition is that the slopes are equal."},{"Start":"01:06.245 ","End":"01:10.280","Text":"The slope in case of a curve is the derivative at the point p,"},{"Start":"01:10.280 ","End":"01:13.980","Text":"is equal to the slope of the tangent."},{"Start":"01:13.980 ","End":"01:16.630","Text":"Well, the slope of a straight line doesn\u0027t depend on the points."},{"Start":"01:16.630 ","End":"01:18.935","Text":"I\u0027ll just say the slope of the tangent,"},{"Start":"01:18.935 ","End":"01:22.850","Text":"at the point p. I\u0027ll certainly need the derivative."},{"Start":"01:22.850 ","End":"01:27.860","Text":"I\u0027ll just write that here y prime equals 2kx."},{"Start":"01:27.860 ","End":"01:31.655","Text":"Now let\u0027s see how these conditions translate."},{"Start":"01:31.655 ","End":"01:36.795","Text":"The first one says that x_1, y_1 is on this curve."},{"Start":"01:36.795 ","End":"01:41.410","Text":"So y_1 is equal to kx_1 squared."},{"Start":"01:41.410 ","End":"01:43.535","Text":"That\u0027s from here by substitution."},{"Start":"01:43.535 ","End":"01:45.770","Text":"Secondly, it\u0027s on the tangent."},{"Start":"01:45.770 ","End":"01:51.995","Text":"We can substitute here and get that y_1 is 2x_1 minus 2."},{"Start":"01:51.995 ","End":"01:54.515","Text":"Now the thing about the slopes being equal,"},{"Start":"01:54.515 ","End":"01:56.825","Text":"the derivative at p,"},{"Start":"01:56.825 ","End":"01:59.830","Text":"p meaning x_1, y_1,"},{"Start":"01:59.830 ","End":"02:03.345","Text":"we only need the x of the points, which is x_1."},{"Start":"02:03.345 ","End":"02:06.760","Text":"This left-hand side is 2kx_1,"},{"Start":"02:06.760 ","End":"02:11.135","Text":"and the slope of the tangent is just the coefficient of x."},{"Start":"02:11.135 ","End":"02:13.895","Text":"Or you could differentiate it and get 2."},{"Start":"02:13.895 ","End":"02:15.590","Text":"That\u0027s equal to 2."},{"Start":"02:15.590 ","End":"02:18.110","Text":"So 3 equations and 3 unknowns."},{"Start":"02:18.110 ","End":"02:20.660","Text":"Then we want to solve for k."},{"Start":"02:20.660 ","End":"02:22.370","Text":"We don\u0027t really need the x_1, y_1."},{"Start":"02:22.370 ","End":"02:23.480","Text":"It wasn\u0027t asked for."},{"Start":"02:23.480 ","End":"02:25.450","Text":"We could find them if we wanted to."},{"Start":"02:25.450 ","End":"02:27.695","Text":"From the last one, I\u0027ll extract k,"},{"Start":"02:27.695 ","End":"02:31.385","Text":"divide both sides by 2kx_1 is 1."},{"Start":"02:31.385 ","End":"02:35.975","Text":"That gives me that k equals 1 over x_1."},{"Start":"02:35.975 ","End":"02:39.645","Text":"Now I\u0027ll substitute k into here."},{"Start":"02:39.645 ","End":"02:49.140","Text":"That will give me that y_1 is equal to 1 over x_1 times x_1 squared."},{"Start":"02:49.140 ","End":"02:50.270","Text":"If you simplify this,"},{"Start":"02:50.270 ","End":"02:52.295","Text":"this is just equal to x_1."},{"Start":"02:52.295 ","End":"02:54.545","Text":"So y_1 equals x_1."},{"Start":"02:54.545 ","End":"03:01.140","Text":"If we do that here, we\u0027ll get that x_1 equals 2x_1 minus 2."},{"Start":"03:01.140 ","End":"03:02.745","Text":"If we solve this,"},{"Start":"03:02.745 ","End":"03:06.349","Text":"easily see that x_1 is equal to 2."},{"Start":"03:06.349 ","End":"03:08.345","Text":"If x_1 is equal to 2,"},{"Start":"03:08.345 ","End":"03:10.490","Text":"then I\u0027ll plug that in here."},{"Start":"03:10.490 ","End":"03:12.430","Text":"I\u0027ll get y_1 is x_1."},{"Start":"03:12.430 ","End":"03:15.360","Text":"So y_1 equals 2."},{"Start":"03:15.360 ","End":"03:16.620","Text":"I don\u0027t really need that,"},{"Start":"03:16.620 ","End":"03:18.420","Text":"but good to have."},{"Start":"03:18.420 ","End":"03:22.400","Text":"Really all I should do is put x_1 down here"},{"Start":"03:22.400 ","End":"03:26.975","Text":"so that k is equal to 1 over x_1 is 1/2."},{"Start":"03:26.975 ","End":"03:29.820","Text":"I can say that k equals 1/2,"},{"Start":"03:29.820 ","End":"03:32.120","Text":"and so we\u0027re done."},{"Start":"03:32.120 ","End":"03:34.309","Text":"But I could say the curve."},{"Start":"03:34.309 ","End":"03:36.215","Text":"If I was asked more stuff,"},{"Start":"03:36.215 ","End":"03:42.620","Text":"I could answer in full in saying that the curve is y equals 1/2x squared,"},{"Start":"03:42.620 ","End":"03:50.430","Text":"and the point which is x_1, y_1 is just 2, 2."},{"Start":"03:50.430 ","End":"03:58.320","Text":"I even know that the tangent to this curve at this point is y equals 2x minus 2."},{"Start":"03:58.320 ","End":"04:02.870","Text":"My recommendation is to take a separate exercise and say,"},{"Start":"04:02.870 ","End":"04:08.390","Text":"find the equation of the tangent to this curve at this point and see if you get this."},{"Start":"04:08.390 ","End":"04:11.735","Text":"That would be an extra check and extra practice if you need it."},{"Start":"04:11.735 ","End":"04:14.670","Text":"Meanwhile, we\u0027re done here."}],"ID":10572},{"Watched":false,"Name":"Exercise 1","Duration":"6m 7s","ChapterTopicVideoID":10247,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to determine the constant k such that"},{"Start":"00:04.590 ","End":"00:10.214","Text":"this line given by this equation is tangent to the curve given by this equation."},{"Start":"00:10.214 ","End":"00:11.985","Text":"It is the case here."},{"Start":"00:11.985 ","End":"00:14.535","Text":"That\u0027s the constant that we\u0027re talking about."},{"Start":"00:14.535 ","End":"00:17.745","Text":"To remind you about curves and tangents,"},{"Start":"00:17.745 ","End":"00:21.210","Text":"a little sketch might help although it\u0027s not necessary."},{"Start":"00:21.210 ","End":"00:22.950","Text":"We have, let\u0027s say,"},{"Start":"00:22.950 ","End":"00:25.500","Text":"the y-axis and the x-axis."},{"Start":"00:25.500 ","End":"00:30.405","Text":"I also notice that the curve is not defined for x equals 1."},{"Start":"00:30.405 ","End":"00:34.610","Text":"I\u0027m just going to write note that x is not equal to minus 1,"},{"Start":"00:34.610 ","End":"00:38.095","Text":"it\u0027s not in the domain of this curve here."},{"Start":"00:38.095 ","End":"00:46.250","Text":"So the curve, something like this and the line will be something like this."},{"Start":"00:46.250 ","End":"00:48.935","Text":"There\u0027s a point here which"},{"Start":"00:48.935 ","End":"00:52.745","Text":"notice that if the point is both on the curve and on the line,"},{"Start":"00:52.745 ","End":"00:57.980","Text":"and the slope of the curve is equal to the slope of the line."},{"Start":"00:57.980 ","End":"01:00.620","Text":"But we won\u0027t talk about necessarily the slope of the curve,"},{"Start":"01:00.620 ","End":"01:02.210","Text":"we\u0027ll talk about the derivative."},{"Start":"01:02.210 ","End":"01:05.300","Text":"The theory says that in order for this to happen,"},{"Start":"01:05.300 ","End":"01:08.390","Text":"what we have to have is 3 things."},{"Start":"01:08.390 ","End":"01:10.800","Text":"We have to have a,"},{"Start":"01:11.020 ","End":"01:14.300","Text":"just forgot to mark this point."},{"Start":"01:14.300 ","End":"01:20.040","Text":"It\u0027s just some x, y could really have called it x_1, y_1 point."},{"Start":"01:20.040 ","End":"01:22.700","Text":"I\u0027ll just indicate what is obvious,"},{"Start":"01:22.700 ","End":"01:25.654","Text":"this is the curve and this is the tangent line."},{"Start":"01:25.654 ","End":"01:30.625","Text":"So 3 conditions have to hold for this particular x, y."},{"Start":"01:30.625 ","End":"01:35.300","Text":"A, x, y is on the line,"},{"Start":"01:35.300 ","End":"01:40.290","Text":"which means that y equals minus x plus 3."},{"Start":"01:40.290 ","End":"01:42.965","Text":"Talking now about this particular x, y."},{"Start":"01:42.965 ","End":"01:45.980","Text":"The second thing, which is kind of obvious is x,"},{"Start":"01:45.980 ","End":"01:48.220","Text":"y is on the curve."},{"Start":"01:48.220 ","End":"01:52.475","Text":"All that means is that it satisfies the equation."},{"Start":"01:52.475 ","End":"01:54.200","Text":"So we have i-e,"},{"Start":"01:54.200 ","End":"01:59.990","Text":"y is equal to k over x plus 1."},{"Start":"01:59.990 ","End":"02:09.665","Text":"The third thing is that the derivative of the curve at this particular point,"},{"Start":"02:09.665 ","End":"02:11.675","Text":"at x, y,"},{"Start":"02:11.675 ","End":"02:17.825","Text":"is equal to the slope of the tangent line."},{"Start":"02:17.825 ","End":"02:23.705","Text":"The derivative of the curve happens to be the slope of the curve at that point."},{"Start":"02:23.705 ","End":"02:27.650","Text":"I mean geometrically, it\u0027s clear that if the slope of"},{"Start":"02:27.650 ","End":"02:32.330","Text":"the curve has to be just the same thing as the slope of the line,"},{"Start":"02:32.330 ","End":"02:34.830","Text":"they have to have the same slope."},{"Start":"02:34.830 ","End":"02:36.650","Text":"In the case of a curve,"},{"Start":"02:36.650 ","End":"02:37.810","Text":"the slope is the derivative,"},{"Start":"02:37.810 ","End":"02:39.245","Text":"and in the case of a line,"},{"Start":"02:39.245 ","End":"02:41.795","Text":"it\u0027s simply the coefficient of x."},{"Start":"02:41.795 ","End":"02:48.020","Text":"So c could be written as y prime at our particular x,"},{"Start":"02:48.020 ","End":"02:51.470","Text":"y is equal to minus 1,"},{"Start":"02:51.470 ","End":"02:53.465","Text":"because that\u0027s the coefficient of x."},{"Start":"02:53.465 ","End":"02:55.880","Text":"That\u0027s the derivative of the curve."},{"Start":"02:55.880 ","End":"02:58.220","Text":"Okay, so let\u0027s get some equations written."},{"Start":"02:58.220 ","End":"03:00.620","Text":"We should get 3 equations and 3 unknowns."},{"Start":"03:00.620 ","End":"03:02.900","Text":"What we need is this particular x,"},{"Start":"03:02.900 ","End":"03:05.930","Text":"y, which should really be called x_1, y_1,"},{"Start":"03:05.930 ","End":"03:11.090","Text":"but just be messy to keep writing the 1 all the time and k. X,"},{"Start":"03:11.090 ","End":"03:12.800","Text":"y, and k are the variables,"},{"Start":"03:12.800 ","End":"03:15.275","Text":"the 3 equations and 3 unknowns."},{"Start":"03:15.275 ","End":"03:17.915","Text":"I want to expand on the last 1,"},{"Start":"03:17.915 ","End":"03:22.130","Text":"the last 1, y prime just have to find the derivative."},{"Start":"03:22.130 ","End":"03:27.875","Text":"We\u0027re talking about y prime of the curve is what I\u0027m referring to."},{"Start":"03:27.875 ","End":"03:36.155","Text":"So let\u0027s continue with C. So y prime is just k. The k stays because it\u0027s constant,"},{"Start":"03:36.155 ","End":"03:43.160","Text":"and the derivative of 1 over x plus 1 is just minus 1 over x plus 1."},{"Start":"03:43.160 ","End":"03:46.400","Text":"The reason that we don\u0027t have an internal derivative, what we do,"},{"Start":"03:46.400 ","End":"03:49.190","Text":"I should really write it as times 1,"},{"Start":"03:49.190 ","End":"03:52.445","Text":"which is the inner derivative of x plus 1."},{"Start":"03:52.445 ","End":"03:56.170","Text":"That has got to equal minus 1."},{"Start":"03:56.300 ","End":"03:59.805","Text":"Got to attack these 3 things."},{"Start":"03:59.805 ","End":"04:01.250","Text":"I\u0027ll simplify this a bit."},{"Start":"04:01.250 ","End":"04:04.490","Text":"If we multiply both sides by minus 1,"},{"Start":"04:04.490 ","End":"04:06.605","Text":"we get rid of the minus and put this here."},{"Start":"04:06.605 ","End":"04:12.605","Text":"So basically we will get them again as a set of equations."},{"Start":"04:12.605 ","End":"04:18.770","Text":"So first of all, we have that y equals minus x plus 3."},{"Start":"04:18.770 ","End":"04:23.900","Text":"Then we have that y equals k over x plus 1."},{"Start":"04:23.900 ","End":"04:29.430","Text":"Finally, we have that k equals x plus 1 squared."},{"Start":"04:29.430 ","End":"04:32.370","Text":"So what I\u0027m suggesting we\u0027ll do,"},{"Start":"04:32.370 ","End":"04:37.250","Text":"is to take this k from here and put it in here."},{"Start":"04:37.250 ","End":"04:40.390","Text":"So now we\u0027ll get only 2 equations,"},{"Start":"04:40.390 ","End":"04:42.035","Text":"and we\u0027ll keep the last 1."},{"Start":"04:42.035 ","End":"04:50.005","Text":"From this 1, I get that y equals x plus 1 squared over x plus 1."},{"Start":"04:50.005 ","End":"04:53.180","Text":"Note that x is not equal to minus 1,"},{"Start":"04:53.180 ","End":"04:54.705","Text":"it\u0027s not in the domain,"},{"Start":"04:54.705 ","End":"04:58.400","Text":"so we can divide top and bottom by x plus 1."},{"Start":"04:58.400 ","End":"05:03.140","Text":"So this gives us that y equals x plus 1."},{"Start":"05:03.140 ","End":"05:06.320","Text":"Now we have this equation and this equation,"},{"Start":"05:06.320 ","End":"05:09.125","Text":"but y also equals x plus 3."},{"Start":"05:09.125 ","End":"05:11.555","Text":"So from here and here,"},{"Start":"05:11.555 ","End":"05:15.845","Text":"we\u0027ll now get this y here is minus x plus 3."},{"Start":"05:15.845 ","End":"05:19.310","Text":"I\u0027m going up here, minus x plus 3,"},{"Start":"05:19.310 ","End":"05:22.625","Text":"which is y is going to equal to x plus 1."},{"Start":"05:22.625 ","End":"05:28.760","Text":"If I solve this I\u0027d see 2x on this side is equal to 2."},{"Start":"05:28.760 ","End":"05:30.695","Text":"So x equals 1."},{"Start":"05:30.695 ","End":"05:33.785","Text":"This gives us x equals 1."},{"Start":"05:33.785 ","End":"05:37.144","Text":"Now I can put that in here,"},{"Start":"05:37.144 ","End":"05:39.840","Text":"and y equals 2,"},{"Start":"05:39.840 ","End":"05:41.325","Text":"because x is 1."},{"Start":"05:41.325 ","End":"05:45.155","Text":"The other thing I do is look at k, because now I have x."},{"Start":"05:45.155 ","End":"05:48.485","Text":"So I\u0027ll put this x equals 1 over here,"},{"Start":"05:48.485 ","End":"05:56.485","Text":"and I\u0027ll get that k equals 1 plus 1 squared, which equals 4."},{"Start":"05:56.485 ","End":"05:59.720","Text":"That\u0027s, I believe what the question asked for,"},{"Start":"05:59.720 ","End":"06:07.500","Text":"it\u0027s just for k. The answer is k equals 4, and that\u0027s it."}],"ID":10573},{"Watched":false,"Name":"Exercise 2","Duration":"5m 3s","ChapterTopicVideoID":10248,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we have to determine the constant n"},{"Start":"00:03.870 ","End":"00:07.485","Text":"such that the line y equals minus x plus n,"},{"Start":"00:07.485 ","End":"00:08.895","Text":"that\u0027s this n here,"},{"Start":"00:08.895 ","End":"00:11.490","Text":"is tangent to this curve."},{"Start":"00:11.490 ","End":"00:13.035","Text":"I\u0027ll draw a quick sketch,"},{"Start":"00:13.035 ","End":"00:16.590","Text":"but you definitely do not need a sketch to be able to solve this,"},{"Start":"00:16.590 ","End":"00:18.975","Text":"so just as a visual aid,"},{"Start":"00:18.975 ","End":"00:23.565","Text":"here we have the y-axis, the x-axis."},{"Start":"00:23.565 ","End":"00:27.135","Text":"Y equals 1 over x, I\u0027m familiar with."},{"Start":"00:27.135 ","End":"00:31.095","Text":"It\u0027s a hyperbola actually has another side to it as well."},{"Start":"00:31.095 ","End":"00:32.790","Text":"Minus x plus n,"},{"Start":"00:32.790 ","End":"00:36.180","Text":"I can already imagine slope of minus 1."},{"Start":"00:36.180 ","End":"00:38.500","Text":"It should be something like,"},{"Start":"00:38.500 ","End":"00:42.260","Text":"I\u0027m going to make it hit somewhere over here,"},{"Start":"00:42.260 ","End":"00:43.895","Text":"let\u0027s say is the point."},{"Start":"00:43.895 ","End":"00:45.604","Text":"Here we have the curve,"},{"Start":"00:45.604 ","End":"00:49.880","Text":"the tangent line continues, of course."},{"Start":"00:49.880 ","End":"00:51.365","Text":"This is the point,"},{"Start":"00:51.365 ","End":"00:53.230","Text":"let\u0027s call it x, y."},{"Start":"00:53.230 ","End":"00:55.635","Text":"For these to be tangent,"},{"Start":"00:55.635 ","End":"00:57.810","Text":"the 3 things have to hold true,"},{"Start":"00:57.810 ","End":"01:00.210","Text":"either way x cannot equal 0,"},{"Start":"01:00.210 ","End":"01:05.120","Text":"as domain of this case it comes up maybe I better write it now."},{"Start":"01:05.120 ","End":"01:08.940","Text":"We need the 3 things are going to hold about,"},{"Start":"01:08.940 ","End":"01:10.940","Text":"we have 3 variables, x, y,"},{"Start":"01:10.940 ","End":"01:13.985","Text":"and n, and we\u0027re going to need 3 equations."},{"Start":"01:13.985 ","End":"01:20.460","Text":"A is that, I don\u0027t even need to write that,"},{"Start":"01:20.460 ","End":"01:22.500","Text":"obviously the point is on the curve,"},{"Start":"01:22.500 ","End":"01:26.495","Text":"which means that it satisfies the equation of the curve."},{"Start":"01:26.495 ","End":"01:29.585","Text":"First of all, we have y equals 1 over x."},{"Start":"01:29.585 ","End":"01:30.890","Text":"The point is on the curve."},{"Start":"01:30.890 ","End":"01:32.450","Text":"The point is on the line,"},{"Start":"01:32.450 ","End":"01:34.555","Text":"on the tangent line."},{"Start":"01:34.555 ","End":"01:40.425","Text":"That means that y equals minus x plus n. The third 1,"},{"Start":"01:40.425 ","End":"01:42.145","Text":"and that\u0027s the trickier 1,"},{"Start":"01:42.145 ","End":"01:46.745","Text":"is that we have to have the slopes being equal."},{"Start":"01:46.745 ","End":"01:54.380","Text":"Which means that the slope in the case of the line is just the coefficient of x."},{"Start":"01:54.380 ","End":"01:56.910","Text":"The slope is minus 1."},{"Start":"01:57.580 ","End":"02:00.035","Text":"Let me just say that here,"},{"Start":"02:00.035 ","End":"02:04.160","Text":"the slope of the line is known to be the coefficient of the x,"},{"Start":"02:04.160 ","End":"02:06.395","Text":"so it\u0027s minus 1."},{"Start":"02:06.395 ","End":"02:12.305","Text":"What we have to have is the derivative y prime at this point."},{"Start":"02:12.305 ","End":"02:14.485","Text":"Y prime of the curve,"},{"Start":"02:14.485 ","End":"02:19.865","Text":"I\u0027ll just say this equal to the slope of line,"},{"Start":"02:19.865 ","End":"02:22.865","Text":"which means that y prime,"},{"Start":"02:22.865 ","End":"02:26.285","Text":"which is 1 minus 1 over x squared."},{"Start":"02:26.285 ","End":"02:28.880","Text":"In other words, minus 1 over x squared,"},{"Start":"02:28.880 ","End":"02:31.250","Text":"which is y prime of the curve,"},{"Start":"02:31.250 ","End":"02:32.990","Text":"is equal to the slope of the line,"},{"Start":"02:32.990 ","End":"02:35.885","Text":"which as I said, is the co-efficient of x."},{"Start":"02:35.885 ","End":"02:38.885","Text":"When you have y equals ax plus b,"},{"Start":"02:38.885 ","End":"02:41.825","Text":"the a is the slope."},{"Start":"02:41.825 ","End":"02:44.135","Text":"That equals minus 1."},{"Start":"02:44.135 ","End":"02:49.370","Text":"That gives us 3 equations in 3 variables, x,"},{"Start":"02:49.370 ","End":"02:55.175","Text":"y, and n. Let\u0027s start with the last 1."},{"Start":"02:55.175 ","End":"02:58.074","Text":"If you multiply both sides,"},{"Start":"02:58.074 ","End":"03:00.420","Text":"first of all, we can drop the minus off both sides."},{"Start":"03:00.420 ","End":"03:02.605","Text":"If 1 over x squared is 1,"},{"Start":"03:02.605 ","End":"03:07.430","Text":"then if I multiply both sides by x squared then I get that x squared is 1."},{"Start":"03:07.430 ","End":"03:11.315","Text":"This gives us that x squared is equal to 1."},{"Start":"03:11.315 ","End":"03:13.130","Text":"It looks like there might actually be"},{"Start":"03:13.130 ","End":"03:17.865","Text":"more than 1 solution because x is going to be plus or minus 1."},{"Start":"03:17.865 ","End":"03:19.805","Text":"Yes, if we look at this,"},{"Start":"03:19.805 ","End":"03:23.150","Text":"we see that there actually is another place over"},{"Start":"03:23.150 ","End":"03:26.450","Text":"here which also might have slope of minus 1."},{"Start":"03:26.450 ","End":"03:29.090","Text":"So actually there could be 2 solutions."},{"Start":"03:29.090 ","End":"03:30.620","Text":"In fact, in D2,"},{"Start":"03:30.620 ","End":"03:31.910","Text":"we\u0027ll see that there are."},{"Start":"03:31.910 ","End":"03:33.455","Text":"Let\u0027s split them up."},{"Start":"03:33.455 ","End":"03:41.290","Text":"Let\u0027s call it solution, maybe Roman 1."},{"Start":"03:41.290 ","End":"03:46.370","Text":"What we have is the x is take the plus first x is,"},{"Start":"03:46.370 ","End":"03:48.515","Text":"just writing plus for emphasis,"},{"Start":"03:48.515 ","End":"03:50.480","Text":"x equals plus 1."},{"Start":"03:50.480 ","End":"03:52.550","Text":"If x is plus 1,"},{"Start":"03:52.550 ","End":"03:54.475","Text":"then we have from a,"},{"Start":"03:54.475 ","End":"03:58.385","Text":"this gives us from because of a,"},{"Start":"03:58.385 ","End":"03:59.975","Text":"that y is 1 over x."},{"Start":"03:59.975 ","End":"04:02.990","Text":"So y equals 1, also."},{"Start":"04:02.990 ","End":"04:05.780","Text":"This gives us from be,"},{"Start":"04:05.780 ","End":"04:07.590","Text":"that if x is 1,"},{"Start":"04:07.590 ","End":"04:08.940","Text":"and y is 1,"},{"Start":"04:08.940 ","End":"04:13.300","Text":"then n is y plus x, so n is 2."},{"Start":"04:13.400 ","End":"04:17.645","Text":"If we do it also for solution number 2,"},{"Start":"04:17.645 ","End":"04:23.690","Text":"then we get that x is this time the minus case, minus 1."},{"Start":"04:23.690 ","End":"04:27.450","Text":"Again from a, y is also minus 1,"},{"Start":"04:27.450 ","End":"04:29.700","Text":"because it\u0027s 1 over minus 1,"},{"Start":"04:29.700 ","End":"04:37.355","Text":"from part B still x plus y equals n. So n is going to equal minus 2."},{"Start":"04:37.355 ","End":"04:42.230","Text":"Basically, they just want the n,"},{"Start":"04:42.230 ","End":"04:43.980","Text":"so there\u0027s 2 answers,"},{"Start":"04:43.980 ","End":"04:47.025","Text":"but there\u0027s also 2 separate points."},{"Start":"04:47.025 ","End":"04:54.915","Text":"Suppose you could just say n is 2 or minus 2."},{"Start":"04:54.915 ","End":"04:58.910","Text":"N is equal to plus or you can even combine it this way,"},{"Start":"04:58.910 ","End":"05:00.470","Text":"n is plus or minus 2,"},{"Start":"05:00.470 ","End":"05:01.865","Text":"and that\u0027s the answer."},{"Start":"05:01.865 ","End":"05:04.380","Text":"2 separate solutions."}],"ID":10574},{"Watched":false,"Name":"Exercise 3","Duration":"7m 44s","ChapterTopicVideoID":10249,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"In this exercise, we have to determine the constants a and c,"},{"Start":"00:03.885 ","End":"00:06.495","Text":"that\u0027s a over here and c over there,"},{"Start":"00:06.495 ","End":"00:09.150","Text":"such that the line y equals ax"},{"Start":"00:09.150 ","End":"00:15.180","Text":"plus a half is tangent to the curve y equals 2 over x plus c,"},{"Start":"00:15.180 ","End":"00:17.420","Text":"the point where x equals 0."},{"Start":"00:17.420 ","End":"00:22.505","Text":"I brought with me a little sketch which I made and not to waste your time sketching."},{"Start":"00:22.505 ","End":"00:26.960","Text":"There are 3 main quantities or entities concerned in this problem."},{"Start":"00:26.960 ","End":"00:30.250","Text":"There\u0027s a curve, there\u0027s a line, and there\u0027s a point."},{"Start":"00:30.250 ","End":"00:34.610","Text":"This line is tangent to this curve at this point,"},{"Start":"00:34.610 ","End":"00:38.215","Text":"and when that happens in general, 3 things hold,"},{"Start":"00:38.215 ","End":"00:43.095","Text":"which I remember just mnemonically without any mathematics to it."},{"Start":"00:43.095 ","End":"00:45.615","Text":"1 is that the point is on the curve,"},{"Start":"00:45.615 ","End":"00:48.645","Text":"2, the point is on the line."},{"Start":"00:48.645 ","End":"00:51.330","Text":"This point, I didn\u0027t draw it so exact,"},{"Start":"00:51.330 ","End":"00:53.865","Text":"but it\u0027s on the curve and it\u0027s on the line,"},{"Start":"00:53.865 ","End":"00:56.790","Text":"and what makes it special that to makes it"},{"Start":"00:56.790 ","End":"01:00.015","Text":"tangent is that actually the slopes here are the same,"},{"Start":"01:00.015 ","End":"01:01.140","Text":"only in the case of the curve,"},{"Start":"01:01.140 ","End":"01:05.580","Text":"the slope is the derivative so that the derivative at the point,"},{"Start":"01:05.580 ","End":"01:06.770","Text":"and when I say derivative,"},{"Start":"01:06.770 ","End":"01:09.465","Text":"I\u0027m talking about the curve, not the line,"},{"Start":"01:09.465 ","End":"01:15.755","Text":"derivative of the curve at the point is equal to the slope of the line."},{"Start":"01:15.755 ","End":"01:19.190","Text":"These are 3 things to remember qualitatively, not quantitatively."},{"Start":"01:19.190 ","End":"01:21.425","Text":"The points on the curve, the points on the line,"},{"Start":"01:21.425 ","End":"01:25.665","Text":"and the line and the curve have equal slopes."},{"Start":"01:25.665 ","End":"01:27.545","Text":"In the case of a line,"},{"Start":"01:27.545 ","End":"01:29.465","Text":"if line is ax plus b,"},{"Start":"01:29.465 ","End":"01:31.640","Text":"slope of line which equals a,"},{"Start":"01:31.640 ","End":"01:33.230","Text":"presuming that we have the line,"},{"Start":"01:33.230 ","End":"01:36.600","Text":"is of the general form y equals ax plus b."},{"Start":"01:36.600 ","End":"01:40.520","Text":"That\u0027s the a I mean, the coefficient of the x and the line equation."},{"Start":"01:40.520 ","End":"01:45.490","Text":"Let\u0027s see how to interpret this in our case."},{"Start":"01:45.490 ","End":"01:48.215","Text":"Since we\u0027re going to be needing the derivative,"},{"Start":"01:48.215 ","End":"01:53.375","Text":"why don\u0027t I go ahead and differentiate right at the start?"},{"Start":"01:53.375 ","End":"01:56.285","Text":"I find that it\u0027s a good idea in general."},{"Start":"01:56.285 ","End":"02:00.230","Text":"If y equals 2 over x plus c,"},{"Start":"02:00.230 ","End":"02:02.135","Text":"and that\u0027s this curve here,"},{"Start":"02:02.135 ","End":"02:04.820","Text":"then y prime is equal."},{"Start":"02:04.820 ","End":"02:07.840","Text":"Well, it\u0027s just like the derivative of 1 over x,"},{"Start":"02:07.840 ","End":"02:11.825","Text":"except that it\u0027s 2 here and the x has got a plus c. But we start off by"},{"Start":"02:11.825 ","End":"02:15.940","Text":"putting a minus 1 over x plus c squared."},{"Start":"02:15.940 ","End":"02:18.080","Text":"Well, the 2 was here, so it stays."},{"Start":"02:18.080 ","End":"02:20.100","Text":"But also, there\u0027s an internal derivative,"},{"Start":"02:20.100 ","End":"02:22.455","Text":"so I have to multiply by 1."},{"Start":"02:22.455 ","End":"02:25.100","Text":"The 2 is there because it was there."},{"Start":"02:25.100 ","End":"02:27.860","Text":"The minus 1 over something squared because it\u0027s like 1"},{"Start":"02:27.860 ","End":"02:30.695","Text":"over x and the 1 is the internal derivative."},{"Start":"02:30.695 ","End":"02:32.590","Text":"Now, we have y prime."},{"Start":"02:32.590 ","End":"02:37.050","Text":"We can reinterpret this in our case, so a,"},{"Start":"02:37.050 ","End":"02:41.480","Text":"b, and c, which is going to be a reinterpretation of what we have above."},{"Start":"02:41.480 ","End":"02:47.925","Text":"The point is on the curve just is the equation of the curve y equals 2 over,"},{"Start":"02:47.925 ","End":"02:50.150","Text":"at this particular x and y,"},{"Start":"02:50.150 ","End":"02:56.540","Text":"y is going to equal to over x plus c for our x and y."},{"Start":"02:56.540 ","End":"02:59.825","Text":"As for b, the point is on the line to the point,"},{"Start":"02:59.825 ","End":"03:02.150","Text":"if we call, we even have the line."},{"Start":"03:02.150 ","End":"03:03.655","Text":"The line is this,"},{"Start":"03:03.655 ","End":"03:07.770","Text":"so y equals ax plus 0.5,"},{"Start":"03:07.770 ","End":"03:09.990","Text":"and let\u0027s assume that the line is of the form ax"},{"Start":"03:09.990 ","End":"03:12.030","Text":"plus b so we\u0027ll be looking for b afterwards,"},{"Start":"03:12.030 ","End":"03:13.725","Text":"which is 1 of our unknowns."},{"Start":"03:13.725 ","End":"03:15.660","Text":"The 3rd equation,"},{"Start":"03:15.660 ","End":"03:19.980","Text":"now you\u0027re going to get it confused with this c and this a, nothing to do."},{"Start":"03:19.980 ","End":"03:24.485","Text":"Thirdly, the derivative at the point on the curve,"},{"Start":"03:24.485 ","End":"03:26.330","Text":"which is y prime,"},{"Start":"03:26.330 ","End":"03:31.235","Text":"is equal to the slope of the line, which is a."},{"Start":"03:31.235 ","End":"03:34.025","Text":"In fact, since I already have y prime,"},{"Start":"03:34.025 ","End":"03:39.545","Text":"why don\u0027t I just erase it and write what it really is. There we are."},{"Start":"03:39.545 ","End":"03:41.975","Text":"The derivative, which is"},{"Start":"03:41.975 ","End":"03:47.520","Text":"the y prime at that point is equal to the slope of the line. I\u0027m in."},{"Start":"03:48.970 ","End":"03:53.060","Text":"There\u0027s 3 equations, but there\u0027s 4 unknowns."},{"Start":"03:53.060 ","End":"03:54.695","Text":"I mean, we have x and y,"},{"Start":"03:54.695 ","End":"03:56.560","Text":"and a and c,"},{"Start":"03:56.560 ","End":"03:59.165","Text":"so what piece of information have we missed?"},{"Start":"03:59.165 ","End":"04:02.510","Text":"Well, what we\u0027ve missed is that x equals 0."},{"Start":"04:02.510 ","End":"04:05.780","Text":"I can either write it as a 4th condition"},{"Start":"04:05.780 ","End":"04:09.505","Text":"or I can just translate these by putting x equals 0."},{"Start":"04:09.505 ","End":"04:12.200","Text":"Let\u0027s do the second. At the place where x is 0,"},{"Start":"04:12.200 ","End":"04:13.370","Text":"in fact, in my picture,"},{"Start":"04:13.370 ","End":"04:18.050","Text":"I tried to make it come out on the y-axis so that x would be 0."},{"Start":"04:18.050 ","End":"04:20.165","Text":"Where x equals 0,"},{"Start":"04:20.165 ","End":"04:26.395","Text":"here we get that y is equal to 2/c,"},{"Start":"04:26.395 ","End":"04:32.880","Text":"and from here, I get that y equals 0.5."},{"Start":"04:32.880 ","End":"04:35.655","Text":"Lastly, minus 2,"},{"Start":"04:35.655 ","End":"04:40.890","Text":"so all we\u0027re left with is the c squared and this is equal to a."},{"Start":"04:40.890 ","End":"04:44.640","Text":"Now, 3 equations and 3 unknowns,"},{"Start":"04:44.640 ","End":"04:47.655","Text":"y, c, and a."},{"Start":"04:47.655 ","End":"04:53.150","Text":"We got from 4 equations down to 3 by putting x equals 0,"},{"Start":"04:53.150 ","End":"04:57.380","Text":"we also know why y has to be 0.5."},{"Start":"04:57.380 ","End":"05:01.835","Text":"In fact, I could have said here that the picture xy is just mentioned,"},{"Start":"05:01.835 ","End":"05:04.130","Text":"0 comma 0.5,"},{"Start":"05:04.130 ","End":"05:07.635","Text":"if anyone asked us for the tangent line, and do they?"},{"Start":"05:07.635 ","End":"05:09.690","Text":"No, they don\u0027t, just ask for the constants."},{"Start":"05:09.690 ","End":"05:12.320","Text":"But if they wanted the x and y also,"},{"Start":"05:12.320 ","End":"05:17.000","Text":"we say that it\u0027s 0 comma 0.5."},{"Start":"05:17.000 ","End":"05:24.895","Text":"Now, back to here where we said we\u0027re going to substitute y equals 0.5,"},{"Start":"05:24.895 ","End":"05:34.190","Text":"let\u0027s remember it, I\u0027ll highlight it and say that y is equal to 0.5."},{"Start":"05:34.190 ","End":"05:36.150","Text":"Just as before we,"},{"Start":"05:36.150 ","End":"05:40.055","Text":"highlighted, we should have copied it to really."},{"Start":"05:40.055 ","End":"05:43.310","Text":"I\u0027ve tidied up a bit taking the underlining out of"},{"Start":"05:43.310 ","End":"05:46.520","Text":"there and written the fresh x equals 0 over here,"},{"Start":"05:46.520 ","End":"05:49.625","Text":"which is another variable I\u0027ve discovered, were trying to discover."},{"Start":"05:49.625 ","End":"05:51.170","Text":"I took all the ones we don\u0027t need,"},{"Start":"05:51.170 ","End":"05:52.895","Text":"we need a and c, and meanwhile,"},{"Start":"05:52.895 ","End":"05:59.750","Text":"all we\u0027ve got is x and y. I just rewrote that I took x equals 0 and plugged it into a,"},{"Start":"05:59.750 ","End":"06:02.015","Text":"b, and c, and I got these 3 equations."},{"Start":"06:02.015 ","End":"06:06.525","Text":"Now, I\u0027m going to take the y equals 0.5 and plug it in to,"},{"Start":"06:06.525 ","End":"06:10.570","Text":"in fact, there is only 1 where I have to plug it into. It\u0027s over here."},{"Start":"06:10.570 ","End":"06:14.540","Text":"Now, I get a couple of new equations besides the x and y."},{"Start":"06:14.540 ","End":"06:19.115","Text":"I\u0027ll just copy the last 1 again and the first 1 substituted."},{"Start":"06:19.115 ","End":"06:21.060","Text":"Now, we have, I don\u0027t know,"},{"Start":"06:21.060 ","End":"06:25.215","Text":"2 equations lets call them 1 and 2."},{"Start":"06:25.215 ","End":"06:32.880","Text":"I put y in here and get 2/c is equal to y, which is 0.5."},{"Start":"06:32.880 ","End":"06:36.345","Text":"Here, a equals minus 2/c^2."},{"Start":"06:36.345 ","End":"06:41.330","Text":"Now, what I think would be best is to isolate c as a number from here,"},{"Start":"06:41.330 ","End":"06:43.115","Text":"and then we can plug it into there."},{"Start":"06:43.115 ","End":"06:47.750","Text":"This should give us, if we multiply both sides by 2,"},{"Start":"06:47.750 ","End":"06:49.385","Text":"we get 4/c,"},{"Start":"06:49.385 ","End":"06:55.065","Text":"is 1, then it has to be that c equals 4,"},{"Start":"06:55.065 ","End":"06:57.225","Text":"so c equals 4."},{"Start":"06:57.225 ","End":"06:59.025","Text":"We\u0027ve got another 1, xy."},{"Start":"06:59.025 ","End":"07:02.565","Text":"Now, we\u0027ve got c and I highlight that,"},{"Start":"07:02.565 ","End":"07:04.800","Text":"and all we\u0027re left with is a."},{"Start":"07:04.800 ","End":"07:07.425","Text":"If we put c equals 4 here,"},{"Start":"07:07.425 ","End":"07:12.735","Text":"this equals minus 2/c squared is 4 squared."},{"Start":"07:12.735 ","End":"07:14.055","Text":"Put 4 for c,"},{"Start":"07:14.055 ","End":"07:18.765","Text":"that\u0027s 16, that\u0027s minus 1/8."},{"Start":"07:18.765 ","End":"07:22.200","Text":"Again, we\u0027ll highlight that also."},{"Start":"07:22.200 ","End":"07:23.910","Text":"Now, we\u0027ve got all 4,"},{"Start":"07:23.910 ","End":"07:26.100","Text":"just making sure I got everything they asked for,"},{"Start":"07:26.100 ","End":"07:29.395","Text":"it\u0027s a bad thing to do if you don\u0027t answer the complete question."},{"Start":"07:29.395 ","End":"07:34.760","Text":"Determine the constants a and c. To really summarize it,"},{"Start":"07:34.760 ","End":"07:39.065","Text":"a equals minus 1/8,"},{"Start":"07:39.065 ","End":"07:41.465","Text":"c equals 4,"},{"Start":"07:41.465 ","End":"07:44.400","Text":"and that is our answer."}],"ID":10575},{"Watched":false,"Name":"Exercise 4","Duration":"8m 22s","ChapterTopicVideoID":10250,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this exercise, we have to write the equation of the straight line,"},{"Start":"00:04.005 ","End":"00:09.705","Text":"passing through the point minus 1,0 and tangent to the curve,"},{"Start":"00:09.705 ","End":"00:12.660","Text":"f of x equals the square root of x."},{"Start":"00:12.660 ","End":"00:15.780","Text":"Finally, sketch the curve and the line."},{"Start":"00:15.780 ","End":"00:20.865","Text":"Well, I\u0027ve saved you a bit of work and I brought the sketch with me."},{"Start":"00:20.865 ","End":"00:23.205","Text":"So x and y axis,"},{"Start":"00:23.205 ","End":"00:27.855","Text":"we have a curve which was given to us above."},{"Start":"00:27.855 ","End":"00:29.670","Text":"Here we have the line,"},{"Start":"00:29.670 ","End":"00:33.165","Text":"tangent line, and here we have the tangent point."},{"Start":"00:33.165 ","End":"00:35.790","Text":"It\u0027s hard to tell exactly where it is,"},{"Start":"00:35.790 ","End":"00:38.300","Text":"but this is the point."},{"Start":"00:38.300 ","End":"00:39.500","Text":"We have a line, a curve,"},{"Start":"00:39.500 ","End":"00:41.915","Text":"and a point in this question."},{"Start":"00:41.915 ","End":"00:48.395","Text":"Here\u0027s the point minus 1,0 that the curve goes through. That\u0027s about it."},{"Start":"00:48.395 ","End":"00:54.020","Text":"Notice that the curve is only in the upper quadrant."},{"Start":"00:54.020 ","End":"00:56.290","Text":"It\u0027s positive, non-zero,"},{"Start":"00:56.290 ","End":"00:58.910","Text":"and it\u0027s also defined only for non-zero because it\u0027s"},{"Start":"00:58.910 ","End":"01:02.690","Text":"the square root of x. That\u0027s about all."},{"Start":"01:02.690 ","End":"01:05.225","Text":"Now, to get back to the algebra,"},{"Start":"01:05.225 ","End":"01:06.890","Text":"you don\u0027t need to sketch curves,"},{"Start":"01:06.890 ","End":"01:09.865","Text":"but it sometimes helps to see it."},{"Start":"01:09.865 ","End":"01:13.535","Text":"I\u0027ll also call this, I\u0027ll write it here."},{"Start":"01:13.535 ","End":"01:18.295","Text":"Y equals square root of x instead of f of x to call it y sometimes."},{"Start":"01:18.295 ","End":"01:24.185","Text":"In general, there are 3 things that determine this problem in of all involving the point,"},{"Start":"01:24.185 ","End":"01:25.565","Text":"the curve, and the line."},{"Start":"01:25.565 ","End":"01:27.275","Text":"There are 3 facts,"},{"Start":"01:27.275 ","End":"01:30.940","Text":"and I\u0027ll write them not in mathematical language in just plain language."},{"Start":"01:30.940 ","End":"01:33.140","Text":"We have property, if you like,"},{"Start":"01:33.140 ","End":"01:37.669","Text":"a, is that the point is on the line,"},{"Start":"01:37.669 ","End":"01:41.630","Text":"let\u0027s telegraphically write that point on line,"},{"Start":"01:41.630 ","End":"01:46.105","Text":"meaning the tangent point is on the tangent line."},{"Start":"01:46.105 ","End":"01:48.740","Text":"Property B, the point is on the curve."},{"Start":"01:48.740 ","End":"01:51.880","Text":"The tangent point happens to lie on the curve."},{"Start":"01:51.880 ","End":"01:55.310","Text":"The third property is that the slopes are equal, well,"},{"Start":"01:55.310 ","End":"01:57.380","Text":"the case of the line,"},{"Start":"01:57.380 ","End":"02:01.265","Text":"It\u0027s a slope and in the case of the curve, it\u0027s the derivative."},{"Start":"02:01.265 ","End":"02:06.340","Text":"Derivative and this is for the curve equals the slope,"},{"Start":"02:06.340 ","End":"02:09.225","Text":"and the slope is for the line."},{"Start":"02:09.225 ","End":"02:14.420","Text":"What I mean by slope is that if I write my line as, say,"},{"Start":"02:14.420 ","End":"02:17.150","Text":"give it some letters like ax plus b,"},{"Start":"02:17.150 ","End":"02:19.070","Text":"which is the typical letters we use,"},{"Start":"02:19.070 ","End":"02:24.865","Text":"then slope in this case is going to equal a."},{"Start":"02:24.865 ","End":"02:26.460","Text":"Those are the 3 things."},{"Start":"02:26.460 ","End":"02:28.560","Text":"The point is on the line,"},{"Start":"02:28.560 ","End":"02:30.030","Text":"it\u0027s also on the curve,"},{"Start":"02:30.030 ","End":"02:31.325","Text":"and at that point,"},{"Start":"02:31.325 ","End":"02:37.110","Text":"the derivative of the curve is equal to the slope of the tangent line."},{"Start":"02:37.110 ","End":"02:41.185","Text":"Actually the derivative is also a slope of the curve."},{"Start":"02:41.185 ","End":"02:43.395","Text":"With these 3 thing,"},{"Start":"02:43.395 ","End":"02:46.590","Text":"we can go ahead and do the algebra."},{"Start":"02:46.590 ","End":"02:49.535","Text":"1 last thing is customary when you\u0027re looking at a function"},{"Start":"02:49.535 ","End":"02:52.430","Text":"to consider its domain because it might come in useful."},{"Start":"02:52.430 ","End":"02:57.590","Text":"In this case, this function is only defined for x bigger or equal to 0."},{"Start":"02:57.590 ","End":"03:00.245","Text":"We should mention that you never know and it could be useful."},{"Start":"03:00.245 ","End":"03:04.505","Text":"Let\u0027s rewrite these 3 things in mathematical terms,"},{"Start":"03:04.505 ","End":"03:06.240","Text":"not just in general terms."},{"Start":"03:06.240 ","End":"03:08.204","Text":"A, the point is on the line,"},{"Start":"03:08.204 ","End":"03:11.265","Text":"means that our point will be,"},{"Start":"03:11.265 ","End":"03:12.745","Text":"I\u0027ll call it x,y."},{"Start":"03:12.745 ","End":"03:17.210","Text":"Or some people like to use a different x,y like x1, y1 to distinguish."},{"Start":"03:17.210 ","End":"03:20.030","Text":"X,y is on the line is just the equation of the line."},{"Start":"03:20.030 ","End":"03:21.140","Text":"It\u0027s okay, I wrote it out."},{"Start":"03:21.140 ","End":"03:22.745","Text":"Y is ax plus b."},{"Start":"03:22.745 ","End":"03:28.010","Text":"This is not the same a as this that\u0027s just the number of the formula, part b."},{"Start":"03:28.010 ","End":"03:29.815","Text":"The point is on the curve,"},{"Start":"03:29.815 ","End":"03:31.430","Text":"the curve is f of x,"},{"Start":"03:31.430 ","End":"03:35.240","Text":"so y equals the square root of x."},{"Start":"03:35.240 ","End":"03:41.750","Text":"The third thing is that the slope of the derivative is equal to a,"},{"Start":"03:41.750 ","End":"03:44.150","Text":"but we don\u0027t have the derivative yet."},{"Start":"03:44.150 ","End":"03:47.495","Text":"Well, let me just write it meanwhile as y-prime."},{"Start":"03:47.495 ","End":"03:50.975","Text":"At the end, I\u0027ll write that this is equal"},{"Start":"03:50.975 ","End":"03:54.260","Text":"to a and then we\u0027ll figure it out. We\u0027ll replace it."},{"Start":"03:54.260 ","End":"03:57.350","Text":"Why don\u0027t we do that? Y prime,"},{"Start":"03:57.350 ","End":"03:59.450","Text":"if y is the square root of x,"},{"Start":"03:59.450 ","End":"04:02.915","Text":"y-prime is 1 of those elementary 1s that you should know by heart."},{"Start":"04:02.915 ","End":"04:12.485","Text":"Anyway, it\u0027s 1 over twice the square root of x. I\u0027ll just replace this here."},{"Start":"04:12.485 ","End":"04:17.315","Text":"I just did a quick copy paste from here to here, that\u0027s y prime."},{"Start":"04:17.315 ","End":"04:20.050","Text":"These are 3 equations."},{"Start":"04:20.050 ","End":"04:23.225","Text":"We have how many unknowns?"},{"Start":"04:23.225 ","End":"04:27.595","Text":"Y, x, a, and b?"},{"Start":"04:27.595 ","End":"04:30.300","Text":"4 unknowns and only 3 equations,"},{"Start":"04:30.300 ","End":"04:32.335","Text":"what factor we missing?"},{"Start":"04:32.335 ","End":"04:36.365","Text":"That it goes through the point minus 1,0."},{"Start":"04:36.365 ","End":"04:39.995","Text":"Anyway, I could call that fourth equation,"},{"Start":"04:39.995 ","End":"04:42.645","Text":"minus 1,0 is on the curve."},{"Start":"04:42.645 ","End":"04:46.610","Text":"That means that if I put y equals minus 1,"},{"Start":"04:46.610 ","End":"04:49.860","Text":"perhaps I\u0027ll just write it in words first,"},{"Start":"04:50.200 ","End":"04:54.300","Text":"minus1,0 not on the curve,"},{"Start":"04:54.300 ","End":"04:56.025","Text":"it\u0027s on the tangent."},{"Start":"04:56.025 ","End":"04:58.850","Text":"It\u0027s a straight line that passes through minus 10,"},{"Start":"04:58.850 ","End":"05:00.559","Text":"so it\u0027s on the line."},{"Start":"05:00.559 ","End":"05:04.130","Text":"What line is that? Ax plus b. Y equals ax plus b."},{"Start":"05:04.130 ","End":"05:07.295","Text":"Well, I have it written over here also I could have anyway."},{"Start":"05:07.295 ","End":"05:13.100","Text":"If y is 0 and x is minus 1,"},{"Start":"05:13.100 ","End":"05:17.810","Text":"we get a times minus 1 and let\u0027s put up this in brackets"},{"Start":"05:17.810 ","End":"05:23.435","Text":"also to show I substituted them from the minus 1,0 plus b,"},{"Start":"05:23.435 ","End":"05:29.555","Text":"which basically, I can straight away say that what this gives us that a equals b."},{"Start":"05:29.555 ","End":"05:31.620","Text":"That\u0027s our fourth equation."},{"Start":"05:31.620 ","End":"05:34.280","Text":"Basically, if we just find x,"},{"Start":"05:34.280 ","End":"05:36.260","Text":"y, and a, then we\u0027ll get b also,"},{"Start":"05:36.260 ","End":"05:39.140","Text":"because a equals b. I stared at"},{"Start":"05:39.140 ","End":"05:43.040","Text":"this a bit and I think I have a strategy, a winning strategy."},{"Start":"05:43.040 ","End":"05:45.525","Text":"If I leave x as my only unknown,"},{"Start":"05:45.525 ","End":"05:47.995","Text":"I\u0027ve pretty much got everything in terms of x."},{"Start":"05:47.995 ","End":"05:50.570","Text":"Because I\u0027ve got a in terms of x from here,"},{"Start":"05:50.570 ","End":"05:52.410","Text":"y in terms of x from here,"},{"Start":"05:52.410 ","End":"06:00.260","Text":"and I could put also copy that y in terms of x here and the a also from here and b,"},{"Start":"06:00.260 ","End":"06:03.740","Text":"I\u0027ll just put the same thing as a. I\u0027ll get this the first equation all"},{"Start":"06:03.740 ","End":"06:07.895","Text":"in terms of x. I\u0027ll show you what I mean more specifically."},{"Start":"06:07.895 ","End":"06:10.295","Text":"I\u0027ll put this here,"},{"Start":"06:10.295 ","End":"06:14.375","Text":"and I\u0027ll put y in terms of square root of x."},{"Start":"06:14.375 ","End":"06:16.115","Text":"This is the square root of x."},{"Start":"06:16.115 ","End":"06:21.485","Text":"Here I\u0027ve used the formula b is equal to a,"},{"Start":"06:21.485 ","End":"06:24.575","Text":"which is 1 over the 2 square root of x."},{"Start":"06:24.575 ","End":"06:27.725","Text":"This is from formula C,"},{"Start":"06:27.725 ","End":"06:32.360","Text":"and I\u0027ll put them in double brackets and be even more clear that these are not variables."},{"Start":"06:32.360 ","End":"06:35.630","Text":"They\u0027re just numbering the equations."},{"Start":"06:35.630 ","End":"06:41.965","Text":"1 over 2 square root of x times x plus b is the same thing as a."},{"Start":"06:41.965 ","End":"06:47.690","Text":"It\u0027s also 1 over same thing as a was 1 over twice the square root of x."},{"Start":"06:47.690 ","End":"06:52.615","Text":"Now we have an equation that\u0027s just in terms of square root of x."},{"Start":"06:52.615 ","End":"06:54.500","Text":"This last 1, as I say,"},{"Start":"06:54.500 ","End":"06:57.350","Text":"is because this is a which equals b."},{"Start":"06:57.350 ","End":"07:01.400","Text":"I\u0027ve given reasonings for each of the pieces that I\u0027ve done here."},{"Start":"07:01.400 ","End":"07:05.600","Text":"Let\u0027s put it over a common denominator."},{"Start":"07:05.600 ","End":"07:07.250","Text":"Clearly x is not 0."},{"Start":"07:07.250 ","End":"07:09.305","Text":"This equation wouldn\u0027t make sense."},{"Start":"07:09.305 ","End":"07:12.890","Text":"That\u0027s because the slope is not going to be 0, it\u0027s going to be a number."},{"Start":"07:12.890 ","End":"07:15.590","Text":"I mean, a has to be proper number,"},{"Start":"07:15.590 ","End":"07:18.980","Text":"so x can\u0027t be 0 because that would make a infinity."},{"Start":"07:18.980 ","End":"07:27.380","Text":"Multiplying by both sides left and right will multiply by twice the square root of x,"},{"Start":"07:27.380 ","End":"07:29.870","Text":"and then we\u0027ll get rid of the fraction."},{"Start":"07:29.870 ","End":"07:33.860","Text":"This times twice square root of x is just 2x,"},{"Start":"07:33.860 ","End":"07:37.825","Text":"this times twice square root of x is just x,"},{"Start":"07:37.825 ","End":"07:41.450","Text":"and this times twice the square root of x is just 1,"},{"Start":"07:41.450 ","End":"07:42.905","Text":"now we\u0027re almost there."},{"Start":"07:42.905 ","End":"07:45.170","Text":"Just bring x over to the other side,"},{"Start":"07:45.170 ","End":"07:48.650","Text":"and we get that x is equal to i."},{"Start":"07:48.650 ","End":"07:52.415","Text":"Once we have that x equals 1, then from here,"},{"Start":"07:52.415 ","End":"07:55.325","Text":"a is 1 over twice the square root of x,"},{"Start":"07:55.325 ","End":"07:59.375","Text":"which means that a is equal to 1.5,"},{"Start":"07:59.375 ","End":"08:02.000","Text":"and b equals a."},{"Start":"08:02.000 ","End":"08:05.565","Text":"That\u0027s also b equals 1.5."},{"Start":"08:05.565 ","End":"08:15.615","Text":"That gives us the equation of the line because it\u0027s y equals ax plus b as 1.5x plus 1.5."},{"Start":"08:15.615 ","End":"08:18.590","Text":"This is the answer for the straight line,"},{"Start":"08:18.590 ","End":"08:22.560","Text":"which is the tangent to the curve. We\u0027re done."}],"ID":10576},{"Watched":false,"Name":"Exercise 5","Duration":"9m 1s","ChapterTopicVideoID":10251,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this exercise, we have to write the equation of"},{"Start":"00:02.820 ","End":"00:06.360","Text":"the straight lines passing through 1.5, 0,"},{"Start":"00:06.360 ","End":"00:08.820","Text":"which are tangent to the curve,"},{"Start":"00:08.820 ","End":"00:13.635","Text":"Y equals or f of x equals 1/4 x squared plus 1."},{"Start":"00:13.635 ","End":"00:16.530","Text":"As a couple of strange things."},{"Start":"00:16.530 ","End":"00:20.505","Text":"Because, well, let\u0027s read the second path and it\u0027ll see what strange."},{"Start":"00:20.505 ","End":"00:24.480","Text":"Sketch the curve and the lines and prove that the lines are perpendicular."},{"Start":"00:24.480 ","End":"00:26.220","Text":"Well, already in the first part,"},{"Start":"00:26.220 ","End":"00:28.290","Text":"notice that the straight lines,"},{"Start":"00:28.290 ","End":"00:31.215","Text":"there\u0027s an S here which imply that there\u0027s more than 1."},{"Start":"00:31.215 ","End":"00:34.320","Text":"We don\u0027t know that yet or we wouldn\u0027t know it."},{"Start":"00:34.320 ","End":"00:37.890","Text":"The second part, to show that they\u0027re perpendicular,"},{"Start":"00:37.890 ","End":"00:40.949","Text":"perpendicular applies only to 2 lines."},{"Start":"00:40.949 ","End":"00:43.740","Text":"We actually know to expect 2 straight lines,"},{"Start":"00:43.740 ","End":"00:46.430","Text":"2 solutions to the question of finding"},{"Start":"00:46.430 ","End":"00:49.640","Text":"a tangent to this curve that goes through this point."},{"Start":"00:49.640 ","End":"00:51.800","Text":"I\u0027ve brought with me a sketch."},{"Start":"00:51.800 ","End":"00:55.370","Text":"There it is, and this actually shows us what is really happening."},{"Start":"00:55.370 ","End":"00:57.545","Text":"Here\u0027s the curve which is a parabola,"},{"Start":"00:57.545 ","End":"01:03.690","Text":"and there are 2 tangent to this curve that passes through the given point, 1.5, 0."},{"Start":"01:03.690 ","End":"01:06.120","Text":"We expect 2 solutions."},{"Start":"01:06.120 ","End":"01:08.120","Text":"Not necessary to draw a curve,"},{"Start":"01:08.120 ","End":"01:09.665","Text":"but it gives you an idea."},{"Start":"01:09.665 ","End":"01:13.220","Text":"Because the 2 solutions we can check if they\u0027re perpendicular or not."},{"Start":"01:13.220 ","End":"01:17.630","Text":"Now, we don\u0027t know what the form that the line takes."},{"Start":"01:17.630 ","End":"01:21.785","Text":"Let\u0027s say that the tangent is going to be, in general,"},{"Start":"01:21.785 ","End":"01:24.724","Text":"a tangent line is going to be aligned,"},{"Start":"01:24.724 ","End":"01:25.925","Text":"which means that it\u0027s,"},{"Start":"01:25.925 ","End":"01:28.235","Text":"let\u0027s just give it a name."},{"Start":"01:28.235 ","End":"01:34.920","Text":"A and B are common letters and we have the usual 3 conditions and this is"},{"Start":"01:34.920 ","End":"01:42.500","Text":"our point so that number 1 just says that the point is on the line."},{"Start":"01:42.500 ","End":"01:46.880","Text":"I\u0027m just writing abbreviated pointers on line number 2,"},{"Start":"01:46.880 ","End":"01:48.515","Text":"the point is on the curve."},{"Start":"01:48.515 ","End":"01:50.930","Text":"The point minus 1.50,"},{"Start":"01:50.930 ","End":"01:52.180","Text":"it\u0027s got to be on this,"},{"Start":"01:52.180 ","End":"01:54.120","Text":"it\u0027s a parabola actually."},{"Start":"01:54.120 ","End":"01:57.470","Text":"Number 3 is that we have to have the same slopes"},{"Start":"01:57.470 ","End":"02:00.920","Text":"when a line hits a curve and it\u0027s tangent,"},{"Start":"02:00.920 ","End":"02:03.140","Text":"then its slope is the same."},{"Start":"02:03.140 ","End":"02:06.375","Text":"Other words slopes are equal,"},{"Start":"02:06.375 ","End":"02:09.120","Text":"which means in the case of the line,"},{"Start":"02:09.120 ","End":"02:12.600","Text":"for a line it\u0027s just A,"},{"Start":"02:12.600 ","End":"02:13.965","Text":"this A here,"},{"Start":"02:13.965 ","End":"02:16.700","Text":"and for a curve which is the function,"},{"Start":"02:16.700 ","End":"02:20.180","Text":"the parabola, it\u0027s y prime at the given point."},{"Start":"02:20.180 ","End":"02:24.949","Text":"The derivative has an interpretation of being the slope of the tangent."},{"Start":"02:24.949 ","End":"02:28.190","Text":"Why don\u0027t we compute this y prime for the curve?"},{"Start":"02:28.190 ","End":"02:30.815","Text":"It\u0027s on this y here."},{"Start":"02:30.815 ","End":"02:36.195","Text":"Y equals 1/4 x squared plus 1,"},{"Start":"02:36.195 ","End":"02:42.815","Text":"That\u0027s the y for the curve and the y prime will equal 2 times 1/4 is 1.5x,"},{"Start":"02:42.815 ","End":"02:45.170","Text":"and that will be it."},{"Start":"02:45.170 ","End":"02:48.890","Text":"I just want to point out by the way that we have in this 1 fact that we"},{"Start":"02:48.890 ","End":"02:53.285","Text":"haven\u0027t used is that the line passes through 1.50."},{"Start":"02:53.285 ","End":"02:56.945","Text":"Now we\u0027re going to need this fact because we can\u0027t do without it."},{"Start":"02:56.945 ","End":"02:58.490","Text":"These are 3 standard ones."},{"Start":"02:58.490 ","End":"03:02.794","Text":"I\u0027ll just make a note here already that this is on the line."},{"Start":"03:02.794 ","End":"03:06.290","Text":"Let\u0027s say that 1.5,"},{"Start":"03:06.290 ","End":"03:09.860","Text":"0 is on the line,"},{"Start":"03:09.860 ","End":"03:11.120","Text":"on the tangent line."},{"Start":"03:11.120 ","End":"03:12.640","Text":"It\u0027s on this line here."},{"Start":"03:12.640 ","End":"03:15.165","Text":"Let\u0027s get this more mathematically."},{"Start":"03:15.165 ","End":"03:18.615","Text":"We\u0027ll reinterpret number 1."},{"Start":"03:18.615 ","End":"03:20.955","Text":"The point is on the line,"},{"Start":"03:20.955 ","End":"03:23.510","Text":"so we just copy the equation of the line,"},{"Start":"03:23.510 ","End":"03:26.805","Text":"y equals ax plus b."},{"Start":"03:26.805 ","End":"03:31.109","Text":"Then the points on the curve, so y equals,"},{"Start":"03:31.109 ","End":"03:36.300","Text":"I can see it here, 1/4 x squared plus 1."},{"Start":"03:36.300 ","End":"03:39.155","Text":"Number 3, the slopes are equal."},{"Start":"03:39.155 ","End":"03:42.200","Text":"The slope on the 1 hand of the curve is y prime,"},{"Start":"03:42.200 ","End":"03:43.880","Text":"which is 1.5 X."},{"Start":"03:43.880 ","End":"03:46.925","Text":"On the other hand, it\u0027s equal to the slope of the line which is a."},{"Start":"03:46.925 ","End":"03:52.890","Text":"Number 4, is this extra thing that the point 1.5,"},{"Start":"03:52.890 ","End":"03:54.780","Text":"0 is on the line,"},{"Start":"03:54.780 ","End":"03:57.440","Text":"and if the tangent line is ax plus b,"},{"Start":"03:57.440 ","End":"04:02.960","Text":"that means if I substitute x as 1.5, y is 0."},{"Start":"04:02.960 ","End":"04:05.330","Text":"The left-hand side is y, which is 0,"},{"Start":"04:05.330 ","End":"04:09.765","Text":"is a times 1.5 plus b."},{"Start":"04:09.765 ","End":"04:12.960","Text":"This is now 4 equations and 4 unknowns,"},{"Start":"04:12.960 ","End":"04:14.515","Text":"a, b, x, and y."},{"Start":"04:14.515 ","End":"04:17.045","Text":"Let\u0027s see what we can make of this."},{"Start":"04:17.045 ","End":"04:22.610","Text":"We can get b in terms of a and x in terms of a and then we\u0027ll be able to"},{"Start":"04:22.610 ","End":"04:28.595","Text":"substitute b and x and then just be left with 2 equations with 2 unknowns, y and a."},{"Start":"04:28.595 ","End":"04:33.980","Text":"Here\u0027s what I mean, from number 4 I get that b is equal to"},{"Start":"04:33.980 ","End":"04:40.145","Text":"minus 1.5 a and that x is equal to 2a."},{"Start":"04:40.145 ","End":"04:42.020","Text":"This is from 3 and 4."},{"Start":"04:42.020 ","End":"04:44.990","Text":"Now if I substitute these into 1 and 2,"},{"Start":"04:44.990 ","End":"04:47.165","Text":"essentially from 1 and 2,"},{"Start":"04:47.165 ","End":"04:53.465","Text":"I can get ax plus b is equal to 1/4 x squared plus 1."},{"Start":"04:53.465 ","End":"04:57.945","Text":"This really means that the line and the curve, they intersect here."},{"Start":"04:57.945 ","End":"05:02.285","Text":"Then I put in b from here and x from here."},{"Start":"05:02.285 ","End":"05:06.920","Text":"Essentially we substitute these into here and we\u0027ll get a,"},{"Start":"05:06.920 ","End":"05:09.320","Text":"which is, well, a is just a,"},{"Start":"05:09.320 ","End":"05:15.290","Text":"but x is 2a and plus b is minus 1.5a,"},{"Start":"05:15.290 ","End":"05:19.235","Text":"an all this equals 1/4 of x,"},{"Start":"05:19.235 ","End":"05:22.910","Text":"which is 2a squared plus 1."},{"Start":"05:22.910 ","End":"05:24.425","Text":"Just got an equation in a,"},{"Start":"05:24.425 ","End":"05:28.355","Text":"because y drop that also when I compared these."},{"Start":"05:28.355 ","End":"05:32.975","Text":"What have we got? We\u0027ve got let\u0027s do the whole thing in 1 go."},{"Start":"05:32.975 ","End":"05:37.150","Text":"This is 2a squared and this is a squared because 2 squared is 4."},{"Start":"05:37.150 ","End":"05:41.230","Text":"2a squared minus a squared is just a squared."},{"Start":"05:41.230 ","End":"05:47.760","Text":"Here we have minus 1.5a minus 1 equals 0."},{"Start":"05:47.760 ","End":"05:49.075","Text":"If I double it,"},{"Start":"05:49.075 ","End":"05:51.130","Text":"I can get rid of the,"},{"Start":"05:51.130 ","End":"05:53.020","Text":"I don\u0027t like fractions here,"},{"Start":"05:53.020 ","End":"06:01.090","Text":"so 2a squared minus 3a minus 2 is equal to 0."},{"Start":"06:01.090 ","End":"06:05.065","Text":"Now let\u0027s do the quadratic equation formula."},{"Start":"06:05.065 ","End":"06:08.860","Text":"So a is equal to minus b,"},{"Start":"06:08.860 ","End":"06:14.660","Text":"which is 3, plus or minus the square root of b squared, which is 9,"},{"Start":"06:14.660 ","End":"06:18.180","Text":"plus minus 4ac,"},{"Start":"06:18.180 ","End":"06:22.730","Text":"which is 4 times 2 times minus 2,"},{"Start":"06:22.730 ","End":"06:26.240","Text":"is plus 16, all over 4,"},{"Start":"06:26.240 ","End":"06:33.090","Text":"which is equal to 3 plus or minus 5 over 4."},{"Start":"06:33.090 ","End":"06:34.880","Text":"Which gives us 2 possibilities."},{"Start":"06:34.880 ","End":"06:37.205","Text":"If I take the plus or if I take the minus."},{"Start":"06:37.205 ","End":"06:40.895","Text":"If I take the plus, it\u0027s 3 plus 5 is 8/4 is 2."},{"Start":"06:40.895 ","End":"06:42.349","Text":"If I take the minus,"},{"Start":"06:42.349 ","End":"06:43.730","Text":"it\u0027s 3 minus 5,"},{"Start":"06:43.730 ","End":"06:46.250","Text":"which is minus 2 over 4,"},{"Start":"06:46.250 ","End":"06:48.770","Text":"which is minus 1/2."},{"Start":"06:48.770 ","End":"06:50.945","Text":"There\u0027s 2 possible solutions."},{"Start":"06:50.945 ","End":"06:54.010","Text":"Let\u0027s call them a_1 and a_2."},{"Start":"06:54.010 ","End":"06:56.985","Text":"That\u0027s for the 2 different lines."},{"Start":"06:56.985 ","End":"07:02.250","Text":"That gives us the 2as that will give rise to 2 different bs."},{"Start":"07:02.250 ","End":"07:05.130","Text":"B is minus 1.5a,"},{"Start":"07:05.130 ","End":"07:06.755","Text":"and if that\u0027s the case,"},{"Start":"07:06.755 ","End":"07:11.830","Text":"so then we\u0027ll get, this implies that b_1."},{"Start":"07:11.830 ","End":"07:19.320","Text":"It\u0027ll be 2 different bs for 2 different lines and we got b_2 is equal to minus 1.5a."},{"Start":"07:19.320 ","End":"07:24.710","Text":"This would be my times minus 1.5 would give us minus 3,"},{"Start":"07:24.710 ","End":"07:28.845","Text":"and if we take minus 1/2 times minus 1.5,"},{"Start":"07:28.845 ","End":"07:30.930","Text":"we get plus 3/4."},{"Start":"07:30.930 ","End":"07:33.870","Text":"That means that we have 2 different tangent lines,"},{"Start":"07:33.870 ","End":"07:37.040","Text":"and what\u0027s the question they asked? Let\u0027s see."},{"Start":"07:37.040 ","End":"07:43.580","Text":"We were asked to write the equations and then to prove that they\u0027re perpendicular."},{"Start":"07:43.580 ","End":"07:45.934","Text":"Let\u0027s do the equation part first."},{"Start":"07:45.934 ","End":"07:48.170","Text":"We have 2 lines, let\u0027s see,"},{"Start":"07:48.170 ","End":"07:52.820","Text":"we have line number 1 and that will be using a_1 and b_1."},{"Start":"07:52.820 ","End":"07:56.515","Text":"It\u0027s y equals ax plus b."},{"Start":"07:56.515 ","End":"08:03.765","Text":"It\u0027s 2x minus 3 and line number 2 will take the bottom part."},{"Start":"08:03.765 ","End":"08:08.490","Text":"It\u0027s y equals a_2 x plus b_2."},{"Start":"08:08.490 ","End":"08:16.440","Text":"It will be minus 1.5x plus 3/4 thus the 2 lines."},{"Start":"08:16.440 ","End":"08:21.485","Text":"The last thing we were asked to show is that the 2 lines are perpendicular."},{"Start":"08:21.485 ","End":"08:26.105","Text":"Actually the condition for perpendicular relates to the slopes."},{"Start":"08:26.105 ","End":"08:27.755","Text":"The slopes of the 2 lines,"},{"Start":"08:27.755 ","End":"08:29.510","Text":"the product has to be minus 1."},{"Start":"08:29.510 ","End":"08:32.770","Text":"In other words, we have to check if a_1 times a_2,"},{"Start":"08:32.770 ","End":"08:35.210","Text":"does this equal minus 1."},{"Start":"08:35.210 ","End":"08:37.070","Text":"That\u0027s the condition of perpendicularity,"},{"Start":"08:37.070 ","End":"08:39.950","Text":"if 2 lines is the product of the slopes is minus 1."},{"Start":"08:39.950 ","End":"08:42.935","Text":"Let\u0027s see, a_1 times a_2."},{"Start":"08:42.935 ","End":"08:47.065","Text":"It\u0027s 2 times minus 1/2."},{"Start":"08:47.065 ","End":"08:48.855","Text":"Yes, it is minus 1."},{"Start":"08:48.855 ","End":"08:51.170","Text":"I can put a check mark and say yes,"},{"Start":"08:51.170 ","End":"08:54.080","Text":"they are perpendicular because the condition is met."},{"Start":"08:54.080 ","End":"08:56.540","Text":"That\u0027s the first line,"},{"Start":"08:56.540 ","End":"09:02.100","Text":"equation of the second line and proof of perpendicularity and we\u0027re done."}],"ID":10577},{"Watched":false,"Name":"Exercise 6","Duration":"11m 44s","ChapterTopicVideoID":10252,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.795","Text":"In this exercise, we have to find all lines that can be drawn tangent to this curve."},{"Start":"00:06.795 ","End":"00:11.295","Text":"I\u0027d like to just also mark it as y"},{"Start":"00:11.295 ","End":"00:16.980","Text":"equals x squared minus 2x plus 1 from the point 2 comma minus 3."},{"Start":"00:16.980 ","End":"00:20.144","Text":"Then we have to sketch the curve and the lines."},{"Start":"00:20.144 ","End":"00:27.030","Text":"Well, what I did was I prepared in advance a sketch that will save you doing this part."},{"Start":"00:27.030 ","End":"00:31.500","Text":"Somewhere down here, there\u0027s a sketch, here it is."},{"Start":"00:31.500 ","End":"00:33.270","Text":"This gives you a general idea."},{"Start":"00:33.270 ","End":"00:34.439","Text":"It\u0027s not to scale,"},{"Start":"00:34.439 ","End":"00:38.820","Text":"but here\u0027s the curve and here\u0027s the point and it\u0027s outside."},{"Start":"00:38.820 ","End":"00:41.955","Text":"This is a parabola, this is below the parabola."},{"Start":"00:41.955 ","End":"00:47.285","Text":"You can see that there are 2 possible tangents that could be drawn visually you can see."},{"Start":"00:47.285 ","End":"00:50.065","Text":"We don\u0027t know this yet algebraically."},{"Start":"00:50.065 ","End":"00:53.070","Text":"Well, you need the sketch because they\u0027ve asked for the sketch but"},{"Start":"00:53.070 ","End":"00:56.670","Text":"to find the equation of the tangents,"},{"Start":"00:56.670 ","End":"00:58.065","Text":"you don\u0027t need the sketch."},{"Start":"00:58.065 ","End":"01:00.535","Text":"Okay. Let\u0027s get back up and do the algebra."},{"Start":"01:00.535 ","End":"01:06.920","Text":"Okay. We\u0027re going to use the usual thing in writing equations."},{"Start":"01:06.920 ","End":"01:09.110","Text":"I have the mnemonic for the equations,"},{"Start":"01:09.110 ","End":"01:11.240","Text":"there\u0027s 3 conditions that have to hold."},{"Start":"01:11.240 ","End":"01:14.000","Text":"The first one is at the point of contact."},{"Start":"01:14.000 ","End":"01:16.850","Text":"When I say the point, I don\u0027t mean this point."},{"Start":"01:16.850 ","End":"01:21.290","Text":"This point is not the point of contact between the tangent and the curve,"},{"Start":"01:21.290 ","End":"01:24.200","Text":"but the point, let\u0027s call it X, Y,"},{"Start":"01:24.200 ","End":"01:25.655","Text":"the general point of x_1,"},{"Start":"01:25.655 ","End":"01:28.775","Text":"y_1 is on the line."},{"Start":"01:28.775 ","End":"01:32.555","Text":"It\u0027s also on the curve where the tangent hits the curve."},{"Start":"01:32.555 ","End":"01:36.289","Text":"That point is on the curve also."},{"Start":"01:36.289 ","End":"01:39.440","Text":"And the other condition is that the slopes are equal."},{"Start":"01:39.440 ","End":"01:41.450","Text":"I\u0027ll call it equal slopes."},{"Start":"01:41.450 ","End":"01:43.175","Text":"This is just a mnemonic."},{"Start":"01:43.175 ","End":"01:44.660","Text":"Let\u0027s spell it out a bit more."},{"Start":"01:44.660 ","End":"01:47.420","Text":"In the case of the line,"},{"Start":"01:47.420 ","End":"01:49.700","Text":"we don\u0027t have an equation for the line,"},{"Start":"01:49.700 ","End":"01:51.455","Text":"so let\u0027s write it somewhere here."},{"Start":"01:51.455 ","End":"01:55.400","Text":"The line just to give it a name or an equation,"},{"Start":"01:55.400 ","End":"02:01.850","Text":"the general line is y equals ax plus b for some constants a and b."},{"Start":"02:01.850 ","End":"02:07.490","Text":"So for the line the slope is just a and for the curve,"},{"Start":"02:07.490 ","End":"02:12.200","Text":"it\u0027s y-prime at the particular X, Y."},{"Start":"02:12.200 ","End":"02:13.750","Text":"Well, while we\u0027re at it, yes,"},{"Start":"02:13.750 ","End":"02:16.130","Text":"what is y prime for the curve?"},{"Start":"02:16.130 ","End":"02:22.385","Text":"Well, the curve is given by y equals x squared minus 2x plus 1,"},{"Start":"02:22.385 ","End":"02:27.455","Text":"and so y prime is equal 2x minus 2."},{"Start":"02:27.455 ","End":"02:30.505","Text":"So let\u0027s write these things out mathematically."},{"Start":"02:30.505 ","End":"02:33.410","Text":"I\u0027m going to leave space because there will be a number 4,"},{"Start":"02:33.410 ","End":"02:37.340","Text":"you\u0027ll see in a minute, which is particular to this problem and not in general."},{"Start":"02:37.340 ","End":"02:39.905","Text":"Okay. So number 1, I\u0027m rewriting."},{"Start":"02:39.905 ","End":"02:42.350","Text":"You just write the equation of the line."},{"Start":"02:42.350 ","End":"02:46.985","Text":"Just copy it. Y equals ax plus b."},{"Start":"02:46.985 ","End":"02:51.035","Text":"Number 2. Just copy the equation of the curve."},{"Start":"02:51.035 ","End":"02:56.940","Text":"Y equals x squared minus 2x plus 1."},{"Start":"02:56.940 ","End":"03:03.200","Text":"Number 3, the equal slopes is that y prime equals a,"},{"Start":"03:03.200 ","End":"03:07.745","Text":"which is 2x minus 2 equals a."},{"Start":"03:07.745 ","End":"03:10.640","Text":"Now, what\u0027s this number 4 I\u0027m talking about?"},{"Start":"03:10.640 ","End":"03:12.500","Text":"Well, there\u0027s something missing. There must be."},{"Start":"03:12.500 ","End":"03:15.770","Text":"Because look, we have 4 variables,"},{"Start":"03:15.770 ","End":"03:18.290","Text":"x, y, a, and b with 3 equations."},{"Start":"03:18.290 ","End":"03:19.655","Text":"What have we missed out?"},{"Start":"03:19.655 ","End":"03:22.670","Text":"Well, we\u0027ve missed out the 2 minus 3."},{"Start":"03:22.670 ","End":"03:25.745","Text":"Not only there\u0027s the point of contact on the tangent,"},{"Start":"03:25.745 ","End":"03:29.900","Text":"but also this 2 minus 3 is on the tangent."},{"Start":"03:29.900 ","End":"03:36.500","Text":"Which means that if I put y equals minus 3 and x equals 2, it will hold."},{"Start":"03:36.500 ","End":"03:46.410","Text":"Minus 3 is equal to a times 2 for x, and plus b."},{"Start":"03:46.410 ","End":"03:50.285","Text":"Now, 4 equations, 4 unknowns."},{"Start":"03:50.285 ","End":"03:52.520","Text":"Let\u0027s do some algebra."},{"Start":"03:52.520 ","End":"03:54.680","Text":"Oh yeah, just for completeness,"},{"Start":"03:54.680 ","End":"03:56.360","Text":"I wanted to say what 4 is,"},{"Start":"03:56.360 ","End":"03:58.850","Text":"is that that\u0027s special, is it\u0027s special to our case,"},{"Start":"03:58.850 ","End":"04:02.285","Text":"not the usual number 4."},{"Start":"04:02.285 ","End":"04:10.335","Text":"Is that 2 minus 3 is on the line."},{"Start":"04:10.335 ","End":"04:17.270","Text":"Okay. Then we went from the verbal description to the mathematical description,"},{"Start":"04:17.270 ","End":"04:22.595","Text":"and now we, have to start doing some algebra."},{"Start":"04:22.595 ","End":"04:26.795","Text":"My suggestion is, and I\u0027ve done this before,"},{"Start":"04:26.795 ","End":"04:33.265","Text":"is to write both b and x in terms of a and then substitute"},{"Start":"04:33.265 ","End":"04:40.450","Text":"here and we can even get rid of y by comparing the 2 right-hand sides."},{"Start":"04:40.450 ","End":"04:43.045","Text":"I\u0027ll show you what I mean. From here,"},{"Start":"04:43.045 ","End":"04:45.670","Text":"I\u0027ve got the b from 4 and 3."},{"Start":"04:45.670 ","End":"04:50.370","Text":"I\u0027ll get b equals minus 3 minus 2a,"},{"Start":"04:50.370 ","End":"04:52.180","Text":"and from 3,"},{"Start":"04:52.180 ","End":"04:58.730","Text":"I\u0027ve got that x is equal to a plus 2 over 2."},{"Start":"04:58.730 ","End":"05:04.840","Text":"Now, if I take b and x and put them in 1 and 2,"},{"Start":"05:04.840 ","End":"05:08.480","Text":"then, well say before that, before I substitute,"},{"Start":"05:08.480 ","End":"05:11.360","Text":"I can just equate the right-hand side,"},{"Start":"05:11.360 ","End":"05:13.130","Text":"if y equals both of these,"},{"Start":"05:13.130 ","End":"05:20.840","Text":"then I have ax plus b is equal to x squared minus 2x plus 1."},{"Start":"05:20.840 ","End":"05:23.585","Text":"The line and the curve intersect."},{"Start":"05:23.585 ","End":"05:25.430","Text":"That\u0027s really what this one means."},{"Start":"05:25.430 ","End":"05:30.185","Text":"Now, I\u0027ll put b into here and x into here."},{"Start":"05:30.185 ","End":"05:34.610","Text":"So b and x will give us everything in terms of a."},{"Start":"05:34.610 ","End":"05:37.280","Text":"So we have a times x,"},{"Start":"05:37.280 ","End":"05:42.170","Text":"which is a plus 2 over 2 plus b,"},{"Start":"05:42.170 ","End":"05:45.775","Text":"which is minus 3 minus 2a,"},{"Start":"05:45.775 ","End":"05:48.140","Text":"is equal to x squared,"},{"Start":"05:48.140 ","End":"05:53.220","Text":"which is a plus 2 over 2 squared"},{"Start":"05:53.220 ","End":"06:00.635","Text":"minus 2x minus twice a plus 2 over 2 plus 1."},{"Start":"06:00.635 ","End":"06:05.015","Text":"One equation in a. I\u0027m going to both open the brackets,"},{"Start":"06:05.015 ","End":"06:06.635","Text":"open the brackets first."},{"Start":"06:06.635 ","End":"06:15.260","Text":"So I\u0027ve got a squared over 2 plus a times 2 over 2."},{"Start":"06:15.260 ","End":"06:19.655","Text":"This is just plus a minus 3,"},{"Start":"06:19.655 ","End":"06:27.185","Text":"minus 2a equals a plus 2 squared is a squared."},{"Start":"06:27.185 ","End":"06:29.720","Text":"But I\u0027m going to also divide by 4,"},{"Start":"06:29.720 ","End":"06:33.920","Text":"so it\u0027s a squared over 4 and then the second term in this,"},{"Start":"06:33.920 ","End":"06:39.395","Text":"a plus something squared is a squared plus twice this times this,"},{"Start":"06:39.395 ","End":"06:42.870","Text":"which is 4a but over 4."},{"Start":"06:42.870 ","End":"06:47.480","Text":"Everything\u0027s going to be over 4 because I\u0027m doing the 2 squared here."},{"Start":"06:47.480 ","End":"06:50.885","Text":"So it\u0027s a squared plus 4a plus 4,"},{"Start":"06:50.885 ","End":"06:57.270","Text":"which is a plus 2 squared minus twice a plus 2 over 2,"},{"Start":"06:57.270 ","End":"07:01.245","Text":"which is minus 2a over 2,"},{"Start":"07:01.245 ","End":"07:06.035","Text":"minus 4 over 2 or already I said 2 over 2 is 1,"},{"Start":"07:06.035 ","End":"07:10.495","Text":"which is minus 2, yeah, plus 1."},{"Start":"07:10.495 ","End":"07:13.280","Text":"Here is a bit of a mess,"},{"Start":"07:13.280 ","End":"07:16.190","Text":"but we\u0027re going to collect together the terms with a squared,"},{"Start":"07:16.190 ","End":"07:18.950","Text":"terms with a and just constants."},{"Start":"07:18.950 ","End":"07:21.095","Text":"How does that go?"},{"Start":"07:21.095 ","End":"07:22.610","Text":"Let\u0027s see, you know what,"},{"Start":"07:22.610 ","End":"07:24.670","Text":"we\u0027ll multiply also by 4."},{"Start":"07:24.670 ","End":"07:27.060","Text":"Maybe I\u0027ll first of all, multiply by 4,"},{"Start":"07:27.060 ","End":"07:28.725","Text":"so we get rid of the fractions."},{"Start":"07:28.725 ","End":"07:31.725","Text":"Okay. Bring everything to the left-hand side."},{"Start":"07:31.725 ","End":"07:37.310","Text":"Let\u0027s see, the a squared terms are only here and here."},{"Start":"07:37.310 ","End":"07:39.845","Text":"So it\u0027s 2a squared minus a squared,"},{"Start":"07:39.845 ","End":"07:41.960","Text":"which is a squared."},{"Start":"07:41.960 ","End":"07:44.060","Text":"Let\u0027s see, go collecting a\u0027s,"},{"Start":"07:44.060 ","End":"07:47.465","Text":"plus 4a minus 8a is minus 4a,"},{"Start":"07:47.465 ","End":"07:52.000","Text":"plus 4a we\u0027re up to 0, minus 4a."},{"Start":"07:52.000 ","End":"07:55.515","Text":"Now, let\u0027s collect the constants, minus 12."},{"Start":"07:55.515 ","End":"08:01.760","Text":"I see here a plus 4 is minus 8 and another minus 8 is"},{"Start":"08:01.760 ","End":"08:08.990","Text":"minus 16 and plus 4 is minus 12 equals 0."},{"Start":"08:08.990 ","End":"08:12.860","Text":"Okay. I don\u0027t want to do the solution of a quadratic equation here."},{"Start":"08:12.860 ","End":"08:16.955","Text":"I did it at the side and I can tell you that I got 2 solutions,"},{"Start":"08:16.955 ","End":"08:24.135","Text":"a_1 is 6 and the other a was equal to minus 2."},{"Start":"08:24.135 ","End":"08:26.450","Text":"I know you know how to do this sort of thing."},{"Start":"08:26.450 ","End":"08:30.805","Text":"So you\u0027ve got 2 different a\u0027s and for each a,"},{"Start":"08:30.805 ","End":"08:33.600","Text":"I have a corresponding b."},{"Start":"08:33.600 ","End":"08:35.405","Text":"This is going to determine 2 lines."},{"Start":"08:35.405 ","End":"08:41.015","Text":"Each one of these a\u0027s from the a will get a b and an x,"},{"Start":"08:41.015 ","End":"08:42.890","Text":"and then once we have a, b, and x,"},{"Start":"08:42.890 ","End":"08:46.250","Text":"we can just compute y in either one of them."},{"Start":"08:46.250 ","End":"08:50.480","Text":"So let\u0027s see how that works and we\u0027ll do it for each one of them."},{"Start":"08:50.480 ","End":"08:54.950","Text":"So underneath, I\u0027ll write b_1 and b_2,"},{"Start":"08:54.950 ","End":"08:57.585","Text":"and b_2 is equal to,"},{"Start":"08:57.585 ","End":"09:01.470","Text":"this we use this 1 minus 3 minus 2a."},{"Start":"09:01.470 ","End":"09:10.275","Text":"Let\u0027s see. Minus 3 minus 12 is minus 15."},{"Start":"09:10.275 ","End":"09:13.760","Text":"If I take minus 3 minus 2a from here,"},{"Start":"09:13.760 ","End":"09:17.585","Text":"it\u0027s minus 3 plus 4,"},{"Start":"09:17.585 ","End":"09:21.750","Text":"which is 1, that\u0027s from the minus 3 minus 2a."},{"Start":"09:21.750 ","End":"09:24.975","Text":"Then we have x as a plus 2 over 2."},{"Start":"09:24.975 ","End":"09:28.305","Text":"So there\u0027ll be x_1 and there\u0027ll be x_2."},{"Start":"09:28.305 ","End":"09:29.655","Text":"There\u0027ll be 2 points,"},{"Start":"09:29.655 ","End":"09:31.965","Text":"x_1, y_1 and x_2, y_2."},{"Start":"09:31.965 ","End":"09:39.005","Text":"x_1 being a plus 2 over 2 is 6 plus 2 over 2,"},{"Start":"09:39.005 ","End":"09:41.435","Text":"which is 8 over 2, which is 4,"},{"Start":"09:41.435 ","End":"09:43.115","Text":"and the x_2,"},{"Start":"09:43.115 ","End":"09:45.110","Text":"we take the other a."},{"Start":"09:45.110 ","End":"09:52.145","Text":"That gives us 0 and Y, y_1 will equal."},{"Start":"09:52.145 ","End":"09:53.570","Text":"Now let\u0027s see,"},{"Start":"09:53.570 ","End":"09:55.520","Text":"where do I get y_1 from?"},{"Start":"09:55.520 ","End":"09:58.700","Text":"I get it from either one of these."},{"Start":"09:58.700 ","End":"10:05.445","Text":"Let\u0027s take it from number 2 is x squared minus 2x plus 1,"},{"Start":"10:05.445 ","End":"10:07.880","Text":"which just happens to be,"},{"Start":"10:07.880 ","End":"10:11.710","Text":"I think it\u0027ll be easier algebraically to compute x minus 1 squared,"},{"Start":"10:11.710 ","End":"10:13.640","Text":"although you could just plug it straight into here."},{"Start":"10:13.640 ","End":"10:15.395","Text":"I\u0027ll find this easier."},{"Start":"10:15.395 ","End":"10:18.635","Text":"So x minus 1 squared."},{"Start":"10:18.635 ","End":"10:20.010","Text":"So here\u0027s x,"},{"Start":"10:20.010 ","End":"10:25.065","Text":"less 1 is 3 and since squared is 9,"},{"Start":"10:25.065 ","End":"10:29.410","Text":"and y_2 will be x_2 minus 1,"},{"Start":"10:29.410 ","End":"10:30.560","Text":"sorry, and then squared,"},{"Start":"10:30.560 ","End":"10:33.640","Text":"so it\u0027s minus 1 squared, so it\u0027s 1."},{"Start":"10:33.640 ","End":"10:37.415","Text":"What were we asked? Let\u0027s look at the original question."},{"Start":"10:37.415 ","End":"10:40.895","Text":"The question is find the lines."},{"Start":"10:40.895 ","End":"10:44.740","Text":"Okay. So for the lines we need the a and the b."},{"Start":"10:44.740 ","End":"10:46.660","Text":"It\u0027s good that we found the x_1,"},{"Start":"10:46.660 ","End":"10:51.010","Text":"y_1 also because this shows that one intersection point is 4."},{"Start":"10:51.010 ","End":"10:52.380","Text":"This is an extra,"},{"Start":"10:52.380 ","End":"10:54.640","Text":"well, you know what, I\u0027ll leave it since it\u0027s an extra,"},{"Start":"10:54.640 ","End":"10:59.840","Text":"but I\u0027ll just say it that one of the tangent cuts the curve at 4,"},{"Start":"10:59.840 ","End":"11:02.480","Text":"9 and the other one at 0, 1."},{"Start":"11:02.480 ","End":"11:04.490","Text":"But the tangent lines is what we want,"},{"Start":"11:04.490 ","End":"11:07.040","Text":"which is the ax plus b part."},{"Start":"11:07.040 ","End":"11:11.390","Text":"So we have tangent number 1,"},{"Start":"11:11.390 ","End":"11:13.995","Text":"y equals ax plus b."},{"Start":"11:13.995 ","End":"11:20.820","Text":"So it\u0027s y equals 6x minus 15."},{"Start":"11:20.820 ","End":"11:30.900","Text":"Number 2 is y equals minus 2x plus 1 tangent. These are all of them."},{"Start":"11:30.900 ","End":"11:33.345","Text":"We said find all those 2 of them."},{"Start":"11:33.345 ","End":"11:36.585","Text":"That\u0027s basically it."},{"Start":"11:36.585 ","End":"11:38.430","Text":"Here\u0027s 1 and here\u0027s 2."},{"Start":"11:38.430 ","End":"11:39.870","Text":"There are 2 solutions,"},{"Start":"11:39.870 ","End":"11:44.740","Text":"and this is 1 and this is the other and I guess we\u0027re done."}],"ID":10578},{"Watched":false,"Name":"Exercise 7","Duration":"6m 8s","ChapterTopicVideoID":10253,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we have to determine the constant b"},{"Start":"00:03.930 ","End":"00:08.070","Text":"such that the line y equals 3x is tangent to the curve"},{"Start":"00:08.070 ","End":"00:10.620","Text":"y equals x square root of x plus b."},{"Start":"00:10.620 ","End":"00:13.005","Text":"There\u0027s the b that we have to find."},{"Start":"00:13.005 ","End":"00:17.235","Text":"Now, a quick sketch just to give you an idea,"},{"Start":"00:17.235 ","End":"00:23.055","Text":"not accurate, just a general sketch of what it means to be a tangent."},{"Start":"00:23.055 ","End":"00:26.040","Text":"There are 3 ingredients"},{"Start":"00:26.040 ","End":"00:27.420","Text":"if you want to call it that."},{"Start":"00:27.420 ","End":"00:30.420","Text":"A curve, we have a tangent line,"},{"Start":"00:30.420 ","End":"00:32.115","Text":"and we have this point,"},{"Start":"00:32.115 ","End":"00:34.320","Text":"I call it the point of contact."},{"Start":"00:34.320 ","End":"00:38.310","Text":"It\u0027s the point of contact between the tangent line and the curve."},{"Start":"00:38.310 ","End":"00:42.150","Text":"There are 3 conditions to be met."},{"Start":"00:42.150 ","End":"00:47.475","Text":"The points got to be on the curve."},{"Start":"00:47.475 ","End":"00:50.435","Text":"Also, the slopes are going to be equal."},{"Start":"00:50.435 ","End":"00:52.190","Text":"The slope of the line is going to be"},{"Start":"00:52.190 ","End":"00:54.200","Text":"the slope of the curve here,"},{"Start":"00:54.200 ","End":"00:56.960","Text":"and the slope happens to be the derivative."},{"Start":"00:56.960 ","End":"00:59.420","Text":"That\u0027s the idea. First of all,"},{"Start":"00:59.420 ","End":"01:02.720","Text":"I\u0027m going to write 3 things that we have to hold."},{"Start":"01:02.720 ","End":"01:08.750","Text":"That is that the point of contact has got to be on the line, a tangent line."},{"Start":"01:08.750 ","End":"01:11.810","Text":"I\u0027m just writing this briefly as a mnemonic."},{"Start":"01:11.810 ","End":"01:14.420","Text":"Then the point is got to be on the curve."},{"Start":"01:14.420 ","End":"01:17.000","Text":"Thirdly, equal slopes."},{"Start":"01:17.000 ","End":"01:20.285","Text":"The slope of the line,"},{"Start":"01:20.285 ","End":"01:24.290","Text":"in general, a line is ax plus b."},{"Start":"01:24.290 ","End":"01:27.980","Text":"In this case, it\u0027s y equals 3x."},{"Start":"01:27.980 ","End":"01:32.195","Text":"But in general, when the line is ax plus b,"},{"Start":"01:32.195 ","End":"01:35.690","Text":"the slope for line is a."},{"Start":"01:35.690 ","End":"01:37.835","Text":"I\u0027m just talking about the general case."},{"Start":"01:37.835 ","End":"01:39.620","Text":"Here we have y equals 3x,"},{"Start":"01:39.620 ","End":"01:44.495","Text":"but a line is ax plus b in general."},{"Start":"01:44.495 ","End":"01:51.815","Text":"For the curve, the slope is given by the derivative y prime at this point."},{"Start":"01:51.815 ","End":"01:55.490","Text":"Why don\u0027t we compute the derivative? We\u0027ll need it."},{"Start":"01:55.490 ","End":"01:57.110","Text":"Let\u0027s compute this over here."},{"Start":"01:57.110 ","End":"02:06.550","Text":"We have the curve is y equals x square root of x plus b."},{"Start":"02:06.550 ","End":"02:09.065","Text":"What is y prime?"},{"Start":"02:09.065 ","End":"02:11.400","Text":"I\u0027m not going to write down the product rule,"},{"Start":"02:11.400 ","End":"02:12.845","Text":"you can look it up."},{"Start":"02:12.845 ","End":"02:15.095","Text":"Product x times square root of x."},{"Start":"02:15.095 ","End":"02:18.920","Text":"It\u0027s the first one derived times the second as is,"},{"Start":"02:18.920 ","End":"02:20.120","Text":"and then vice versa,"},{"Start":"02:20.120 ","End":"02:23.330","Text":"the first as is and the second one derived."},{"Start":"02:23.330 ","End":"02:26.780","Text":"We have first of all the x derived, which is 1,"},{"Start":"02:26.780 ","End":"02:32.750","Text":"and the square root of x as is plus x times the second one derived."},{"Start":"02:32.750 ","End":"02:34.010","Text":"This is a standard one,"},{"Start":"02:34.010 ","End":"02:37.940","Text":"square root of x gives us 1 over twice square root of x,"},{"Start":"02:37.940 ","End":"02:40.104","Text":"and the constant goes to nothing."},{"Start":"02:40.104 ","End":"02:45.010","Text":"Now, let\u0027s interpret these mathematically."},{"Start":"02:45.010 ","End":"02:49.720","Text":"Number 1 and 2 are always pretty much the same pattern,"},{"Start":"02:49.720 ","End":"02:54.255","Text":"is that we just write the original equations."},{"Start":"02:54.255 ","End":"02:56.890","Text":"The points got to be on the line."},{"Start":"02:56.890 ","End":"02:59.620","Text":"That means that y equals 3x,"},{"Start":"02:59.620 ","End":"03:01.915","Text":"just basically copying it."},{"Start":"03:01.915 ","End":"03:04.630","Text":"The point has to be on the curve."},{"Start":"03:04.630 ","End":"03:10.405","Text":"We have to have the y equals x square root of x plus b."},{"Start":"03:10.405 ","End":"03:12.985","Text":"The third condition about the slopes,"},{"Start":"03:12.985 ","End":"03:14.290","Text":"so that the derivative,"},{"Start":"03:14.290 ","End":"03:22.850","Text":"this is y prime, but it\u0027s equal to,"},{"Start":"03:22.850 ","End":"03:25.579","Text":"I should have simplify this."},{"Start":"03:26.370 ","End":"03:32.629","Text":"This is x over square root of x is square root of x over 2,"},{"Start":"03:32.629 ","End":"03:37.189","Text":"1 of something plus 0.5 of something gives 1.5 of something."},{"Start":"03:37.189 ","End":"03:40.820","Text":"It\u0027s 3/2 square root of x."},{"Start":"03:40.820 ","End":"03:45.865","Text":"I\u0027ll just erase the extra bits."},{"Start":"03:45.865 ","End":"03:47.960","Text":"It\u0027s not that."},{"Start":"03:47.960 ","End":"03:50.740","Text":"I\u0027m just writing the y prime. I\u0027m not really using it."},{"Start":"03:50.740 ","End":"03:57.345","Text":"I\u0027m just reminding you what the 3/2 square root of x comes from."},{"Start":"03:57.345 ","End":"04:01.240","Text":"That\u0027s got to equal a slope of the line."},{"Start":"04:01.240 ","End":"04:03.970","Text":"Okay, 3 equations and 3 unknowns,"},{"Start":"04:03.970 ","End":"04:06.730","Text":"y, x, and b."},{"Start":"04:06.730 ","End":"04:09.770","Text":"Well, this a is equal to 3."},{"Start":"04:09.770 ","End":"04:14.660","Text":"This b is just the b from here, that\u0027s highlighted,"},{"Start":"04:14.660 ","End":"04:18.495","Text":"that this b here is this b here."},{"Start":"04:18.495 ","End":"04:20.000","Text":"In the case of the line,"},{"Start":"04:20.000 ","End":"04:22.325","Text":"we know that this a is 3,"},{"Start":"04:22.325 ","End":"04:25.600","Text":"so this is equal to 3."},{"Start":"04:25.600 ","End":"04:27.390","Text":"The rest of it is all fine."},{"Start":"04:27.390 ","End":"04:29.795","Text":"Here, we have the equation of the line."},{"Start":"04:29.795 ","End":"04:32.014","Text":"Here, we have the curve."},{"Start":"04:32.014 ","End":"04:36.715","Text":"Here, we have that the derivative is equal to the slope of the line."},{"Start":"04:36.715 ","End":"04:40.430","Text":"We\u0027ve got that. There\u0027s no real need for the y prime."},{"Start":"04:40.430 ","End":"04:42.905","Text":"It was just to help us to see what\u0027s going on."},{"Start":"04:42.905 ","End":"04:45.740","Text":"I\u0027m actually going to erase that."},{"Start":"04:45.740 ","End":"04:49.895","Text":"So 3 equations and 3 unknowns,"},{"Start":"04:49.895 ","End":"04:51.575","Text":"x, y, and b."},{"Start":"04:51.575 ","End":"04:53.630","Text":"How should we go about it?"},{"Start":"04:53.630 ","End":"04:56.780","Text":"Well, it seems to me that 1 quick thing we can do is"},{"Start":"04:56.780 ","End":"04:59.810","Text":"figure out what x is from this equation."},{"Start":"04:59.810 ","End":"05:08.810","Text":"From this equation, I get that square root of x is equal to 3 divided by 3/2,"},{"Start":"05:08.810 ","End":"05:10.985","Text":"which is like multiplying by 2/3,"},{"Start":"05:10.985 ","End":"05:12.200","Text":"which is 2,"},{"Start":"05:12.200 ","End":"05:16.550","Text":"which makes x equal to 4."},{"Start":"05:16.550 ","End":"05:19.315","Text":"Now that we have x,"},{"Start":"05:19.315 ","End":"05:22.100","Text":"then we can compare."},{"Start":"05:22.100 ","End":"05:24.555","Text":"See y equals this and y equals this."},{"Start":"05:24.555 ","End":"05:29.510","Text":"These 2 things are going to be equal amongst themselves because they\u0027re both equal to y."},{"Start":"05:29.510 ","End":"05:37.040","Text":"If I write that 3x is equal to x square root of x plus b,"},{"Start":"05:37.040 ","End":"05:40.070","Text":"I know x is equal to 4,"},{"Start":"05:40.070 ","End":"05:50.355","Text":"so I get that 12 is equal to 4 square root of 4 is 4 times 2 is 8 plus b,"},{"Start":"05:50.355 ","End":"05:55.660","Text":"which makes b equal to 4."},{"Start":"05:55.880 ","End":"05:58.505","Text":"What is it they asked for?"},{"Start":"05:58.505 ","End":"06:03.260","Text":"The question is determine b, that\u0027s all."},{"Start":"06:03.260 ","End":"06:05.885","Text":"That\u0027s all we\u0027re asked to do."},{"Start":"06:05.885 ","End":"06:09.270","Text":"b equals 4, and we\u0027re done."}],"ID":10579},{"Watched":false,"Name":"Exercise 8","Duration":"8m 49s","ChapterTopicVideoID":10254,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this exercise, we have to determine the constant c,"},{"Start":"00:03.810 ","End":"00:05.310","Text":"that\u0027s the 1 over here,"},{"Start":"00:05.310 ","End":"00:09.420","Text":"such that the curve y equals so and so,"},{"Start":"00:09.420 ","End":"00:13.350","Text":"is tangent to this curve y equals whatever."},{"Start":"00:13.350 ","End":"00:18.105","Text":"We have to find that the tangent point and the joint tangent."},{"Start":"00:18.105 ","End":"00:19.900","Text":"I have a little illustration below,"},{"Start":"00:19.900 ","End":"00:21.645","Text":"and show you in a moment."},{"Start":"00:21.645 ","End":"00:24.090","Text":"These are the 2 functions,"},{"Start":"00:24.090 ","End":"00:26.160","Text":"we were given 2 separate equations."},{"Start":"00:26.160 ","End":"00:28.965","Text":"Doesn\u0027t matter which was which, I\u0027m just illustrating."},{"Start":"00:28.965 ","End":"00:33.045","Text":"These 2 curves happen to touch and where they touch,"},{"Start":"00:33.045 ","End":"00:36.435","Text":"they have a similar meaning, identical slope."},{"Start":"00:36.435 ","End":"00:39.840","Text":"This is the joint tangent,"},{"Start":"00:39.840 ","End":"00:42.165","Text":"this is the tangent point,"},{"Start":"00:42.165 ","End":"00:44.870","Text":"and this is the curve."},{"Start":"00:44.870 ","End":"00:47.150","Text":"One of them, say curve f,"},{"Start":"00:47.150 ","End":"00:55.370","Text":"and this is curve g. 2 curves with a common tangent and a tangent point."},{"Start":"00:55.370 ","End":"00:56.900","Text":"Now all of this can be, as I said,"},{"Start":"00:56.900 ","End":"00:58.160","Text":"can be done algebraically."},{"Start":"00:58.160 ","End":"01:01.865","Text":"We don\u0027t need the picture but it helps to see what we\u0027re talking about."},{"Start":"01:01.865 ","End":"01:04.310","Text":"I\u0027ll just rewrite it."},{"Start":"01:04.310 ","End":"01:14.415","Text":"The first 1, y is f of x is minus 0.5x squared plus C. That\u0027s what we have to discover."},{"Start":"01:14.415 ","End":"01:18.740","Text":"The other 1 is y equals say g of"},{"Start":"01:18.740 ","End":"01:23.420","Text":"x. I sometimes like to use the functional notation rather than just y equals of."},{"Start":"01:23.420 ","End":"01:24.890","Text":"It\u0027s very useful."},{"Start":"01:24.890 ","End":"01:27.800","Text":"Is equal to just 1 over x."},{"Start":"01:27.800 ","End":"01:33.440","Text":"Now, if you remember that tangent point was on both of these functions with"},{"Start":"01:33.440 ","End":"01:39.320","Text":"the common point between f of x and g of x can be found by just equating these 2."},{"Start":"01:39.320 ","End":"01:48.360","Text":"In general, we will write the equation f of x is equal to g of x in general."},{"Start":"01:48.360 ","End":"01:49.895","Text":"In this particular case,"},{"Start":"01:49.895 ","End":"01:54.870","Text":"we get minus 0.5x squared"},{"Start":"01:54.870 ","End":"02:00.485","Text":"plus c is equal to 1 over x to 1 equation."},{"Start":"02:00.485 ","End":"02:03.290","Text":"The other one, I\u0027ll write it already here."},{"Start":"02:03.290 ","End":"02:05.750","Text":"The other equation that we\u0027ll get will be"},{"Start":"02:05.750 ","End":"02:08.735","Text":"that they have the same slope at that point of intersection."},{"Start":"02:08.735 ","End":"02:12.725","Text":"F prime of x equals g prime of x."},{"Start":"02:12.725 ","End":"02:14.119","Text":"These are so elementary,"},{"Start":"02:14.119 ","End":"02:16.220","Text":"we can do it right away in here,"},{"Start":"02:16.220 ","End":"02:17.735","Text":"F prime of x,"},{"Start":"02:17.735 ","End":"02:23.990","Text":"so we get minus 1x, c disappears."},{"Start":"02:23.990 ","End":"02:28.410","Text":"What we get on the other side is derivative of 1 over x."},{"Start":"02:28.410 ","End":"02:31.670","Text":"That\u0027s well known, it\u0027s minus 1 over x squared."},{"Start":"02:31.670 ","End":"02:36.020","Text":"This is 2 equations and 2 unknowns x and c,"},{"Start":"02:36.020 ","End":"02:40.190","Text":"looks like it\u0027s best to deal with the last equation first,"},{"Start":"02:40.190 ","End":"02:44.450","Text":"if I multiply by minus x squared."},{"Start":"02:44.450 ","End":"02:49.300","Text":"I\u0027ll just get that x cubed equals 1."},{"Start":"02:49.300 ","End":"02:53.140","Text":"This is from this last 1,"},{"Start":"02:53.140 ","End":"02:55.820","Text":"I get x cubed equals 1 and therefore x equals 1."},{"Start":"02:55.820 ","End":"02:57.620","Text":"There\u0027s no plus or minuses here,"},{"Start":"02:57.620 ","End":"02:59.405","Text":"it\u0027s just equal to 1."},{"Start":"02:59.405 ","End":"03:01.595","Text":"If x equals 1,"},{"Start":"03:01.595 ","End":"03:05.890","Text":"then I put that up into here, x equals 1,"},{"Start":"03:05.890 ","End":"03:14.280","Text":"0.5 times 1 plus c equals 1 over 1, which is 1."},{"Start":"03:14.280 ","End":"03:18.105","Text":"C is equal to 1.5,"},{"Start":"03:18.105 ","End":"03:21.030","Text":"3 over 2, 1.5,"},{"Start":"03:21.030 ","End":"03:22.425","Text":"I prefer the fraction."},{"Start":"03:22.425 ","End":"03:27.365","Text":"There we have x and c. Let\u0027s put in the x,"},{"Start":"03:27.365 ","End":"03:30.230","Text":"which equals 1, not something useful."},{"Start":"03:30.230 ","End":"03:32.875","Text":"I\u0027ll maybe, perhaps highlight that."},{"Start":"03:32.875 ","End":"03:36.030","Text":"Now we also have c equals. That\u0027s good."},{"Start":"03:36.030 ","End":"03:37.775","Text":"I will highlight that."},{"Start":"03:37.775 ","End":"03:39.440","Text":"See what else we can find."},{"Start":"03:39.440 ","End":"03:45.880","Text":"If I put x equals 1 and this 1 looks easier and it\u0027s got less."},{"Start":"03:45.880 ","End":"03:48.890","Text":"Also I don\u0027t have the c,"},{"Start":"03:48.890 ","End":"03:50.680","Text":"so I\u0027ll better plug into this 1."},{"Start":"03:50.680 ","End":"03:52.550","Text":"But x equals 1,"},{"Start":"03:52.550 ","End":"03:58.325","Text":"then I\u0027ll get that y equals 1 over 1, which is 1."},{"Start":"03:58.325 ","End":"04:02.810","Text":"I get that y equals 1 over 1,"},{"Start":"04:02.810 ","End":"04:05.450","Text":"which is 1, y is 1."},{"Start":"04:05.450 ","End":"04:11.360","Text":"That\u0027s another 1 worth highlighting its one of those basic quantities we\u0027re looking for."},{"Start":"04:11.360 ","End":"04:13.760","Text":"We have x, we have y."},{"Start":"04:13.760 ","End":"04:15.575","Text":"What else do we need?"},{"Start":"04:15.575 ","End":"04:17.885","Text":"Well, we have the tangent point,"},{"Start":"04:17.885 ","End":"04:22.610","Text":"saying that the tangent point is equal to x equals 1,"},{"Start":"04:22.610 ","End":"04:25.010","Text":"y equals 1, that\u0027s 1, 1."},{"Start":"04:25.010 ","End":"04:28.630","Text":"What we\u0027re left is the joint tangent."},{"Start":"04:28.630 ","End":"04:30.420","Text":"Don\u0027t know anything about it."},{"Start":"04:30.420 ","End":"04:32.075","Text":"Let\u0027s write the general form,"},{"Start":"04:32.075 ","End":"04:34.730","Text":"y equals ax plus b."},{"Start":"04:34.730 ","End":"04:39.380","Text":"However, we do know that slope of this line is"},{"Start":"04:39.380 ","End":"04:45.095","Text":"a and that the common slopes are actually also equal to this a."},{"Start":"04:45.095 ","End":"04:49.335","Text":"Now f prime at the point 1, this line here,"},{"Start":"04:49.335 ","End":"04:54.080","Text":"this equation is the equation of the y prime and so should come out the same."},{"Start":"04:54.080 ","End":"04:58.190","Text":"If I put x is 1 here I get minus 1 over 1,"},{"Start":"04:58.190 ","End":"05:00.290","Text":"and here I get minus 1 times 1."},{"Start":"05:00.290 ","End":"05:03.410","Text":"Either case it comes out to minus 1."},{"Start":"05:03.410 ","End":"05:05.390","Text":"These 2 come out to minus 1,"},{"Start":"05:05.390 ","End":"05:09.415","Text":"which means that a is minus 1 by substitution."},{"Start":"05:09.415 ","End":"05:12.560","Text":"That\u0027s also worth highlighting."},{"Start":"05:12.560 ","End":"05:15.965","Text":"I guess the only thing missing is b."},{"Start":"05:15.965 ","End":"05:18.440","Text":"If I want to get b,"},{"Start":"05:18.440 ","End":"05:26.360","Text":"and then I\u0027ll just put that into here together with all the others together with the x,"},{"Start":"05:26.360 ","End":"05:30.945","Text":"which we have and we have a, we have x."},{"Start":"05:30.945 ","End":"05:32.190","Text":"Just do it this way."},{"Start":"05:32.190 ","End":"05:36.030","Text":"Y we have from here."},{"Start":"05:36.030 ","End":"05:38.610","Text":"We put 1 in for the."},{"Start":"05:38.610 ","End":"05:42.100","Text":"A we have from here as minus 1."},{"Start":"05:42.100 ","End":"05:45.330","Text":"X we have from here is 1."},{"Start":"05:45.330 ","End":"05:48.300","Text":"B is what we\u0027re trying to find out."},{"Start":"05:48.300 ","End":"05:51.720","Text":"That\u0027s a question mark. What do we get?"},{"Start":"05:51.720 ","End":"05:56.260","Text":"1 is equal to minus 1 plus something,"},{"Start":"05:56.260 ","End":"05:58.865","Text":"that something has got to be equal to 2."},{"Start":"05:58.865 ","End":"06:04.730","Text":"We get that b equals 2. We\u0027ve got everything."},{"Start":"06:04.730 ","End":"06:05.870","Text":"We got x, y, a,"},{"Start":"06:05.870 ","End":"06:08.255","Text":"b, and c. Now,"},{"Start":"06:08.255 ","End":"06:14.495","Text":"just to make sure that I\u0027m answering the question they ask to determine the constant."},{"Start":"06:14.495 ","End":"06:16.880","Text":"Let\u0027s just see what they have to find."},{"Start":"06:16.880 ","End":"06:18.770","Text":"We have to find out here,"},{"Start":"06:18.770 ","End":"06:27.415","Text":"we need to determine the constant C and the tangent point and the joint tangent."},{"Start":"06:27.415 ","End":"06:29.910","Text":"Let\u0027s see what we have so far."},{"Start":"06:29.910 ","End":"06:33.990","Text":"So far that the tangent point is 1, 1."},{"Start":"06:33.990 ","End":"06:36.905","Text":"It\u0027s 1 of the things that we\u0027re asked for, that\u0027s this."},{"Start":"06:36.905 ","End":"06:38.300","Text":"What else are we answering?"},{"Start":"06:38.300 ","End":"06:42.455","Text":"C is equal to this."},{"Start":"06:42.455 ","End":"06:47.345","Text":"I just left the yellow with a bit of green around so that we have C,"},{"Start":"06:47.345 ","End":"06:50.815","Text":"we have the tangent point,"},{"Start":"06:50.815 ","End":"06:53.870","Text":"and all we need is the joint tangent."},{"Start":"06:53.870 ","End":"06:55.940","Text":"It will come out of this form,"},{"Start":"06:55.940 ","End":"07:01.310","Text":"but we need a formula from analytical geometry and that formula"},{"Start":"07:01.310 ","End":"07:07.765","Text":"says that if you\u0027re given a line that passes through a given point with a given slope."},{"Start":"07:07.765 ","End":"07:12.405","Text":"Our slope is a. Let\u0027s say this."},{"Start":"07:12.405 ","End":"07:14.540","Text":"If, I write at the side here,"},{"Start":"07:14.540 ","End":"07:16.925","Text":"the formula from analytical geometry,"},{"Start":"07:16.925 ","End":"07:20.965","Text":"we have a line through, say,"},{"Start":"07:20.965 ","End":"07:26.580","Text":"x_1, y_1 plus slope a."},{"Start":"07:26.580 ","End":"07:35.775","Text":"That\u0027s given by y minus y_1 is equal to a x minus x_1."},{"Start":"07:35.775 ","End":"07:42.120","Text":"We have all the quantities y_1 and a and x_1."},{"Start":"07:42.120 ","End":"07:46.610","Text":"We\u0027ll get the line to be in our case,"},{"Start":"07:46.610 ","End":"07:51.980","Text":"2y minus y_1 is 1,"},{"Start":"07:51.980 ","End":"07:56.149","Text":"is equal to a, which is minus 1."},{"Start":"07:56.149 ","End":"08:01.350","Text":"X minus x1 is 1."},{"Start":"08:01.350 ","End":"08:06.710","Text":"The x_1, y_1 just mean the x and the y for our particular case,"},{"Start":"08:06.710 ","End":"08:10.895","Text":"this is, you could say this is x_1, y_1."},{"Start":"08:10.895 ","End":"08:13.970","Text":"Anyway, to simplify this,"},{"Start":"08:13.970 ","End":"08:20.245","Text":"y equals minus x plus 1 plus 1 is plus 2."},{"Start":"08:20.245 ","End":"08:24.319","Text":"Y is equal to x plus 2."},{"Start":"08:24.319 ","End":"08:27.530","Text":"That gives us our last thing,"},{"Start":"08:27.530 ","End":"08:29.900","Text":"which is the joint tangents."},{"Start":"08:29.900 ","End":"08:35.430","Text":"Now I can put a checkmark on here. We\u0027ve done everything."},{"Start":"08:35.430 ","End":"08:44.375","Text":"Once again, C is equal to 3 over 2 and the tangent point is here,"},{"Start":"08:44.375 ","End":"08:47.060","Text":"and the joint tangent is here."},{"Start":"08:47.060 ","End":"08:49.890","Text":"Now we\u0027re finally done."}],"ID":10580},{"Watched":false,"Name":"Exercise 9","Duration":"6m 50s","ChapterTopicVideoID":10244,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have to determine the constant c"},{"Start":"00:04.080 ","End":"00:08.489","Text":"such that this parabola is tangent to this parabola."},{"Start":"00:08.489 ","End":"00:13.140","Text":"We also have to find the tangent point and the joint tangent."},{"Start":"00:13.140 ","End":"00:18.990","Text":"Just to help, I brought with me illustration a sketch."},{"Start":"00:18.990 ","End":"00:21.900","Text":"This is parabola number 1,"},{"Start":"00:21.900 ","End":"00:24.000","Text":"parabola number 2,"},{"Start":"00:24.000 ","End":"00:26.860","Text":"the point of contact."},{"Start":"00:28.940 ","End":"00:33.540","Text":"The missing thing was the joint tangent."},{"Start":"00:33.540 ","End":"00:37.215","Text":"Basically what we have is a curve,"},{"Start":"00:37.215 ","End":"00:42.105","Text":"another curve, here we have the joint tangent."},{"Start":"00:42.105 ","End":"00:47.655","Text":"This point here is tangent point, point of contact."},{"Start":"00:47.655 ","End":"00:50.510","Text":"This is tangent both to this and to this."},{"Start":"00:50.510 ","End":"00:54.710","Text":"Our strategy is going to be to find a point that is common to all 3,"},{"Start":"00:54.710 ","End":"00:57.710","Text":"or at least a point that\u0027s common to these 2."},{"Start":"00:57.710 ","End":"01:05.135","Text":"Then we\u0027ll check if the slope of this line is the same as the slope of the curve here."},{"Start":"01:05.135 ","End":"01:07.355","Text":"Similarly, for here,"},{"Start":"01:07.355 ","End":"01:09.820","Text":"all 3 slopes have to be equal."},{"Start":"01:09.820 ","End":"01:12.740","Text":"Slope means 1 thing in the case of a line,"},{"Start":"01:12.740 ","End":"01:15.755","Text":"another thing in the case of a curve,"},{"Start":"01:15.755 ","End":"01:17.735","Text":"or will we shall see."},{"Start":"01:17.735 ","End":"01:20.285","Text":"Back to the exercise."},{"Start":"01:20.285 ","End":"01:23.420","Text":"The first thing is to find this tangent point,"},{"Start":"01:23.420 ","End":"01:25.130","Text":"which is this point of contact."},{"Start":"01:25.130 ","End":"01:28.970","Text":"It\u0027s exactly where this curve hits this curve."},{"Start":"01:28.970 ","End":"01:31.175","Text":"Well, just want to say this in general."},{"Start":"01:31.175 ","End":"01:36.695","Text":"In general, we might have y equals f of x as 1 curve,"},{"Start":"01:36.695 ","End":"01:41.450","Text":"and also y equals g of x is another curve."},{"Start":"01:41.450 ","End":"01:48.155","Text":"What we do to find a point of contact is to make f of x equals g of x."},{"Start":"01:48.155 ","End":"01:50.540","Text":"Solve for that, then we have an x and then we can"},{"Start":"01:50.540 ","End":"01:52.760","Text":"substitute and either one will get the same thing,"},{"Start":"01:52.760 ","End":"01:55.010","Text":"will get the y also."},{"Start":"01:55.010 ","End":"01:57.440","Text":"F of x equals g of x."},{"Start":"01:57.440 ","End":"02:02.325","Text":"Let\u0027s say this one is f and this is f of x just to label it."},{"Start":"02:02.325 ","End":"02:05.080","Text":"This one will be g of x."},{"Start":"02:05.630 ","End":"02:14.490","Text":"Minus x^2 plus c is equal to x^2 minus 4x plus 6."},{"Start":"02:15.500 ","End":"02:17.570","Text":"We know several things."},{"Start":"02:17.570 ","End":"02:22.670","Text":"One of the other things we know is that they have the same slope at the point of contact."},{"Start":"02:22.670 ","End":"02:24.905","Text":"The slope is the derivative."},{"Start":"02:24.905 ","End":"02:31.685","Text":"F prime of x equals g prime of x also at that point."},{"Start":"02:31.685 ","End":"02:35.525","Text":"F prime of x is minus 2x."},{"Start":"02:35.525 ","End":"02:44.565","Text":"This g prime of x is equal to plus 2x minus 4."},{"Start":"02:44.565 ","End":"02:48.230","Text":"Now we have 2 equations and 2 unknowns,"},{"Start":"02:48.230 ","End":"02:52.890","Text":"x and c. I\u0027ll start with this one because doesn\u0027t have any c in it,"},{"Start":"02:52.890 ","End":"02:54.630","Text":"so I can just find x."},{"Start":"02:54.630 ","End":"02:59.070","Text":"If I bring the minus 2x to this side and the 4 to this side to get 4 equals 4x."},{"Start":"02:59.070 ","End":"03:02.350","Text":"X is equal to 1."},{"Start":"03:02.350 ","End":"03:04.445","Text":"Once I have x equals 1,"},{"Start":"03:04.445 ","End":"03:13.795","Text":"then I can plug that in here and get minus 1 plus c is equal to 1 minus 4 plus 6."},{"Start":"03:13.795 ","End":"03:16.830","Text":"Altogether c is 4."},{"Start":"03:16.830 ","End":"03:18.540","Text":"Now we\u0027re already getting close."},{"Start":"03:18.540 ","End":"03:23.040","Text":"We have c. This answers for one thing is answers"},{"Start":"03:23.040 ","End":"03:28.365","Text":"the one of the questions that was asked is that c is 4."},{"Start":"03:28.365 ","End":"03:33.020","Text":"This was where the constant c that we\u0027re looking for is 4."},{"Start":"03:33.020 ","End":"03:37.355","Text":"That\u0027s the one thing. Then we have 2 more things to find the tangent point."},{"Start":"03:37.355 ","End":"03:39.200","Text":"Well, for the tangent point."},{"Start":"03:39.200 ","End":"03:41.145","Text":"It\u0027s what we\u0027re going to do now."},{"Start":"03:41.145 ","End":"03:44.630","Text":"We just substitute x in either one of these."},{"Start":"03:44.630 ","End":"03:50.185","Text":"That will bring me that if x equals 1, then y equals,"},{"Start":"03:50.185 ","End":"03:57.270","Text":"let\u0027s see f of x is minus x squared plus c. Minus x squared plus"},{"Start":"03:57.270 ","End":"04:06.060","Text":"c is minus 1 plus c, which is 3."},{"Start":"04:06.060 ","End":"04:09.785","Text":"These 2 together give us the tangent point."},{"Start":"04:09.785 ","End":"04:14.395","Text":"Tangent point is x, y is 1,3."},{"Start":"04:14.395 ","End":"04:18.040","Text":"The joint tangents is what we have to find now."},{"Start":"04:18.040 ","End":"04:22.160","Text":"What I need is, in general,"},{"Start":"04:22.160 ","End":"04:27.905","Text":"the equation of the line is y equals ax plus b."},{"Start":"04:27.905 ","End":"04:31.340","Text":"Now these 2 things together,"},{"Start":"04:31.340 ","End":"04:34.580","Text":"f prime and g prime are actually equal to the slope,"},{"Start":"04:34.580 ","End":"04:36.335","Text":"and we\u0027ll call that a,"},{"Start":"04:36.335 ","End":"04:40.400","Text":"assuming that we take the form of the line is y equals ax plus b."},{"Start":"04:40.400 ","End":"04:42.110","Text":"This is a general form of a line."},{"Start":"04:42.110 ","End":"04:44.435","Text":"2 constants usually call them a and b,"},{"Start":"04:44.435 ","End":"04:46.870","Text":"and this is the same a."},{"Start":"04:46.870 ","End":"04:51.070","Text":"This tangent has to pass through the tangent point."},{"Start":"04:51.070 ","End":"04:54.335","Text":"In other words, the tangent point,"},{"Start":"04:54.335 ","End":"05:00.950","Text":"which was 1,3 is on this equation which is a,"},{"Start":"05:00.950 ","End":"05:02.945","Text":"we know already is."},{"Start":"05:02.945 ","End":"05:07.430","Text":"There was a, a was equal to the f prime of x,"},{"Start":"05:07.430 ","End":"05:12.820","Text":"g prime of x. I didn\u0027t substitute a."},{"Start":"05:12.820 ","End":"05:15.825","Text":"This was f prime and this was g prime."},{"Start":"05:15.825 ","End":"05:18.725","Text":"If I substitute 1 in either of these,"},{"Start":"05:18.725 ","End":"05:21.555","Text":"I will get minus 2."},{"Start":"05:21.555 ","End":"05:24.905","Text":"The other case 2 minus 4 is also minus 2."},{"Start":"05:24.905 ","End":"05:28.400","Text":"This comes out to be minus 2."},{"Start":"05:28.400 ","End":"05:34.035","Text":"That has to be a. I\u0027ll just highlight that also,"},{"Start":"05:34.035 ","End":"05:38.640","Text":"a is minus 2 and it\u0027s the same a as this."},{"Start":"05:38.640 ","End":"05:42.780","Text":"This has got to be 1,3 is on the line."},{"Start":"05:42.780 ","End":"05:46.290","Text":"This is the point and it\u0027s on the line y equals a,"},{"Start":"05:46.290 ","End":"05:50.355","Text":"which is minus 2x plus b."},{"Start":"05:50.355 ","End":"05:53.790","Text":"Which means if I put this for y and this for x,"},{"Start":"05:53.790 ","End":"05:59.760","Text":"3 equals minus 2 times 1 plus b."},{"Start":"05:59.760 ","End":"06:08.430","Text":"That gives us that b is 3 plus 2 is 5, b equals 5."},{"Start":"06:08.430 ","End":"06:12.045","Text":"That means that the joint tangent, which is this."},{"Start":"06:12.045 ","End":"06:22.260","Text":"This comes out to be that y equals minus 2x plus 5."},{"Start":"06:22.260 ","End":"06:25.080","Text":"This is my b,"},{"Start":"06:25.080 ","End":"06:28.620","Text":"the joint tangent is this."},{"Start":"06:28.620 ","End":"06:32.040","Text":"This c was 4,"},{"Start":"06:32.040 ","End":"06:36.210","Text":"tangent point was 1,3."},{"Start":"06:36.210 ","End":"06:39.285","Text":"Joint tangent this line here."},{"Start":"06:39.285 ","End":"06:41.900","Text":"It\u0027s always good to go back to the beginning and"},{"Start":"06:41.900 ","End":"06:44.660","Text":"see if you\u0027ve actually answered the question."},{"Start":"06:44.660 ","End":"06:50.670","Text":"Sometimes there\u0027s more than one part or you may have misread. We\u0027re done."}],"ID":10581},{"Watched":false,"Name":"Exercise 10","Duration":"15m 46s","ChapterTopicVideoID":10245,"CourseChapterTopicPlaylistID":18296,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have to determine the constant c,"},{"Start":"00:03.780 ","End":"00:05.310","Text":"that\u0027s the one over here,"},{"Start":"00:05.310 ","End":"00:11.085","Text":"such that the curve so and so is tangent to the curve so and so."},{"Start":"00:11.085 ","End":"00:14.055","Text":"Then we also have to find the tangent points,"},{"Start":"00:14.055 ","End":"00:18.600","Text":"maybe more than 1, so we put an s here and the joint tangents."},{"Start":"00:18.600 ","End":"00:20.700","Text":"That\u0027s 3 things to find,"},{"Start":"00:20.700 ","End":"00:24.630","Text":"constant c, tangent point, joint tangent."},{"Start":"00:24.630 ","End":"00:27.240","Text":"Though these may have more than 1."},{"Start":"00:27.240 ","End":"00:29.540","Text":"What I\u0027ve actually done,"},{"Start":"00:29.540 ","End":"00:31.190","Text":"which is not necessary,"},{"Start":"00:31.190 ","End":"00:33.200","Text":"is to make a sketch."},{"Start":"00:33.200 ","End":"00:34.640","Text":"I\u0027ve brought a sketch with me,"},{"Start":"00:34.640 ","End":"00:35.780","Text":"it\u0027s somewhere down below."},{"Start":"00:35.780 ","End":"00:38.020","Text":"Let\u0027s go and take a look."},{"Start":"00:38.020 ","End":"00:40.784","Text":"This is the general idea."},{"Start":"00:40.784 ","End":"00:43.440","Text":"This looks like 3 curves,"},{"Start":"00:43.440 ","End":"00:48.315","Text":"but this one and this one are part of 1 curve."},{"Start":"00:48.315 ","End":"00:51.770","Text":"Just I presume that 0 is not in the domain or something."},{"Start":"00:51.770 ","End":"00:54.550","Text":"This is a curve,"},{"Start":"00:54.550 ","End":"00:57.000","Text":"and this is also continued,"},{"Start":"00:57.000 ","End":"00:58.575","Text":"say curve 1,"},{"Start":"00:58.575 ","End":"01:00.960","Text":"this is also curve 1."},{"Start":"01:00.960 ","End":"01:03.260","Text":"This one is curve 2."},{"Start":"01:03.260 ","End":"01:07.000","Text":"I don\u0027t know if that means that\u0027s the order they appeared above."},{"Start":"01:07.000 ","End":"01:10.075","Text":"This looks like an inverted parabola."},{"Start":"01:10.075 ","End":"01:12.130","Text":"This one looks like a hyperbola."},{"Start":"01:12.130 ","End":"01:13.780","Text":"Maybe it is, maybe it isn\u0027t."},{"Start":"01:13.780 ","End":"01:15.250","Text":"Anyway, we see here there are"},{"Start":"01:15.250 ","End":"01:20.155","Text":"2 tangent points and corresponding to each there\u0027s a tangent line."},{"Start":"01:20.155 ","End":"01:23.095","Text":"We are expecting 2 solutions."},{"Start":"01:23.095 ","End":"01:26.500","Text":"Let\u0027s call it a tangent point,"},{"Start":"01:26.500 ","End":"01:29.425","Text":"and this is also a tangent point,"},{"Start":"01:29.425 ","End":"01:35.550","Text":"and this is 1 of the tangent lines."},{"Start":"01:35.550 ","End":"01:40.065","Text":"This is also, I\u0027ll also call it another TL, tangent line."},{"Start":"01:40.065 ","End":"01:42.730","Text":"Okay. This should give you an idea of what to expect,"},{"Start":"01:42.730 ","End":"01:46.370","Text":"but don\u0027t put too much emphasis on the sketch."},{"Start":"01:46.370 ","End":"01:54.020","Text":"1 of the things about the way we approach finding the tangent point is to compare."},{"Start":"01:54.020 ","End":"01:56.470","Text":"Usually, you get 2 functions, this one and this one."},{"Start":"01:56.470 ","End":"02:00.415","Text":"In general, you have y equals f of x,"},{"Start":"02:00.415 ","End":"02:02.280","Text":"this is f, which case?"},{"Start":"02:02.280 ","End":"02:05.525","Text":"Actually, that was the lower one, the inverted parabola."},{"Start":"02:05.525 ","End":"02:10.610","Text":"We also have y equals g of x."},{"Start":"02:10.610 ","End":"02:16.200","Text":"What we do is to find the common point to the 2 curves,"},{"Start":"02:16.200 ","End":"02:19.205","Text":"you compare f of x equals g of x."},{"Start":"02:19.205 ","End":"02:26.455","Text":"In our case, we have minus 8x squared plus c"},{"Start":"02:26.455 ","End":"02:33.575","Text":"is going to equal 1 plus 6x squared over 2x squared."},{"Start":"02:33.575 ","End":"02:36.260","Text":"Well, that\u0027s just 1 equation because,"},{"Start":"02:36.260 ","End":"02:37.789","Text":"I mean, we have 2 variables,"},{"Start":"02:37.789 ","End":"02:40.530","Text":"x and c. This is not the general x,"},{"Start":"02:40.530 ","End":"02:43.910","Text":"is the x of the specific tangent point."},{"Start":"02:43.910 ","End":"02:45.780","Text":"We need another equation,"},{"Start":"02:45.780 ","End":"02:53.480","Text":"and that other equation is that f prime of x is also equal to g prime of x."},{"Start":"02:53.480 ","End":"02:56.135","Text":"They have the same slope at that point."},{"Start":"02:56.135 ","End":"03:01.480","Text":"The slope is the derivative that will give us another equation."},{"Start":"03:01.580 ","End":"03:08.075","Text":"I\u0027m assuming that this one is my f and this one is the g,"},{"Start":"03:08.075 ","End":"03:14.110","Text":"so f prime of x is minus 16x,"},{"Start":"03:14.110 ","End":"03:22.765","Text":"has got to equal g prime of x. G prime of x can be done using the quotient rule,"},{"Start":"03:22.765 ","End":"03:25.280","Text":"but I think if we do some division,"},{"Start":"03:25.280 ","End":"03:26.720","Text":"we can simplify it."},{"Start":"03:26.720 ","End":"03:31.885","Text":"Let me at the side do what the derivative of g of x is."},{"Start":"03:31.885 ","End":"03:34.590","Text":"G of x, if I rewrite it,"},{"Start":"03:34.590 ","End":"03:37.405","Text":"I can put it as 1 over 2x squared."},{"Start":"03:37.405 ","End":"03:42.855","Text":"I\u0027d rather write it as 1/2 times 1 over x squared,"},{"Start":"03:42.855 ","End":"03:46.190","Text":"so 1 over this plus 6x squared over this,"},{"Start":"03:46.190 ","End":"03:47.855","Text":"I break up the fraction."},{"Start":"03:47.855 ","End":"03:54.475","Text":"1 over 2x squared is this and 6x squared over 2x squared is just 3."},{"Start":"03:54.475 ","End":"03:57.890","Text":"I\u0027m presuming that we have to observe the domain of"},{"Start":"03:57.890 ","End":"04:02.645","Text":"definition that x is not equal to 0 here in the same domain."},{"Start":"04:02.645 ","End":"04:04.355","Text":"All this makes sense."},{"Start":"04:04.355 ","End":"04:08.325","Text":"Now, g prime of x is equal to 1/2."},{"Start":"04:08.325 ","End":"04:11.025","Text":"Is a multiplicative constant, it stays."},{"Start":"04:11.025 ","End":"04:16.174","Text":"1 over x squared, the derivative is minus 2 over x cubed, that\u0027s pretty standard."},{"Start":"04:16.174 ","End":"04:18.170","Text":"It\u0027s x to the minus 2,"},{"Start":"04:18.170 ","End":"04:21.190","Text":"so it\u0027s minus 2x to the minus 3,"},{"Start":"04:21.190 ","End":"04:27.285","Text":"so times minus 2 over x cubed."},{"Start":"04:27.285 ","End":"04:28.860","Text":"Because it\u0027s minus,"},{"Start":"04:28.860 ","End":"04:31.500","Text":"this thing is minus 2x to the minus 3."},{"Start":"04:31.500 ","End":"04:32.970","Text":"The 3 [inaudible] to nothing."},{"Start":"04:32.970 ","End":"04:35.670","Text":"What we\u0027re left with is the 2 and the 2 cancel,"},{"Start":"04:35.670 ","End":"04:39.000","Text":"it\u0027s minus 1 over x cubed."},{"Start":"04:39.000 ","End":"04:42.680","Text":"16x is minus 1 over x cubed."},{"Start":"04:42.680 ","End":"04:48.065","Text":"If I get rid of the minus on both sides and multiply by x cubed,"},{"Start":"04:48.065 ","End":"04:55.460","Text":"then I\u0027ve got 16x to the 4th is equal to 1."},{"Start":"04:55.460 ","End":"04:59.000","Text":"If we take the square root of this,"},{"Start":"04:59.000 ","End":"05:08.135","Text":"then we get the plus or minus 4x squared is equal to 1."},{"Start":"05:08.135 ","End":"05:13.700","Text":"We have to conclude that it\u0027s 4x squared that equals 1."},{"Start":"05:13.700 ","End":"05:15.600","Text":"I can\u0027t have the minus here,"},{"Start":"05:15.600 ","End":"05:19.280","Text":"that\u0027s ruled out because it was minus 4x squared that has"},{"Start":"05:19.280 ","End":"05:23.265","Text":"to be non-positive and it can\u0027t equal 1."},{"Start":"05:23.265 ","End":"05:25.740","Text":"It has to be that the 4x squared is 1,"},{"Start":"05:25.740 ","End":"05:29.310","Text":"which means that this is 2x all squared,"},{"Start":"05:29.310 ","End":"05:33.045","Text":"so 2x has to be plus or minus 1."},{"Start":"05:33.045 ","End":"05:38.730","Text":"X is equal to plus or minus 1/2."},{"Start":"05:38.730 ","End":"05:41.160","Text":"That\u0027s what\u0027s going to give us the 2 solutions,"},{"Start":"05:41.160 ","End":"05:44.895","Text":"is that we have a plus 1/2 or a minus 1/2 for x."},{"Start":"05:44.895 ","End":"05:46.870","Text":"Once we have x,"},{"Start":"05:46.870 ","End":"05:51.605","Text":"we can get another variable which is c. Let\u0027s do it in a column."},{"Start":"05:51.605 ","End":"05:53.255","Text":"I\u0027ll call them 1 and 2."},{"Start":"05:53.255 ","End":"06:01.500","Text":"X1 is equal to 1/2 and x2 is minus 1/2."},{"Start":"06:01.500 ","End":"06:06.735","Text":"Now, if I take the plus 1/2 and put it here,"},{"Start":"06:06.735 ","End":"06:10.155","Text":"let\u0027s see, we\u0027ll do it at the side somewhere."},{"Start":"06:10.155 ","End":"06:12.930","Text":"Let\u0027s try doing this exercise."},{"Start":"06:12.930 ","End":"06:16.515","Text":"Let\u0027s see, minus 8,"},{"Start":"06:16.515 ","End":"06:24.300","Text":"let\u0027s take the plus 1/2 first of 1/2 squared plus c equals 1 plus"},{"Start":"06:24.300 ","End":"06:33.030","Text":"6 times 1/2 squared over twice 1/2 squared."},{"Start":"06:33.030 ","End":"06:34.845","Text":"What do we get here?"},{"Start":"06:34.845 ","End":"06:37.875","Text":"We get 1/2 squared is 1/4."},{"Start":"06:37.875 ","End":"06:44.460","Text":"That\u0027s minus 2 plus c equals 1/2 squared is 1/4,"},{"Start":"06:44.460 ","End":"06:48.255","Text":"1/4 times 6 is 3 over 2,"},{"Start":"06:48.255 ","End":"06:55.695","Text":"is 1 plus 3 over 2 all over twice 1/2 squared."},{"Start":"06:55.695 ","End":"06:58.230","Text":"Twice 1/4 is 1/2."},{"Start":"06:58.230 ","End":"07:01.250","Text":"Here we\u0027ll multiply top and bottom by 2."},{"Start":"07:01.250 ","End":"07:06.275","Text":"Let\u0027s multiply top and bottom by 2 and then we\u0027ll get 5."},{"Start":"07:06.275 ","End":"07:09.905","Text":"That gives us that c is equal to 7."},{"Start":"07:09.905 ","End":"07:13.110","Text":"Here we have c1 equals 7."},{"Start":"07:13.110 ","End":"07:16.550","Text":"Now, we have x and c. In the other case,"},{"Start":"07:16.550 ","End":"07:19.065","Text":"if it was minus 1/2,"},{"Start":"07:19.065 ","End":"07:21.330","Text":"well, everything here was squared."},{"Start":"07:21.330 ","End":"07:23.955","Text":"X squared is a 1/4 regardless,"},{"Start":"07:23.955 ","End":"07:27.525","Text":"so c2 has got to also equal 7."},{"Start":"07:27.525 ","End":"07:31.550","Text":"That gives us x and c. For 1 thing,"},{"Start":"07:31.550 ","End":"07:33.710","Text":"we\u0027ve answered the constant c,"},{"Start":"07:33.710 ","End":"07:35.720","Text":"that came out to be plural."},{"Start":"07:35.720 ","End":"07:39.690","Text":"We\u0027ve actually done this part of the question where we want"},{"Start":"07:39.690 ","End":"07:44.570","Text":"the constant c but there is only 1 value of c. Basically,"},{"Start":"07:44.570 ","End":"07:48.210","Text":"I can just write it as c equals 7."},{"Start":"07:49.790 ","End":"07:51.900","Text":"That\u0027s the common c,"},{"Start":"07:51.900 ","End":"07:53.325","Text":"so that is 7."},{"Start":"07:53.325 ","End":"07:55.350","Text":"Now, we\u0027d really like to find everything."},{"Start":"07:55.350 ","End":"07:58.305","Text":"Actually, we should find y also."},{"Start":"07:58.305 ","End":"08:04.110","Text":"To find y, we just put it into f of x because it\u0027s easier."},{"Start":"08:04.110 ","End":"08:07.215","Text":"This one looks messier. Let\u0027s see."},{"Start":"08:07.215 ","End":"08:12.680","Text":"Y1 and we\u0027re also going to get a y2. Very good."},{"Start":"08:12.680 ","End":"08:15.590","Text":"Again, x only appears as x squared,"},{"Start":"08:15.590 ","End":"08:18.275","Text":"so will both get the same y."},{"Start":"08:18.275 ","End":"08:23.025","Text":"If we put 1/2 squared is 1/4,"},{"Start":"08:23.025 ","End":"08:27.570","Text":"minus 8 times 1/4 is minus 2,"},{"Start":"08:27.570 ","End":"08:31.260","Text":"minus 2 plus c is 7."},{"Start":"08:31.260 ","End":"08:35.015","Text":"If c is 7, I\u0027ll just write that again."},{"Start":"08:35.015 ","End":"08:39.610","Text":"If c is 7, that\u0027s a stopping point,"},{"Start":"08:39.610 ","End":"08:45.265","Text":"but then I add that x equals plus or minus 1/2."},{"Start":"08:45.265 ","End":"08:53.620","Text":"That gives us that y equals minus 8 times 1/4,"},{"Start":"08:53.620 ","End":"08:56.470","Text":"because whether it\u0027s minus 1/2 or plus 1/2,"},{"Start":"08:56.470 ","End":"08:59.125","Text":"the x squared will equal 1/4,"},{"Start":"08:59.125 ","End":"09:01.075","Text":"plus the 7,"},{"Start":"09:01.075 ","End":"09:05.440","Text":"which equals minus 2 plus 7, which is 5."},{"Start":"09:05.440 ","End":"09:07.165","Text":"That means that,"},{"Start":"09:07.165 ","End":"09:11.905","Text":"in both cases, the y is equal to 5."},{"Start":"09:11.905 ","End":"09:14.725","Text":"The x, c, y."},{"Start":"09:14.725 ","End":"09:17.680","Text":"There are actually 2 other variables that we haven\u0027t"},{"Start":"09:17.680 ","End":"09:21.385","Text":"mentioned yet because we\u0027re talking about the joint tangents,"},{"Start":"09:21.385 ","End":"09:24.190","Text":"and we know that the tangent is a line."},{"Start":"09:24.190 ","End":"09:30.670","Text":"The tangent, we shall call y equals ax plus b,"},{"Start":"09:30.670 ","End":"09:32.860","Text":"is the general equation of the tangent,"},{"Start":"09:32.860 ","End":"09:34.645","Text":"and we\u0027re missing a and b."},{"Start":"09:34.645 ","End":"09:36.865","Text":"Now, how do I find a and b?"},{"Start":"09:36.865 ","End":"09:42.130","Text":"Well, a is easy because it turns out that the slope of the tangent,"},{"Start":"09:42.130 ","End":"09:44.665","Text":"which is the slope of a line which is a,"},{"Start":"09:44.665 ","End":"09:47.530","Text":"is the same as this common slope of the curves."},{"Start":"09:47.530 ","End":"09:49.075","Text":"In fact, these 3 things are equal,"},{"Start":"09:49.075 ","End":"09:50.650","Text":"3 slopes are equal,1 curve,"},{"Start":"09:50.650 ","End":"09:51.970","Text":"the other curve, and the tangent,"},{"Start":"09:51.970 ","End":"09:53.320","Text":"so this equals a."},{"Start":"09:53.320 ","End":"10:00.430","Text":"All I have to do to find a is to plug in x into either f prime or g prime."},{"Start":"10:00.430 ","End":"10:02.950","Text":"Let\u0027s see which looks easier."},{"Start":"10:02.950 ","End":"10:06.320","Text":"The f prime we did first."},{"Start":"10:07.680 ","End":"10:10.780","Text":"This was f prime and this was g prime."},{"Start":"10:10.780 ","End":"10:12.235","Text":"Now, let see. What did I want?"},{"Start":"10:12.235 ","End":"10:17.575","Text":"We wanted to put in x is 1/2 or minus 1/2."},{"Start":"10:17.575 ","End":"10:20.485","Text":"We\u0027ll get different results."},{"Start":"10:20.485 ","End":"10:23.875","Text":"We have an a_1 and we\u0027ll have an a_2."},{"Start":"10:23.875 ","End":"10:26.785","Text":"Get a_1, put x equals 1/2."},{"Start":"10:26.785 ","End":"10:28.930","Text":"Here x is 1/2."},{"Start":"10:28.930 ","End":"10:32.560","Text":"Then it comes out to minus 8."},{"Start":"10:32.560 ","End":"10:35.050","Text":"If x is 1/2, also here,"},{"Start":"10:35.050 ","End":"10:38.230","Text":"we have minus 1 over 1/8,"},{"Start":"10:38.230 ","End":"10:40.615","Text":"which is also minus 8."},{"Start":"10:40.615 ","End":"10:42.040","Text":"These are odd functions,"},{"Start":"10:42.040 ","End":"10:47.655","Text":"and if we put x is equal to minus 1/2,"},{"Start":"10:47.655 ","End":"10:50.130","Text":"then we\u0027re going to get plus 8."},{"Start":"10:50.130 ","End":"10:52.875","Text":"The last thing is the b."},{"Start":"10:52.875 ","End":"10:57.095","Text":"It\u0027ll also be a b_1 and b_2."},{"Start":"10:57.095 ","End":"10:59.185","Text":"How do we get this b?"},{"Start":"10:59.185 ","End":"11:06.120","Text":"Well, this tangent line can simplify because we have a."},{"Start":"11:06.120 ","End":"11:07.785","Text":"Because we have a,"},{"Start":"11:07.785 ","End":"11:10.905","Text":"which is either minus 8 or an 8,"},{"Start":"11:10.905 ","End":"11:13.360","Text":"in the case 1,"},{"Start":"11:15.210 ","End":"11:19.090","Text":"we have a is minus 8,"},{"Start":"11:19.090 ","End":"11:24.085","Text":"so y equals minus 8x plus b."},{"Start":"11:24.085 ","End":"11:26.665","Text":"Then the other case, case 2,"},{"Start":"11:26.665 ","End":"11:31.840","Text":"y equals plus 8x plus b."},{"Start":"11:31.840 ","End":"11:33.970","Text":"Now, the other thing we know is that"},{"Start":"11:33.970 ","End":"11:37.060","Text":"the tangent point has got to be on the tangent line."},{"Start":"11:37.060 ","End":"11:39.639","Text":"Let\u0027s get to the tangent point."},{"Start":"11:39.639 ","End":"11:42.055","Text":"Tangent point is x, y."},{"Start":"11:42.055 ","End":"11:47.480","Text":"For tangent point, we have x_1, y_1."},{"Start":"11:48.330 ","End":"11:50.545","Text":"Scroll a little bit."},{"Start":"11:50.545 ","End":"11:56.800","Text":"What I want to do here is also write the tangent point and the tangent lines."},{"Start":"11:56.800 ","End":"12:02.410","Text":"This space I\u0027ll reserve for the tangent point and this for the tangent line."},{"Start":"12:02.410 ","End":"12:05.065","Text":"A tangent point is just x, y."},{"Start":"12:05.065 ","End":"12:10.405","Text":"Over here, it\u0027s going to be 1/2, 5."},{"Start":"12:10.405 ","End":"12:12.250","Text":"That\u0027s the tangent point."},{"Start":"12:12.250 ","End":"12:17.065","Text":"Here the tangent point is going to be minus 1/2,"},{"Start":"12:17.065 ","End":"12:19.210","Text":"y, which again,"},{"Start":"12:19.210 ","End":"12:20.965","Text":"comes out to 5."},{"Start":"12:20.965 ","End":"12:26.395","Text":"The tangent point, which we were doing in green,"},{"Start":"12:26.395 ","End":"12:27.700","Text":"that\u0027s the tangent point,"},{"Start":"12:27.700 ","End":"12:29.155","Text":"so that\u0027s 1 case,"},{"Start":"12:29.155 ","End":"12:30.745","Text":"and that\u0027s another case."},{"Start":"12:30.745 ","End":"12:34.135","Text":"There was only 1 answer for the constant c,"},{"Start":"12:34.135 ","End":"12:37.615","Text":"but now how about the tangent line?"},{"Start":"12:37.615 ","End":"12:39.715","Text":"For the tangent line,"},{"Start":"12:39.715 ","End":"12:43.580","Text":"we have to find what b is."},{"Start":"12:44.100 ","End":"12:49.900","Text":"I should have said that we actually know the value of y."},{"Start":"12:49.900 ","End":"12:52.540","Text":"The y\u0027s in both cases came out to be 5,"},{"Start":"12:52.540 ","End":"12:56.080","Text":"so either 5 is equal y."},{"Start":"12:56.080 ","End":"12:58.690","Text":"We have a. All we\u0027re missing is b."},{"Start":"12:58.690 ","End":"13:02.875","Text":"In case 1, we have a set of y and a,"},{"Start":"13:02.875 ","End":"13:04.360","Text":"5 and minus 8,"},{"Start":"13:04.360 ","End":"13:06.850","Text":"and in the other case, we have 5 and plus 8."},{"Start":"13:06.850 ","End":"13:08.440","Text":"B is still unknown."},{"Start":"13:08.440 ","End":"13:10.240","Text":"In the first case,"},{"Start":"13:10.240 ","End":"13:13.435","Text":"we put x equals 1/2."},{"Start":"13:13.435 ","End":"13:15.340","Text":"If x is 1/2,"},{"Start":"13:15.340 ","End":"13:21.750","Text":"here I put x as 1/2 and here I put x as minus 1/2."},{"Start":"13:21.750 ","End":"13:25.455","Text":"If it\u0027s 1/2, it\u0027s minus 4 here,"},{"Start":"13:25.455 ","End":"13:30.050","Text":"which goes together with the 5 and makes b equals 9."},{"Start":"13:30.050 ","End":"13:32.455","Text":"B equals 9 here."},{"Start":"13:32.455 ","End":"13:34.720","Text":"Here if I put x as minus 1/2,"},{"Start":"13:34.720 ","End":"13:37.705","Text":"yes, here it\u0027s minus 8 times 1/2,"},{"Start":"13:37.705 ","End":"13:39.160","Text":"which is minus 4,"},{"Start":"13:39.160 ","End":"13:41.605","Text":"and here it\u0027s 8 times minus 1/2,"},{"Start":"13:41.605 ","End":"13:43.375","Text":"which is also minus 4."},{"Start":"13:43.375 ","End":"13:46.615","Text":"Again, it gives us b equals 9."},{"Start":"13:46.615 ","End":"13:49.450","Text":"B equals 9 or b equals 9,"},{"Start":"13:49.450 ","End":"13:53.065","Text":"in that case, I\u0027ll take b equals 9."},{"Start":"13:53.065 ","End":"13:56.410","Text":"Some of them can be written independently without the 1 or 2."},{"Start":"13:56.410 ","End":"13:58.270","Text":"For example, here they were both the same,"},{"Start":"13:58.270 ","End":"14:00.430","Text":"and I could have just said y equals 5,"},{"Start":"14:00.430 ","End":"14:04.870","Text":"and here I could say b equals 9 because it\u0027s a common thing."},{"Start":"14:04.870 ","End":"14:10.240","Text":"Now, we have b and we have different possibilities for a."},{"Start":"14:10.240 ","End":"14:13.660","Text":"Ultimately, even though we have b equals 9,"},{"Start":"14:13.660 ","End":"14:16.600","Text":"we still have 2 different tangents."},{"Start":"14:16.600 ","End":"14:18.355","Text":"Let me put these,"},{"Start":"14:18.355 ","End":"14:20.095","Text":"the joint tangents,"},{"Start":"14:20.095 ","End":"14:23.510","Text":"anyway, I\u0027ll write them in here first."},{"Start":"14:25.170 ","End":"14:31.240","Text":"What they are is we already said that it\u0027s y equals ax plus b,"},{"Start":"14:31.240 ","End":"14:33.835","Text":"but this time it\u0027s the general y."},{"Start":"14:33.835 ","End":"14:35.860","Text":"A and b are the specific values."},{"Start":"14:35.860 ","End":"14:38.770","Text":"Y and x are not the specific y and x,"},{"Start":"14:38.770 ","End":"14:40.465","Text":"but they are the general."},{"Start":"14:40.465 ","End":"14:45.340","Text":"We have that y equals ax plus b,"},{"Start":"14:45.340 ","End":"14:46.840","Text":"which is a general shape."},{"Start":"14:46.840 ","End":"14:54.910","Text":"Y equals a, which is 8x plus b is plus 9."},{"Start":"14:54.910 ","End":"14:56.785","Text":"That was the first one."},{"Start":"14:56.785 ","End":"15:00.145","Text":"The second one is y,"},{"Start":"15:00.145 ","End":"15:04.795","Text":"it\u0027s taking this a, and this b."},{"Start":"15:04.795 ","End":"15:06.280","Text":"Here it was minus 8, 9."},{"Start":"15:06.280 ","End":"15:08.035","Text":"Here it\u0027s 8, 9."},{"Start":"15:08.035 ","End":"15:10.720","Text":"Here it\u0027s y."},{"Start":"15:10.720 ","End":"15:12.400","Text":"It should be a minus."},{"Start":"15:12.400 ","End":"15:14.035","Text":"I\u0027ll fix it in a second."},{"Start":"15:14.035 ","End":"15:19.015","Text":"This is y equals ax plus b, 8x plus 9."},{"Start":"15:19.015 ","End":"15:20.620","Text":"This one, well,"},{"Start":"15:20.620 ","End":"15:23.425","Text":"it\u0027s not an 8, it\u0027s a minus 8."},{"Start":"15:23.425 ","End":"15:28.120","Text":"Fixed it. I think we have everything now."},{"Start":"15:28.120 ","End":"15:30.624","Text":"Let\u0027s just read the question again."},{"Start":"15:30.624 ","End":"15:33.520","Text":"Determine the constant c, c is 7,"},{"Start":"15:33.520 ","End":"15:37.015","Text":"such that the curve is tangent to this curve,"},{"Start":"15:37.015 ","End":"15:39.235","Text":"and then find the tangent points."},{"Start":"15:39.235 ","End":"15:43.705","Text":"Well, there\u0027s 2 tangent points giving rise to 2 joint tangents,"},{"Start":"15:43.705 ","End":"15:47.030","Text":"and that\u0027s it. We\u0027re done."}],"ID":10582}],"Thumbnail":null,"ID":18296},{"Name":"Tangent and Normal Lines of Implicit Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Implicit Functions","Duration":"3m 51s","ChapterTopicVideoID":8260,"CourseChapterTopicPlaylistID":1666,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this clip, we\u0027re continuing with equations of tangents and normals to a curve,"},{"Start":"00:05.190 ","End":"00:08.670","Text":"but sometimes the curve is given by an implicit function."},{"Start":"00:08.670 ","End":"00:10.170","Text":"I\u0027ll give an example."},{"Start":"00:10.170 ","End":"00:19.005","Text":"I want to find the equations of both the tangent and the normal to the curve."},{"Start":"00:19.005 ","End":"00:21.285","Text":"Let\u0027s take a circle of radius 5,"},{"Start":"00:21.285 ","End":"00:29.325","Text":"x squared plus y squared equals 5 squared at the point 4, 3."},{"Start":"00:29.325 ","End":"00:32.955","Text":"Well, before we even start the exercise or at the very beginning,"},{"Start":"00:32.955 ","End":"00:35.620","Text":"we should really check that this is on the curve."},{"Start":"00:35.620 ","End":"00:37.400","Text":"You can mentally check that,"},{"Start":"00:37.400 ","End":"00:43.925","Text":"indeed that 4 squared plus 3 squared radii equals 25."},{"Start":"00:43.925 ","End":"00:46.580","Text":"That\u0027s true because it\u0027s 16 plus 9."},{"Start":"00:46.580 ","End":"00:48.335","Text":"Just making a note to that."},{"Start":"00:48.335 ","End":"00:50.615","Text":"Let\u0027s see how we solve this."},{"Start":"00:50.615 ","End":"00:54.140","Text":"The first thing I want to do is to find the derivative only,"},{"Start":"00:54.140 ","End":"00:56.630","Text":"we have to use implicit differentiation."},{"Start":"00:56.630 ","End":"01:02.960","Text":"From this equation, I can get that the derivative x squared gave us 2x,"},{"Start":"01:02.960 ","End":"01:06.950","Text":"y squared gives us not just 2y, well,"},{"Start":"01:06.950 ","End":"01:10.615","Text":"because it\u0027s a function of y times y prime,"},{"Start":"01:10.615 ","End":"01:13.370","Text":"and the derivative of 25 is 0."},{"Start":"01:13.370 ","End":"01:16.460","Text":"Then from here, I can extract y prime,"},{"Start":"01:16.460 ","End":"01:18.110","Text":"and divide by 2."},{"Start":"01:18.110 ","End":"01:21.425","Text":"From here, if I bring this to the other side, and divide by y,"},{"Start":"01:21.425 ","End":"01:28.475","Text":"I get that y prime is equal to minus x over y."},{"Start":"01:28.475 ","End":"01:30.110","Text":"At our particular point,"},{"Start":"01:30.110 ","End":"01:33.060","Text":"at the point 4, 3,"},{"Start":"01:33.060 ","End":"01:38.660","Text":"we get that y prime equals minus 4 over 3,"},{"Start":"01:38.660 ","End":"01:44.995","Text":"and this has the interpretation that it\u0027s the slope of the tangent."},{"Start":"01:44.995 ","End":"01:48.755","Text":"If I want the slope of the normal,"},{"Start":"01:48.755 ","End":"01:51.260","Text":"then I take the negative reciprocal."},{"Start":"01:51.260 ","End":"01:56.075","Text":"It\u0027s minus 1 divided by minus 4/3,"},{"Start":"01:56.075 ","End":"01:59.030","Text":"is equal to plus 3/4."},{"Start":"01:59.030 ","End":"02:02.280","Text":"3/4 is the slope of the normal."},{"Start":"02:02.280 ","End":"02:03.640","Text":"We\u0027re going to use both."},{"Start":"02:03.640 ","End":"02:11.840","Text":"The equation of the tangent at that point will be y minus the y of the point,"},{"Start":"02:11.840 ","End":"02:16.430","Text":"which is 3 equals slope minus 4/3,"},{"Start":"02:16.430 ","End":"02:19.325","Text":"x minus the x of the point,"},{"Start":"02:19.325 ","End":"02:26.795","Text":"and parallel of the normal will just be also y minus 3,"},{"Start":"02:26.795 ","End":"02:29.870","Text":"and also something x minus 4."},{"Start":"02:29.870 ","End":"02:34.730","Text":"But instead of minus 4/3, I put 3/4."},{"Start":"02:34.730 ","End":"02:36.560","Text":"Let\u0027s simplify each one."},{"Start":"02:36.560 ","End":"02:41.810","Text":"This 1 gives me that y equals minus 4/3x,"},{"Start":"02:41.810 ","End":"02:47.220","Text":"then I get minus 4/3 times minus 4 is plus 16 over 3."},{"Start":"02:47.220 ","End":"02:50.175","Text":"16 over 3 is 5 and 1/3,"},{"Start":"02:50.175 ","End":"02:52.110","Text":"but I also have to add the 3,"},{"Start":"02:52.110 ","End":"02:54.075","Text":"so it\u0027s 8 and 1/3."},{"Start":"02:54.075 ","End":"02:57.250","Text":"Let me write that as 25 over 3 for consistency."},{"Start":"02:57.250 ","End":"03:00.365","Text":"If I put an improper fraction here, I\u0027ll do so here."},{"Start":"03:00.365 ","End":"03:02.480","Text":"That\u0027s part 1 is the tangent,"},{"Start":"03:02.480 ","End":"03:03.620","Text":"and for the normal,"},{"Start":"03:03.620 ","End":"03:07.445","Text":"we get that y equals 3/4x."},{"Start":"03:07.445 ","End":"03:09.530","Text":"3/4 times 4 is 3,"},{"Start":"03:09.530 ","End":"03:13.285","Text":"so it\u0027s minus 3 plus the 3, so that\u0027s all."},{"Start":"03:13.285 ","End":"03:16.880","Text":"This is one of the solution for the tangent,"},{"Start":"03:16.880 ","End":"03:19.430","Text":"and this is the solution for the normal."},{"Start":"03:19.430 ","End":"03:22.730","Text":"Notice that this is a line through the origin."},{"Start":"03:22.730 ","End":"03:24.020","Text":"We could stop here,"},{"Start":"03:24.020 ","End":"03:27.245","Text":"but I\u0027d like to show you some intuition of why this is so."},{"Start":"03:27.245 ","End":"03:29.735","Text":"Let\u0027s draw y-axis and an x-axis,"},{"Start":"03:29.735 ","End":"03:32.575","Text":"and a circle of radius 5."},{"Start":"03:32.575 ","End":"03:35.430","Text":"This might be the point 4, 3,"},{"Start":"03:35.430 ","End":"03:38.120","Text":"so the tangent is somewhere here,"},{"Start":"03:38.120 ","End":"03:43.310","Text":"but the normal to a circle always passes through the origin."},{"Start":"03:43.310 ","End":"03:47.000","Text":"This is the answer. There are several more exercises"},{"Start":"03:47.000 ","End":"03:51.600","Text":"like this following this brief tutorial. Done."}],"ID":8421},{"Watched":false,"Name":"Exercise 1","Duration":"6m 51s","ChapterTopicVideoID":4387,"CourseChapterTopicPlaylistID":1666,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"This exercise is for the more advanced students and you know who you are."},{"Start":"00:05.070 ","End":"00:12.495","Text":"What we have to do is to find the equation of the line tangent to this at the point 2, 2."},{"Start":"00:12.495 ","End":"00:14.250","Text":"Now what is this?"},{"Start":"00:14.250 ","End":"00:16.140","Text":"This is an implicit function."},{"Start":"00:16.140 ","End":"00:19.500","Text":"It\u0027s not a function where y is a function of x."},{"Start":"00:19.500 ","End":"00:22.735","Text":"It\u0027s implicit, but it still defines a curve."},{"Start":"00:22.735 ","End":"00:27.260","Text":"At this point there will be a tangent and we have to find its equation."},{"Start":"00:27.260 ","End":"00:31.200","Text":"Now, strictly speaking I want to check that this really is on the curve."},{"Start":"00:31.200 ","End":"00:33.650","Text":"I did it in my head just earlier."},{"Start":"00:33.650 ","End":"00:35.870","Text":"I put x equals 2y equals 2,"},{"Start":"00:35.870 ","End":"00:38.260","Text":"and I got 8 equals 8. So that\u0027s fine."},{"Start":"00:38.260 ","End":"00:40.040","Text":"But it\u0027s a minor point,"},{"Start":"00:40.040 ","End":"00:41.750","Text":"but it should be checked because someone made"},{"Start":"00:41.750 ","End":"00:44.155","Text":"a mistake and this point is not on this curve."},{"Start":"00:44.155 ","End":"00:48.970","Text":"I have a sketch which might help to understand what\u0027s going on here."},{"Start":"00:48.970 ","End":"00:51.125","Text":"What we have here is the curve,"},{"Start":"00:51.125 ","End":"00:52.220","Text":"which is not a function."},{"Start":"00:52.220 ","End":"00:53.600","Text":"How can we tell it\u0027s not a function?"},{"Start":"00:53.600 ","End":"00:56.695","Text":"A vertical line can cut it more than once."},{"Start":"00:56.695 ","End":"01:00.950","Text":"This is the curve given by the equation."},{"Start":"01:00.950 ","End":"01:02.525","Text":"I\u0027m not going to repeat it here."},{"Start":"01:02.525 ","End":"01:05.090","Text":"This must be the 0.22,"},{"Start":"01:05.090 ","End":"01:07.260","Text":"although it doesn\u0027t look to scale."},{"Start":"01:07.260 ","End":"01:08.975","Text":"The sketches not to scale."},{"Start":"01:08.975 ","End":"01:12.175","Text":"At this point, this will be the tangent line."},{"Start":"01:12.175 ","End":"01:17.330","Text":"There is not much more to say except that should be easy to find the tangent."},{"Start":"01:17.330 ","End":"01:20.540","Text":"The strategy will be we already have a point and all we"},{"Start":"01:20.540 ","End":"01:24.350","Text":"need is the slope of this line because some analytical geometry,"},{"Start":"01:24.350 ","End":"01:28.565","Text":"we know how to find a line from a point and a slope."},{"Start":"01:28.565 ","End":"01:33.395","Text":"Now the slope as you know is tied into the derivative."},{"Start":"01:33.395 ","End":"01:39.620","Text":"The derivative at a point is actually equal to the slope of the tangent to the curve."},{"Start":"01:39.620 ","End":"01:43.730","Text":"Let\u0027s differentiate, but remember this is implicit differentiation."},{"Start":"01:43.730 ","End":"01:48.770","Text":"Let me just copy it first and I\u0027ll also copy the other detail that x,"},{"Start":"01:48.770 ","End":"01:50.510","Text":"y it\u0027s equal to x_1, y_1,"},{"Start":"01:50.510 ","End":"01:54.110","Text":"not the general x, y is equal to 2, 2."},{"Start":"01:54.110 ","End":"01:57.050","Text":"All we need to do, I\u0027ll just remark is to say what"},{"Start":"01:57.050 ","End":"02:00.950","Text":"the slope at that point equals and then we can get the equation."},{"Start":"02:00.950 ","End":"02:06.390","Text":"Differentiating 2x plus 2y and don\u0027t"},{"Start":"02:06.390 ","End":"02:12.215","Text":"forget the y prime implicit differentiation is equal to 2."},{"Start":"02:12.215 ","End":"02:14.105","Text":"Here we have a product."},{"Start":"02:14.105 ","End":"02:17.810","Text":"Maybe I\u0027ll write the product formula in case someone has forgotten it."},{"Start":"02:17.810 ","End":"02:20.795","Text":"If you have 2 functions of x, say u and v,"},{"Start":"02:20.795 ","End":"02:25.790","Text":"then their derivative is equal to the first one derived second as is,"},{"Start":"02:25.790 ","End":"02:30.635","Text":"and then the first one as is times the second one derived."},{"Start":"02:30.635 ","End":"02:38.990","Text":"So x is going to be the u and y is going to be v. We have derivative of x,"},{"Start":"02:38.990 ","End":"02:49.115","Text":"which is 1, y is plus x and the derivative of y minus the derivative of 2x is 2."},{"Start":"02:49.115 ","End":"02:56.880","Text":"The derivative of y is y prime and the 6 doesn\u0027t give us anything, it\u0027s a constant."},{"Start":"02:56.890 ","End":"03:01.610","Text":"Let\u0027s open the brackets and put everything on."},{"Start":"03:01.610 ","End":"03:06.440","Text":"Also as we do so we\u0027ll put it on the left-hand side and do all that in one go."},{"Start":"03:06.440 ","End":"03:11.205","Text":"2x plus 2yy prime."},{"Start":"03:11.205 ","End":"03:16.080","Text":"Then 2 times y is 2y so it\u0027ll go over as minus 2y."},{"Start":"03:16.080 ","End":"03:18.785","Text":"Then minus 2xy prime,"},{"Start":"03:18.785 ","End":"03:23.509","Text":"minus 2 minus y prime is equal to 0."},{"Start":"03:23.509 ","End":"03:27.200","Text":"Then we\u0027ll collect together the bits with y prime."},{"Start":"03:27.200 ","End":"03:29.030","Text":"This is as a y prime,"},{"Start":"03:29.030 ","End":"03:32.545","Text":"there is a y prime and there is a y prime."},{"Start":"03:32.545 ","End":"03:37.070","Text":"What I\u0027m going to do is to keep the stuff with"},{"Start":"03:37.070 ","End":"03:41.750","Text":"y prime on this side and move the rest to the other side of the equation."},{"Start":"03:41.750 ","End":"03:44.120","Text":"How many y prime do we have?"},{"Start":"03:44.120 ","End":"03:51.300","Text":"Well, we have 2y from here and then minus 2x from here and minus 1 from here,"},{"Start":"03:51.300 ","End":"03:56.220","Text":"all these y prime is equal to the rest onto the other side,"},{"Start":"03:56.220 ","End":"04:02.280","Text":"and I have minus 2x plus 2y plus 2."},{"Start":"04:02.280 ","End":"04:07.600","Text":"What that gives us is y prime by just dividing this by this,"},{"Start":"04:07.600 ","End":"04:10.505","Text":"so we have now that y prime."},{"Start":"04:10.505 ","End":"04:15.200","Text":"Now I want to emphasize that y prime is a function of x and y."},{"Start":"04:15.200 ","End":"04:17.810","Text":"Normally we use the regular function y prime"},{"Start":"04:17.810 ","End":"04:20.780","Text":"is just a function of x. I want to emphasize it,"},{"Start":"04:20.780 ","End":"04:24.620","Text":"I\u0027ll just add this reminder that it\u0027s a function of x"},{"Start":"04:24.620 ","End":"04:28.910","Text":"and y in implicit differentiation as opposed to what we usually have."},{"Start":"04:28.910 ","End":"04:30.140","Text":"This is just for emphasis."},{"Start":"04:30.140 ","End":"04:32.215","Text":"That\u0027s y prime is equal to,"},{"Start":"04:32.215 ","End":"04:34.610","Text":"maybe I will keep things in the same order,"},{"Start":"04:34.610 ","End":"04:36.140","Text":"I\u0027ll put the y before the x,"},{"Start":"04:36.140 ","End":"04:41.075","Text":"so I\u0027ll just change the order here just for neatness 2y minus 2x,"},{"Start":"04:41.075 ","End":"04:45.720","Text":"also I don\u0027t like starting with a minus, plus 2 over."},{"Start":"04:45.720 ","End":"04:49.690","Text":"Let\u0027s see what else 2y minus 2x minus 1."},{"Start":"04:50.000 ","End":"04:54.750","Text":"Now what we need is the slope of the line and"},{"Start":"04:54.750 ","End":"05:01.200","Text":"the slope is equal to y prime at our particular point is 2,"},{"Start":"05:01.200 ","End":"05:03.480","Text":"2, which equals,"},{"Start":"05:03.480 ","End":"05:04.860","Text":"now if I put 2,"},{"Start":"05:04.860 ","End":"05:07.695","Text":"2 in here, let\u0027s see."},{"Start":"05:07.695 ","End":"05:13.420","Text":"2y minus 2x is just nothing because y is equal to x."},{"Start":"05:13.420 ","End":"05:17.160","Text":"We just end up with getting 2 over minus 1,"},{"Start":"05:17.160 ","End":"05:22.060","Text":"which is minus 2.That gives us the slope because they are missing ingredient."},{"Start":"05:22.060 ","End":"05:24.815","Text":"Now the equation from analytical geometry,"},{"Start":"05:24.815 ","End":"05:29.625","Text":"equation of line through x_1,"},{"Start":"05:29.625 ","End":"05:32.840","Text":"y_1 with slope equals,"},{"Start":"05:32.840 ","End":"05:39.400","Text":"let\u0027s give it some letter say m is a commonly used letter for slope is given by y"},{"Start":"05:39.400 ","End":"05:47.170","Text":"minus y_1 is equal to m times x minus x_1."},{"Start":"05:47.170 ","End":"05:51.455","Text":"Now what we have is that we have y_1 there it is,"},{"Start":"05:51.455 ","End":"05:54.050","Text":"and the slope we now know is minus 2,"},{"Start":"05:54.050 ","End":"05:56.675","Text":"so I\u0027m going to erase this question mark."},{"Start":"05:56.675 ","End":"05:58.430","Text":"I wrote it for you, slope,"},{"Start":"05:58.430 ","End":"06:01.650","Text":"which is denoted by m is minus 2."},{"Start":"06:01.650 ","End":"06:04.360","Text":"All the relevant information we need is here."},{"Start":"06:04.360 ","End":"06:08.930","Text":"What I\u0027m going to do is just plug these values into the formula,"},{"Start":"06:08.930 ","End":"06:12.115","Text":"and that will give us the equation of our line."},{"Start":"06:12.115 ","End":"06:14.810","Text":"What we have is, in our case,"},{"Start":"06:14.810 ","End":"06:20.565","Text":"y minus y_1 is this 2 is equal to m"},{"Start":"06:20.565 ","End":"06:27.435","Text":"which is minus 2 times x minus x_1 which is also 2."},{"Start":"06:27.435 ","End":"06:29.000","Text":"That\u0027s basically the answer,"},{"Start":"06:29.000 ","End":"06:33.410","Text":"but more customary to just put it as y in terms of x."},{"Start":"06:33.410 ","End":"06:35.495","Text":"In other words, just simplify a bit."},{"Start":"06:35.495 ","End":"06:41.440","Text":"We get that y on its own is equal to minus 2x,"},{"Start":"06:41.440 ","End":"06:47.490","Text":"minus 2 times minus 2 is plus 4 add another 2 is plus 6."},{"Start":"06:47.490 ","End":"06:51.730","Text":"This is the answer and we\u0027re done."}],"ID":4396},{"Watched":false,"Name":"Exercise 2","Duration":"8m 12s","ChapterTopicVideoID":4845,"CourseChapterTopicPlaylistID":1666,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.669","Text":"In this exercise, we\u0027re given a curve in implicit form"},{"Start":"00:05.669 ","End":"00:12.795","Text":"and we have to find the equation of the tangent line at the point on the curve,"},{"Start":"00:12.795 ","End":"00:18.015","Text":"where x equals 1 and y is negative."},{"Start":"00:18.015 ","End":"00:21.885","Text":"Now, a sketch can help."},{"Start":"00:21.885 ","End":"00:24.480","Text":"It\u0027s not an essential, but it helps,"},{"Start":"00:24.480 ","End":"00:26.730","Text":"and I\u0027ve already pre-prepared 1,"},{"Start":"00:26.730 ","End":"00:28.960","Text":"so let\u0027s go to it."},{"Start":"00:29.150 ","End":"00:33.870","Text":"Here we are, what a nice picture this is?"},{"Start":"00:33.870 ","End":"00:42.400","Text":"This is the point where x equals 1 and as you may notice,"},{"Start":"00:42.400 ","End":"00:51.809","Text":"there are actually 3 places where the vertical line x equals 1 cuts the curve."},{"Start":"00:51.809 ","End":"00:55.215","Text":"You see it cuts here,"},{"Start":"00:55.215 ","End":"00:59.590","Text":"and here and this 1 here."},{"Start":"00:59.590 ","End":"01:01.730","Text":"But at this point,"},{"Start":"01:01.730 ","End":"01:04.335","Text":"y is bigger than 0,"},{"Start":"01:04.335 ","End":"01:06.900","Text":"and here y equals 0,"},{"Start":"01:06.900 ","End":"01:10.025","Text":"and in this 1 y is less than 0,"},{"Start":"01:10.025 ","End":"01:14.680","Text":"which is why this is the 1 we want and this is the tangent line."},{"Start":"01:14.680 ","End":"01:19.865","Text":"Just label them, curve and tangent."},{"Start":"01:19.865 ","End":"01:25.100","Text":"Now we\u0027ll go back to where we were as if we didn\u0027t have a picture,"},{"Start":"01:25.100 ","End":"01:28.620","Text":"but the picture certainly helps."},{"Start":"01:28.700 ","End":"01:32.690","Text":"Here we are back up here again."},{"Start":"01:32.690 ","End":"01:35.510","Text":"We want to find the point x, y,"},{"Start":"01:35.510 ","End":"01:39.510","Text":"first of all, where this holds."},{"Start":"01:39.510 ","End":"01:43.925","Text":"We know the x and we just need to find the y because x is 1."},{"Start":"01:43.925 ","End":"01:50.295","Text":"What we do is just put 1 in this equation and get 1 cubed plus"},{"Start":"01:50.295 ","End":"01:58.930","Text":"4 times 1 times y minus 4y cubed equals 1."},{"Start":"02:00.020 ","End":"02:02.130","Text":"1 cubed is 1,"},{"Start":"02:02.130 ","End":"02:03.795","Text":"that cancels with that."},{"Start":"02:03.795 ","End":"02:10.510","Text":"We have 4y minus 4y cubed equal 0."},{"Start":"02:10.700 ","End":"02:21.585","Text":"I can factorize this as 4y times 1 minus y squared equals 0,"},{"Start":"02:21.585 ","End":"02:24.735","Text":"and this factorizes as a difference of squares."},{"Start":"02:24.735 ","End":"02:28.269","Text":"4y, 1 minus y,"},{"Start":"02:28.269 ","End":"02:32.180","Text":"1 plus y equals 0."},{"Start":"02:32.180 ","End":"02:38.630","Text":"Constant, I don\u0027t need divide both sides by 4 but essentially this gives me 3 solutions."},{"Start":"02:38.630 ","End":"02:42.200","Text":"Either this is 0 or this is 0, or this is 0."},{"Start":"02:42.200 ","End":"02:50.520","Text":"In which case y is equal to either 0 or 1 or minus 1."},{"Start":"02:50.520 ","End":"02:53.020","Text":"Now, out of these 3."},{"Start":"02:53.020 ","End":"02:56.270","Text":"Only 1 satisfies y less than 0,"},{"Start":"02:56.270 ","End":"02:58.465","Text":"which is this 1."},{"Start":"02:58.465 ","End":"03:00.820","Text":"Since x is 1,"},{"Start":"03:00.820 ","End":"03:06.215","Text":"we come to the conclusion that our point xy is the point 1,"},{"Start":"03:06.215 ","End":"03:09.940","Text":"minus 1, that is the point on the curve."},{"Start":"03:09.940 ","End":"03:11.615","Text":"That\u0027s the first step."},{"Start":"03:11.615 ","End":"03:13.550","Text":"Now for a tangent line,"},{"Start":"03:13.550 ","End":"03:16.040","Text":"we need a point and the slope."},{"Start":"03:16.040 ","End":"03:20.990","Text":"I mean, if this is, let\u0027s say x_1 and this is y_1,"},{"Start":"03:20.990 ","End":"03:25.424","Text":"then we know that the equation we want is y"},{"Start":"03:25.424 ","End":"03:32.280","Text":"minus y_1 equals m, x minus x_1."},{"Start":"03:32.280 ","End":"03:34.800","Text":"In fact, we know x_1 and y_1,"},{"Start":"03:34.800 ","End":"03:37.380","Text":"so why don\u0027t I just put them straight here?"},{"Start":"03:37.380 ","End":"03:39.500","Text":"y minus minus 1,"},{"Start":"03:39.500 ","End":"03:41.120","Text":"which is y plus 1,"},{"Start":"03:41.120 ","End":"03:44.550","Text":"is m, x minus 1."},{"Start":"03:44.550 ","End":"03:52.415","Text":"The thing is that we don\u0027t have m. What we\u0027re going to do is an implicit differentiation."},{"Start":"03:52.415 ","End":"03:55.835","Text":"Find y prime in terms of x and y,"},{"Start":"03:55.835 ","End":"03:58.520","Text":"and then substitute 1 and minus 1."},{"Start":"03:58.520 ","End":"03:59.990","Text":"That\u0027s the plan of action."},{"Start":"03:59.990 ","End":"04:04.350","Text":"Let\u0027s start from this and differentiate it."},{"Start":"04:06.290 ","End":"04:12.190","Text":"We\u0027re going to do an implicit differentiation on this equation."},{"Start":"04:12.190 ","End":"04:22.370","Text":"We get 3x squared plus 4."},{"Start":"04:22.370 ","End":"04:24.380","Text":"Now the derivative of xy,"},{"Start":"04:24.380 ","End":"04:25.909","Text":"I\u0027ll do it by a product."},{"Start":"04:25.909 ","End":"04:31.800","Text":"The derivative of x is 1 times y as is,"},{"Start":"04:31.800 ","End":"04:37.100","Text":"and then this undifferentiated and the other derived,"},{"Start":"04:37.100 ","End":"04:43.600","Text":"which gives me y prime minus then from 4y cubed,"},{"Start":"04:43.600 ","End":"04:45.820","Text":"I get,4 times 3 is 12,"},{"Start":"04:45.820 ","End":"04:52.115","Text":"minus 12y squared, but times y prime, it\u0027s implicit differentiation."},{"Start":"04:52.115 ","End":"04:56.030","Text":"All this equals derivative of 1 which is 0."},{"Start":"04:56.030 ","End":"04:58.669","Text":"Let me just multiply out the brackets."},{"Start":"04:58.669 ","End":"05:04.765","Text":"We get 3x squared plus 4y"},{"Start":"05:04.765 ","End":"05:12.945","Text":"plus 4xy prime minus 12y squared,"},{"Start":"05:12.945 ","End":"05:15.990","Text":"y prime equals 0."},{"Start":"05:15.990 ","End":"05:19.140","Text":"Everything with y prime we\u0027ll stay on this side."},{"Start":"05:19.140 ","End":"05:21.465","Text":"Put everything else on the other side."},{"Start":"05:21.465 ","End":"05:24.970","Text":"In fact, I\u0027ll take out the brackets and I\u0027ll get"},{"Start":"05:24.970 ","End":"05:34.065","Text":"4x minus 12y squared times y prime equals,"},{"Start":"05:34.065 ","End":"05:43.620","Text":"these 2 I put on the other side and get minus 3x squared minus 4y."},{"Start":"05:43.760 ","End":"05:49.205","Text":"Then when we divide by this,"},{"Start":"05:49.205 ","End":"05:59.030","Text":"we can now get an equation that y prime is equal to minus 3x squared minus"},{"Start":"05:59.030 ","End":"06:02.420","Text":"4y over"},{"Start":"06:02.420 ","End":"06:10.735","Text":"4x minus 12y squared."},{"Start":"06:10.735 ","End":"06:13.640","Text":"It changed its color. This is important."},{"Start":"06:13.640 ","End":"06:17.615","Text":"This is the derivative y prime in terms of x and y."},{"Start":"06:17.615 ","End":"06:22.830","Text":"Then we were looking for m. m is going to"},{"Start":"06:22.830 ","End":"06:28.625","Text":"equal what y prime is when I put x equals 1 and y equals minus 1."},{"Start":"06:28.625 ","End":"06:35.885","Text":"It\u0027s minus 3, 1 squared minus 4 times minus 1"},{"Start":"06:35.885 ","End":"06:45.205","Text":"over 4 times 1 minus 12 times minus 1 squared."},{"Start":"06:45.205 ","End":"06:48.210","Text":"Let\u0027s see now, 1 squared is 1."},{"Start":"06:48.210 ","End":"06:50.265","Text":"That\u0027s minus 3."},{"Start":"06:50.265 ","End":"06:52.140","Text":"Minus 3,"},{"Start":"06:52.140 ","End":"06:56.175","Text":"minus minus is plus, that\u0027s plus 4."},{"Start":"06:56.175 ","End":"06:58.710","Text":"This is 4,"},{"Start":"06:58.710 ","End":"07:02.700","Text":"and this is minus 1 squared is plus 1,"},{"Start":"07:02.700 ","End":"07:04.890","Text":"so it\u0027s minus 12."},{"Start":"07:04.890 ","End":"07:09.350","Text":"What I get is minus 3 plus 4 is 1,"},{"Start":"07:09.350 ","End":"07:11.945","Text":"4 minus 12 is minus 8."},{"Start":"07:11.945 ","End":"07:14.760","Text":"I get minus 1/8."},{"Start":"07:15.400 ","End":"07:20.180","Text":"That\u0027s another important finding and I\u0027ll highlight that."},{"Start":"07:20.180 ","End":"07:24.035","Text":"I\u0027m running out of space here so let me erase some stuff."},{"Start":"07:24.035 ","End":"07:27.295","Text":"I can erase these calculations."},{"Start":"07:27.295 ","End":"07:32.030","Text":"What we need to do is put m equals minus an 1/8."},{"Start":"07:32.030 ","End":"07:39.970","Text":"I get y plus 1 equals minus 1/8, x minus 1."},{"Start":"07:39.970 ","End":"07:48.835","Text":"Now I just tidy up a bit and say that y equals minus 1/8x,"},{"Start":"07:48.835 ","End":"07:53.350","Text":"minus an 1/8 times minus 1 is plus an 1/8."},{"Start":"07:53.660 ","End":"08:04.770","Text":"But plus an 1/8 minus 1 is going to give me minus 7/8."},{"Start":"08:04.770 ","End":"08:08.195","Text":"This is the answer that we\u0027re looking for,"},{"Start":"08:08.195 ","End":"08:10.145","Text":"and I\u0027ll highlight it,"},{"Start":"08:10.145 ","End":"08:12.870","Text":"and we are done."}],"ID":4845},{"Watched":false,"Name":"Exercise 3","Duration":"8m 14s","ChapterTopicVideoID":4389,"CourseChapterTopicPlaylistID":1666,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"This exercise is for the more advanced students and you know who you are."},{"Start":"00:04.710 ","End":"00:08.610","Text":"In it, we have to find the equation of the line tangent"},{"Start":"00:08.610 ","End":"00:13.410","Text":"to this thing at the point on the curve where x equals a."},{"Start":"00:13.410 ","End":"00:15.195","Text":"What do I mean by this thing?"},{"Start":"00:15.195 ","End":"00:17.310","Text":"Well, it\u0027s an implicit function."},{"Start":"00:17.310 ","End":"00:19.470","Text":"It isn\u0027t y given in terms of x,"},{"Start":"00:19.470 ","End":"00:22.395","Text":"but an equation involving x and y,"},{"Start":"00:22.395 ","End":"00:24.540","Text":"which defines a curve,"},{"Start":"00:24.540 ","End":"00:29.090","Text":"and at any given point there will be a tangent line."},{"Start":"00:29.090 ","End":"00:32.750","Text":"I also have to add that a is got to be bigger than 0,"},{"Start":"00:32.750 ","End":"00:35.330","Text":"otherwise, it doesn\u0027t make sense."},{"Start":"00:35.330 ","End":"00:38.120","Text":"I\u0027ve drawn a sketch for you so you can have"},{"Start":"00:38.120 ","End":"00:40.970","Text":"an idea of what we\u0027re talking about. What do we see here?"},{"Start":"00:40.970 ","End":"00:44.780","Text":"We see the y and the x-axis,"},{"Start":"00:44.780 ","End":"00:46.905","Text":"and we see a curve,"},{"Start":"00:46.905 ","End":"00:50.870","Text":"this is the curve which is given by that equation,"},{"Start":"00:50.870 ","End":"00:55.305","Text":"and this is the particular point that\u0027s on the curve."},{"Start":"00:55.305 ","End":"00:56.775","Text":"If you actually check,"},{"Start":"00:56.775 ","End":"00:58.740","Text":"and you should check, the point a,"},{"Start":"00:58.740 ","End":"01:01.280","Text":"a is on the curve because square root of x plus square root of"},{"Start":"01:01.280 ","End":"01:04.250","Text":"y is twice the square root of a, and it works."},{"Start":"01:04.250 ","End":"01:06.950","Text":"Notice it\u0027s only to the right of"},{"Start":"01:06.950 ","End":"01:10.430","Text":"the y-axis because the sum of 2 square roots can\u0027t be negative."},{"Start":"01:10.430 ","End":"01:13.670","Text":"Basically, this is the tangent and that\u0027s why we have"},{"Start":"01:13.670 ","End":"01:17.765","Text":"to find the equation of the tangent line at that point."},{"Start":"01:17.765 ","End":"01:20.945","Text":"We\u0027re in good shape because we have a point on the line,"},{"Start":"01:20.945 ","End":"01:22.160","Text":"although it\u0027s a parameter,"},{"Start":"01:22.160 ","End":"01:25.040","Text":"but still a known point a is unknown,"},{"Start":"01:25.040 ","End":"01:27.530","Text":"because it\u0027s a supposedly unknown quantity."},{"Start":"01:27.530 ","End":"01:32.075","Text":"All that we\u0027ll need really is to find the slope of this line,"},{"Start":"01:32.075 ","End":"01:36.975","Text":"because a point and a slope determine uniquely a line and there\u0027s a formula for it."},{"Start":"01:36.975 ","End":"01:41.644","Text":"Let\u0027s get back up there and see if we can find the slope of this tangent."},{"Start":"01:41.644 ","End":"01:45.650","Text":"Now, 1 of the interpretations of the derivative,"},{"Start":"01:45.650 ","End":"01:47.405","Text":"in this case, y prime,"},{"Start":"01:47.405 ","End":"01:52.865","Text":"is that if you\u0027ve had a point and you know y prime at that point,"},{"Start":"01:52.865 ","End":"01:58.690","Text":"that y prime is exactly the slope of the tangent line there."},{"Start":"01:58.690 ","End":"02:04.520","Text":"So we have to find y prime and what we\u0027ll do is implicit differentiation."},{"Start":"02:04.520 ","End":"02:07.040","Text":"If we take the original function,"},{"Start":"02:07.040 ","End":"02:10.025","Text":"square root of x plus square root of y"},{"Start":"02:10.025 ","End":"02:14.480","Text":"equals twice square root of a and we differentiate it."},{"Start":"02:14.480 ","End":"02:16.010","Text":"I see we need a formula."},{"Start":"02:16.010 ","End":"02:18.830","Text":"Let\u0027s just pause and write down a formula."},{"Start":"02:18.830 ","End":"02:20.745","Text":"Some of you might know this,"},{"Start":"02:20.745 ","End":"02:23.855","Text":"it\u0027s a template formula for square roots."},{"Start":"02:23.855 ","End":"02:26.720","Text":"You have the square root of something and that something is"},{"Start":"02:26.720 ","End":"02:29.660","Text":"a function of x or may depend on x,"},{"Start":"02:29.660 ","End":"02:31.715","Text":"and you want to know its derivative."},{"Start":"02:31.715 ","End":"02:33.965","Text":"What you do is you take 1 over,"},{"Start":"02:33.965 ","End":"02:35.450","Text":"hadn\u0027t written the 1 deliberately,"},{"Start":"02:35.450 ","End":"02:41.060","Text":"1 over twice the square root of that same thing, the function of x,"},{"Start":"02:41.060 ","End":"02:43.370","Text":"times the internal derivative,"},{"Start":"02:43.370 ","End":"02:45.860","Text":"which is, say this is a box prime,"},{"Start":"02:45.860 ","End":"02:51.650","Text":"but I prefer to write it above the dividing line rather than 1 over times it like this."},{"Start":"02:51.650 ","End":"02:53.810","Text":"In this case, we\u0027ll use it twice,"},{"Start":"02:53.810 ","End":"02:55.895","Text":"once here and once here."},{"Start":"02:55.895 ","End":"02:59.090","Text":"Continuing, we differentiate,"},{"Start":"02:59.090 ","End":"03:01.025","Text":"we get for this bit,"},{"Start":"03:01.025 ","End":"03:06.185","Text":"1 over twice the square root of x times x prime,"},{"Start":"03:06.185 ","End":"03:09.665","Text":"which is 1, or this is also an immediate formula,"},{"Start":"03:09.665 ","End":"03:15.350","Text":"plus something over twice the square root of y and its derivative,"},{"Start":"03:15.350 ","End":"03:17.269","Text":"which is y prime."},{"Start":"03:17.269 ","End":"03:19.145","Text":"This is a constant."},{"Start":"03:19.145 ","End":"03:21.440","Text":"We don\u0027t know what a is, but it\u0027s not a variable,"},{"Start":"03:21.440 ","End":"03:24.125","Text":"it\u0027s a parameter, it\u0027s a constant."},{"Start":"03:24.125 ","End":"03:26.225","Text":"Some number, 7 maybe,"},{"Start":"03:26.225 ","End":"03:28.850","Text":"I don\u0027t know, 342,"},{"Start":"03:28.850 ","End":"03:31.940","Text":"whatever, as long as it\u0027s a constant,"},{"Start":"03:31.940 ","End":"03:34.030","Text":"so that equals 0."},{"Start":"03:34.030 ","End":"03:39.035","Text":"What we have to do next is isolate y prime in terms of x and y,"},{"Start":"03:39.035 ","End":"03:41.915","Text":"just as y prime is what we want."},{"Start":"03:41.915 ","End":"03:48.350","Text":"Let\u0027s multiply both sides by the common denominator."},{"Start":"03:48.350 ","End":"03:53.105","Text":"I\u0027ll just write down, we\u0027re going to multiply left and right by twice square root of x,"},{"Start":"03:53.105 ","End":"03:54.180","Text":"square root of y,"},{"Start":"03:54.180 ","End":"03:56.180","Text":"that should take care of both of these."},{"Start":"03:56.180 ","End":"04:00.220","Text":"The missing ingredient here will be the square root of y,"},{"Start":"04:00.220 ","End":"04:07.070","Text":"and the missing ingredient to square root of y is missing square root of x times y prime."},{"Start":"04:07.070 ","End":"04:08.450","Text":"Here it was times 1,"},{"Start":"04:08.450 ","End":"04:10.415","Text":"I\u0027m multiplying the numerator of course,"},{"Start":"04:10.415 ","End":"04:15.995","Text":"so equals still 0 because you multiply 0 by anything is 0."},{"Start":"04:15.995 ","End":"04:18.320","Text":"Now I can isolate y prime."},{"Start":"04:18.320 ","End":"04:21.290","Text":"I\u0027ll take the square root of y to the right-hand side"},{"Start":"04:21.290 ","End":"04:24.320","Text":"and divide it by the square root of x and I\u0027ll get"},{"Start":"04:24.320 ","End":"04:30.860","Text":"that y prime is the square root of y minus over the square root of x,"},{"Start":"04:30.860 ","End":"04:36.680","Text":"and it may be more useful to us to combine the square root of sum of a fraction,"},{"Start":"04:36.680 ","End":"04:41.765","Text":"y over x, whichever of these forms is more convenient for us."},{"Start":"04:41.765 ","End":"04:44.060","Text":"Now we\u0027re proceeding very well,"},{"Start":"04:44.060 ","End":"04:45.815","Text":"I said something about a slope."},{"Start":"04:45.815 ","End":"04:48.845","Text":"It turns out that, and you\u0027ve studied this,"},{"Start":"04:48.845 ","End":"04:54.850","Text":"the geometrical meaning of the derivative is the slope of the tangent."},{"Start":"04:54.850 ","End":"05:02.945","Text":"In our case, I want to emphasize that y prime is a function of both x and y."},{"Start":"05:02.945 ","End":"05:05.200","Text":"Normally, when you have a function,"},{"Start":"05:05.200 ","End":"05:07.520","Text":"and y prime is given in terms of x,"},{"Start":"05:07.520 ","End":"05:10.115","Text":"but here y prime, because it\u0027s an implicit function,"},{"Start":"05:10.115 ","End":"05:12.335","Text":"is given in terms of x and y."},{"Start":"05:12.335 ","End":"05:14.405","Text":"So this is equal to,"},{"Start":"05:14.405 ","End":"05:16.990","Text":"again, same thing as the above."},{"Start":"05:16.990 ","End":"05:22.400","Text":"All I\u0027m saying here is that bear in mind that y prime is a function of x and y,"},{"Start":"05:22.400 ","End":"05:25.700","Text":"and I\u0027m going to use either form. Change your mind."},{"Start":"05:25.700 ","End":"05:33.250","Text":"I\u0027ll just put it as a little note that y prime is actually a function of both x and y."},{"Start":"05:33.250 ","End":"05:41.960","Text":"Now what we need is y prime at our particular point and our particular point is a, a."},{"Start":"05:41.960 ","End":"05:46.010","Text":"So y prime of a,"},{"Start":"05:46.010 ","End":"05:49.425","Text":"a is equal to,"},{"Start":"05:49.425 ","End":"05:51.195","Text":"and I\u0027ll look at the formula,"},{"Start":"05:51.195 ","End":"05:52.630","Text":"I\u0027ll use this 1,"},{"Start":"05:52.630 ","End":"05:58.760","Text":"minus the square root of a over a and a is not 0."},{"Start":"05:58.760 ","End":"06:02.570","Text":"I already said that a is bigger than 0."},{"Start":"06:02.570 ","End":"06:05.645","Text":"We said that just above somewhere,"},{"Start":"06:05.645 ","End":"06:07.325","Text":"a is bigger than 0,"},{"Start":"06:07.325 ","End":"06:08.930","Text":"so no problem there,"},{"Start":"06:08.930 ","End":"06:11.560","Text":"so this is equal to minus 1."},{"Start":"06:11.560 ","End":"06:14.345","Text":"Now that this minus 1, as I said,"},{"Start":"06:14.345 ","End":"06:21.295","Text":"geometrically means this is the slope of the tangent at a,"},{"Start":"06:21.295 ","End":"06:24.845","Text":"a, the derivative is the slope."},{"Start":"06:24.845 ","End":"06:27.695","Text":"I\u0027m just summarizing the info we\u0027ve collected."},{"Start":"06:27.695 ","End":"06:30.980","Text":"It goes through the point a,"},{"Start":"06:30.980 ","End":"06:36.565","Text":"a and has slope minus 1."},{"Start":"06:36.565 ","End":"06:38.885","Text":"Now how do we get the equation of that?"},{"Start":"06:38.885 ","End":"06:42.230","Text":"Well, the tangent is a line and there\u0027s an equation in"},{"Start":"06:42.230 ","End":"06:46.720","Text":"analytical geometry for the line passing through a point with a given slope."},{"Start":"06:46.720 ","End":"06:48.830","Text":"I\u0027ll write that down the side here."},{"Start":"06:48.830 ","End":"06:55.310","Text":"The formula for a line through a point x_1,"},{"Start":"06:55.310 ","End":"06:58.695","Text":"y_1, and slope,"},{"Start":"06:58.695 ","End":"07:03.265","Text":"let\u0027s call it m, that\u0027s the letter I\u0027ve seen most often used for slopes,"},{"Start":"07:03.265 ","End":"07:12.515","Text":"and the slope equals m. That formula is that y minus y_1 is equal to m,"},{"Start":"07:12.515 ","End":"07:15.970","Text":"the slope, times x minus x_1."},{"Start":"07:15.970 ","End":"07:17.570","Text":"Let\u0027s do that here,"},{"Start":"07:17.570 ","End":"07:19.100","Text":"I mean, we have everything."},{"Start":"07:19.100 ","End":"07:21.515","Text":"We have x_1 and y_1,"},{"Start":"07:21.515 ","End":"07:23.210","Text":"so in other words,"},{"Start":"07:23.210 ","End":"07:27.055","Text":"in this formula, this will be x_1,"},{"Start":"07:27.055 ","End":"07:29.325","Text":"this will be y_1,"},{"Start":"07:29.325 ","End":"07:31.095","Text":"and this will be m,"},{"Start":"07:31.095 ","End":"07:35.570","Text":"and all I have to do is plug this in this green formula,"},{"Start":"07:35.570 ","End":"07:42.170","Text":"and then what we get is y minus a is"},{"Start":"07:42.170 ","End":"07:49.715","Text":"equal to minus 1 times x minus the other a, which is still a."},{"Start":"07:49.715 ","End":"07:53.570","Text":"That\u0027s basically the answer except the cleaning up a bit."},{"Start":"07:53.570 ","End":"08:02.570","Text":"We\u0027ll just clean up a bit and say that y equals minus x plus a,"},{"Start":"08:02.570 ","End":"08:03.920","Text":"because it\u0027s minus times minus,"},{"Start":"08:03.920 ","End":"08:06.200","Text":"and plus another a when moving this over,"},{"Start":"08:06.200 ","End":"08:09.700","Text":"so it\u0027s minus x plus 2a."},{"Start":"08:09.700 ","End":"08:11.940","Text":"That\u0027s the equation of that tangent,"},{"Start":"08:11.940 ","End":"08:14.440","Text":"and we are done."}],"ID":4398}],"Thumbnail":null,"ID":1666},{"Name":"Tangent and Normal lines - Parametric Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Parametric Functions","Duration":"3m 41s","ChapterTopicVideoID":8436,"CourseChapterTopicPlaylistID":1668,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This time we\u0027re going to talk about another curve."},{"Start":"00:02.820 ","End":"00:05.985","Text":"We already had implicit functions."},{"Start":"00:05.985 ","End":"00:07.785","Text":"Now we have parametric."},{"Start":"00:07.785 ","End":"00:13.800","Text":"That\u0027s when we have both x and y as functions of another variable,"},{"Start":"00:13.800 ","End":"00:18.120","Text":"usually t. Let\u0027s say I have x equals some function of"},{"Start":"00:18.120 ","End":"00:22.650","Text":"t and y is another function of t. To find the tangent and normal,"},{"Start":"00:22.650 ","End":"00:26.125","Text":"I\u0027m going to need the derivative y prime."},{"Start":"00:26.125 ","End":"00:31.070","Text":"Only y prime doesn\u0027t quite make sense because y is a function of t,"},{"Start":"00:31.070 ","End":"00:34.085","Text":"but I also want to look at it as a function of x."},{"Start":"00:34.085 ","End":"00:38.210","Text":"I would say that the derivative is y prime with little x."},{"Start":"00:38.210 ","End":"00:48.185","Text":"The formula for this is that it\u0027s equal to y prime with respect to t. That\u0027s g prime of t"},{"Start":"00:48.185 ","End":"00:54.545","Text":"over x prime with respect to t. Or if you have the names of the functions that would be"},{"Start":"00:54.545 ","End":"01:02.780","Text":"g prime of t over f prime of t. There\u0027s also the Leibnitz notation."},{"Start":"01:02.780 ","End":"01:04.340","Text":"This is the Newton notation."},{"Start":"01:04.340 ","End":"01:06.710","Text":"If we write the derivative as"},{"Start":"01:06.710 ","End":"01:10.850","Text":"dy over dx and then there\u0027s no confusion as y with respect to x."},{"Start":"01:10.850 ","End":"01:14.425","Text":"It turns out that this is equal to dy over"},{"Start":"01:14.425 ","End":"01:19.790","Text":"dt over dx by dt and it cancels as if it was a fraction."},{"Start":"01:19.790 ","End":"01:22.310","Text":"That\u0027s 1 of the nice things about this notation."},{"Start":"01:22.310 ","End":"01:28.950","Text":"Let\u0027s take a specific example where I have x equals 2t,"},{"Start":"01:28.950 ","End":"01:32.430","Text":"y equals t squared,"},{"Start":"01:32.430 ","End":"01:41.735","Text":"and I want both the tangent and the normal when t is equal to 1, for example."},{"Start":"01:41.735 ","End":"01:46.550","Text":"That means that x is 2 and y is 1."},{"Start":"01:46.550 ","End":"01:48.875","Text":"To be more precise the point 2, 1,"},{"Start":"01:48.875 ","End":"01:51.320","Text":"but often we just give the value of the parameter."},{"Start":"01:51.320 ","End":"01:53.495","Text":"What we want is the derivative,"},{"Start":"01:53.495 ","End":"01:57.350","Text":"which is y derivative with respect to x,"},{"Start":"01:57.350 ","End":"02:00.860","Text":"it\u0027s a curve, but you look at it as if y was a function of x."},{"Start":"02:00.860 ","End":"02:05.225","Text":"This is equal to the derivative of y."},{"Start":"02:05.225 ","End":"02:09.435","Text":"In other words, is a derivative of t squared, which is 2t."},{"Start":"02:09.435 ","End":"02:14.260","Text":"The derivative of 2t is just 2."},{"Start":"02:14.260 ","End":"02:17.290","Text":"This is the derivative of t squared,"},{"Start":"02:17.290 ","End":"02:20.955","Text":"is derivative of 2t, that\u0027s how I get it."},{"Start":"02:20.955 ","End":"02:28.970","Text":"What I get is that this is equal to 2t over 2 is just t. When t equals 1,"},{"Start":"02:28.970 ","End":"02:35.240","Text":"I get that this y prime with respect to x is just equal to 1,"},{"Start":"02:35.240 ","End":"02:40.390","Text":"and that is the slope of the tangent."},{"Start":"02:40.390 ","End":"02:49.055","Text":"The slope of the normal at that point is going to be equal to minus 1 over 1,"},{"Start":"02:49.055 ","End":"02:50.645","Text":"which is minus 1."},{"Start":"02:50.645 ","End":"02:54.900","Text":"In both cases, the point is 2, 1."},{"Start":"02:54.900 ","End":"02:57.665","Text":"The equation of the tangent,"},{"Start":"02:57.665 ","End":"03:01.625","Text":"just using the regular formula for our point in slope,"},{"Start":"03:01.625 ","End":"03:07.130","Text":"would be y minus the y of the point equals the slope,"},{"Start":"03:07.130 ","End":"03:12.005","Text":"which is 1 times x minus the x of the point."},{"Start":"03:12.005 ","End":"03:13.520","Text":"As for the normal,"},{"Start":"03:13.520 ","End":"03:16.640","Text":"I just used the negative reciprocal slope, this 1."},{"Start":"03:16.640 ","End":"03:19.025","Text":"I also get y minus 1,"},{"Start":"03:19.025 ","End":"03:20.210","Text":"x minus 2,"},{"Start":"03:20.210 ","End":"03:23.370","Text":"but instead of 1, I have minus 1."},{"Start":"03:23.370 ","End":"03:29.180","Text":"If I just simplify this comes out to be y equals x minus 1."},{"Start":"03:29.180 ","End":"03:35.840","Text":"The normal comes out to be y equals minus x plus 3."},{"Start":"03:35.840 ","End":"03:37.430","Text":"That\u0027s the answer for the tangent,"},{"Start":"03:37.430 ","End":"03:42.210","Text":"that\u0027s the answer for the normal, and we\u0027re done."}],"ID":8633},{"Watched":false,"Name":"Exercise 1","Duration":"6m 44s","ChapterTopicVideoID":1800,"CourseChapterTopicPlaylistID":1668,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.210","Text":"In this exercise, we\u0027re given a curve in parametric form where both x and y are functions"},{"Start":"00:06.210 ","End":"00:13.090","Text":"of t. We get some curve in the plane."},{"Start":"00:13.550 ","End":"00:15.570","Text":"In most cases,"},{"Start":"00:15.570 ","End":"00:16.590","Text":"at any given point,"},{"Start":"00:16.590 ","End":"00:20.970","Text":"we could take y as a function of x, look at it as a graph."},{"Start":"00:20.970 ","End":"00:24.555","Text":"Unless it\u0027s vertical, something like this,"},{"Start":"00:24.555 ","End":"00:27.150","Text":"then you couldn\u0027t get y as a function of x,"},{"Start":"00:27.150 ","End":"00:30.509","Text":"especially near a point where the tangent is vertical."},{"Start":"00:30.509 ","End":"00:40.375","Text":"Turns out, this behaves like a fraction that this is dy by dt over dx by dt."},{"Start":"00:40.375 ","End":"00:45.050","Text":"I could rewrite this in newton form y-prime."},{"Start":"00:45.050 ","End":"00:46.250","Text":"Only with y-prime,"},{"Start":"00:46.250 ","End":"00:48.665","Text":"you can get confused because y is also a function of"},{"Start":"00:48.665 ","End":"00:51.590","Text":"t. You might want to write a little x here."},{"Start":"00:51.590 ","End":"00:56.245","Text":"You can even omit the prime and it will be understood, will equal."},{"Start":"00:56.245 ","End":"00:59.630","Text":"I could say y prime with respect to t,"},{"Start":"00:59.630 ","End":"01:01.970","Text":"but we have already the name of the function."},{"Start":"01:01.970 ","End":"01:08.120","Text":"It\u0027s g prime of t over"},{"Start":"01:08.120 ","End":"01:15.435","Text":"f prime of t. We\u0027ll use this in both parts."},{"Start":"01:15.435 ","End":"01:20.035","Text":"There\u0027s 2 separate parametric forms."},{"Start":"01:20.035 ","End":"01:22.605","Text":"2 questions, 1 and 2,"},{"Start":"01:22.605 ","End":"01:29.840","Text":"and that\u0027s 2 questions and then we have 2 more because part B gives us also to"},{"Start":"01:29.840 ","End":"01:37.385","Text":"find the equation of the tangent line when t is 2 for each of these parametric curves."},{"Start":"01:37.385 ","End":"01:44.450","Text":"We\u0027ll start with 1."},{"Start":"01:44.450 ","End":"01:55.020","Text":"In 1, I\u0027ll just label this f of t and this g of t. I can just use the formula."},{"Start":"01:55.020 ","End":"02:01.110","Text":"In question 1, let\u0027s see 1a,"},{"Start":"02:01.110 ","End":"02:05.250","Text":"we have that dy by dx,"},{"Start":"02:05.250 ","End":"02:08.660","Text":"just giving it in the original form it was requested,"},{"Start":"02:08.660 ","End":"02:11.300","Text":"is equal to, I can just go to here,"},{"Start":"02:11.300 ","End":"02:18.845","Text":"g prime of t over f prime of t. G prime is"},{"Start":"02:18.845 ","End":"02:28.880","Text":"8t plus 4 the derivative of this divided by f prime of t is, okay."},{"Start":"02:28.880 ","End":"02:30.995","Text":"Now I have a square root."},{"Start":"02:30.995 ","End":"02:35.795","Text":"Now in general, the derivative of the square root of some function of x,"},{"Start":"02:35.795 ","End":"02:38.080","Text":"that\u0027s a square root of box."},{"Start":"02:38.080 ","End":"02:39.920","Text":"I want to differentiate it."},{"Start":"02:39.920 ","End":"02:42.560","Text":"We start off with the derivative of the square root,"},{"Start":"02:42.560 ","End":"02:46.895","Text":"which is 1 over twice the square root of that thing."},{"Start":"02:46.895 ","End":"02:48.980","Text":"But then we need the inner derivative."},{"Start":"02:48.980 ","End":"02:52.415","Text":"So we have to multiply by box prime,"},{"Start":"02:52.415 ","End":"02:55.775","Text":"and often we just put it on the numerator."},{"Start":"02:55.775 ","End":"03:00.035","Text":"I\u0027ll just make this 1 into a box and then prime."},{"Start":"03:00.035 ","End":"03:02.045","Text":"If I use that here,"},{"Start":"03:02.045 ","End":"03:05.210","Text":"then the derivative of f of t would be on"},{"Start":"03:05.210 ","End":"03:09.920","Text":"the denominator twice the square root of the box,"},{"Start":"03:09.920 ","End":"03:12.315","Text":"2t squared plus 1."},{"Start":"03:12.315 ","End":"03:16.025","Text":"On the numerator, the derivative of box,"},{"Start":"03:16.025 ","End":"03:24.420","Text":"which was 2t squared plus 1 is just for t. This is the derivative,"},{"Start":"03:24.420 ","End":"03:26.635","Text":"we just have to simplify it a bit."},{"Start":"03:26.635 ","End":"03:29.770","Text":"I could cancel top and bottom here by 4."},{"Start":"03:29.770 ","End":"03:31.660","Text":"If I cancel this full,"},{"Start":"03:31.660 ","End":"03:37.160","Text":"I can make this just 2t plus 1,"},{"Start":"03:37.370 ","End":"03:42.380","Text":"then I can bring the denominator of the denominator into the numerator."},{"Start":"03:42.380 ","End":"03:45.330","Text":"I get that, see, the 2 here."},{"Start":"03:45.330 ","End":"03:48.125","Text":"Then I\u0027ll write the 2t plus 1,"},{"Start":"03:48.125 ","End":"03:51.880","Text":"then the square root of 2 t squared plus 1,"},{"Start":"03:51.880 ","End":"03:56.340","Text":"and then a t in the denominator."},{"Start":"03:56.340 ","End":"04:00.220","Text":"Well, that\u0027s part a."},{"Start":"04:00.590 ","End":"04:05.220","Text":"Now let\u0027s get on to part b."},{"Start":"04:05.220 ","End":"04:11.600","Text":"We want an equation of the tangent line hen t is 2,"},{"Start":"04:11.600 ","End":"04:14.815","Text":"just make a note, this is part B."},{"Start":"04:14.815 ","End":"04:17.125","Text":"When t equals 2,"},{"Start":"04:17.125 ","End":"04:24.230","Text":"what we get is that I\u0027ve lost my f and g. Here they are, sorry."},{"Start":"04:24.230 ","End":"04:29.000","Text":"We get that x is equal to"},{"Start":"04:29.040 ","End":"04:37.120","Text":"2 squared is 4 times 2 is 8 plus 1 is 9."},{"Start":"04:37.120 ","End":"04:41.830","Text":"Square root of 9 is 3 and y is equal"},{"Start":"04:41.830 ","End":"04:47.985","Text":"to 4 times 4 is 16,"},{"Start":"04:47.985 ","End":"04:52.004","Text":"plus 4 times 2 is 8."},{"Start":"04:52.004 ","End":"04:56.950","Text":"That\u0027s 24 plus 1 is 25."},{"Start":"04:57.500 ","End":"05:02.945","Text":"I also need the derivative,"},{"Start":"05:02.945 ","End":"05:08.430","Text":"which is actually going to be the slope of the tangent line."},{"Start":"05:08.430 ","End":"05:12.875","Text":"I\u0027ll just write it as saying that m, the derivative,"},{"Start":"05:12.875 ","End":"05:15.830","Text":"which is dy by dx,"},{"Start":"05:15.830 ","End":"05:18.935","Text":"is equal to, from here,"},{"Start":"05:18.935 ","End":"05:20.990","Text":"when t is 2,"},{"Start":"05:20.990 ","End":"05:23.180","Text":"I\u0027ve got twice,"},{"Start":"05:23.180 ","End":"05:26.930","Text":"and then 2 t plus 1 is 5,"},{"Start":"05:26.930 ","End":"05:32.270","Text":"and then square root,2t squared plus 1 square root."},{"Start":"05:32.270 ","End":"05:38.055","Text":"Didn\u0027t we already compute that and got that as 3."},{"Start":"05:38.055 ","End":"05:40.490","Text":"No need to do the work again,"},{"Start":"05:40.490 ","End":"05:42.050","Text":"that\u0027s times 3,"},{"Start":"05:42.050 ","End":"05:46.610","Text":"and then T, which is 2, the 2s cancel."},{"Start":"05:46.610 ","End":"05:52.985","Text":"So this comes out to be 5 times 3 is 15."},{"Start":"05:52.985 ","End":"05:56.360","Text":"Now we have a point and a slope,"},{"Start":"05:56.360 ","End":"06:00.920","Text":"so we use the point-slope form of a line,"},{"Start":"06:00.920 ","End":"06:02.860","Text":"and we get that,"},{"Start":"06:02.860 ","End":"06:06.920","Text":"I\u0027ll use a different y for the line, for the tangent."},{"Start":"06:06.920 ","End":"06:11.974","Text":"I\u0027ll make it a capital Y. Y minus the y of the point,"},{"Start":"06:11.974 ","End":"06:21.320","Text":"which is 25, is equal to the slope 15 times x minus the x of the point."},{"Start":"06:21.320 ","End":"06:26.905","Text":"That would be the tangent line."},{"Start":"06:26.905 ","End":"06:28.910","Text":"You could simplify it."},{"Start":"06:28.910 ","End":"06:33.095","Text":"You could bring everything to open the brackets and put everything on the right,"},{"Start":"06:33.095 ","End":"06:44.100","Text":"and we would get y equals 15x minus 45 plus 25 is minus 20."}],"ID":1812},{"Watched":false,"Name":"Exercise 2","Duration":"6m 55s","ChapterTopicVideoID":1801,"CourseChapterTopicPlaylistID":1668,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"We just did the first 1 of 2."},{"Start":"00:03.105 ","End":"00:05.370","Text":"Now we\u0027ll come to part 2."},{"Start":"00:05.370 ","End":"00:07.455","Text":"In part 1,"},{"Start":"00:07.455 ","End":"00:14.580","Text":"we said that this is equal to g prime of t over f prime of t."},{"Start":"00:14.580 ","End":"00:20.810","Text":"This, in our case, is our f of t and this is our g of t."},{"Start":"00:20.810 ","End":"00:24.675","Text":"We need to do some differentiation."},{"Start":"00:24.675 ","End":"00:27.960","Text":"Let me write these as follows."},{"Start":"00:27.960 ","End":"00:33.405","Text":"I\u0027ll write f of t as something to the power of 1/2,"},{"Start":"00:33.405 ","End":"00:38.175","Text":"2t cubed plus 5t squared to the power of 1/2,"},{"Start":"00:38.175 ","End":"00:44.175","Text":"and g of t is 4t to the power of 1/3."},{"Start":"00:44.175 ","End":"00:47.580","Text":"Here we used the formula for square root in the previous exercise."},{"Start":"00:47.580 ","End":"00:52.855","Text":"For the consistency, I\u0027m going to make them both the fractional exponents."},{"Start":"00:52.855 ","End":"00:59.880","Text":"Now I get that f prime of t is equal"},{"Start":"00:59.880 ","End":"01:07.940","Text":"to 1/2 of this thing to the power of minus 1/2."},{"Start":"01:07.940 ","End":"01:15.665","Text":"I\u0027m just copying it now, 2t cubed plus 5t squared times the inner derivative,"},{"Start":"01:15.665 ","End":"01:21.570","Text":"which is going to be 3 times 2 is 6t squared."},{"Start":"01:22.480 ","End":"01:26.690","Text":"Then 2 times 5 is 10 plus 10t,"},{"Start":"01:26.690 ","End":"01:29.015","Text":"and I need brackets,"},{"Start":"01:29.015 ","End":"01:36.250","Text":"and g prime of t is equal to, this time I have 1/3,"},{"Start":"01:36.250 ","End":"01:44.335","Text":"so it\u0027s 1/3 times 4t to the power of minus 2/3 if I subtract 1 from here,"},{"Start":"01:44.335 ","End":"01:48.385","Text":"and the inner derivative is just 4."},{"Start":"01:48.385 ","End":"01:54.400","Text":"I have that dy by dx is,"},{"Start":"01:54.400 ","End":"01:56.335","Text":"according to this formula,"},{"Start":"01:56.335 ","End":"02:00.025","Text":"the g prime over f prime."},{"Start":"02:00.025 ","End":"02:03.190","Text":"I\u0027ll write it as a fraction."},{"Start":"02:03.190 ","End":"02:05.715","Text":"Now, here, I have a 1/3,"},{"Start":"02:05.715 ","End":"02:07.380","Text":"so I\u0027ll put the 3 in the bottom,"},{"Start":"02:07.380 ","End":"02:09.540","Text":"but I\u0027ll leave the 4 at the top."},{"Start":"02:09.540 ","End":"02:11.040","Text":"The negative power,"},{"Start":"02:11.040 ","End":"02:13.425","Text":"I prefer to put it in the denominator."},{"Start":"02:13.425 ","End":"02:18.810","Text":"I\u0027ll put 4t to the positive 2/3 here."},{"Start":"02:18.810 ","End":"02:21.710","Text":"f prime of t is in the denominator,"},{"Start":"02:21.710 ","End":"02:23.555","Text":"so I have to reverse everything."},{"Start":"02:23.555 ","End":"02:27.075","Text":"The 1/2 becomes a 2,"},{"Start":"02:27.075 ","End":"02:31.555","Text":"the minus 1/2 becomes plus 1/2."},{"Start":"02:31.555 ","End":"02:39.185","Text":"I\u0027ve got 2t cubed plus 5t squared to the power of plus 1/2."},{"Start":"02:39.185 ","End":"02:41.765","Text":"This one goes in the denominator,"},{"Start":"02:41.765 ","End":"02:45.720","Text":"6t squared plus 10t."},{"Start":"02:46.540 ","End":"02:50.080","Text":"It looks a mess, but not so bad."},{"Start":"02:50.080 ","End":"02:54.510","Text":"We would like this when t equals 2."},{"Start":"02:54.510 ","End":"02:58.760","Text":"Let\u0027s just substitute t equals 2 here and see what we get,"},{"Start":"02:58.760 ","End":"03:01.360","Text":"4 times 2 is 8."},{"Start":"03:01.360 ","End":"03:09.185","Text":"Then t cubed is 8 times 2 is 16."},{"Start":"03:09.185 ","End":"03:14.120","Text":"16 plus, this comes out 20, 36."},{"Start":"03:14.120 ","End":"03:21.960","Text":"I\u0027ve got times 36 to the 1/2 over 3."},{"Start":"03:21.960 ","End":"03:24.960","Text":"Now, 4t is 8."},{"Start":"03:24.960 ","End":"03:28.710","Text":"8 to the power of 2/3 is,"},{"Start":"03:28.710 ","End":"03:31.080","Text":"you can take it as 8 squared then the cube root"},{"Start":"03:31.080 ","End":"03:34.980","Text":"or cube root first and then squared, you get 4."},{"Start":"03:34.980 ","End":"03:43.245","Text":"Here, t squared is 4 times 6 is 24,"},{"Start":"03:43.245 ","End":"03:52.620","Text":"plus 10 times 2 is 20, so that\u0027s 44."},{"Start":"03:52.620 ","End":"03:57.285","Text":"So 3 times 4 times 44."},{"Start":"03:57.285 ","End":"04:00.860","Text":"Let\u0027s see if we can simplify the square root."},{"Start":"04:00.860 ","End":"04:03.270","Text":"I mean, to the power of 1/2 is a square root."},{"Start":"04:03.270 ","End":"04:04.380","Text":"This thing is,"},{"Start":"04:04.380 ","End":"04:07.450","Text":"square root of 36 is 6."},{"Start":"04:07.970 ","End":"04:11.115","Text":"Now, let\u0027s see what else."},{"Start":"04:11.115 ","End":"04:15.970","Text":"3 into 6 goes twice."},{"Start":"04:18.200 ","End":"04:23.670","Text":"Then 4 into 8 goes twice."},{"Start":"04:23.670 ","End":"04:27.240","Text":"Now I have 2 times 2 is 4."},{"Start":"04:27.240 ","End":"04:32.560","Text":"4 into 44 goes 11."},{"Start":"04:34.490 ","End":"04:38.330","Text":"What we have left on the denominator is 11."},{"Start":"04:38.330 ","End":"04:39.410","Text":"If you have nothing left,"},{"Start":"04:39.410 ","End":"04:40.880","Text":"that means it\u0027s a 1."},{"Start":"04:40.880 ","End":"04:43.800","Text":"So we get 1 over 11."},{"Start":"04:44.050 ","End":"04:47.930","Text":"This 1/11th, the derivative is going to be the slope."},{"Start":"04:47.930 ","End":"04:52.340","Text":"I\u0027ll just call it m. Now we want the point which t equals to"},{"Start":"04:52.340 ","End":"04:59.655","Text":"the point is x, y. I need x to be f of 2,"},{"Start":"04:59.655 ","End":"05:05.170","Text":"and f of 2 is what happens when I put 2 in here."},{"Start":"05:05.420 ","End":"05:08.850","Text":"2t cubed plus 5t squared,"},{"Start":"05:08.850 ","End":"05:11.355","Text":"square root, I think we\u0027ve done that here."},{"Start":"05:11.355 ","End":"05:18.015","Text":"This came out to be 36, and it\u0027s 6."},{"Start":"05:18.015 ","End":"05:19.410","Text":"We can compute it again."},{"Start":"05:19.410 ","End":"05:23.085","Text":"I mean, 2 cubed is 8 times 2 is 16."},{"Start":"05:23.085 ","End":"05:24.600","Text":"Here, 5 times 4 is 20,"},{"Start":"05:24.600 ","End":"05:27.730","Text":"36 square root is 6."},{"Start":"05:28.030 ","End":"05:32.775","Text":"Now y is g of 2."},{"Start":"05:32.775 ","End":"05:35.180","Text":"Again, we\u0027ve done this before somewhere."},{"Start":"05:35.180 ","End":"05:36.455","Text":"Well, maybe not quite."},{"Start":"05:36.455 ","End":"05:40.115","Text":"But anyway, 4 times 2 is 8,"},{"Start":"05:40.115 ","End":"05:42.305","Text":"cube root of 8 is 2."},{"Start":"05:42.305 ","End":"05:43.715","Text":"So that\u0027s 2."},{"Start":"05:43.715 ","End":"05:47.555","Text":"Now we have a point 6, 2"},{"Start":"05:47.555 ","End":"05:51.710","Text":"and a slope which is 1/11,"},{"Start":"05:51.710 ","End":"05:54.410","Text":"that\u0027s the m. We have a standard formula"},{"Start":"05:54.410 ","End":"05:57.710","Text":"for equation of a line through a point with a slope."},{"Start":"05:57.710 ","End":"06:01.440","Text":"I\u0027ll use different x and y. Big X and Y."},{"Start":"06:02.080 ","End":"06:07.879","Text":"We start with Y minus the y of the point, which is 2,"},{"Start":"06:07.879 ","End":"06:17.735","Text":"equals the slope 1/11th times X minus the x of the point."},{"Start":"06:17.735 ","End":"06:20.035","Text":"We could leave it like this,"},{"Start":"06:20.035 ","End":"06:22.750","Text":"I prefer to write it as Y in terms of x."},{"Start":"06:22.750 ","End":"06:33.450","Text":"Y equals 1/11th x minus 6 over 11 plus 2."},{"Start":"06:33.450 ","End":"06:35.430","Text":"2 is 22 over 11,"},{"Start":"06:35.430 ","End":"06:37.530","Text":"22 minus 6 is,"},{"Start":"06:37.530 ","End":"06:39.870","Text":"what is 22 minus 6?"},{"Start":"06:39.870 ","End":"06:48.450","Text":"16. It\u0027s minus 16/11."},{"Start":"06:48.450 ","End":"06:51.130","Text":"That would be our answer."},{"Start":"06:51.890 ","End":"06:55.300","Text":"We\u0027re done with part 2."}],"ID":1813}],"Thumbnail":null,"ID":1668},{"Name":"The Angle Between Two Curves","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Angle between Two Curves","Duration":"5m 58s","ChapterTopicVideoID":8261,"CourseChapterTopicPlaylistID":1669,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this clip, we\u0027re going to be talking about the angle between 2 curves,"},{"Start":"00:03.690 ","End":"00:08.565","Text":"which is very much related to the tangent to a curve at a point."},{"Start":"00:08.565 ","End":"00:10.905","Text":"What is the angle between 2 curves?"},{"Start":"00:10.905 ","End":"00:13.200","Text":"I\u0027ll bring in a little sketch I made,"},{"Start":"00:13.200 ","End":"00:14.820","Text":"here are 2 curves,"},{"Start":"00:14.820 ","End":"00:16.950","Text":"they intersect at some point."},{"Start":"00:16.950 ","End":"00:20.805","Text":"I want to know what is the angle between these 2 curves at this point."},{"Start":"00:20.805 ","End":"00:23.475","Text":"I draw in the tangent lines,"},{"Start":"00:23.475 ","End":"00:25.920","Text":"here is the tangent to the green curve,"},{"Start":"00:25.920 ","End":"00:28.815","Text":"and here\u0027s the tangent to the blue curve."},{"Start":"00:28.815 ","End":"00:31.980","Text":"What I need is the angle between the 2 tangents,"},{"Start":"00:31.980 ","End":"00:35.200","Text":"and that\u0027s defined to be the angle between the curves."},{"Start":"00:35.200 ","End":"00:40.525","Text":"But I have to remind you what is the angle between 2 lines, there\u0027s a formula."},{"Start":"00:40.525 ","End":"00:48.390","Text":"Suppose 1 of the lines is y equals ax and generally x plus b,"},{"Start":"00:48.390 ","End":"00:50.340","Text":"I better make it a_1 and b_1,"},{"Start":"00:50.340 ","End":"00:55.380","Text":"and the other line will be a_2x plus b2,"},{"Start":"00:55.380 ","End":"00:58.860","Text":"let say the angle between them is Alpha,"},{"Start":"00:58.860 ","End":"01:02.090","Text":"then I have a formula not for Alpha exactly,"},{"Start":"01:02.090 ","End":"01:03.895","Text":"but for tangent Alpha."},{"Start":"01:03.895 ","End":"01:08.360","Text":"Of course, we can afterwards take the arc tangent or the inverse tangent,"},{"Start":"01:08.360 ","End":"01:13.880","Text":"and it only relates to the slopes to a_1 and a_2,"},{"Start":"01:13.880 ","End":"01:15.560","Text":"the b\u0027s don\u0027t come into it."},{"Start":"01:15.560 ","End":"01:24.660","Text":"It\u0027s a_1 minus a_2 over 1 plus a_1, a_2."},{"Start":"01:24.660 ","End":"01:26.550","Text":"See when 2 lines intersect,"},{"Start":"01:26.550 ","End":"01:28.615","Text":"there\u0027s actually 4 angles here."},{"Start":"01:28.615 ","End":"01:31.640","Text":"These 2 are the same and these 2 are the same."},{"Start":"01:31.640 ","End":"01:36.965","Text":"But the smaller of the angles is the 1 that\u0027s less than 90 degrees."},{"Start":"01:36.965 ","End":"01:38.930","Text":"That\u0027s the 1 I\u0027ll call Alpha,"},{"Start":"01:38.930 ","End":"01:40.415","Text":"and this is also Alpha."},{"Start":"01:40.415 ","End":"01:45.080","Text":"What we have to do to make sure that we get the smaller of the 2 to use that convention"},{"Start":"01:45.080 ","End":"01:50.420","Text":"is to put absolute value because when the angle is less than 90 degrees,"},{"Start":"01:50.420 ","End":"01:52.085","Text":"the tangent is positive."},{"Start":"01:52.085 ","End":"01:55.825","Text":"So if we put the absolute value we\u0027re guaranteed to get the smaller 1."},{"Start":"01:55.825 ","End":"01:59.810","Text":"In the example, I need to give you 2 curves,"},{"Start":"01:59.810 ","End":"02:02.210","Text":"I\u0027ll make it easy I\u0027ll give it an explicit form,"},{"Start":"02:02.210 ","End":"02:03.590","Text":"y in terms of x."},{"Start":"02:03.590 ","End":"02:05.930","Text":"Let\u0027s say I have the 2 curves,"},{"Start":"02:05.930 ","End":"02:09.980","Text":"y equals x squared and the other curve,"},{"Start":"02:09.980 ","End":"02:14.320","Text":"y equals 3x minus 2."},{"Start":"02:14.320 ","End":"02:17.840","Text":"The first thing to do is to find the point of intersection."},{"Start":"02:17.840 ","End":"02:20.299","Text":"To get the intersection, I would just compare,"},{"Start":"02:20.299 ","End":"02:23.105","Text":"solve 2 equations and look for x,"},{"Start":"02:23.105 ","End":"02:28.330","Text":"I just get x squared equals 3x minus 2,"},{"Start":"02:28.330 ","End":"02:31.520","Text":"from which we get a quadratic equation,"},{"Start":"02:31.520 ","End":"02:36.565","Text":"x squared minus 3x minus 2 equals 0."},{"Start":"02:36.565 ","End":"02:38.745","Text":"Use the formula on this,"},{"Start":"02:38.745 ","End":"02:43.220","Text":"and say x equals 3 plus or minus the square root of b squared is"},{"Start":"02:43.220 ","End":"02:51.245","Text":"9 minus 4 times 1 times minus 2 all over 2,"},{"Start":"02:51.245 ","End":"02:54.725","Text":"which gives us 3 plus or minus,"},{"Start":"02:54.725 ","End":"03:00.310","Text":"this is a plus 2 and therefore this is a plus 2,"},{"Start":"03:00.310 ","End":"03:04.470","Text":"and so under the square root I have 9 minus 8 which is 1,"},{"Start":"03:04.470 ","End":"03:07.760","Text":"so it\u0027s 3 plus or minus 1 over 2,"},{"Start":"03:07.760 ","End":"03:10.805","Text":"which gives me 4 over 2 is 2,"},{"Start":"03:10.805 ","End":"03:12.895","Text":"and 2 over 2 is 1."},{"Start":"03:12.895 ","End":"03:15.405","Text":"So actually I have 2 solutions,"},{"Start":"03:15.405 ","End":"03:18.330","Text":"what this means is that there\u0027s 2 different angles,"},{"Start":"03:18.330 ","End":"03:20.705","Text":"like 1 curve is a parabola,"},{"Start":"03:20.705 ","End":"03:26.550","Text":"the other one\u0027s a straight line and it intersects at 2 different points,"},{"Start":"03:26.550 ","End":"03:29.655","Text":"so there\u0027s going to be 2 different angles,"},{"Start":"03:29.655 ","End":"03:33.420","Text":"this 1 would correspond to x equals 1,"},{"Start":"03:33.420 ","End":"03:37.490","Text":"this would correspond to x equals 2."},{"Start":"03:37.490 ","End":"03:39.860","Text":"But I don\u0027t want to waste too much time,"},{"Start":"03:39.860 ","End":"03:41.510","Text":"let\u0027s just choose 1 of them."},{"Start":"03:41.510 ","End":"03:46.185","Text":"Let\u0027s just say that we want the x equals 2, just arbitrate,"},{"Start":"03:46.185 ","End":"03:48.620","Text":"choose 1 of them not to do too much work"},{"Start":"03:48.620 ","End":"03:51.350","Text":"and I\u0027ll leave you the other 1 to do on your own."},{"Start":"03:51.350 ","End":"03:53.105","Text":"Notice that in this formula,"},{"Start":"03:53.105 ","End":"03:55.655","Text":"I don\u0027t actually need the tangent lines."},{"Start":"03:55.655 ","End":"03:58.325","Text":"I only need the slopes,"},{"Start":"03:58.325 ","End":"04:01.925","Text":"the slope of the tangent is the derivative."},{"Start":"04:01.925 ","End":"04:04.895","Text":"So what I need to do is get the 2 derivatives,"},{"Start":"04:04.895 ","End":"04:08.330","Text":"for the first 1, I have y equals 2x."},{"Start":"04:08.330 ","End":"04:12.400","Text":"Second 1 the derivative y prime equals 3,"},{"Start":"04:12.400 ","End":"04:16.965","Text":"and so when x equals 2,"},{"Start":"04:16.965 ","End":"04:21.540","Text":"let\u0027s say this is the 1st curve and the 2nd curve."},{"Start":"04:21.540 ","End":"04:26.375","Text":"The a_1 is the slope of this 1 which would"},{"Start":"04:26.375 ","End":"04:31.130","Text":"be what happens when we let x equals 2 here into the derivative."},{"Start":"04:31.130 ","End":"04:34.670","Text":"Let me just say that it\u0027s equal to twice 2,"},{"Start":"04:34.670 ","End":"04:39.180","Text":"which is equal to 4 and a_2,"},{"Start":"04:39.180 ","End":"04:43.105","Text":"well, there\u0027s nothing to substitute so it\u0027s equal to 3."},{"Start":"04:43.105 ","End":"04:47.180","Text":"Now I can get that the tangent of the angle between"},{"Start":"04:47.180 ","End":"04:51.320","Text":"them is equal to absolute value of a_1,"},{"Start":"04:51.320 ","End":"05:00.815","Text":"which is 4 minus 3 over 1 plus 4 times 3,"},{"Start":"05:00.815 ","End":"05:05.120","Text":"and this comes out to be 1 over 13."},{"Start":"05:05.120 ","End":"05:08.480","Text":"On the calculator, I do the arctangent,"},{"Start":"05:08.480 ","End":"05:14.155","Text":"so it comes out approximately to 4.4 degrees,"},{"Start":"05:14.155 ","End":"05:15.875","Text":"and that would be our answer,"},{"Start":"05:15.875 ","End":"05:19.080","Text":"except if you want to also do it for x equals 1,"},{"Start":"05:19.080 ","End":"05:22.740","Text":"and then you would get a_1 equals 2 and a_2 equals 3,"},{"Start":"05:22.740 ","End":"05:24.720","Text":"and you\u0027d get very similar."},{"Start":"05:24.720 ","End":"05:27.275","Text":"I just wanted to give you an introduction."},{"Start":"05:27.275 ","End":"05:34.220","Text":"The main thing is that the angle between curves is the angle between the tangents."},{"Start":"05:34.220 ","End":"05:36.110","Text":"I should have really written that."},{"Start":"05:36.110 ","End":"05:37.610","Text":"Maybe it\u0027s not too late."},{"Start":"05:37.610 ","End":"05:42.410","Text":"The angle between 2 curves equals the angle between"},{"Start":"05:42.410 ","End":"05:48.295","Text":"the tangent at the point of intersection, of course."},{"Start":"05:48.295 ","End":"05:50.460","Text":"That\u0027s the general idea,"},{"Start":"05:50.460 ","End":"05:54.110","Text":"and then we have a formula for the angle between tangents because these are"},{"Start":"05:54.110 ","End":"05:59.280","Text":"lines and we only need the slopes. That\u0027s enough."}],"ID":8423},{"Watched":false,"Name":"Exercise 1","Duration":"8m 51s","ChapterTopicVideoID":1802,"CourseChapterTopicPlaylistID":1669,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.729","Text":"In this exercise, we\u0027re given 2 curves."},{"Start":"00:02.729 ","End":"00:05.220","Text":"Here\u0027s 1 and here\u0027s the other."},{"Start":"00:05.220 ","End":"00:06.495","Text":"We have to show,"},{"Start":"00:06.495 ","End":"00:10.575","Text":"well that they intersect at right angles and I\u0027ve"},{"Start":"00:10.575 ","End":"00:15.405","Text":"brought a little sketch with me first just to help you understand what\u0027s going on."},{"Start":"00:15.405 ","End":"00:20.110","Text":"Here\u0027s 1 of the curves it happens to be an ellipse,"},{"Start":"00:20.110 ","End":"00:22.490","Text":"here\u0027s the other curve."},{"Start":"00:22.490 ","End":"00:26.780","Text":"These are 2 branches of a hyperbola."},{"Start":"00:26.780 ","End":"00:29.070","Text":"I\u0027ve labeled them curve 2,"},{"Start":"00:29.070 ","End":"00:32.295","Text":"and here\u0027s a point of intersection,"},{"Start":"00:32.295 ","End":"00:36.230","Text":"and there are actually 4 but I think we\u0027ll just deal with"},{"Start":"00:36.230 ","End":"00:41.280","Text":"1 and then explain why we expect the other 3 to act the same."},{"Start":"00:41.560 ","End":"00:45.230","Text":"When we say they intersect at right angles,"},{"Start":"00:45.230 ","End":"00:49.340","Text":"we mean that the tangents at that point intersect at right angles to"},{"Start":"00:49.340 ","End":"00:54.080","Text":"this line and this line have to intersect at 90 degrees."},{"Start":"00:54.080 ","End":"00:55.970","Text":"In this simple way of testing for that,"},{"Start":"00:55.970 ","End":"00:58.310","Text":"we take the slope of this and the slope of this,"},{"Start":"00:58.310 ","End":"01:01.970","Text":"multiply them together and if we get minus 1, that\u0027s good."},{"Start":"01:01.970 ","End":"01:04.580","Text":"Anyway, I\u0027ll repeat all that later I just wanted to give you"},{"Start":"01:04.580 ","End":"01:07.480","Text":"a general idea what it looks like and what the strategy is going to be."},{"Start":"01:07.480 ","End":"01:09.230","Text":"We\u0027re going to find the intersection,"},{"Start":"01:09.230 ","End":"01:11.000","Text":"find the 2 tangents,"},{"Start":"01:11.000 ","End":"01:15.395","Text":"find the slopes, and then apply our formula for perpendicularity."},{"Start":"01:15.395 ","End":"01:19.535","Text":"Backup to the algebra and the hard work."},{"Start":"01:19.535 ","End":"01:22.610","Text":"Now, these are not regular functions,"},{"Start":"01:22.610 ","End":"01:24.400","Text":"these are implicit functions."},{"Start":"01:24.400 ","End":"01:30.005","Text":"In order to get slopes will need to use implicit differentiation."},{"Start":"01:30.005 ","End":"01:32.720","Text":"Before we do the implicit differentiation,"},{"Start":"01:32.720 ","End":"01:36.110","Text":"we might want to find the points of intersection we could do it in"},{"Start":"01:36.110 ","End":"01:39.950","Text":"either order let\u0027s do the points of intersection first."},{"Start":"01:39.950 ","End":"01:44.995","Text":"To find the points of intersection is like getting 2 equations in 2 unknowns."},{"Start":"01:44.995 ","End":"01:55.910","Text":"We have x squared plus 2y squared equals 8 and x squared minus y squared equals 2."},{"Start":"01:55.910 ","End":"01:59.735","Text":"This is like 2 equations in 2 unknowns,"},{"Start":"01:59.735 ","End":"02:02.600","Text":"linear equations if you look at x squared and y"},{"Start":"02:02.600 ","End":"02:06.485","Text":"squared units anyway, same techniques apply."},{"Start":"02:06.485 ","End":"02:08.765","Text":"Subtract the second from the first,"},{"Start":"02:08.765 ","End":"02:10.825","Text":"and I will get,"},{"Start":"02:10.825 ","End":"02:13.200","Text":"okay, just wrote it all out."},{"Start":"02:13.200 ","End":"02:18.530","Text":"These 2 together by some fraction give us this y squared therefore equals 2 by"},{"Start":"02:18.530 ","End":"02:23.510","Text":"division and then y must equal plus or minus the square root of 2."},{"Start":"02:23.510 ","End":"02:26.090","Text":"If I plug the y squared equals 2 in here,"},{"Start":"02:26.090 ","End":"02:28.855","Text":"I\u0027ve got x squared minus 2 equals 2, 2x squared equals 4."},{"Start":"02:28.855 ","End":"02:31.000","Text":"X is plus or minus 2."},{"Start":"02:31.000 ","End":"02:34.370","Text":"There are actually 4 points of intersection."},{"Start":"02:34.370 ","End":"02:36.490","Text":"We have, if we do it by the first,"},{"Start":"02:36.490 ","End":"02:38.675","Text":"second, third, fourth quadrants,"},{"Start":"02:38.675 ","End":"02:42.340","Text":"we have 2 square root of 2,"},{"Start":"02:42.340 ","End":"02:46.905","Text":"we have minus 2 square root of 2."},{"Start":"02:46.905 ","End":"02:48.545","Text":"In the third quadrant,"},{"Start":"02:48.545 ","End":"02:53.720","Text":"we have minus 2 minus square root of 2 and in the fourth quadrant we"},{"Start":"02:53.720 ","End":"02:59.465","Text":"have plus 2 and minus the square root of 2."},{"Start":"02:59.465 ","End":"03:01.205","Text":"Everything symmetrical."},{"Start":"03:01.205 ","End":"03:05.060","Text":"Next thing we have to do now that we found the points of intersection"},{"Start":"03:05.060 ","End":"03:08.720","Text":"is to find the slopes for these curves that we"},{"Start":"03:08.720 ","End":"03:12.350","Text":"can substitute and I think we\u0027ll probably just go with lighter"},{"Start":"03:12.350 ","End":"03:16.805","Text":"with this 1 of these points and say that similarly all the rest of them."},{"Start":"03:16.805 ","End":"03:19.345","Text":"First 1, the ellipse,"},{"Start":"03:19.345 ","End":"03:25.474","Text":"x squared plus 2y squared is equal to 8."},{"Start":"03:25.474 ","End":"03:30.830","Text":"Differentiating, we get 2x plus 4y and don\u0027t"},{"Start":"03:30.830 ","End":"03:35.930","Text":"forget the y prime equals 0 and if we divide this by 2,"},{"Start":"03:35.930 ","End":"03:39.840","Text":"we get x plus 2y,"},{"Start":"03:39.840 ","End":"03:45.469","Text":"y prime equals 0 and if isolate y prime,"},{"Start":"03:45.469 ","End":"03:46.700","Text":"the x over to the other side,"},{"Start":"03:46.700 ","End":"03:49.100","Text":"it\u0027s minus x over 2y."},{"Start":"03:49.100 ","End":"03:53.555","Text":"We have that y prime is equal to"},{"Start":"03:53.555 ","End":"04:01.385","Text":"minus x over 2y and that\u0027s for the first curve, the ellipse."},{"Start":"04:01.385 ","End":"04:03.740","Text":"Now y prime at this,"},{"Start":"04:03.740 ","End":"04:07.160","Text":"at our point, let\u0027s just take this 1 as our point."},{"Start":"04:07.160 ","End":"04:10.245","Text":"This will be the tangent point."},{"Start":"04:10.245 ","End":"04:11.990","Text":"At our tangent point,"},{"Start":"04:11.990 ","End":"04:16.280","Text":"y prime is equal to minus x,"},{"Start":"04:16.280 ","End":"04:20.995","Text":"which is minus 2 over 2y,"},{"Start":"04:20.995 ","End":"04:26.780","Text":"which is twice square root of 2 which just equals minus"},{"Start":"04:26.780 ","End":"04:33.670","Text":"1 over the square root of 2 and for the other curve,"},{"Start":"04:33.670 ","End":"04:40.575","Text":"we get, copy it x squared minus y squared equals 2."},{"Start":"04:40.575 ","End":"04:46.160","Text":"So 2x minus 2y times y prime don\u0027t forget"},{"Start":"04:46.160 ","End":"04:52.485","Text":"that is equal to 0 dividing by 2x minus y,"},{"Start":"04:52.485 ","End":"04:58.940","Text":"y prime equals 0 and then if I switch sides and divide,"},{"Start":"04:58.940 ","End":"05:05.570","Text":"you\u0027ll see that I get that y prime.Say I put the this over on this side and y,"},{"Start":"05:05.570 ","End":"05:06.875","Text":"y prime is x,"},{"Start":"05:06.875 ","End":"05:11.390","Text":"so y prime is x over y and for our particular curve,"},{"Start":"05:11.390 ","End":"05:12.935","Text":"which is the hyperbola,"},{"Start":"05:12.935 ","End":"05:16.340","Text":"we get that y prime is equal to,"},{"Start":"05:16.340 ","End":"05:20.600","Text":"we have x 2 over y,"},{"Start":"05:20.600 ","End":"05:26.915","Text":"which is the square root of 2 and this just equals the square root of 2."},{"Start":"05:26.915 ","End":"05:29.630","Text":"You can see that 2 is like the square root of"},{"Start":"05:29.630 ","End":"05:33.630","Text":"2 times the square root of 2 and anyway, this works out."},{"Start":"05:33.630 ","End":"05:36.500","Text":"This is the slope of the curve 1,"},{"Start":"05:36.500 ","End":"05:39.515","Text":"and this is this whole thing belongs to curve 1."},{"Start":"05:39.515 ","End":"05:43.180","Text":"All this belongs to curve 2,"},{"Start":"05:43.180 ","End":"05:47.625","Text":"that\u0027s the hyperbola and here we get slope 2,"},{"Start":"05:47.625 ","End":"05:51.710","Text":"this is actually the a that we would use in the tangent line,"},{"Start":"05:51.710 ","End":"05:54.635","Text":"but we don\u0027t need the tangent line itself,"},{"Start":"05:54.635 ","End":"05:56.854","Text":"we just need the slope."},{"Start":"05:56.854 ","End":"06:00.155","Text":"So just highlighting them."},{"Start":"06:00.155 ","End":"06:02.450","Text":"We\u0027ve got for the first curve,"},{"Start":"06:02.450 ","End":"06:05.790","Text":"the slope is minus 1 over the square root of 2"},{"Start":"06:05.790 ","End":"06:09.860","Text":"and for the second curve is just square root of 2."},{"Start":"06:09.860 ","End":"06:14.735","Text":"How can we tell when 2 line are perpendicular?"},{"Start":"06:14.735 ","End":"06:21.285","Text":"Let\u0027s say this is, lets give it a letter let\u0027s say this was a_1,"},{"Start":"06:21.285 ","End":"06:24.595","Text":"say, and this 1 is a_2."},{"Start":"06:24.595 ","End":"06:29.915","Text":"In general, lines are perpendicular if the product of the slopes is minus 1,"},{"Start":"06:29.915 ","End":"06:31.460","Text":"I have to check now,"},{"Start":"06:31.460 ","End":"06:37.155","Text":"does a_1 times a_2 equal minus 1?"},{"Start":"06:37.155 ","End":"06:38.340","Text":"If the answer is yes,"},{"Start":"06:38.340 ","End":"06:39.400","Text":"they are perpendicular,"},{"Start":"06:39.400 ","End":"06:41.930","Text":"otherwise not. So let\u0027s see."},{"Start":"06:41.930 ","End":"06:50.070","Text":"Minus 1 over square root of 2 times square root of 2 is equal to minus 1."},{"Start":"06:50.070 ","End":"06:53.525","Text":"Yes. The answer is that perpendicular."},{"Start":"06:53.525 ","End":"06:58.865","Text":"Now we\u0027ve only done it for this particular tangent point."},{"Start":"06:58.865 ","End":"07:01.880","Text":"We don\u0027t have to actually put in the numbers."},{"Start":"07:01.880 ","End":"07:07.085","Text":"Because notice that not just yet anyway,"},{"Start":"07:07.085 ","End":"07:11.495","Text":"what I\u0027d like to show you is that if I highlight,"},{"Start":"07:11.495 ","End":"07:16.865","Text":"this is the slope for 1 curve and this is a slope for the other curve."},{"Start":"07:16.865 ","End":"07:21.320","Text":"Now, what happens if I multiply these 2 together at any particular point x,"},{"Start":"07:21.320 ","End":"07:23.630","Text":"y, that where they intersect."},{"Start":"07:23.630 ","End":"07:25.955","Text":"Then I will get,"},{"Start":"07:25.955 ","End":"07:29.510","Text":"let\u0027s call this in general a_1, I mean,"},{"Start":"07:29.510 ","End":"07:36.374","Text":"this could just as well be our a_1 and this could just as well be our A_2 in general."},{"Start":"07:36.374 ","End":"07:47.074","Text":"So a_1 times a2 is minus x over 2y times x over y,"},{"Start":"07:47.074 ","End":"07:56.705","Text":"which equals minus x squared over 2y squared and now, if I look,"},{"Start":"07:56.705 ","End":"08:04.955","Text":"I can see that I have x squared over here and it\u0027s full and I have y squared,"},{"Start":"08:04.955 ","End":"08:08.825","Text":"which is over here, and it\u0027s 2."},{"Start":"08:08.825 ","End":"08:13.250","Text":"If I substitute, what I\u0027m going to get is minus x"},{"Start":"08:13.250 ","End":"08:19.485","Text":"squared becomes minus 4 over 2y squared,"},{"Start":"08:19.485 ","End":"08:22.980","Text":"which is twice 2, 2 times 2."},{"Start":"08:22.980 ","End":"08:27.440","Text":"So it\u0027s equal to minus 1 in general and I see I didn\u0027t have to"},{"Start":"08:27.440 ","End":"08:32.360","Text":"actually plug the specific slopes at this particular point."},{"Start":"08:32.360 ","End":"08:35.660","Text":"It could be for any point of intersection meaning for all the other 3,"},{"Start":"08:35.660 ","End":"08:42.565","Text":"it would work also and you can actually try plugging the numbers in and checking it."},{"Start":"08:42.565 ","End":"08:51.910","Text":"So was that they asked for just to show yes and we have shown it admirably, that\u0027s it."}],"ID":1814},{"Watched":false,"Name":"Exercise 2 part a","Duration":"4m 51s","ChapterTopicVideoID":1803,"CourseChapterTopicPlaylistID":1669,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.345","Text":"In this exercise, we have to find the angles between the following pairs of curves."},{"Start":"00:05.345 ","End":"00:07.010","Text":"First 2 separate exercises."},{"Start":"00:07.010 ","End":"00:08.330","Text":"There\u0027s a and there\u0027s b,"},{"Start":"00:08.330 ","End":"00:10.400","Text":"and each of them were given 2 curves,"},{"Start":"00:10.400 ","End":"00:13.110","Text":"got to find the angles between the pair."},{"Start":"00:13.110 ","End":"00:14.490","Text":"We\u0027ll start with a."},{"Start":"00:14.490 ","End":"00:16.249","Text":"I\u0027ll copy the exercise."},{"Start":"00:16.249 ","End":"00:21.215","Text":"y equals x squared and y equals 1 over x."},{"Start":"00:21.215 ","End":"00:26.150","Text":"Now, I just happened to bring a sketch with me that might help a visual aid."},{"Start":"00:26.150 ","End":"00:29.960","Text":"This is the y equals x squared curve."},{"Start":"00:29.960 ","End":"00:35.980","Text":"This is a curve number 1 and this is curve number 2."},{"Start":"00:35.980 ","End":"00:39.155","Text":"This is the point of intersection,"},{"Start":"00:39.155 ","End":"00:41.660","Text":"the tangent point, the curve 2,"},{"Start":"00:41.660 ","End":"00:44.570","Text":"and it\u0027s also the tangent point to curve a."},{"Start":"00:44.570 ","End":"00:46.449","Text":"Just call it the point."},{"Start":"00:46.449 ","End":"00:51.210","Text":"Here we have the tangent to curve 1 at the point,"},{"Start":"00:51.210 ","End":"00:56.695","Text":"I\u0027ll call it tangent 1, and this line here is tangent 2."},{"Start":"00:56.695 ","End":"01:00.530","Text":"The definition of the angle between curves is"},{"Start":"01:00.530 ","End":"01:04.865","Text":"simply the angle between the tangents and it\u0027s denoted as Alpha."},{"Start":"01:04.865 ","End":"01:06.770","Text":"There\u0027s 4 different angles,"},{"Start":"01:06.770 ","End":"01:09.050","Text":"but this one is equal to this one,"},{"Start":"01:09.050 ","End":"01:10.580","Text":"and this one is equal to this one."},{"Start":"01:10.580 ","End":"01:15.845","Text":"We take the one which is acute between 0 and 90 degrees."},{"Start":"01:15.845 ","End":"01:18.875","Text":"This gives us a general idea of what\u0027s going on."},{"Start":"01:18.875 ","End":"01:21.755","Text":"Actually, all we need is the slopes,"},{"Start":"01:21.755 ","End":"01:26.165","Text":"slope of tangent 1 and the slope of tangent 2."},{"Start":"01:26.165 ","End":"01:30.649","Text":"Then there\u0027s a formula for finding the angle between 2 slopes."},{"Start":"01:30.649 ","End":"01:33.190","Text":"Back to the top there."},{"Start":"01:33.190 ","End":"01:37.325","Text":"In a, we have to, first of all, find the point of intersection."},{"Start":"01:37.325 ","End":"01:39.155","Text":"For the point of intersection,"},{"Start":"01:39.155 ","End":"01:43.250","Text":"we just equate x squared equals 1 over x."},{"Start":"01:43.250 ","End":"01:45.440","Text":"If x squared is 1 over x,"},{"Start":"01:45.440 ","End":"01:50.375","Text":"that gives us that x cubed is 1."},{"Start":"01:50.375 ","End":"01:53.449","Text":"That gives us that x equals 1."},{"Start":"01:53.449 ","End":"01:55.325","Text":"If x equals 1,"},{"Start":"01:55.325 ","End":"01:58.340","Text":"then y is equal to also 1 either from here,"},{"Start":"01:58.340 ","End":"02:00.610","Text":"1 squared or 1 over 1."},{"Start":"02:00.610 ","End":"02:05.485","Text":"We know that the point is 1, 1."},{"Start":"02:05.485 ","End":"02:11.665","Text":"Now we need in each of these to find the slope at 1, 1."},{"Start":"02:11.665 ","End":"02:15.950","Text":"We just have to differentiate for y equals x squared."},{"Start":"02:15.950 ","End":"02:19.335","Text":"This is actually the parabola, this is a hyperbola."},{"Start":"02:19.335 ","End":"02:21.425","Text":"For y equals x squared,"},{"Start":"02:21.425 ","End":"02:25.820","Text":"we get y prime is equal to 2x."},{"Start":"02:25.820 ","End":"02:31.595","Text":"The slope is just 2x, just say that y prime at the point,"},{"Start":"02:31.595 ","End":"02:35.390","Text":"we\u0027re not going to use the y here, just use the x."},{"Start":"02:35.390 ","End":"02:41.165","Text":"But anyway, y prime at this point is equal to 2 times 1, which is 2."},{"Start":"02:41.165 ","End":"02:43.610","Text":"The slope is 2."},{"Start":"02:43.610 ","End":"02:47.375","Text":"For the other one, for y equals 1 over x,"},{"Start":"02:47.375 ","End":"02:53.625","Text":"then we get y prime is minus 1 over x squared."},{"Start":"02:53.625 ","End":"02:56.930","Text":"y prime at the point 1, 1,"},{"Start":"02:56.930 ","End":"02:58.550","Text":"that point of intersection."},{"Start":"02:58.550 ","End":"03:02.855","Text":"Again, put x equals 1 so minus 1 over 1 squared,"},{"Start":"03:02.855 ","End":"03:04.805","Text":"which is minus 1."},{"Start":"03:04.805 ","End":"03:07.010","Text":"That\u0027s the slope, lets call this,"},{"Start":"03:07.010 ","End":"03:11.435","Text":"say, slope 1 and slope 2 is minus 1."},{"Start":"03:11.435 ","End":"03:13.640","Text":"Often, don\u0027t use the word slope."},{"Start":"03:13.640 ","End":"03:17.608","Text":"I\u0027ll just use the letter a to just label the 2 curves."},{"Start":"03:17.608 ","End":"03:20.690","Text":"Let\u0027s say, this is number 1 and this is number 2,"},{"Start":"03:20.690 ","End":"03:26.505","Text":"then they will call this slope a_1 and this slope a_2."},{"Start":"03:26.505 ","End":"03:28.185","Text":"We have the 2 slopes."},{"Start":"03:28.185 ","End":"03:30.770","Text":"We weren\u0027t asked to find the equation of the lines,"},{"Start":"03:30.770 ","End":"03:34.610","Text":"but the line would be y equals ax plus something,"},{"Start":"03:34.610 ","End":"03:36.935","Text":"and this would be the a on each of them."},{"Start":"03:36.935 ","End":"03:39.800","Text":"All we have to do is find the angle between them."},{"Start":"03:39.800 ","End":"03:43.580","Text":"The formula for the angle given these 2 slopes."},{"Start":"03:43.580 ","End":"03:51.590","Text":"That formula is the tangent of the angle between these 2 tangents,"},{"Start":"03:51.590 ","End":"03:53.085","Text":"whose slopes are these,"},{"Start":"03:53.085 ","End":"03:56.925","Text":"is the tangent of Alpha is a_2."},{"Start":"03:56.925 ","End":"04:06.120","Text":"In general, a_2 minus a_1 over 1 plus a_1 times a_2"},{"Start":"04:06.120 ","End":"04:08.480","Text":"and in absolute value to make it positive."},{"Start":"04:08.480 ","End":"04:16.300","Text":"In our case, it\u0027s equal to the absolute value of minus 1, which is this,"},{"Start":"04:16.300 ","End":"04:26.450","Text":"minus a_1 minus 2 over 1 plus minus 1 times 2."},{"Start":"04:26.450 ","End":"04:30.095","Text":"This is equal to minus 3,"},{"Start":"04:30.095 ","End":"04:34.760","Text":"1 plus minus 2 is minus 1,"},{"Start":"04:34.760 ","End":"04:38.495","Text":"so that is equal to 3."},{"Start":"04:38.495 ","End":"04:40.940","Text":"I just looked it up to the exercise."},{"Start":"04:40.940 ","End":"04:44.975","Text":"I got 71.57 degrees unless I made a mistake."},{"Start":"04:44.975 ","End":"04:48.185","Text":"That\u0027s the answer for part a,"},{"Start":"04:48.185 ","End":"04:51.180","Text":"but we still have part b."}],"ID":1815},{"Watched":false,"Name":"Exercise 2 part b","Duration":"6m 7s","ChapterTopicVideoID":1804,"CourseChapterTopicPlaylistID":1669,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:10.320","Text":"Part B, 2 curves are x squared plus y squared equals 8 and y squared equals 2x."},{"Start":"00:10.320 ","End":"00:16.844","Text":"I know that this is an equation of a circle and this is a parabola on its side."},{"Start":"00:16.844 ","End":"00:21.000","Text":"As before, the first thing to do is to find the point"},{"Start":"00:21.000 ","End":"00:25.170","Text":"of intersection of 2 equations and 2 unknowns,"},{"Start":"00:25.170 ","End":"00:33.705","Text":"x squared plus y squared equals 8 and y squared equals 2x."},{"Start":"00:33.705 ","End":"00:38.935","Text":"What I think we can do here is to subtract the second from the first."},{"Start":"00:38.935 ","End":"00:44.640","Text":"The y squareds will cancel and I find these 2 solutions."},{"Start":"00:44.640 ","End":"00:48.835","Text":"Say x_1 is 2."},{"Start":"00:48.835 ","End":"00:51.020","Text":"I made a slight mistake here."},{"Start":"00:51.020 ","End":"00:52.910","Text":"This should be plus and this should be minus,"},{"Start":"00:52.910 ","End":"00:54.754","Text":"I\u0027ll fix it at once."},{"Start":"00:54.754 ","End":"00:58.640","Text":"Anyway, the answers I got when I did the calculation,"},{"Start":"00:58.640 ","End":"00:59.750","Text":"I didn\u0027t make this mistake,"},{"Start":"00:59.750 ","End":"01:04.820","Text":"and I got x_1 equals 2 and x_2 is minus 4."},{"Start":"01:04.820 ","End":"01:09.500","Text":"Now, I actually can rule one of these out because if you look at this here,"},{"Start":"01:09.500 ","End":"01:11.990","Text":"y squared is 2x."},{"Start":"01:11.990 ","End":"01:14.415","Text":"X can\u0027t be negative,"},{"Start":"01:14.415 ","End":"01:18.860","Text":"so this solution I can rule out because of the y"},{"Start":"01:18.860 ","End":"01:25.250","Text":"squared equals 2x forces x to be bigger or equal to 0."},{"Start":"01:25.250 ","End":"01:29.780","Text":"All we\u0027re left with is that x is equal to 2,"},{"Start":"01:29.780 ","End":"01:38.825","Text":"so we can definitely say from here that x equals 2 and now we need to find y."},{"Start":"01:38.825 ","End":"01:41.180","Text":"If x equals 2,"},{"Start":"01:41.180 ","End":"01:42.650","Text":"I can put it in here,"},{"Start":"01:42.650 ","End":"01:45.180","Text":"y squared equals 4,"},{"Start":"01:45.180 ","End":"01:50.480","Text":"so that means that y is plus or minus 2."},{"Start":"01:50.480 ","End":"01:53.030","Text":"Note that we got 2 solutions."},{"Start":"01:53.030 ","End":"01:57.850","Text":"If I draw a quick sketch here, you\u0027ll see why."},{"Start":"01:57.850 ","End":"01:59.280","Text":"We don\u0027t have to draw a sketch,"},{"Start":"01:59.280 ","End":"02:00.695","Text":"we can just continue."},{"Start":"02:00.695 ","End":"02:05.000","Text":"But notice that each of these contains y squared,"},{"Start":"02:05.000 ","End":"02:07.025","Text":"you don\u0027t have y on its own."},{"Start":"02:07.025 ","End":"02:10.800","Text":"If I put y instead of minus y,"},{"Start":"02:11.600 ","End":"02:15.560","Text":"it will hold if I replace y by minus y."},{"Start":"02:15.560 ","End":"02:19.670","Text":"That means we have symmetry about the x-axis."},{"Start":"02:19.670 ","End":"02:23.465","Text":"In actual fact, one of them looks like a circle,"},{"Start":"02:23.465 ","End":"02:27.350","Text":"and the other one is a parabola on its side,"},{"Start":"02:27.350 ","End":"02:30.995","Text":"so we actually have 2 points of intersection,"},{"Start":"02:30.995 ","End":"02:32.960","Text":"this one and this one."},{"Start":"02:32.960 ","End":"02:34.895","Text":"But because of the symmetry,"},{"Start":"02:34.895 ","End":"02:37.000","Text":"the angle will be the same at both,"},{"Start":"02:37.000 ","End":"02:39.850","Text":"so let\u0027s just make life easier and say,"},{"Start":"02:39.850 ","End":"02:44.690","Text":"we\u0027ll take that y equals 2 because it\u0027ll be the same on the other side."},{"Start":"02:44.690 ","End":"02:47.940","Text":"This will actually be the point 2, 2,"},{"Start":"02:47.940 ","End":"02:52.385","Text":"but the point 2, minus 2 will also give you the same angle."},{"Start":"02:52.385 ","End":"02:53.580","Text":"It\u0027s a mirror image,"},{"Start":"02:53.580 ","End":"02:55.860","Text":"so the angle won\u0027t change."},{"Start":"02:55.860 ","End":"03:01.065","Text":"Anyway, let\u0027s call this curve number 1,"},{"Start":"03:01.065 ","End":"03:03.050","Text":"and this will be curve number 2."},{"Start":"03:03.050 ","End":"03:05.365","Text":"That\u0027s where we\u0027ll use subscripts,"},{"Start":"03:05.365 ","End":"03:07.170","Text":"and we\u0027ll have a_1,"},{"Start":"03:07.170 ","End":"03:10.010","Text":"and a_2, and so on for the slopes,"},{"Start":"03:10.010 ","End":"03:12.350","Text":"Let\u0027s do some implicit differentiation."},{"Start":"03:12.350 ","End":"03:13.640","Text":"We\u0027re going to need the slope,"},{"Start":"03:13.640 ","End":"03:15.170","Text":"so we\u0027re going to need derivatives."},{"Start":"03:15.170 ","End":"03:16.570","Text":"For the first one,"},{"Start":"03:16.570 ","End":"03:17.895","Text":"again I get,"},{"Start":"03:17.895 ","End":"03:21.045","Text":"2x plus 2y,"},{"Start":"03:21.045 ","End":"03:24.605","Text":"and don\u0027t forget the y prime equals 0."},{"Start":"03:24.605 ","End":"03:27.660","Text":"We can divide both sides by 2,"},{"Start":"03:27.660 ","End":"03:29.160","Text":"so that doesn\u0027t appear."},{"Start":"03:29.160 ","End":"03:36.935","Text":"What we\u0027re left with is y prime is minus x over y. Y prime is minus x over y,"},{"Start":"03:36.935 ","End":"03:39.800","Text":"and y prime in our particular point 2,"},{"Start":"03:39.800 ","End":"03:44.760","Text":"2 is minus 2 over 2, is minus 1."},{"Start":"03:44.760 ","End":"03:47.445","Text":"That\u0027s what we\u0027ll call our a_1."},{"Start":"03:47.445 ","End":"03:48.810","Text":"We\u0027ll say a_1,"},{"Start":"03:48.810 ","End":"03:51.295","Text":"slope number 1 is minus 1."},{"Start":"03:51.295 ","End":"03:55.085","Text":"That actually means because the first curve is the circle,"},{"Start":"03:55.085 ","End":"03:59.585","Text":"this is the first slope supposed to go through here."},{"Start":"03:59.585 ","End":"04:03.140","Text":"The second slope will be what we get from there,"},{"Start":"04:03.140 ","End":"04:06.525","Text":"will be the slope somewhere through here,"},{"Start":"04:06.525 ","End":"04:10.710","Text":"and then we need the angle between these 2."},{"Start":"04:11.620 ","End":"04:14.575","Text":"Let\u0027s go to number 2."},{"Start":"04:14.575 ","End":"04:17.880","Text":"We\u0027re all right now. The curve number 2,"},{"Start":"04:17.880 ","End":"04:19.590","Text":"y squared equals 2x."},{"Start":"04:19.590 ","End":"04:22.520","Text":"So 2y, again,"},{"Start":"04:22.520 ","End":"04:30.255","Text":"don\u0027t forget the times y prime is equal to derivative of 2x is 2."},{"Start":"04:30.255 ","End":"04:35.280","Text":"We could divide by the 2 and we just get a 1 here."},{"Start":"04:35.280 ","End":"04:41.385","Text":"Then y prime is 1 over y. Y prime is 1 over y."},{"Start":"04:41.385 ","End":"04:43.670","Text":"At the point 2,"},{"Start":"04:43.670 ","End":"04:48.145","Text":"2, this is equal to 1 over 2."},{"Start":"04:48.145 ","End":"04:53.730","Text":"The other slope, a_2 is 1/2."},{"Start":"04:53.730 ","End":"04:57.860","Text":"Now, we just get the formula for the angle between,"},{"Start":"04:57.860 ","End":"05:01.575","Text":"that\u0027s Alpha here between these 2."},{"Start":"05:01.575 ","End":"05:08.690","Text":"We get that the tangent of Alpha is the absolute value of difference in"},{"Start":"05:08.690 ","End":"05:17.375","Text":"the slopes over 1 plus a_1a_2, in absolute value."},{"Start":"05:17.375 ","End":"05:20.360","Text":"That makes sure that we get the acute angle because"},{"Start":"05:20.360 ","End":"05:24.600","Text":"an acute angle always has a positive tangent."},{"Start":"05:24.640 ","End":"05:30.450","Text":"Let\u0027s see, a_2 minus a_1 is 3 over 2,"},{"Start":"05:30.450 ","End":"05:34.200","Text":"a_1 times a_2 is minus 1/2."},{"Start":"05:34.200 ","End":"05:36.675","Text":"It\u0027s 1 plus minus 1/2,"},{"Start":"05:36.675 ","End":"05:41.044","Text":"it\u0027s over 1/2 in absolute value,"},{"Start":"05:41.044 ","End":"05:46.670","Text":"which is the absolute value of 3, which is 3."},{"Start":"05:46.670 ","End":"05:49.165","Text":"Tangent Alpha is 3,"},{"Start":"05:49.165 ","End":"05:50.990","Text":"and to get Alpha,"},{"Start":"05:50.990 ","End":"05:53.540","Text":"you just use a scientific calculator,"},{"Start":"05:53.540 ","End":"05:54.560","Text":"and enter 3,"},{"Start":"05:54.560 ","End":"05:57.080","Text":"and then do Shift Tangent."},{"Start":"05:57.080 ","End":"06:04.625","Text":"I get 71.57 something degrees."},{"Start":"06:04.625 ","End":"06:08.040","Text":"That\u0027s what we had to find, so we\u0027re done."}],"ID":1816}],"Thumbnail":null,"ID":1669},{"Name":"Vertical Tangents and Cusps","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vertical Tangents","Duration":"7m 38s","ChapterTopicVideoID":8263,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this clip, I\u0027ll be talking about vertical tangents."},{"Start":"00:03.240 ","End":"00:06.180","Text":"Contrast that with, say, horizontal tangents,"},{"Start":"00:06.180 ","End":"00:11.550","Text":"which are very easy to determine because a horizontal tangent is 1 where the slope is 0,"},{"Start":"00:11.550 ","End":"00:13.275","Text":"so the derivative is 0."},{"Start":"00:13.275 ","End":"00:15.690","Text":"There\u0027s a problem with vertical tangents,"},{"Start":"00:15.690 ","End":"00:18.540","Text":"is that a vertical line doesn\u0027t have a slope."},{"Start":"00:18.540 ","End":"00:21.150","Text":"The slope is undefined for a vertical line."},{"Start":"00:21.150 ","End":"00:25.015","Text":"How do we decide when we have a vertical tangent?"},{"Start":"00:25.015 ","End":"00:26.660","Text":"I\u0027m going to write a definition."},{"Start":"00:26.660 ","End":"00:29.460","Text":"We\u0027re only going to be talking about continuous functions."},{"Start":"00:29.460 ","End":"00:33.110","Text":"Let\u0027s say we have a continuous function f of x,"},{"Start":"00:33.110 ","End":"00:40.160","Text":"and we say that it has a vertical tangent at some point where,"},{"Start":"00:40.160 ","End":"00:42.664","Text":"say, x equals a."},{"Start":"00:42.664 ","End":"00:44.540","Text":"Let\u0027s give a definition."},{"Start":"00:44.540 ","End":"00:45.830","Text":"What might the condition be?"},{"Start":"00:45.830 ","End":"00:49.250","Text":"We can\u0027t talk about f prime of a because at x equals a,"},{"Start":"00:49.250 ","End":"00:50.510","Text":"the derivative won\u0027t exist."},{"Start":"00:50.510 ","End":"00:53.480","Text":"We can talk about the limit, and that\u0027s what we do."},{"Start":"00:53.480 ","End":"00:59.525","Text":"If the limit as x goes to a of f prime of x,"},{"Start":"00:59.525 ","End":"01:04.555","Text":"it\u0027s equal to infinity, that\u0027s 1 possibility."},{"Start":"01:04.555 ","End":"01:07.105","Text":"Or very similar,"},{"Start":"01:07.105 ","End":"01:14.795","Text":"limit as x goes to a of f prime of x is equal to minus infinity."},{"Start":"01:14.795 ","End":"01:20.275","Text":"There is a modification of this if a happens to be the endpoint of the domain."},{"Start":"01:20.275 ","End":"01:26.270","Text":"I\u0027ll just say if a is the endpoint of the domain,"},{"Start":"01:26.270 ","End":"01:31.100","Text":"then we just replace in the definition the 2-sided limit,"},{"Start":"01:31.100 ","End":"01:37.350","Text":"the limit as x goes to a by limit 1-sided,"},{"Start":"01:37.350 ","End":"01:41.695","Text":"x goes to a from the left if it\u0027s a right endpoint,"},{"Start":"01:41.695 ","End":"01:50.075","Text":"or limit as x goes to a from above if it\u0027s a left endpoint to the domain appropriately,"},{"Start":"01:50.075 ","End":"01:54.230","Text":"like I said, for the right end and this is for the left end of the domain."},{"Start":"01:54.230 ","End":"01:56.665","Text":"I\u0027m going to give 2 examples,"},{"Start":"01:56.665 ","End":"02:02.240","Text":"1 is going to be for the case of a 2-sided limit with a point, not an endpoint."},{"Start":"02:02.240 ","End":"02:04.280","Text":"I\u0027ll give an example of an endpoint."},{"Start":"02:04.280 ","End":"02:07.685","Text":"Example 1 will be as follows."},{"Start":"02:07.685 ","End":"02:15.484","Text":"Let\u0027s take f of x equals the 5th root of 2 minus x."},{"Start":"02:15.484 ","End":"02:21.295","Text":"Now, it\u0027s defined for all x because I can always take a 5th root,"},{"Start":"02:21.295 ","End":"02:22.870","Text":"it\u0027s an odd number."},{"Start":"02:22.870 ","End":"02:25.205","Text":"If it was square root, that would be something else."},{"Start":"02:25.205 ","End":"02:26.995","Text":"It\u0027s defined for all x."},{"Start":"02:26.995 ","End":"02:29.330","Text":"Let\u0027s see what its derivative is."},{"Start":"02:29.330 ","End":"02:33.280","Text":"F prime of x is equal to,"},{"Start":"02:33.280 ","End":"02:39.710","Text":"and I should have said that here in the example, find vertical tangents."},{"Start":"02:39.710 ","End":"02:41.570","Text":"We differentiate, first,"},{"Start":"02:41.570 ","End":"02:44.645","Text":"f prime of x. I\u0027ll give you the answer then I\u0027ll show you why."},{"Start":"02:44.645 ","End":"02:48.615","Text":"It\u0027s equal to minus 1/5 times"},{"Start":"02:48.615 ","End":"02:56.230","Text":"1 over 2 minus x to the power of 4/5."},{"Start":"02:56.230 ","End":"03:00.005","Text":"The reason for that is that if f of x,"},{"Start":"03:00.005 ","End":"03:05.545","Text":"I rewrite it as 2 minus x to the power of 1/5,"},{"Start":"03:05.545 ","End":"03:13.160","Text":"then f prime of x using the rule for powers is 1/5 2 minus x,"},{"Start":"03:13.160 ","End":"03:16.535","Text":"subtract 1 from here and it gives me minus 4/5."},{"Start":"03:16.535 ","End":"03:20.480","Text":"Then there is an internal derivative which is minus 1."},{"Start":"03:20.480 ","End":"03:26.150","Text":"If you rewrite this using the negative exponents to put it to the denominator,"},{"Start":"03:26.150 ","End":"03:28.910","Text":"the minus 1 in front you see that we get this."},{"Start":"03:28.910 ","End":"03:33.080","Text":"Now, I want to know what is the limit as x goes to 2."},{"Start":"03:33.080 ","End":"03:34.535","Text":"Why did I choose 2?"},{"Start":"03:34.535 ","End":"03:37.880","Text":"Because at 2 the derivative doesn\u0027t exist and there is a problem."},{"Start":"03:37.880 ","End":"03:40.940","Text":"Every other value of x has a derivative,"},{"Start":"03:40.940 ","End":"03:43.595","Text":"so we\u0027re not going to get a vertical tangent."},{"Start":"03:43.595 ","End":"03:47.480","Text":"Vertical tangent has to be where there\u0027s a problem with the derivative."},{"Start":"03:47.480 ","End":"03:49.415","Text":"That\u0027s only x equals 2."},{"Start":"03:49.415 ","End":"03:52.670","Text":"When x goes to 2 of f prime of x,"},{"Start":"03:52.670 ","End":"03:54.770","Text":"let\u0027s see what this is equal to."},{"Start":"03:54.770 ","End":"03:58.065","Text":"Now look, when x goes to 2,"},{"Start":"03:58.065 ","End":"04:01.860","Text":"2 minus x goes to 0."},{"Start":"04:01.860 ","End":"04:04.555","Text":"Now, we have here a power of 4/5."},{"Start":"04:04.555 ","End":"04:10.775","Text":"2 minus x to the power of 4 goes to 0 plus,"},{"Start":"04:10.775 ","End":"04:12.800","Text":"meaning it goes to 0 from the right,"},{"Start":"04:12.800 ","End":"04:15.590","Text":"it goes to 0 through positive values."},{"Start":"04:15.590 ","End":"04:21.370","Text":"When I take the 5th root of that 2 minus x to the power of 4/5,"},{"Start":"04:21.370 ","End":"04:25.820","Text":"the 5th root of 0 plus is still 0 plus."},{"Start":"04:25.820 ","End":"04:35.805","Text":"What I get symbolically here is minus 1/5 times 1/0 plus,"},{"Start":"04:35.805 ","End":"04:38.690","Text":"and 1/0 plus is plus infinity."},{"Start":"04:38.690 ","End":"04:43.890","Text":"We have minus 1/5 times infinity which is minus infinity."},{"Start":"04:43.890 ","End":"04:48.830","Text":"Or in the case where we have a limit and it\u0027s equal to minus infinity,"},{"Start":"04:48.830 ","End":"04:56.525","Text":"we have that x equals 2 is a vertical tangent."},{"Start":"04:56.525 ","End":"04:59.060","Text":"When I say x equals 2 is a vertical tangent,"},{"Start":"04:59.060 ","End":"05:02.720","Text":"I don\u0027t just mean that we have a tangent when x is 2,"},{"Start":"05:02.720 ","End":"05:05.300","Text":"which is true, but that\u0027s actually the equation of"},{"Start":"05:05.300 ","End":"05:10.360","Text":"the tangent because the equation of a vertical line is x equals something."},{"Start":"05:10.360 ","End":"05:13.275","Text":"All that\u0027s missing is a picture."},{"Start":"05:13.275 ","End":"05:15.420","Text":"Here\u0027s what it looks like."},{"Start":"05:15.420 ","End":"05:17.525","Text":"I\u0027m not going to say too much about it."},{"Start":"05:17.525 ","End":"05:21.245","Text":"Just you can see that the curve takes a steep dive here,"},{"Start":"05:21.245 ","End":"05:23.025","Text":"goes vertical and continues."},{"Start":"05:23.025 ","End":"05:24.980","Text":"You can\u0027t see the red here,"},{"Start":"05:24.980 ","End":"05:27.290","Text":"but it continues, it\u0027s continuous,"},{"Start":"05:27.290 ","End":"05:29.270","Text":"and at 2 it\u0027s equal to 0,"},{"Start":"05:29.270 ","End":"05:30.650","Text":"as you can see,"},{"Start":"05:30.650 ","End":"05:32.060","Text":"and x equals 2."},{"Start":"05:32.060 ","End":"05:34.610","Text":"The blue line is the vertical tangent."},{"Start":"05:34.610 ","End":"05:37.295","Text":"Now, I\u0027m going to do example 2,"},{"Start":"05:37.295 ","End":"05:40.915","Text":"which is going to be an endpoint."},{"Start":"05:40.915 ","End":"05:48.620","Text":"Now, notice that this is only defined for x bigger or equal to 0."},{"Start":"05:48.620 ","End":"05:51.920","Text":"First thing to do is to take the derivative f prime of"},{"Start":"05:51.920 ","End":"05:56.690","Text":"x is 1 over twice the square root of x."},{"Start":"05:56.690 ","End":"06:01.925","Text":"This is also defined only for x bigger than 0 strictly."},{"Start":"06:01.925 ","End":"06:06.350","Text":"As you see at 0, we have a problem with the denominator,"},{"Start":"06:06.350 ","End":"06:08.885","Text":"which is good because we\u0027re looking for problems,"},{"Start":"06:08.885 ","End":"06:11.060","Text":"because we want to find vertical tangents."},{"Start":"06:11.060 ","End":"06:15.320","Text":"Usually, denominator is 0 or something like that for the derivative."},{"Start":"06:15.320 ","End":"06:18.020","Text":"This time we have to look only for"},{"Start":"06:18.020 ","End":"06:23.505","Text":"a 1-sided limit because we only defined on the non-negative."},{"Start":"06:23.505 ","End":"06:29.830","Text":"We can look as x goes to 0 from the right of f prime of x."},{"Start":"06:29.830 ","End":"06:33.175","Text":"We can\u0027t actually plug in 0 as we saw,"},{"Start":"06:33.175 ","End":"06:35.020","Text":"but we can take the right limit."},{"Start":"06:35.020 ","End":"06:37.030","Text":"If we can get plus or minus infinity,"},{"Start":"06:37.030 ","End":"06:38.260","Text":"that will be good,"},{"Start":"06:38.260 ","End":"06:40.210","Text":"then we\u0027ll have a vertical tangent."},{"Start":"06:40.210 ","End":"06:47.410","Text":"Well, clearly this is equal to plus infinity because when x goes to 0 from the right,"},{"Start":"06:47.410 ","End":"06:51.985","Text":"the square root of x also goes to 0 from the right."},{"Start":"06:51.985 ","End":"06:59.175","Text":"1 over the square root of x goes to infinity."},{"Start":"06:59.175 ","End":"07:09.830","Text":"I have half of infinity which is infinity and so x equals 0 is a vertical tangent."},{"Start":"07:09.830 ","End":"07:13.190","Text":"Not only do we have a vertical tangent when x equals 0,"},{"Start":"07:13.190 ","End":"07:17.435","Text":"but x equals 0 is actually the equation of the vertical tangent."},{"Start":"07:17.435 ","End":"07:19.340","Text":"I\u0027ll bring in a picture."},{"Start":"07:19.340 ","End":"07:21.800","Text":"Notice that as we go towards the 0,"},{"Start":"07:21.800 ","End":"07:24.485","Text":"the slope gets more and more vertical."},{"Start":"07:24.485 ","End":"07:26.565","Text":"That\u0027s example 2."},{"Start":"07:26.565 ","End":"07:30.755","Text":"That\u0027s all I want to say on vertical tangents,"},{"Start":"07:30.755 ","End":"07:36.680","Text":"except that there\u0027s some more in the solved exercises after this tutorial."},{"Start":"07:36.680 ","End":"07:39.360","Text":"Okay. That\u0027s it."}],"ID":8425},{"Watched":false,"Name":"Cusps","Duration":"6m 9s","ChapterTopicVideoID":6437,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.300","Text":"In this clip, I\u0027m going to talk about something called cusps."},{"Start":"00:04.300 ","End":"00:05.915","Text":"This is related to"},{"Start":"00:05.915 ","End":"00:08.789","Text":"but is different from vertical tangents."},{"Start":"00:08.789 ","End":"00:10.320","Text":"Let me contrast."},{"Start":"00:10.320 ","End":"00:12.120","Text":"I\u0027m going to start by bringing"},{"Start":"00:12.120 ","End":"00:15.730","Text":"the definition of vertical tangents from the previous clip."},{"Start":"00:15.730 ","End":"00:19.940","Text":"Basically, if we have the limit as x goes to the point"},{"Start":"00:19.940 ","End":"00:23.750","Text":"and it\u0027s either equal to infinity or it\u0027s equal to minus infinity,"},{"Start":"00:23.750 ","End":"00:25.400","Text":"that\u0027s a vertical tangent."},{"Start":"00:25.400 ","End":"00:27.125","Text":"There\u0027s also a condition."},{"Start":"00:27.125 ","End":"00:30.155","Text":"If it\u0027s an endpoint, then we only have to take the one-sided limit."},{"Start":"00:30.155 ","End":"00:33.320","Text":"Now with cusps, they don\u0027t exist at endpoints."},{"Start":"00:33.320 ","End":"00:34.670","Text":"With a cusp, instead of this,"},{"Start":"00:34.670 ","End":"00:38.180","Text":"we\u0027re going to replace it by a different condition that"},{"Start":"00:38.180 ","End":"00:40.460","Text":"the limit on one side will be infinity"},{"Start":"00:40.460 ","End":"00:42.500","Text":"and on the other side, it\u0027ll be minus infinity."},{"Start":"00:42.500 ","End":"00:46.550","Text":"First of all, change the word vertical tangent to cusp."},{"Start":"00:46.550 ","End":"00:50.239","Text":"Next, I\u0027ll replace the word or by the word and,"},{"Start":"00:50.239 ","End":"00:52.910","Text":"and I\u0027ll make this one-sided limit."},{"Start":"00:52.910 ","End":"00:55.970","Text":"Let\u0027s say this is from the right and this is from the left."},{"Start":"00:55.970 ","End":"00:59.270","Text":"If we have a limit from one side is infinity"},{"Start":"00:59.270 ","End":"01:02.075","Text":"and from the other side is minus infinity, that\u0027s a cusp."},{"Start":"01:02.075 ","End":"01:04.360","Text":"But it could be vice versa."},{"Start":"01:04.360 ","End":"01:06.770","Text":"We could have or."},{"Start":"01:06.770 ","End":"01:07.815","Text":"The other way around,"},{"Start":"01:07.815 ","End":"01:11.975","Text":"notice I\u0027ve replaced the limit from the right and left and left to right."},{"Start":"01:11.975 ","End":"01:14.690","Text":"What it means is that one side, I don\u0027t know which,"},{"Start":"01:14.690 ","End":"01:17.870","Text":"is going to go to infinity and the other side to minus infinity."},{"Start":"01:17.870 ","End":"01:20.285","Text":"Then it\u0027s not a vertical tangent, it\u0027s a cusp."},{"Start":"01:20.285 ","End":"01:23.330","Text":"I repeat, it\u0027s only for the points in the middle of the domain,"},{"Start":"01:23.330 ","End":"01:24.635","Text":"not at the endpoints."},{"Start":"01:24.635 ","End":"01:28.820","Text":"I want to give another definition and I hope this is not going to be confusing."},{"Start":"01:28.820 ","End":"01:32.105","Text":"We had the concept of a vertical tangent,"},{"Start":"01:32.105 ","End":"01:35.960","Text":"but also we have not a vertical tangent,"},{"Start":"01:35.960 ","End":"01:39.115","Text":"but a tangent line which is vertical."},{"Start":"01:39.115 ","End":"01:41.545","Text":"I\u0027m going to define this concept,"},{"Start":"01:41.545 ","End":"01:46.370","Text":"and it\u0027s different from a vertical tangent function f of x,"},{"Start":"01:46.370 ","End":"01:50.035","Text":"also continuous, and we have x equals a."},{"Start":"01:50.035 ","End":"01:53.090","Text":"We say that f of x as a tangent line,"},{"Start":"01:53.090 ","End":"01:55.355","Text":"which is vertical at x equals a,"},{"Start":"01:55.355 ","End":"01:57.830","Text":"if any 1 of 4 conditions hold,"},{"Start":"01:57.830 ","End":"02:02.090","Text":"we have the limit as x goes to a plus,"},{"Start":"02:02.090 ","End":"02:04.820","Text":"could be plus infinity or minus infinity."},{"Start":"02:04.820 ","End":"02:08.905","Text":"Let me just write plus or minus infinity of the f of x, of course,"},{"Start":"02:08.905 ","End":"02:15.560","Text":"or the limit as x goes to a from the left of f of x"},{"Start":"02:15.560 ","End":"02:19.910","Text":"is either plus infinity or minus infinity, I\u0027ll write them both."},{"Start":"02:19.910 ","End":"02:22.175","Text":"If any 1 of these 4 conditions hold"},{"Start":"02:22.175 ","End":"02:24.950","Text":"and the tangent line is vertical,"},{"Start":"02:24.950 ","End":"02:26.720","Text":"notice that at a cusp,"},{"Start":"02:26.720 ","End":"02:29.795","Text":"we definitely have a tangent line which is vertical,"},{"Start":"02:29.795 ","End":"02:32.380","Text":"but it\u0027s still not a vertical tangent."},{"Start":"02:32.380 ","End":"02:34.075","Text":"Because for a vertical tangent,"},{"Start":"02:34.075 ","End":"02:37.340","Text":"we have to have both of these being infinity"},{"Start":"02:37.340 ","End":"02:39.619","Text":"or both of these being minus infinity."},{"Start":"02:39.619 ","End":"02:40.760","Text":"I hope this is clear."},{"Start":"02:40.760 ","End":"02:44.435","Text":"If not, then the exercises will explain it more fully."},{"Start":"02:44.435 ","End":"02:51.045","Text":"The example, f of x is equal to x to the power of 2/5."},{"Start":"02:51.045 ","End":"02:52.250","Text":"The typical exercise,"},{"Start":"02:52.250 ","End":"02:53.300","Text":"we\u0027ll say, first of all,"},{"Start":"02:53.300 ","End":"02:56.000","Text":"to find the points where the tangent line is vertical,"},{"Start":"02:56.000 ","End":"02:58.025","Text":"that would be say part A of an exercise."},{"Start":"02:58.025 ","End":"02:59.675","Text":"Then from amongst these,"},{"Start":"02:59.675 ","End":"03:02.030","Text":"we have to look for cusps,"},{"Start":"03:02.030 ","End":"03:04.655","Text":"or it might say, it doesn\u0027t have cusps."},{"Start":"03:04.655 ","End":"03:08.690","Text":"That would be like part A of a question and part B of a question."},{"Start":"03:08.690 ","End":"03:12.130","Text":"Because at cusps, the tangent line is vertical."},{"Start":"03:12.130 ","End":"03:13.790","Text":"Not a vertical tangent,"},{"Start":"03:13.790 ","End":"03:16.385","Text":"just a tangent line which happens to be vertical."},{"Start":"03:16.385 ","End":"03:18.380","Text":"Find points where tangent line is vertical."},{"Start":"03:18.380 ","End":"03:25.085","Text":"We\u0027ll need the derivative and the derivative f prime of x is equal to 2/5,"},{"Start":"03:25.085 ","End":"03:27.560","Text":"x to the minus 3/5,"},{"Start":"03:27.560 ","End":"03:28.850","Text":"if I subtract 1 here,"},{"Start":"03:28.850 ","End":"03:37.640","Text":"but I prefer to write it as 2/5 times 1 over x to the power of plus 3/5."},{"Start":"03:37.640 ","End":"03:38.885","Text":"It\u0027s easier to handle."},{"Start":"03:38.885 ","End":"03:41.960","Text":"Now notice that when x is 0,"},{"Start":"03:41.960 ","End":"03:44.615","Text":"we have a problem with the denominator."},{"Start":"03:44.615 ","End":"03:46.790","Text":"We can have x equals 0,"},{"Start":"03:46.790 ","End":"03:49.160","Text":"but problems are exactly what we\u0027re looking for,"},{"Start":"03:49.160 ","End":"03:53.105","Text":"because that\u0027s the only place where we could find a tangent which is vertical,"},{"Start":"03:53.105 ","End":"03:55.475","Text":"is where the derivative is not defined."},{"Start":"03:55.475 ","End":"03:57.830","Text":"Let\u0027s go with x equals 0."},{"Start":"03:57.830 ","End":"03:59.360","Text":"Although we can\u0027t substitute it,"},{"Start":"03:59.360 ","End":"04:01.685","Text":"we can take a left limit or a right limit."},{"Start":"04:01.685 ","End":"04:03.575","Text":"Let\u0027s take, for example,"},{"Start":"04:03.575 ","End":"04:07.400","Text":"the right limit of f prime of x."},{"Start":"04:07.400 ","End":"04:11.165","Text":"I claim that this is equal to plus infinity."},{"Start":"04:11.165 ","End":"04:12.620","Text":"The reason is,"},{"Start":"04:12.620 ","End":"04:16.010","Text":"that when x goes to 0 plus,"},{"Start":"04:16.010 ","End":"04:21.260","Text":"then x cubed also goes to 0 plus."},{"Start":"04:21.260 ","End":"04:27.650","Text":"The fifth root of x cubed goes to 0 through the positive numbers."},{"Start":"04:27.650 ","End":"04:28.880","Text":"That\u0027s what this is."},{"Start":"04:28.880 ","End":"04:31.585","Text":"What we get from here is 2/5,"},{"Start":"04:31.585 ","End":"04:33.175","Text":"which is not really relevant,"},{"Start":"04:33.175 ","End":"04:35.260","Text":"1 over 0 plus,"},{"Start":"04:35.260 ","End":"04:38.965","Text":"and because 1 over positive 0 is plus infinity,"},{"Start":"04:38.965 ","End":"04:41.600","Text":"that\u0027s why we get the infinity here."},{"Start":"04:41.600 ","End":"04:44.840","Text":"Now already we know that it\u0027s a vertical tangent,"},{"Start":"04:44.840 ","End":"04:46.390","Text":"but I might just add,"},{"Start":"04:46.390 ","End":"04:49.030","Text":"and besides we\u0027ll need it anyway for checking cusps,"},{"Start":"04:49.030 ","End":"04:51.530","Text":"let\u0027s see what the limit on the other side is."},{"Start":"04:51.530 ","End":"04:55.985","Text":"If x goes to 0 from the left of f prime of x,"},{"Start":"04:55.985 ","End":"04:58.040","Text":"what we get is very similar,"},{"Start":"04:58.040 ","End":"05:00.155","Text":"except with 0 minus,"},{"Start":"05:00.155 ","End":"05:02.790","Text":"just check everything, we get a 0 minus here,"},{"Start":"05:02.790 ","End":"05:05.350","Text":"we\u0027ve got minus infinity."},{"Start":"05:05.350 ","End":"05:09.785","Text":"We have met the condition that on one side, it\u0027s infinity,"},{"Start":"05:09.785 ","End":"05:12.575","Text":"on one side, it\u0027s minus infinity,"},{"Start":"05:12.575 ","End":"05:18.830","Text":"so x equals 0 is a cusp of f of x."},{"Start":"05:18.830 ","End":"05:20.950","Text":"Well, it\u0027s not the point itself,"},{"Start":"05:20.950 ","End":"05:23.450","Text":"we could really say that when x is 0,"},{"Start":"05:23.450 ","End":"05:25.085","Text":"y is also 0."},{"Start":"05:25.085 ","End":"05:28.010","Text":"I could have said 0, 0 precisely,"},{"Start":"05:28.010 ","End":"05:30.275","Text":"but usually, we just give the x of the point."},{"Start":"05:30.275 ","End":"05:32.300","Text":"Finally, just to complete the picture,"},{"Start":"05:32.300 ","End":"05:33.380","Text":"why don\u0027t I give a picture?"},{"Start":"05:33.380 ","End":"05:35.930","Text":"You can see that what happens is,"},{"Start":"05:35.930 ","End":"05:38.105","Text":"that as we go to 0 from the right,"},{"Start":"05:38.105 ","End":"05:40.910","Text":"the slope gets steeper and steeper and steeper,"},{"Start":"05:40.910 ","End":"05:43.610","Text":"but positive goes to plus infinity,"},{"Start":"05:43.610 ","End":"05:47.929","Text":"and on this side, it gets steeper but downward sloping"},{"Start":"05:47.929 ","End":"05:50.110","Text":"down to minus infinity."},{"Start":"05:50.110 ","End":"05:53.825","Text":"The tangent line is this blue line."},{"Start":"05:53.825 ","End":"05:56.390","Text":"This tangent happens to be vertical,"},{"Start":"05:56.390 ","End":"05:59.285","Text":"but it\u0027s not what is called a vertical tangent."},{"Start":"05:59.285 ","End":"06:03.515","Text":"Anyway, this is a cusp and this is the general shape of a cusp."},{"Start":"06:03.515 ","End":"06:05.640","Text":"That\u0027s it for the tutorial."},{"Start":"06:05.640 ","End":"06:09.960","Text":"There are plenty of more solved examples following this."}],"ID":6465},{"Watched":false,"Name":"Exercise 1","Duration":"7m 1s","ChapterTopicVideoID":6446,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.205","Text":"In this exercise, we have to find all the points on this graph,"},{"Start":"00:05.205 ","End":"00:08.310","Text":"y equals the 5th root of 4 minus x,"},{"Start":"00:08.310 ","End":"00:11.610","Text":"where the tangent line is vertical."},{"Start":"00:11.610 ","End":"00:14.400","Text":"Then when we\u0027ve done that, there\u0027s another question."},{"Start":"00:14.400 ","End":"00:17.160","Text":"Does the function have a vertical cusp?"},{"Start":"00:17.160 ","End":"00:19.980","Text":"I\u0027ve brought with me a sketch,"},{"Start":"00:19.980 ","End":"00:23.100","Text":"first of all, that don\u0027t need the picture, but it helps."},{"Start":"00:23.100 ","End":"00:28.110","Text":"That\u0027s the sketch and I have reminded us of the original function."},{"Start":"00:28.110 ","End":"00:31.410","Text":"Something happens at x equals 4."},{"Start":"00:31.410 ","End":"00:35.085","Text":"The curve goes from positive to negative and it looks like"},{"Start":"00:35.085 ","End":"00:40.270","Text":"the tangent is vertical and the tangent line is the x equals 4 line."},{"Start":"00:40.270 ","End":"00:42.320","Text":"One thing we can see about the positive and the"},{"Start":"00:42.320 ","End":"00:45.200","Text":"negative because when x is bigger than 4,"},{"Start":"00:45.200 ","End":"00:49.070","Text":"then what\u0027s under here is negative and 5th root of negative is negative,"},{"Start":"00:49.070 ","End":"00:51.650","Text":"likewise, 5th root of positive is positive."},{"Start":"00:51.650 ","End":"00:53.585","Text":"It looks like it makes sense."},{"Start":"00:53.585 ","End":"00:56.195","Text":"Let\u0027s get back to the algebraic part."},{"Start":"00:56.195 ","End":"00:57.800","Text":"From the picture, at any rate,"},{"Start":"00:57.800 ","End":"01:01.400","Text":"it looks like where tangent line is vertical is x equals 4."},{"Start":"01:01.400 ","End":"01:03.995","Text":"Everywhere else it\u0027s sloppy,"},{"Start":"01:03.995 ","End":"01:05.930","Text":"and here suddenly becomes vertical,"},{"Start":"01:05.930 ","End":"01:08.240","Text":"then flattens out again."},{"Start":"01:08.240 ","End":"01:11.645","Text":"Just note that the domain is all of x,"},{"Start":"01:11.645 ","End":"01:14.690","Text":"which it wouldn\u0027t have been if 5 wasn\u0027t an odd number."},{"Start":"01:14.690 ","End":"01:18.170","Text":"How do we know when the tangent line is vertical?"},{"Start":"01:18.170 ","End":"01:21.860","Text":"When is the tangent line vertical at a particular x,"},{"Start":"01:21.860 ","End":"01:24.130","Text":"say x equals x_1."},{"Start":"01:24.130 ","End":"01:26.190","Text":"When x equals x_1,"},{"Start":"01:26.190 ","End":"01:34.275","Text":"the tangent is vertical if the limit as x goes to x_1,"},{"Start":"01:34.275 ","End":"01:38.520","Text":"and this could be x_1 plus or x_1 minus."},{"Start":"01:38.520 ","End":"01:45.920","Text":"If either one of these limits of the derivative is equal to plus or minus infinity,"},{"Start":"01:45.920 ","End":"01:48.125","Text":"then the tangent line is vertical."},{"Start":"01:48.125 ","End":"01:50.465","Text":"In other words, if the limit on the left is"},{"Start":"01:50.465 ","End":"01:53.180","Text":"plus infinity or minus infinity it\u0027s already vertical."},{"Start":"01:53.180 ","End":"01:56.495","Text":"Similarly, if it\u0027s on the right it\u0027s plus or minus infinity,"},{"Start":"01:56.495 ","End":"01:58.540","Text":"then the tangent line is vertical."},{"Start":"01:58.540 ","End":"02:04.175","Text":"What we need to do next is to find f prime of x before we can continue."},{"Start":"02:04.175 ","End":"02:07.685","Text":"F of x is equal to this,"},{"Start":"02:07.685 ","End":"02:14.290","Text":"which is equal to 4 minus x to the power of 1/5."},{"Start":"02:14.290 ","End":"02:17.915","Text":"I can use the formula for exponent."},{"Start":"02:17.915 ","End":"02:23.985","Text":"F prime of x is equal to 1/5 4 minus x,"},{"Start":"02:23.985 ","End":"02:26.460","Text":"and I take away minus 4/5,"},{"Start":"02:26.460 ","End":"02:28.010","Text":"times internal derivative,"},{"Start":"02:28.010 ","End":"02:29.330","Text":"is minus 1,"},{"Start":"02:29.330 ","End":"02:39.050","Text":"which is equal to minus 1 over 5 times 4 minus x to the 4/5."},{"Start":"02:39.050 ","End":"02:42.560","Text":"If I write it with roots radicals,"},{"Start":"02:42.560 ","End":"02:47.840","Text":"then what we get is that this is equal to minus"},{"Start":"02:47.840 ","End":"02:53.479","Text":"1 over 5 times the 5th root"},{"Start":"02:53.479 ","End":"02:58.760","Text":"of 4 minus x to the power of 4."},{"Start":"02:58.760 ","End":"03:00.710","Text":"It doesn\u0027t matter in which order you do things,"},{"Start":"03:00.710 ","End":"03:03.770","Text":"if you take the power of 4th and the 5th root or vice versa,"},{"Start":"03:03.770 ","End":"03:05.555","Text":"this is what we will get."},{"Start":"03:05.555 ","End":"03:11.960","Text":"Now notice that the only way that f prime of x can be infinity or 10 to infinity,"},{"Start":"03:11.960 ","End":"03:15.215","Text":"is if the denominator goes to 0."},{"Start":"03:15.215 ","End":"03:18.950","Text":"The only place that can happen is at x equals 4."},{"Start":"03:18.950 ","End":"03:21.190","Text":"If x goes to anything other than 4,"},{"Start":"03:21.190 ","End":"03:23.355","Text":"then this will not be 0."},{"Start":"03:23.355 ","End":"03:28.260","Text":"Let\u0027s suggest, is that x_1 equals 4, and we\u0027ll check."},{"Start":"03:28.260 ","End":"03:32.390","Text":"In fact, we only have to check one of the limits on one side,"},{"Start":"03:32.390 ","End":"03:36.050","Text":"but we\u0027ll have to do both sides in order to answer the question."},{"Start":"03:36.050 ","End":"03:39.940","Text":"First of all, let\u0027s see what happens if we put x_1 equals 4."},{"Start":"03:39.940 ","End":"03:43.100","Text":"Then the limit as x goes to 4,"},{"Start":"03:43.100 ","End":"03:45.695","Text":"let\u0027s try it from the right first of all,"},{"Start":"03:45.695 ","End":"03:52.490","Text":"of f prime of x is equal to the limit as x goes to 4 from the right"},{"Start":"03:52.490 ","End":"04:00.350","Text":"of minus 1 over 5 times the 5th root of 4 minus x to the 4th."},{"Start":"04:00.350 ","End":"04:03.770","Text":"What happens? X goes to 4 from the right,"},{"Start":"04:03.770 ","End":"04:08.270","Text":"4 plus 4 minus x is negative 0."},{"Start":"04:08.270 ","End":"04:14.805","Text":"We get minus 1 over 5 times the 5th root,"},{"Start":"04:14.805 ","End":"04:17.655","Text":"0 minus to the 4th,"},{"Start":"04:17.655 ","End":"04:22.410","Text":"which equals, 0 minus to the 4th is just 0 plus."},{"Start":"04:22.410 ","End":"04:25.280","Text":"Let\u0027s split it up, we get minus 1 over 5."},{"Start":"04:25.280 ","End":"04:27.200","Text":"Let\u0027s put that to the side a minute,"},{"Start":"04:27.200 ","End":"04:29.285","Text":"times 1 over,"},{"Start":"04:29.285 ","End":"04:33.255","Text":"the 5th root of 0 plus is just 0 plus."},{"Start":"04:33.255 ","End":"04:35.505","Text":"1 over 0 plus is infinity,"},{"Start":"04:35.505 ","End":"04:39.905","Text":"so it\u0027s minus 1/5 times infinity,"},{"Start":"04:39.905 ","End":"04:42.320","Text":"which is minus infinity."},{"Start":"04:42.320 ","End":"04:46.220","Text":"This is enough to show that the tangent line is"},{"Start":"04:46.220 ","End":"04:50.300","Text":"vertical because if we get infinity or minus infinity on either side,"},{"Start":"04:50.300 ","End":"04:52.400","Text":"that\u0027s enough. But let\u0027s continue."},{"Start":"04:52.400 ","End":"04:57.499","Text":"We can actually show that the two-sided limit is equal to minus infinity."},{"Start":"04:57.499 ","End":"04:59.480","Text":"I\u0027ve done the bit on the right,"},{"Start":"04:59.480 ","End":"05:01.550","Text":"but what if it was the limit to the left?"},{"Start":"05:01.550 ","End":"05:05.765","Text":"Suppose I had here that x was not going to 4 plus,"},{"Start":"05:05.765 ","End":"05:07.870","Text":"but to 4 from the left."},{"Start":"05:07.870 ","End":"05:10.980","Text":"Then 4 from the left would be 4 minus,"},{"Start":"05:10.980 ","End":"05:14.235","Text":"something less than 4 would be 0 plus,"},{"Start":"05:14.235 ","End":"05:16.185","Text":"slightly bigger than 0."},{"Start":"05:16.185 ","End":"05:18.590","Text":"If we had a 0 plus here,"},{"Start":"05:18.590 ","End":"05:20.180","Text":"instead of a 0 minus,"},{"Start":"05:20.180 ","End":"05:25.055","Text":"wouldn\u0027t make any difference because of the power of the 4 is still equal to 0 plus."},{"Start":"05:25.055 ","End":"05:26.800","Text":"Exactly the same limit,"},{"Start":"05:26.800 ","End":"05:28.400","Text":"when we will go from the right."},{"Start":"05:28.400 ","End":"05:30.335","Text":"If we go from the left on the right,"},{"Start":"05:30.335 ","End":"05:32.345","Text":"then that means that that\u0027s the limit."},{"Start":"05:32.345 ","End":"05:41.265","Text":"What we can write is that the limit as x goes to 4 two-sided of f prime of x,"},{"Start":"05:41.265 ","End":"05:44.210","Text":"is equal to minus infinity."},{"Start":"05:44.210 ","End":"05:49.640","Text":"Whenever we have a two-sided limit at our point at say x_1 is less 4,"},{"Start":"05:49.640 ","End":"05:53.300","Text":"it\u0027s either infinity or minus infinity in both cases,"},{"Start":"05:53.300 ","End":"05:56.150","Text":"then it means that we have a vertical tangent."},{"Start":"05:56.150 ","End":"05:59.800","Text":"That means that x equals 4 is a vertical tangent."},{"Start":"05:59.800 ","End":"06:04.295","Text":"There\u0027s no other points to consider for a tangent which is vertical,"},{"Start":"06:04.295 ","End":"06:06.680","Text":"so x_1 is 4 is a vertical tangent,"},{"Start":"06:06.680 ","End":"06:08.945","Text":"it\u0027s the only x that\u0027s interesting."},{"Start":"06:08.945 ","End":"06:10.505","Text":"If it\u0027s a vertical tangent,"},{"Start":"06:10.505 ","End":"06:12.260","Text":"it\u0027s not a vertical cusp."},{"Start":"06:12.260 ","End":"06:13.805","Text":"It can\u0027t be both."},{"Start":"06:13.805 ","End":"06:16.670","Text":"If I want to summarize what I have so far,"},{"Start":"06:16.670 ","End":"06:22.490","Text":"and this is a vertical tangent and x_1 equals 4 is where the tangent is vertical,"},{"Start":"06:22.490 ","End":"06:24.605","Text":"and nowhere is there a vertical cusp,"},{"Start":"06:24.605 ","End":"06:27.980","Text":"because it can\u0027t be both a vertical tangent and a vertical cusp."},{"Start":"06:27.980 ","End":"06:31.315","Text":"That answers part A of the question."},{"Start":"06:31.315 ","End":"06:34.050","Text":"Part A and part B,"},{"Start":"06:34.050 ","End":"06:36.635","Text":"by which I mean that if you look back at what they asked,"},{"Start":"06:36.635 ","End":"06:38.690","Text":"where is the tangent vertical,"},{"Start":"06:38.690 ","End":"06:41.420","Text":"that was part A of the question,"},{"Start":"06:41.420 ","End":"06:43.605","Text":"and does the function has a vertical cusp,"},{"Start":"06:43.605 ","End":"06:44.730","Text":"that was part B,"},{"Start":"06:44.730 ","End":"06:46.620","Text":"so we\u0027ve answered both of them."},{"Start":"06:46.620 ","End":"06:48.780","Text":"Here\u0027s the vertical tangent,"},{"Start":"06:48.780 ","End":"06:51.350","Text":"the vertical tangent is x equals 4,"},{"Start":"06:51.350 ","End":"06:54.100","Text":"and we can see that it\u0027s not a vertical cusp,"},{"Start":"06:54.100 ","End":"06:56.130","Text":"but otherwise it would be something like this,"},{"Start":"06:56.130 ","End":"06:57.755","Text":"would be a sharp corner."},{"Start":"06:57.755 ","End":"07:01.260","Text":"Everything makes sense and that\u0027s it."}],"ID":6473},{"Watched":false,"Name":"Exercise 2","Duration":"4m 27s","ChapterTopicVideoID":6447,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have to find all the points on"},{"Start":"00:03.360 ","End":"00:06.825","Text":"the graph y equals square root of x plus cube root of x,"},{"Start":"00:06.825 ","End":"00:08.969","Text":"where the tangent line is vertical."},{"Start":"00:08.969 ","End":"00:11.295","Text":"When we\u0027ve done that, we have to answer the question,"},{"Start":"00:11.295 ","End":"00:13.860","Text":"does the function have a vertical cusp?"},{"Start":"00:13.860 ","End":"00:18.000","Text":"I wrote it in terms of f of x and I\u0027d like you to note that"},{"Start":"00:18.000 ","End":"00:23.970","Text":"the square root of x has a domain of x bigger or equal to 0."},{"Start":"00:23.970 ","End":"00:28.125","Text":"Let\u0027s take a look at the sketch that I\u0027ve brought with me might help."},{"Start":"00:28.125 ","End":"00:29.625","Text":"Here\u0027s our function."},{"Start":"00:29.625 ","End":"00:32.190","Text":"We noticed that at 0, 0 here,"},{"Start":"00:32.190 ","End":"00:35.415","Text":"the curve straightens out to be vertical."},{"Start":"00:35.415 ","End":"00:38.380","Text":"Here in fact is the tangent line,"},{"Start":"00:38.380 ","End":"00:40.160","Text":"this dotted green line."},{"Start":"00:40.160 ","End":"00:43.850","Text":"You see, it is defined only for x bigger or equal to 0."},{"Start":"00:43.850 ","End":"00:46.745","Text":"First thing to do is to find the derivative."},{"Start":"00:46.745 ","End":"00:49.730","Text":"Let\u0027s find f prime of x, which is equal."},{"Start":"00:49.730 ","End":"00:51.305","Text":"It\u0027s the sum of these 2 things."},{"Start":"00:51.305 ","End":"00:55.200","Text":"The derivative of square root of x is a well-known 1."},{"Start":"00:55.200 ","End":"00:57.545","Text":"1 over twice the square root of x."},{"Start":"00:57.545 ","End":"00:59.690","Text":"This is also known, but if you don\u0027t know it,"},{"Start":"00:59.690 ","End":"01:03.780","Text":"you can use the technique of x to the power of 1/3 and"},{"Start":"01:03.780 ","End":"01:08.125","Text":"differentiate it,1/3 x to the power of minus 2/3."},{"Start":"01:08.125 ","End":"01:09.725","Text":"After you\u0027ve simplified it,"},{"Start":"01:09.725 ","End":"01:14.930","Text":"you get 1/3 square root of x squared."},{"Start":"01:14.930 ","End":"01:18.290","Text":"This holds certainly for x bigger than 0."},{"Start":"01:18.290 ","End":"01:20.885","Text":"It makes no sense to put 0 here."},{"Start":"01:20.885 ","End":"01:23.600","Text":"But already, we see that something happens at x equals 0,"},{"Start":"01:23.600 ","End":"01:25.265","Text":"just like we saw in the picture."},{"Start":"01:25.265 ","End":"01:27.295","Text":"That\u0027s the only interesting point."},{"Start":"01:27.295 ","End":"01:28.610","Text":"If we\u0027re going to find the limit,"},{"Start":"01:28.610 ","End":"01:30.410","Text":"we don\u0027t need x equals 0,"},{"Start":"01:30.410 ","End":"01:32.690","Text":"we only need x close to 0."},{"Start":"01:32.690 ","End":"01:34.610","Text":"Let\u0027s see if we can find the limit,"},{"Start":"01:34.610 ","End":"01:36.605","Text":"either one-sided or two-sided."},{"Start":"01:36.605 ","End":"01:41.555","Text":"X equals 0 is our suspect point for tangent line which is vertical."},{"Start":"01:41.555 ","End":"01:44.254","Text":"Let\u0027s see, this will be so if the derivative"},{"Start":"01:44.254 ","End":"01:46.910","Text":"is either plus or minus infinity on either side."},{"Start":"01:46.910 ","End":"01:51.965","Text":"Let\u0027s take the limit on the right-hand side of f prime of x."},{"Start":"01:51.965 ","End":"01:57.705","Text":"Now as x goes to 0 from this plus side, slight mistake."},{"Start":"01:57.705 ","End":"02:01.175","Text":"It\u0027s a 3 here, but there\u0027s also a cube root here."},{"Start":"02:01.175 ","End":"02:03.890","Text":"Now, x is 0 positive,"},{"Start":"02:03.890 ","End":"02:05.750","Text":"so as x goes to 0,"},{"Start":"02:05.750 ","End":"02:10.685","Text":"this goes to 0 plus the square root of 0 plus 0 plus,"},{"Start":"02:10.685 ","End":"02:14.465","Text":"it\u0027s 1/2 times 0 plus."},{"Start":"02:14.465 ","End":"02:21.110","Text":"This one is 1/3 times x is 0 plus 0 is x squared is equal to 0 plus,"},{"Start":"02:21.110 ","End":"02:23.060","Text":"and the cube root is also 0 plus."},{"Start":"02:23.060 ","End":"02:25.279","Text":"Because 1/0 plus infinity,"},{"Start":"02:25.279 ","End":"02:27.505","Text":"it comes out to be plus infinity."},{"Start":"02:27.505 ","End":"02:29.989","Text":"I just put the plus for emphasis."},{"Start":"02:29.989 ","End":"02:34.010","Text":"That already means if we have one-sided limit which is plus or minus infinity,"},{"Start":"02:34.010 ","End":"02:37.430","Text":"the tangent line is vertical at x equals 0."},{"Start":"02:37.430 ","End":"02:40.310","Text":"Let\u0027s call this part a and this is part b."},{"Start":"02:40.310 ","End":"02:41.765","Text":"We\u0027ve already answered part a."},{"Start":"02:41.765 ","End":"02:49.310","Text":"Part a, the tangent line is vertical at x equals 0."},{"Start":"02:49.310 ","End":"02:51.050","Text":"Now as for part b,"},{"Start":"02:51.050 ","End":"02:56.135","Text":"let\u0027s see if we can find the limit as x goes to 0."},{"Start":"02:56.135 ","End":"03:00.665","Text":"X is only defined on the right-hand side of the 0."},{"Start":"03:00.665 ","End":"03:06.980","Text":"The limit as x goes to 0 of f prime of x does exist,"},{"Start":"03:06.980 ","End":"03:13.700","Text":"and it\u0027s equal to the limit as x goes to 0 plus of f prime of x."},{"Start":"03:13.700 ","End":"03:17.525","Text":"You see, this thing is defined when x is bigger than 0."},{"Start":"03:17.525 ","End":"03:20.015","Text":"It isn\u0027t defined when x is less than 0."},{"Start":"03:20.015 ","End":"03:23.240","Text":"The limit is that x goes to the point,"},{"Start":"03:23.240 ","End":"03:29.630","Text":"but only remaining within the domain forces it to be bigger than 0."},{"Start":"03:29.630 ","End":"03:31.640","Text":"The two-sided limits, so to speak,"},{"Start":"03:31.640 ","End":"03:35.465","Text":"or just the limit is equal to the one-sided limit,"},{"Start":"03:35.465 ","End":"03:39.035","Text":"which we already computed as plus infinity."},{"Start":"03:39.035 ","End":"03:43.505","Text":"This is a condition for there being a vertical tangent."},{"Start":"03:43.505 ","End":"03:46.160","Text":"In actual fact, it has a vertical tangent,"},{"Start":"03:46.160 ","End":"03:48.305","Text":"so it doesn\u0027t have a vertical cusp."},{"Start":"03:48.305 ","End":"03:57.769","Text":"Number b, x equals 0 is actually a vertical tangent and not a vertical cusp."},{"Start":"03:57.769 ","End":"04:01.520","Text":"The other way of looking at it is that there is no limit when x goes"},{"Start":"04:01.520 ","End":"04:05.920","Text":"to 0 minus of f prime of x."},{"Start":"04:05.920 ","End":"04:08.270","Text":"You can\u0027t take the limit as x goes to 0"},{"Start":"04:08.270 ","End":"04:10.760","Text":"from the left because there are no values on the left,"},{"Start":"04:10.760 ","End":"04:11.960","Text":"it\u0027s not defined there."},{"Start":"04:11.960 ","End":"04:15.635","Text":"The answer here is specifically not a vertical cusp."},{"Start":"04:15.635 ","End":"04:18.095","Text":"Since x equals 0 is the only place,"},{"Start":"04:18.095 ","End":"04:21.250","Text":"there is no vertical cusp."},{"Start":"04:21.250 ","End":"04:22.625","Text":"Here\u0027s the 2 answers,"},{"Start":"04:22.625 ","End":"04:24.440","Text":"x equals 0 for tangent line,"},{"Start":"04:24.440 ","End":"04:27.810","Text":"which is vertical and there\u0027s no vertical cusp. We\u0027re done."}],"ID":6474},{"Watched":false,"Name":"Exercise 3","Duration":"3m 17s","ChapterTopicVideoID":6448,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have to find all the points on"},{"Start":"00:03.360 ","End":"00:06.465","Text":"the graph y equals cube root of x squared,"},{"Start":"00:06.465 ","End":"00:08.820","Text":"where the tangent line is vertical."},{"Start":"00:08.820 ","End":"00:11.130","Text":"That\u0027s part a of the problem,"},{"Start":"00:11.130 ","End":"00:12.870","Text":"there\u0027s also a part b,"},{"Start":"00:12.870 ","End":"00:15.435","Text":"does the function have a vertical cusp?"},{"Start":"00:15.435 ","End":"00:17.640","Text":"I like to use the functional notation,"},{"Start":"00:17.640 ","End":"00:19.260","Text":"so I\u0027ll write it like this,"},{"Start":"00:19.260 ","End":"00:23.025","Text":"which is equal to the cube root of x squared,"},{"Start":"00:23.025 ","End":"00:27.785","Text":"and I\u0027d like to give you a sketch to see what the outcome is."},{"Start":"00:27.785 ","End":"00:29.960","Text":"It looks like there\u0027s a dotted line here,"},{"Start":"00:29.960 ","End":"00:32.510","Text":"that tangent is vertical here,"},{"Start":"00:32.510 ","End":"00:34.640","Text":"both on the right and on the left,"},{"Start":"00:34.640 ","End":"00:39.560","Text":"and this is the point where x is 0 and y is 0."},{"Start":"00:39.560 ","End":"00:42.710","Text":"The function is defined for all x and it looks like we have"},{"Start":"00:42.710 ","End":"00:44.990","Text":"a vertical cusp because it looks like we\u0027ll get"},{"Start":"00:44.990 ","End":"00:48.185","Text":"minus infinity here and plus infinity on this side."},{"Start":"00:48.185 ","End":"00:54.410","Text":"All right, then, so y prime is f prime of x and it\u0027s equal to,"},{"Start":"00:54.410 ","End":"00:56.090","Text":"we can do this in our heads, look,"},{"Start":"00:56.090 ","End":"00:59.120","Text":"it\u0027s x to the power of 2 over 3,"},{"Start":"00:59.120 ","End":"01:02.240","Text":"so the derivative is 2 over 3,"},{"Start":"01:02.240 ","End":"01:05.230","Text":"x to the power of minus a 1/3,"},{"Start":"01:05.230 ","End":"01:11.210","Text":"x to the power of minus a 1/3 is 1 over the cube root of x."},{"Start":"01:11.210 ","End":"01:13.070","Text":"The minus puts it in the denominator,"},{"Start":"01:13.070 ","End":"01:15.440","Text":"and the 1/3 makes it a cube root."},{"Start":"01:15.440 ","End":"01:18.005","Text":"As I said, it\u0027s defined for all x."},{"Start":"01:18.005 ","End":"01:25.040","Text":"What we were looking for places where y prime goes to minus infinity or plus infinity,"},{"Start":"01:25.040 ","End":"01:29.450","Text":"and the only possible pace I would think if looking for that is where x is 0,"},{"Start":"01:29.450 ","End":"01:30.665","Text":"I mean everything else,"},{"Start":"01:30.665 ","End":"01:37.090","Text":"no problem, so x equals 0 is a candidate for tangent line is vertical,"},{"Start":"01:37.090 ","End":"01:39.350","Text":"so let\u0027s check what happens at x equals 0."},{"Start":"01:39.350 ","End":"01:42.065","Text":"Let\u0027s take the limit from the left and the limit from the right,"},{"Start":"01:42.065 ","End":"01:44.150","Text":"the limit as x goes to,"},{"Start":"01:44.150 ","End":"01:49.190","Text":"let\u0027s say the right first of f prime of x."},{"Start":"01:49.190 ","End":"01:54.320","Text":"Now if x goes to 0 plus the cube root of plus is plus,"},{"Start":"01:54.320 ","End":"02:02.315","Text":"so it\u0027s also 0 plus and 1 over 0 plus is infinity times the 2/3 still infinity."},{"Start":"02:02.315 ","End":"02:04.789","Text":"I\u0027ll just write it here as plus infinity."},{"Start":"02:04.789 ","End":"02:10.280","Text":"Basically what I did was just substitute x equals 0 plus here,"},{"Start":"02:10.280 ","End":"02:11.765","Text":"and then I got the infinity."},{"Start":"02:11.765 ","End":"02:16.520","Text":"What would happen if I put x goes to 0 from the left,"},{"Start":"02:16.520 ","End":"02:18.800","Text":"0 from the left would mean same thing,"},{"Start":"02:18.800 ","End":"02:21.830","Text":"except that we\u0027d have x equals minus,"},{"Start":"02:21.830 ","End":"02:23.270","Text":"so the whole thing just changes,"},{"Start":"02:23.270 ","End":"02:26.090","Text":"cube root of minus 0 is minus 0,"},{"Start":"02:26.090 ","End":"02:27.875","Text":"we\u0027ll get minus infinity,"},{"Start":"02:27.875 ","End":"02:29.515","Text":"we\u0027ll just write similarly."},{"Start":"02:29.515 ","End":"02:33.905","Text":"This means that we have exactly the conditions for a cusp"},{"Start":"02:33.905 ","End":"02:36.170","Text":"because the limit on one side is plus infinity"},{"Start":"02:36.170 ","End":"02:38.510","Text":"in the limit on the other side is minus infinity,"},{"Start":"02:38.510 ","End":"02:41.605","Text":"or it could have been vice versa, only different signs."},{"Start":"02:41.605 ","End":"02:43.490","Text":"Other words, in part a,"},{"Start":"02:43.490 ","End":"02:49.745","Text":"we have a vertical tangent at x equals 0,"},{"Start":"02:49.745 ","End":"02:55.295","Text":"both 0 plus and 0 minus just to add to that,"},{"Start":"02:55.295 ","End":"03:02.065","Text":"and b, we have a vertical cusp at x equals 0."},{"Start":"03:02.065 ","End":"03:03.949","Text":"Maybe it could be more precise,"},{"Start":"03:03.949 ","End":"03:07.040","Text":"if they say the point not equals x equals 0,"},{"Start":"03:07.040 ","End":"03:11.945","Text":"we can substitute and get y is also 0, i.e."},{"Start":"03:11.945 ","End":"03:14.270","Text":"0,0, just to be more precise."},{"Start":"03:14.270 ","End":"03:18.000","Text":"Other than that, we\u0027re done with this exercise."}],"ID":6475},{"Watched":false,"Name":"Exercise 4","Duration":"3m 54s","ChapterTopicVideoID":6450,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.680","Text":"In this exercise, we have to find all the points on this graph,"},{"Start":"00:03.680 ","End":"00:07.185","Text":"y equals absolute value of x cubed minus 27,"},{"Start":"00:07.185 ","End":"00:09.135","Text":"where the tangent line is vertical."},{"Start":"00:09.135 ","End":"00:10.860","Text":"There\u0027s another part where we\u0027re asked,"},{"Start":"00:10.860 ","End":"00:13.155","Text":"does the function have a vertical cusp?"},{"Start":"00:13.155 ","End":"00:15.285","Text":"Let\u0027s begin with the first part."},{"Start":"00:15.285 ","End":"00:17.280","Text":"Because of the absolute value,"},{"Start":"00:17.280 ","End":"00:20.789","Text":"we should write this as a split definition function."},{"Start":"00:20.789 ","End":"00:25.725","Text":"Something happens around x equals 3, yx equals 3."},{"Start":"00:25.725 ","End":"00:28.260","Text":"Because around the place where it\u0027s 0,"},{"Start":"00:28.260 ","End":"00:32.220","Text":"x cubed minus 27 is 0 when x equals 3."},{"Start":"00:32.220 ","End":"00:34.830","Text":"In actual fact, if x is bigger than 3,"},{"Start":"00:34.830 ","End":"00:39.140","Text":"then x cubed would be bigger than 27, it will be positive,"},{"Start":"00:39.140 ","End":"00:40.490","Text":"and if x is less than 3,"},{"Start":"00:40.490 ","End":"00:42.335","Text":"x cubed will be less than 27,"},{"Start":"00:42.335 ","End":"00:43.714","Text":"the sign will be negative."},{"Start":"00:43.714 ","End":"00:46.165","Text":"I can write it as a split definition."},{"Start":"00:46.165 ","End":"00:48.980","Text":"To say that the absolute value of where it\u0027s"},{"Start":"00:48.980 ","End":"00:52.940","Text":"positive and where it\u0027s positive is bigger than 3,"},{"Start":"00:52.940 ","End":"01:01.625","Text":"it\u0027s equal to just x cubed minus 27 itself when x is bigger than or bigger or equal to 3."},{"Start":"01:01.625 ","End":"01:05.210","Text":"If x is less than 3 or less than or equal to 3,"},{"Start":"01:05.210 ","End":"01:08.765","Text":"what we have is the other way around for x less than 3,"},{"Start":"01:08.765 ","End":"01:10.940","Text":"x cubed is going to be less than 27."},{"Start":"01:10.940 ","End":"01:11.960","Text":"So this is negative,"},{"Start":"01:11.960 ","End":"01:14.530","Text":"so it has to be minus this thing."},{"Start":"01:14.530 ","End":"01:15.760","Text":"You can put a minus in front"},{"Start":"01:15.760 ","End":"01:17.405","Text":"or I can reverse the order,"},{"Start":"01:17.405 ","End":"01:20.355","Text":"I\u0027ll choose to reverse the order."},{"Start":"01:20.355 ","End":"01:22.965","Text":"Here\u0027s x equals 3,"},{"Start":"01:22.965 ","End":"01:24.535","Text":"and an x is 3,"},{"Start":"01:24.535 ","End":"01:27.050","Text":"the function is always positive because the absolute value."},{"Start":"01:27.050 ","End":"01:31.295","Text":"But something behaves according to 1 definition here and to 1 definition here."},{"Start":"01:31.295 ","End":"01:33.890","Text":"Visually, this looks a bit like a cusp,"},{"Start":"01:33.890 ","End":"01:36.155","Text":"but if you look closer,"},{"Start":"01:36.155 ","End":"01:41.300","Text":"this slope of this tangent is not vertical and nor is it here steep,"},{"Start":"01:41.300 ","End":"01:42.800","Text":"but it\u0027s not vertical."},{"Start":"01:42.800 ","End":"01:44.270","Text":"At least according to the picture,"},{"Start":"01:44.270 ","End":"01:47.810","Text":"we\u0027re expecting not to have any vertical tangent lines,"},{"Start":"01:47.810 ","End":"01:49.885","Text":"and therefore, no cusp."},{"Start":"01:49.885 ","End":"01:53.270","Text":"Let\u0027s figure out what f prime is."},{"Start":"01:53.270 ","End":"01:56.240","Text":"f prime of x is equal to,"},{"Start":"01:56.240 ","End":"02:01.790","Text":"and we have a split definition with 1 of these borders or seam lines at x equals 3,"},{"Start":"02:01.790 ","End":"02:04.865","Text":"then we don\u0027t really know the derivative or even if there is 1."},{"Start":"02:04.865 ","End":"02:07.875","Text":"Let\u0027s leave x equals 3 separately."},{"Start":"02:07.875 ","End":"02:10.825","Text":"What we can say is that when x is bigger or equal to 3,"},{"Start":"02:10.825 ","End":"02:12.940","Text":"this is what? 3x squared."},{"Start":"02:12.940 ","End":"02:17.380","Text":"It\u0027s 3x squared when x is bigger than 3"},{"Start":"02:17.380 ","End":"02:22.760","Text":"and x equals 3 exactly because it\u0027s 1 of these borders seam lines,"},{"Start":"02:22.760 ","End":"02:25.190","Text":"we don\u0027t know, I\u0027ll leave it as 3 question marks."},{"Start":"02:25.190 ","End":"02:28.220","Text":"When x is less than 3,"},{"Start":"02:28.220 ","End":"02:30.350","Text":"then we take a function from here."},{"Start":"02:30.350 ","End":"02:33.710","Text":"The derivative is minus 3x squared."},{"Start":"02:33.710 ","End":"02:38.765","Text":"What I need is the limit on the left and the right of 3."},{"Start":"02:38.765 ","End":"02:41.615","Text":"Why do I only consider the 0.3?"},{"Start":"02:41.615 ","End":"02:46.440","Text":"Because everywhere else, 3x squared is never going to be infinity,"},{"Start":"02:46.440 ","End":"02:48.470","Text":"3x squared, whatever it is,"},{"Start":"02:48.470 ","End":"02:51.875","Text":"it\u0027s just a number, it\u0027s not equal to infinity for any x."},{"Start":"02:51.875 ","End":"02:55.445","Text":"Likewise, minus 3x squared is not equal to infinity."},{"Start":"02:55.445 ","End":"02:59.625","Text":"The only place left to look is possibly at x equals 3."},{"Start":"02:59.625 ","End":"03:01.070","Text":"Let\u0027s see what happens."},{"Start":"03:01.070 ","End":"03:03.095","Text":"Take the left and the right limit."},{"Start":"03:03.095 ","End":"03:09.485","Text":"The limit as x goes to 3 from the right of 3x squared."},{"Start":"03:09.485 ","End":"03:13.715","Text":"That equals 3 times 3 squared, which is 27."},{"Start":"03:13.715 ","End":"03:19.160","Text":"The limit as x goes to 3 from the left of f prime of x,"},{"Start":"03:19.160 ","End":"03:24.320","Text":"which is minus 3x squared is minus 27."},{"Start":"03:24.320 ","End":"03:27.575","Text":"This is not equal to plus or minus infinity,"},{"Start":"03:27.575 ","End":"03:31.550","Text":"and this is not equal to plus or minus infinity."},{"Start":"03:31.550 ","End":"03:34.460","Text":"If we don\u0027t have a plus or minus infinity,"},{"Start":"03:34.460 ","End":"03:37.130","Text":"then we don\u0027t have a tangent which is vertical."},{"Start":"03:37.130 ","End":"03:41.160","Text":"We have neither a vertical tangent nor a vertical cusp."},{"Start":"03:41.160 ","End":"03:49.015","Text":"The tangent line is never vertical, no vertical cusp."},{"Start":"03:49.015 ","End":"03:52.010","Text":"That answers both the questions we were asked."},{"Start":"03:52.010 ","End":"03:55.350","Text":"No such points, the answer here is no."}],"ID":6477},{"Watched":false,"Name":"Exercise 5","Duration":"6m 43s","ChapterTopicVideoID":6451,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise we have to find all the points on the graph y equals"},{"Start":"00:04.590 ","End":"00:09.090","Text":"square root of 4 minus x squared where the tangent line is vertical,"},{"Start":"00:09.090 ","End":"00:14.775","Text":"and we also have to answer the question does the function have a vertical cusp?"},{"Start":"00:14.775 ","End":"00:19.440","Text":"I like to use the function form that of y call it also f of x,"},{"Start":"00:19.440 ","End":"00:23.730","Text":"which is the square root of 4 minus x squared."},{"Start":"00:23.730 ","End":"00:27.450","Text":"It\u0027s important to note the domain of definition,"},{"Start":"00:27.450 ","End":"00:30.800","Text":"what under the square root sign has to be bigger or equal to"},{"Start":"00:30.800 ","End":"00:35.120","Text":"0 which means that x squared is going to be less than or equal to 4."},{"Start":"00:35.120 ","End":"00:36.830","Text":"If x squared is less than or equal to 4,"},{"Start":"00:36.830 ","End":"00:41.090","Text":"then x is going to be between minus 2 and 2."},{"Start":"00:41.090 ","End":"00:46.625","Text":"Before we continue, let\u0027s take a look at a sketch that I\u0027ve brought with me."},{"Start":"00:46.625 ","End":"00:50.750","Text":"This is the point where x is minus 2,"},{"Start":"00:50.750 ","End":"00:53.240","Text":"and this is 2, this is x,"},{"Start":"00:53.240 ","End":"00:59.975","Text":"this is y. Y equals square root of 4 minus x squared."},{"Start":"00:59.975 ","End":"01:01.760","Text":"From the picture; any rate,"},{"Start":"01:01.760 ","End":"01:05.480","Text":"the only possibility it looks like 2 and minus 2 and it looks like"},{"Start":"01:05.480 ","End":"01:09.710","Text":"here we have a vertical tangent and here we have a vertical tangent."},{"Start":"01:09.710 ","End":"01:12.850","Text":"Now the function is not even defined outside here,"},{"Start":"01:12.850 ","End":"01:15.530","Text":"so we won\u0027t have a two-sided limit."},{"Start":"01:15.530 ","End":"01:16.850","Text":"Basically, we won\u0027t get"},{"Start":"01:16.850 ","End":"01:21.305","Text":"a vertical cusp because the function has to be defined on both sides."},{"Start":"01:21.305 ","End":"01:24.850","Text":"Alternately, plus and minus infinity here it\u0027s not defined at all,"},{"Start":"01:24.850 ","End":"01:29.410","Text":"it\u0027s like there is a blank area here and a blank area here,"},{"Start":"01:29.410 ","End":"01:32.510","Text":"so we will not have a vertical cusp anywhere."},{"Start":"01:32.510 ","End":"01:35.690","Text":"It looks like we\u0027ll get a vertical tangent line,"},{"Start":"01:35.690 ","End":"01:38.465","Text":"and technically this will be a vertical tangent."},{"Start":"01:38.465 ","End":"01:43.630","Text":"We don\u0027t need a two-sided limit when the function is only defined on one side."},{"Start":"01:43.630 ","End":"01:46.710","Text":"Whatever it is you need the derivative for this question,"},{"Start":"01:46.710 ","End":"01:52.790","Text":"so let\u0027s go with the derivative y prime which is also f prime of x. I get 1"},{"Start":"01:52.790 ","End":"01:59.610","Text":"over twice the square root of the same thing times the internal derivative minus 2x."},{"Start":"01:59.610 ","End":"02:01.540","Text":"What basically this equals to,"},{"Start":"02:01.540 ","End":"02:07.745","Text":"is minus x over the square root of 4 minus x squared."},{"Start":"02:07.745 ","End":"02:10.535","Text":"Everywhere between minus 2 and 2,"},{"Start":"02:10.535 ","End":"02:14.215","Text":"we\u0027re going to get x squared which is less than 4 and is defined."},{"Start":"02:14.215 ","End":"02:16.295","Text":"Whatever it is it\u0027ll give us a number,"},{"Start":"02:16.295 ","End":"02:19.080","Text":"so there\u0027s no infinity in the middle;"},{"Start":"02:19.080 ","End":"02:20.760","Text":"between minus 2 and 2."},{"Start":"02:20.760 ","End":"02:27.350","Text":"Minus 2 and 2 themselves I cannot substitute because if I put x equals minus 2 or 2,"},{"Start":"02:27.350 ","End":"02:31.370","Text":"I get minus x over 0 which doesn\u0027t work."},{"Start":"02:31.370 ","End":"02:35.060","Text":"You have to note that here we have strict inequality;"},{"Start":"02:35.060 ","End":"02:37.700","Text":"less than x, less than 2,"},{"Start":"02:37.700 ","End":"02:41.120","Text":"and here there\u0027s no tangent lines which are vertical."},{"Start":"02:41.120 ","End":"02:46.120","Text":"The only two places left to look are 2 and minus 2 themselves,"},{"Start":"02:46.120 ","End":"02:50.389","Text":"so we\u0027re going to check only at x equals"},{"Start":"02:50.389 ","End":"02:56.215","Text":"2 and x equals minus 2 for possible tangent lines that are vertical."},{"Start":"02:56.215 ","End":"03:02.000","Text":"Let\u0027s take the limit as x goes to 2 of f prime of x."},{"Start":"03:02.000 ","End":"03:07.520","Text":"Since the function is only defined on the left of x equals 2; it\u0027s not defined,"},{"Start":"03:07.520 ","End":"03:09.860","Text":"then limit is considered to be"},{"Start":"03:09.860 ","End":"03:12.600","Text":"the one-sided limit because it\u0027s"},{"Start":"03:12.600 ","End":"03:15.855","Text":"not defined anywhere else and that we can get to from below."},{"Start":"03:15.855 ","End":"03:20.750","Text":"As x goes to 2 from below of f prime of x,"},{"Start":"03:20.750 ","End":"03:27.500","Text":"if I let x equals 2 from below then 2 minus x squared is going to be slightly positive."},{"Start":"03:27.500 ","End":"03:32.120","Text":"For example, if x was 1.9999 and squared,"},{"Start":"03:32.120 ","End":"03:37.620","Text":"it would be 3.99 something and this would be 0 plus."},{"Start":"03:37.620 ","End":"03:42.500","Text":"If I let x equal 2 from below I\u0027m going to get minus 2 over"},{"Start":"03:42.500 ","End":"03:48.980","Text":"the square root of 0 plus.1 over 0 plus is infinity."},{"Start":"03:48.980 ","End":"03:50.890","Text":"Infinity times a negative number,"},{"Start":"03:50.890 ","End":"03:53.270","Text":"this is minus infinity."},{"Start":"03:53.270 ","End":"03:57.700","Text":"At x equals 2, ie,"},{"Start":"03:57.700 ","End":"04:00.120","Text":"the x and the y. Ie at 2,"},{"Start":"04:00.120 ","End":"04:02.630","Text":"0 because if you put x equals 2 here,"},{"Start":"04:02.630 ","End":"04:04.820","Text":"you get y equals 0."},{"Start":"04:04.820 ","End":"04:11.800","Text":"At this point we know because of the infinity that the tangent is vertical."},{"Start":"04:11.800 ","End":"04:14.570","Text":"It doesn\u0027t matter plus infinity or minus infinity,"},{"Start":"04:14.570 ","End":"04:16.400","Text":"it makes it vertical."},{"Start":"04:16.400 ","End":"04:23.765","Text":"Now it\u0027s almost exactly the same when you try it with x equals minus 2."},{"Start":"04:23.765 ","End":"04:28.900","Text":"When x is minus 2 you get the very same thing and you\u0027ll find that;"},{"Start":"04:28.900 ","End":"04:30.605","Text":"I\u0027ll leave it to you to do,"},{"Start":"04:30.605 ","End":"04:32.920","Text":"that x equals minus 2."},{"Start":"04:32.920 ","End":"04:36.375","Text":"Ie point minus 2, 0,"},{"Start":"04:36.375 ","End":"04:38.655","Text":"also the tangent is vertical,"},{"Start":"04:38.655 ","End":"04:40.895","Text":"but we can do it in a very similar way."},{"Start":"04:40.895 ","End":"04:44.180","Text":"When x is minus 2 the limit is actually the limit"},{"Start":"04:44.180 ","End":"04:48.055","Text":"from the right so you get minus 2 from the right,"},{"Start":"04:48.055 ","End":"04:54.110","Text":"but minus 2 from the right would be like minus 1.999."},{"Start":"04:54.110 ","End":"04:55.775","Text":"If you square that,"},{"Start":"04:55.775 ","End":"04:59.089","Text":"it\u0027s going to get something like 3.9 something."},{"Start":"04:59.089 ","End":"05:00.470","Text":"That\u0027s going to be less than 4,"},{"Start":"05:00.470 ","End":"05:02.755","Text":"so this is going to be a plus."},{"Start":"05:02.755 ","End":"05:04.960","Text":"Square root of something has had to be plus anyway,"},{"Start":"05:04.960 ","End":"05:08.315","Text":"so the square root of plus 0 is plus 0."},{"Start":"05:08.315 ","End":"05:13.970","Text":"Anyway, we get plus 0 at the bottom and we get minus 2 at the top which is 2,"},{"Start":"05:13.970 ","End":"05:18.390","Text":"so you get 2 over 0 plus and it\u0027s infinity."},{"Start":"05:18.390 ","End":"05:24.080","Text":"The difference is but the limit as x goes to 2 minus of"},{"Start":"05:24.080 ","End":"05:30.480","Text":"the f prime of x is in contrast to this, it\u0027s plus infinity."},{"Start":"05:30.480 ","End":"05:33.510","Text":"The plus parenthesis is opposed to the minus."},{"Start":"05:33.510 ","End":"05:39.720","Text":"At this point the limit is minus infinity and at this point the limit is plus infinity,"},{"Start":"05:39.720 ","End":"05:44.780","Text":"but in both cases is when the limit exists it\u0027s considered to be a vertical tangent."},{"Start":"05:44.780 ","End":"05:53.270","Text":"At both x equals minus 2 or 2 and there is a vertical tangent,"},{"Start":"05:53.270 ","End":"05:55.195","Text":"but there\u0027s no cusp."},{"Start":"05:55.195 ","End":"06:00.695","Text":"That\u0027s essentially what we have as we\u0027ve answered both parts of the question."},{"Start":"06:00.695 ","End":"06:05.210","Text":"What we\u0027ve said is that the tangent line is vertical."},{"Start":"06:05.210 ","End":"06:08.790","Text":"We found that, that happens at minus 2,"},{"Start":"06:08.790 ","End":"06:10.470","Text":"0 and 2,"},{"Start":"06:10.470 ","End":"06:15.170","Text":"0 which is at x equals 2 and minus 2."},{"Start":"06:15.170 ","End":"06:21.255","Text":"At this point, we have a vertical tangent which is actually not important."},{"Start":"06:21.255 ","End":"06:22.620","Text":"Not what we were asked,"},{"Start":"06:22.620 ","End":"06:24.615","Text":"but I\u0027m just saying it to emphasize."},{"Start":"06:24.615 ","End":"06:28.050","Text":"It\u0027s a vertical tangent but it\u0027s no cusp,"},{"Start":"06:28.050 ","End":"06:32.815","Text":"because a cusp has to have a plus infinity and a minus infinity on each side."},{"Start":"06:32.815 ","End":"06:36.935","Text":"This belongs to the point where x is 2 and this belongs to the point minus 2."},{"Start":"06:36.935 ","End":"06:40.205","Text":"It\u0027s separate points, it\u0027s not infinity and minus infinity on both sides."},{"Start":"06:40.205 ","End":"06:43.770","Text":"That basically answers the question and we\u0027re done."}],"ID":6478},{"Watched":false,"Name":"Exercise 6","Duration":"5m 20s","ChapterTopicVideoID":6452,"CourseChapterTopicPlaylistID":1670,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this exercise, we have to find all the points on the graph f of x"},{"Start":"00:04.860 ","End":"00:10.365","Text":"equals and it\u0027s this piecewise definition where the tangent line is vertical."},{"Start":"00:10.365 ","End":"00:13.350","Text":"After that, we have to answer the question,"},{"Start":"00:13.350 ","End":"00:15.870","Text":"does the function have a vertical cusp?"},{"Start":"00:15.870 ","End":"00:17.460","Text":"It\u0027s piecewise."},{"Start":"00:17.460 ","End":"00:20.955","Text":"It\u0027s defined 1 way for x negative and 0,"},{"Start":"00:20.955 ","End":"00:22.875","Text":"another way for x positive."},{"Start":"00:22.875 ","End":"00:27.030","Text":"Why don\u0027t we take a look at a sketch which I happen to have brought with me?"},{"Start":"00:27.030 ","End":"00:29.265","Text":"Here\u0027s the piecewise function."},{"Start":"00:29.265 ","End":"00:31.035","Text":"Here\u0027s x equals 0."},{"Start":"00:31.035 ","End":"00:33.165","Text":"On the left, it\u0027s defined 1 way."},{"Start":"00:33.165 ","End":"00:35.325","Text":"On the right, it\u0027s defined the other way."},{"Start":"00:35.325 ","End":"00:39.270","Text":"This is the point where they meet y equals 4."},{"Start":"00:39.270 ","End":"00:43.070","Text":"In fact, where x equals 0 if you check in both definitions."},{"Start":"00:43.070 ","End":"00:45.035","Text":"It is continuous."},{"Start":"00:45.035 ","End":"00:49.100","Text":"It certainly looks like there\u0027s a vertical cusp because here"},{"Start":"00:49.100 ","End":"00:53.630","Text":"the slope looks like it\u0027s infinity and it looks like minus infinity."},{"Start":"00:53.630 ","End":"00:56.660","Text":"If this dotted green line is anything to go by,"},{"Start":"00:56.660 ","End":"00:59.515","Text":"I guess it\u0027s intended to be a vertical cusp."},{"Start":"00:59.515 ","End":"01:03.065","Text":"In any event, we need the definition of the derivative."},{"Start":"01:03.065 ","End":"01:05.325","Text":"It is defined for all x, by the way."},{"Start":"01:05.325 ","End":"01:07.430","Text":"Each of these separately, x^1/3,"},{"Start":"01:07.430 ","End":"01:10.160","Text":"which is the cube root of x is defined for all x,"},{"Start":"01:10.160 ","End":"01:12.410","Text":"so it\u0027s certainly defined for less than or equal to 0."},{"Start":"01:12.410 ","End":"01:15.850","Text":"Similarly, the 5th root of x is defined for all x and so on,"},{"Start":"01:15.850 ","End":"01:18.785","Text":"so there\u0027s no actual problem definition."},{"Start":"01:18.785 ","End":"01:21.650","Text":"What we do need though is y prime,"},{"Start":"01:21.650 ","End":"01:24.540","Text":"which is f prime of x because there\u0027s"},{"Start":"01:24.540 ","End":"01:29.450","Text":"a border line or if you like a seam line at x equals 0,"},{"Start":"01:29.450 ","End":"01:33.004","Text":"we don\u0027t know what the derivative is at x equals 0,"},{"Start":"01:33.004 ","End":"01:37.085","Text":"but we will separate 3 cases as x less than 0,"},{"Start":"01:37.085 ","End":"01:39.350","Text":"equal to 0, and bigger than 0."},{"Start":"01:39.350 ","End":"01:44.105","Text":"I don\u0027t know what it is when x equals 0,"},{"Start":"01:44.105 ","End":"01:47.195","Text":"but for x less than or equal to 0,"},{"Start":"01:47.195 ","End":"01:48.980","Text":"it\u0027s a simple differentiation."},{"Start":"01:48.980 ","End":"01:57.680","Text":"The 4 goes to nothing and X^1/3 is 1/3x^minus 2/3."},{"Start":"01:57.680 ","End":"02:02.615","Text":"We\u0027ll be using exponential notation rather than square roots and cube roots and stuff."},{"Start":"02:02.615 ","End":"02:07.759","Text":"The last 1 is when x is bigger than 0."},{"Start":"02:07.759 ","End":"02:12.140","Text":"Sorry, I just noticed here I wrote accidentally less than or equal to."},{"Start":"02:12.140 ","End":"02:15.875","Text":"I meant to write x is less than 0."},{"Start":"02:15.875 ","End":"02:19.850","Text":"Here, it\u0027s minus because it was a minus here."},{"Start":"02:19.850 ","End":"02:23.820","Text":"It\u0027s 1/5x, lower the exponent by 1,"},{"Start":"02:23.820 ","End":"02:26.350","Text":"so it\u0027s minus 4/5."},{"Start":"02:26.360 ","End":"02:29.010","Text":"We need some limits."},{"Start":"02:29.010 ","End":"02:30.770","Text":"X goes to 0,"},{"Start":"02:30.770 ","End":"02:35.040","Text":"let\u0027s say from the right first of f prime of x."},{"Start":"02:35.040 ","End":"02:37.220","Text":"If x is slightly positive,"},{"Start":"02:37.220 ","End":"02:38.660","Text":"we use this definition."},{"Start":"02:38.660 ","End":"02:42.460","Text":"This is minus 1 over,"},{"Start":"02:42.460 ","End":"02:45.820","Text":"dividing is so that we get rid of the minus,"},{"Start":"02:46.520 ","End":"02:50.340","Text":"x^4/5 is the 5th root of x^4."},{"Start":"02:50.340 ","End":"02:54.165","Text":"Let\u0027s see, if x is 0 from the right,"},{"Start":"02:54.165 ","End":"02:57.765","Text":"x^4 is equal to positive 0."},{"Start":"02:57.765 ","End":"03:01.005","Text":"Take the 5th root, it\u0027s still positive 0,"},{"Start":"03:01.005 ","End":"03:05.465","Text":"so it\u0027s minus 1 over positive 0,"},{"Start":"03:05.465 ","End":"03:08.810","Text":"which is equal to minus infinity."},{"Start":"03:08.810 ","End":"03:13.280","Text":"That already shows us that 0 has a vertical tangent."},{"Start":"03:13.280 ","End":"03:22.220","Text":"So already we can tell that we have a tangent line is vertical at x equals 0,"},{"Start":"03:22.220 ","End":"03:24.185","Text":"or if you like, i.e.,"},{"Start":"03:24.185 ","End":"03:26.670","Text":"at the point 0,4,"},{"Start":"03:26.670 ","End":"03:29.285","Text":"if you substitute 0 here you get 4."},{"Start":"03:29.285 ","End":"03:33.540","Text":"That\u0027s already answers this question where the tangent line is vertical."},{"Start":"03:33.540 ","End":"03:34.710","Text":"It can only be at 0."},{"Start":"03:34.710 ","End":"03:37.665","Text":"There\u0027s no problems anywhere else."},{"Start":"03:37.665 ","End":"03:40.704","Text":"Infinity is defined."},{"Start":"03:40.704 ","End":"03:43.535","Text":"0,4 is 1 possibility."},{"Start":"03:43.535 ","End":"03:45.845","Text":"In order to know if it\u0027s a cusp or not,"},{"Start":"03:45.845 ","End":"03:49.760","Text":"we have to check on the other side what is the limit as x"},{"Start":"03:49.760 ","End":"03:53.840","Text":"goes to 0 from the left of f prime of x."},{"Start":"03:53.840 ","End":"03:57.680","Text":"Well, that\u0027s the limit if x goes to 0 from the left term."},{"Start":"03:57.680 ","End":"04:03.500","Text":"This part here is equal to 1/3 at the bottom."},{"Start":"04:03.500 ","End":"04:07.025","Text":"X^2/3 means the cube root of x squared."},{"Start":"04:07.025 ","End":"04:12.720","Text":"Now x is 0 minus because 0 minus squared is still 0 plus."},{"Start":"04:12.720 ","End":"04:16.790","Text":"Essentially, we\u0027re going to get 1/3 times 0 plus,"},{"Start":"04:16.790 ","End":"04:18.830","Text":"which is equal to plus infinity."},{"Start":"04:18.830 ","End":"04:21.185","Text":"I\u0027ll just write plus emphasis."},{"Start":"04:21.185 ","End":"04:25.680","Text":"Because on 1 side we get minus infinity and the other side we get plus infinity,"},{"Start":"04:25.680 ","End":"04:27.440","Text":"it could have been vice versa."},{"Start":"04:27.440 ","End":"04:31.315","Text":"That means that we do have a vertical cusp."},{"Start":"04:31.315 ","End":"04:33.920","Text":"In other words, x equals 0,"},{"Start":"04:33.920 ","End":"04:39.050","Text":"or I should say 0,4 is a vertical cusp,"},{"Start":"04:39.050 ","End":"04:41.899","Text":"besides just having a tangent line which is vertical."},{"Start":"04:41.899 ","End":"04:45.025","Text":"That basically answers all the questions."},{"Start":"04:45.025 ","End":"04:46.375","Text":"In the first part,"},{"Start":"04:46.375 ","End":"04:50.505","Text":"let\u0027s called this 1 part a and part b."},{"Start":"04:50.505 ","End":"04:51.940","Text":"For part a,"},{"Start":"04:51.940 ","End":"04:58.490","Text":"we want to find where the tangent is vertical and where the tangent is vertical is 0,4."},{"Start":"04:58.490 ","End":"05:00.930","Text":"That\u0027s the point on the graph."},{"Start":"05:00.930 ","End":"05:04.970","Text":"Although it\u0027d be acceptable to say x equals 0, either 1."},{"Start":"05:04.970 ","End":"05:08.225","Text":"It\u0027s common to call points just by the x."},{"Start":"05:08.225 ","End":"05:10.370","Text":"As for part b,"},{"Start":"05:10.370 ","End":"05:12.830","Text":"does the function have a vertical cusp?"},{"Start":"05:12.830 ","End":"05:14.890","Text":"The answer is yes,"},{"Start":"05:14.890 ","End":"05:18.330","Text":"0.4 is a vertical cusp."},{"Start":"05:18.330 ","End":"05:21.190","Text":"That\u0027s it. We\u0027re done."}],"ID":6479}],"Thumbnail":null,"ID":1670},{"Name":"Linear Approximation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Linear Approximation","Duration":"9m 23s","ChapterTopicVideoID":1899,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/1899.jpeg","UploadDate":"2019-11-14T07:01:08.5200000","DurationForVideoObject":"PT9M23S","Description":null,"MetaTitle":"Linear Approximation: Video + Workbook | Proprep","MetaDescription":"Tangents, Normal lines and Linear Approximation - Linear Approximation. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/tangents%2c-normal-lines-and-linear-approximation/linear-approximation/vid1911","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this clip, I\u0027m going to introduce you to"},{"Start":"00:02.160 ","End":"00:05.144","Text":"something called the linear approximation formula."},{"Start":"00:05.144 ","End":"00:11.220","Text":"This is a formula that allows us to make calculations without a calculator."},{"Start":"00:11.220 ","End":"00:15.345","Text":"This formula was invented before the days that everyone had a calculator,"},{"Start":"00:15.345 ","End":"00:22.605","Text":"and it enables us to compute stuff like the square root of 5,"},{"Start":"00:22.605 ","End":"00:26.145","Text":"natural logarithm of 3,"},{"Start":"00:26.145 ","End":"00:32.285","Text":"the cube root of 29, and so on."},{"Start":"00:32.285 ","End":"00:36.470","Text":"That normally we\u0027d have no problem in computing today with a calculator"},{"Start":"00:36.470 ","End":"00:40.970","Text":"but in case you\u0027re on a desert island or don\u0027t have your calculator,"},{"Start":"00:40.970 ","End":"00:45.529","Text":"you could use what I will present as the linear approximation formula."},{"Start":"00:45.529 ","End":"00:53.620","Text":"I\u0027ll just say a few words that it is used to make computations without a calculator."},{"Start":"00:53.620 ","End":"00:59.630","Text":"I\u0027d normally need a calculator and calculator would give a more exact value,"},{"Start":"00:59.630 ","End":"01:03.975","Text":"but this gives us something approximate, e.g, this."},{"Start":"01:03.975 ","End":"01:11.435","Text":"The formula is f of x is approximately equal to f"},{"Start":"01:11.435 ","End":"01:20.570","Text":"of x_0 plus f prime of x_0 times x minus x_0."},{"Start":"01:20.570 ","End":"01:25.555","Text":"The function at a value x is approximately equal to f at another value"},{"Start":"01:25.555 ","End":"01:30.985","Text":"x_0 plus the derivative at x_0 times x minus x_0."},{"Start":"01:30.985 ","End":"01:33.800","Text":"It won\u0027t make any sense until we do an example,"},{"Start":"01:33.800 ","End":"01:36.200","Text":"and we\u0027ll do an example in a moment."},{"Start":"01:36.200 ","End":"01:39.560","Text":"I\u0027ll just tell you that f is a function."},{"Start":"01:39.560 ","End":"01:42.590","Text":"For example, here I would take the square root function here,"},{"Start":"01:42.590 ","End":"01:44.495","Text":"the natural logarithm function,"},{"Start":"01:44.495 ","End":"01:46.415","Text":"here the cube root function."},{"Start":"01:46.415 ","End":"01:50.420","Text":"x is taken as the value that we want to know about,"},{"Start":"01:50.420 ","End":"01:53.330","Text":"like 5 or 3 or 29,"},{"Start":"01:53.330 ","End":"01:57.635","Text":"and x_0 is the value that we do know f of it."},{"Start":"01:57.635 ","End":"02:01.160","Text":"For example, here, if I talk x_0 to be 27,"},{"Start":"02:01.160 ","End":"02:04.400","Text":"I do know the cube root of 27 it\u0027s 3,"},{"Start":"02:04.400 ","End":"02:08.465","Text":"but we\u0027re best to do the example and then we\u0027ll see."},{"Start":"02:08.465 ","End":"02:15.395","Text":"In the example, I want to compute approximately the square root of 5."},{"Start":"02:15.395 ","End":"02:17.120","Text":"Let\u0027s see how we do this."},{"Start":"02:17.120 ","End":"02:20.670","Text":"The first thing to do is to choose our function f,"},{"Start":"02:20.670 ","End":"02:22.670","Text":"and since we have the square root here,"},{"Start":"02:22.670 ","End":"02:29.605","Text":"it makes sense to take as f of x to be equal to the square root of x."},{"Start":"02:29.605 ","End":"02:32.420","Text":"Since we have f prime in the formula,"},{"Start":"02:32.420 ","End":"02:36.035","Text":"we also need to know what is the derivative."},{"Start":"02:36.035 ","End":"02:40.280","Text":"We know that f prime of x in general,"},{"Start":"02:40.280 ","End":"02:42.050","Text":"for the square root function,"},{"Start":"02:42.050 ","End":"02:46.155","Text":"is 1 over twice the square root function."},{"Start":"02:46.155 ","End":"02:52.175","Text":"In this case will also take x to be equal to 5,"},{"Start":"02:52.175 ","End":"02:57.680","Text":"and x_0 is a value that we do know the square root of."},{"Start":"02:57.680 ","End":"02:59.390","Text":"Now, what\u0027s closest thing to 5,"},{"Start":"02:59.390 ","End":"03:01.130","Text":"that we know the square root of?"},{"Start":"03:01.130 ","End":"03:04.915","Text":"I would take 4 because I know the square root of 4 is 2."},{"Start":"03:04.915 ","End":"03:07.920","Text":"Let\u0027s see how this formula works."},{"Start":"03:07.920 ","End":"03:13.850","Text":"Now, I can write it more generally for the square root disregarding the 5 and the 4,"},{"Start":"03:13.850 ","End":"03:16.310","Text":"and I could write, for example,"},{"Start":"03:16.310 ","End":"03:18.740","Text":"that the square root of x,"},{"Start":"03:18.740 ","End":"03:20.525","Text":"because f is the square root,"},{"Start":"03:20.525 ","End":"03:26.630","Text":"is approximately equal to the square root of another number x_0,"},{"Start":"03:26.630 ","End":"03:28.430","Text":"which is close to x,"},{"Start":"03:28.430 ","End":"03:32.460","Text":"plus x minus x_0."},{"Start":"03:32.460 ","End":"03:35.705","Text":"This is 1 over twice the square root,"},{"Start":"03:35.705 ","End":"03:37.690","Text":"I\u0027ll just put it on the denominator."},{"Start":"03:37.690 ","End":"03:42.095","Text":"Over twice the square root of x_0."},{"Start":"03:42.095 ","End":"03:46.795","Text":"This will be a more general approximation that\u0027s good for the square root function,"},{"Start":"03:46.795 ","End":"03:48.340","Text":"and in our case,"},{"Start":"03:48.340 ","End":"03:51.295","Text":"we want specifically the square root of 5,"},{"Start":"03:51.295 ","End":"03:55.385","Text":"so what I\u0027ll do is take the square root of 5"},{"Start":"03:55.385 ","End":"03:59.980","Text":"is approximately equal to the square root of 4."},{"Start":"03:59.980 ","End":"04:03.160","Text":"Here I\u0027ve taken these 2, x is 5,"},{"Start":"04:03.160 ","End":"04:06.870","Text":"x_0 is 4 plus x minus x_0,"},{"Start":"04:06.870 ","End":"04:13.000","Text":"5 minus 4 over twice the square root of 4."},{"Start":"04:13.000 ","End":"04:14.815","Text":"You see why we took 4,"},{"Start":"04:14.815 ","End":"04:18.380","Text":"the x_0 as something that we know about otherwise we couldn\u0027t get started."},{"Start":"04:18.380 ","End":"04:20.900","Text":"This is equal to 2."},{"Start":"04:20.900 ","End":"04:26.840","Text":"This is now exactly equal to 2 plus 1 over 2 times,"},{"Start":"04:26.840 ","End":"04:28.655","Text":"and the square root of 4 is 2."},{"Start":"04:28.655 ","End":"04:32.700","Text":"This is equal to 2.25."},{"Start":"04:32.870 ","End":"04:35.465","Text":"If you have a calculator,"},{"Start":"04:35.465 ","End":"04:37.940","Text":"check the square root of 5 in your calculator,"},{"Start":"04:37.940 ","End":"04:40.550","Text":"I believe it\u0027s something like 2.26."},{"Start":"04:40.550 ","End":"04:42.260","Text":"It\u0027s fairly close."},{"Start":"04:42.260 ","End":"04:44.450","Text":"But this doesn\u0027t tell us how close we are,"},{"Start":"04:44.450 ","End":"04:46.745","Text":"just that it\u0027s an approximation."},{"Start":"04:46.745 ","End":"04:48.650","Text":"Just want to go over it a bit again."},{"Start":"04:48.650 ","End":"04:51.950","Text":"We wrote down a formula which is hard to understand by"},{"Start":"04:51.950 ","End":"04:55.220","Text":"itself but you did learn a new symbol,"},{"Start":"04:55.220 ","End":"04:58.725","Text":"approximately equal, which you may have not seen before."},{"Start":"04:58.725 ","End":"05:01.360","Text":"Approximately equal to."},{"Start":"05:01.460 ","End":"05:05.190","Text":"This is a formula in general for a function f,"},{"Start":"05:05.190 ","End":"05:08.390","Text":"and we learned how to use it by means of an example"},{"Start":"05:08.390 ","End":"05:11.630","Text":"that when we had to do the square root of 5,"},{"Start":"05:11.630 ","End":"05:14.540","Text":"we took the function as the square root function."},{"Start":"05:14.540 ","End":"05:17.209","Text":"Then for this particular square root function,"},{"Start":"05:17.209 ","End":"05:20.705","Text":"we could interpret this formula that f of x,"},{"Start":"05:20.705 ","End":"05:23.735","Text":"which is here, is the square root of x."},{"Start":"05:23.735 ","End":"05:26.570","Text":"Then we also have f of x_0,"},{"Start":"05:26.570 ","End":"05:29.990","Text":"which is the same thing but with x_0 instead of x."},{"Start":"05:29.990 ","End":"05:35.005","Text":"Then we just copy the x minus x_0 from here,"},{"Start":"05:35.005 ","End":"05:38.540","Text":"this is just as is if I put the brackets around it."},{"Start":"05:38.540 ","End":"05:41.540","Text":"The f prime of x_0 was,"},{"Start":"05:41.540 ","End":"05:43.700","Text":"well, it had a 1 in it which I omitted,"},{"Start":"05:43.700 ","End":"05:48.645","Text":"but this bit was the f prime of x_0,"},{"Start":"05:48.645 ","End":"05:52.955","Text":"which once we computed f prime of x,"},{"Start":"05:52.955 ","End":"05:55.570","Text":"here, we got 1 over twice the square root of x,"},{"Start":"05:55.570 ","End":"05:58.400","Text":"so we put 1 over twice the square root of x_0."},{"Start":"05:58.400 ","End":"06:03.755","Text":"This is just an interpretation of this for the case where f of x is square root of x."},{"Start":"06:03.755 ","End":"06:06.350","Text":"Now we can use this formula many times,"},{"Start":"06:06.350 ","End":"06:08.720","Text":"not just for the square root of 5,"},{"Start":"06:08.720 ","End":"06:10.220","Text":"but for the square root of 50,"},{"Start":"06:10.220 ","End":"06:12.020","Text":"the square root of 80,"},{"Start":"06:12.020 ","End":"06:13.790","Text":"any number of times."},{"Start":"06:13.790 ","End":"06:17.045","Text":"We just used it once on the number 5."},{"Start":"06:17.045 ","End":"06:21.505","Text":"But when you have to do a particular number that\u0027s your x,"},{"Start":"06:21.505 ","End":"06:26.460","Text":"you want to know 5 and the x_0 is a number close to what"},{"Start":"06:26.460 ","End":"06:31.340","Text":"we want that we do know the square root of or the function of in general."},{"Start":"06:31.340 ","End":"06:34.145","Text":"I hope this explains it a bit better."},{"Start":"06:34.145 ","End":"06:36.305","Text":"Now I\u0027d like to do another example,"},{"Start":"06:36.305 ","End":"06:38.935","Text":"but staying with the square root function."},{"Start":"06:38.935 ","End":"06:44.695","Text":"I\u0027m going to give another example to compute approximately the square root of 8."},{"Start":"06:44.695 ","End":"06:48.800","Text":"Once again, I\u0027ll use this formula that\u0027s in blue,"},{"Start":"06:48.800 ","End":"06:53.180","Text":"but this time when I put the same f of x and the same f prime of x,"},{"Start":"06:53.180 ","End":"06:54.845","Text":"I can reuse those."},{"Start":"06:54.845 ","End":"06:58.060","Text":"But this time my x is 8,"},{"Start":"06:58.060 ","End":"07:00.435","Text":"and so I have to look for x_0,"},{"Start":"07:00.435 ","End":"07:02.230","Text":"that\u0027s close to 8,"},{"Start":"07:02.230 ","End":"07:04.805","Text":"but a number that I do know the square root of."},{"Start":"07:04.805 ","End":"07:07.285","Text":"Well, that should be 9."},{"Start":"07:07.285 ","End":"07:10.535","Text":"If I use that now in this function,"},{"Start":"07:10.535 ","End":"07:14.930","Text":"then I will get that the square root of 8 is"},{"Start":"07:14.930 ","End":"07:20.225","Text":"approximately equal to the square root of 9 plus,"},{"Start":"07:20.225 ","End":"07:25.280","Text":"x minus x_0 is now 8 minus 9,"},{"Start":"07:25.280 ","End":"07:28.830","Text":"over twice the square root of x_0,"},{"Start":"07:28.830 ","End":"07:31.860","Text":"twice the square root of 9."},{"Start":"07:31.860 ","End":"07:36.410","Text":"Now this is equal to square root of 9 is 3."},{"Start":"07:36.410 ","End":"07:39.560","Text":"8 minus 9 is minus 1."},{"Start":"07:39.560 ","End":"07:41.280","Text":"Square root of 9 is 3,"},{"Start":"07:41.280 ","End":"07:44.325","Text":"again, 2 times 3."},{"Start":"07:44.325 ","End":"07:47.475","Text":"I get 3 minus 1/6,"},{"Start":"07:47.475 ","End":"07:51.585","Text":"which is 2 and 5/6."},{"Start":"07:51.585 ","End":"07:54.815","Text":"That will be my approximation as a fraction."},{"Start":"07:54.815 ","End":"07:58.325","Text":"We could divide it out and do it as a decimal also."},{"Start":"07:58.325 ","End":"08:03.610","Text":"Think this will be 2.833 and so on."},{"Start":"08:03.610 ","End":"08:05.325","Text":"That\u0027s another example."},{"Start":"08:05.325 ","End":"08:08.435","Text":"We can reuse the function with different values."},{"Start":"08:08.435 ","End":"08:10.685","Text":"Let\u0027s go for yet another example."},{"Start":"08:10.685 ","End":"08:15.575","Text":"Let\u0027s take the square root of 27."},{"Start":"08:15.575 ","End":"08:23.540","Text":"This time our x is 27 and x_0 I look for something close that we know the square root of,"},{"Start":"08:23.540 ","End":"08:26.070","Text":"that will be 25."},{"Start":"08:26.070 ","End":"08:33.950","Text":"This time we would get that the square root of 27 is approximately equal"},{"Start":"08:33.950 ","End":"08:41.840","Text":"to square root of 25 plus 27 minus 25,"},{"Start":"08:41.840 ","End":"08:43.760","Text":"that\u0027s our x, that\u0027s our x_0,"},{"Start":"08:43.760 ","End":"08:48.965","Text":"over twice the square root of x_0, which is 25."},{"Start":"08:48.965 ","End":"08:54.650","Text":"This gives us 5 plus 2 over 2 times"},{"Start":"08:54.650 ","End":"09:01.390","Text":"5 plus 20 over 10, which is 5.2."},{"Start":"09:01.390 ","End":"09:03.710","Text":"Again, I\u0027ll leave it to you to check on"},{"Start":"09:03.710 ","End":"09:09.020","Text":"the calculator what the exact values are of square root of 8,"},{"Start":"09:09.020 ","End":"09:10.790","Text":"which I approximated like this,"},{"Start":"09:10.790 ","End":"09:12.500","Text":"the square root of 5,"},{"Start":"09:12.500 ","End":"09:14.885","Text":"which we approximated like this,"},{"Start":"09:14.885 ","End":"09:18.950","Text":"square root of 27, and see what the exact values are."},{"Start":"09:18.950 ","End":"09:21.155","Text":"That\u0027s yet another example."},{"Start":"09:21.155 ","End":"09:23.740","Text":"Meanwhile, that\u0027s it."}],"ID":1911},{"Watched":false,"Name":"Linear Approximation (continued)","Duration":"6m 15s","ChapterTopicVideoID":1900,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"This clip is a continuation of the previous clip where we"},{"Start":"00:03.600 ","End":"00:07.574","Text":"talked about the linear approximation formula."},{"Start":"00:07.574 ","End":"00:13.290","Text":"I copied the formula from the previous clip. Here it is."},{"Start":"00:13.290 ","End":"00:16.770","Text":"In the previous clip we practiced with this formula,"},{"Start":"00:16.770 ","End":"00:21.530","Text":"how to use it in the case where f of x is the square root of x."},{"Start":"00:21.530 ","End":"00:25.205","Text":"This time we\u0027ll be taking another couple of functions;"},{"Start":"00:25.205 ","End":"00:29.225","Text":"natural log and the cube root, I believe."},{"Start":"00:29.225 ","End":"00:34.460","Text":"Let\u0027s start with the following exercise for example,"},{"Start":"00:34.460 ","End":"00:44.430","Text":"where we\u0027re asked to compute the natural log of 1.3 approximately."},{"Start":"00:44.430 ","End":"00:47.660","Text":"In this case, what I do is first of all,"},{"Start":"00:47.660 ","End":"00:50.525","Text":"I notice that the function is the natural logarithm."},{"Start":"00:50.525 ","End":"00:52.110","Text":"The number is less important."},{"Start":"00:52.110 ","End":"00:58.810","Text":"We want to find the corresponding formula for the natural logarithm."},{"Start":"00:58.810 ","End":"01:06.260","Text":"What we get, if we put f of x equals natural log of x,"},{"Start":"01:06.260 ","End":"01:10.385","Text":"then this formula becomes f of x,"},{"Start":"01:10.385 ","End":"01:12.920","Text":"which is natural log of x,"},{"Start":"01:12.920 ","End":"01:16.730","Text":"is approximately equal to f of x_0,"},{"Start":"01:16.730 ","End":"01:18.470","Text":"natural log of x_0."},{"Start":"01:18.470 ","End":"01:20.705","Text":"We don\u0027t know x and x_0 yet,"},{"Start":"01:20.705 ","End":"01:24.680","Text":"plus the derivative and the derivative,"},{"Start":"01:24.680 ","End":"01:29.165","Text":"I can get in general as 1 over x."},{"Start":"01:29.165 ","End":"01:31.400","Text":"If I go back there,"},{"Start":"01:31.400 ","End":"01:33.950","Text":"this will be 1 over x,"},{"Start":"01:33.950 ","End":"01:36.065","Text":"but not quite x, x_0,"},{"Start":"01:36.065 ","End":"01:39.885","Text":"and then x minus x_0."},{"Start":"01:39.885 ","End":"01:44.840","Text":"This will be the general formula for the natural log approximation."},{"Start":"01:44.840 ","End":"01:48.335","Text":"Now, we wanted specifically, in our case,"},{"Start":"01:48.335 ","End":"01:53.565","Text":"we want x to be equal to 1.3."},{"Start":"01:53.565 ","End":"01:55.525","Text":"What are we going to take as x_0?"},{"Start":"01:55.525 ","End":"02:02.620","Text":"Well, the guideline is that we take as x_0 is close as we can to the x that we want,"},{"Start":"02:02.620 ","End":"02:08.990","Text":"but a value that has a nice round value that we even can compute it for f of x."},{"Start":"02:08.990 ","End":"02:14.030","Text":"In other words, I want to be able to compute natural log of something close to 1.3."},{"Start":"02:14.030 ","End":"02:18.650","Text":"Then I remember natural log of 1 is very simple to compute."},{"Start":"02:18.650 ","End":"02:20.345","Text":"In fact, it\u0027s just 0."},{"Start":"02:20.345 ","End":"02:23.045","Text":"We\u0027ll take this as x, this is x_0."},{"Start":"02:23.045 ","End":"02:25.535","Text":"Then what we\u0027ll get,"},{"Start":"02:25.535 ","End":"02:27.350","Text":"in our particular case,"},{"Start":"02:27.350 ","End":"02:36.105","Text":"is that the natural log of 1.3 is approximately equal to the natural log of 1."},{"Start":"02:36.105 ","End":"02:38.455","Text":"Where I see x, I put 1.3,"},{"Start":"02:38.455 ","End":"02:46.620","Text":"x_0 is 1 plus 1 over 1 times 1.3 minus 1."},{"Start":"02:46.620 ","End":"02:48.340","Text":"Let\u0027s continue with this."},{"Start":"02:48.340 ","End":"02:51.380","Text":"This is equal to natural log of 1 is 0,"},{"Start":"02:51.380 ","End":"02:59.200","Text":"1 over 1 is just 1 and 1.3 minus 1 is 0.3."},{"Start":"02:59.200 ","End":"03:03.765","Text":"All in all, what I get is 0.3."},{"Start":"03:03.765 ","End":"03:09.620","Text":"That\u0027s going to be an approximation for natural log of 1.3."},{"Start":"03:09.620 ","End":"03:12.970","Text":"That\u0027s an example with natural logarithm."},{"Start":"03:12.970 ","End":"03:15.080","Text":"In the next example,"},{"Start":"03:15.080 ","End":"03:23.090","Text":"I want to get an approximation to the cube root of 9."},{"Start":"03:23.090 ","End":"03:29.660","Text":"In this case, I\u0027m going to let my function be the cube root function."},{"Start":"03:29.660 ","End":"03:33.620","Text":"I\u0027d like to know also how much is the derivative."},{"Start":"03:33.620 ","End":"03:38.825","Text":"I\u0027ll need it in the formula f prime of x in general will equal."},{"Start":"03:38.825 ","End":"03:40.370","Text":"I\u0027m not going to do the computation."},{"Start":"03:40.370 ","End":"03:41.420","Text":"I just remember it."},{"Start":"03:41.420 ","End":"03:48.280","Text":"1 over 3 times the cube root of x squared."},{"Start":"03:48.280 ","End":"03:50.780","Text":"Let\u0027s do the general formula for"},{"Start":"03:50.780 ","End":"03:55.015","Text":"cube root and then we\u0027ll see what happens in our specific case."},{"Start":"03:55.015 ","End":"03:57.485","Text":"We\u0027ll get for this function,"},{"Start":"03:57.485 ","End":"03:59.150","Text":"the linear approximation formula,"},{"Start":"03:59.150 ","End":"04:01.445","Text":"this is it and now I\u0027m going to apply it,"},{"Start":"04:01.445 ","End":"04:11.890","Text":"so we get that the cube root of x is approximately equal to the cube root of x_0 plus"},{"Start":"04:11.890 ","End":"04:17.010","Text":"1 over 3 times the cube root of"},{"Start":"04:17.010 ","End":"04:23.570","Text":"x_0 squared times x minus x_0."},{"Start":"04:23.570 ","End":"04:27.530","Text":"Let\u0027s see what will happen in our particular case."},{"Start":"04:27.530 ","End":"04:32.920","Text":"We\u0027re going to choose x equals 9 because that\u0027s the number whose cube root we want."},{"Start":"04:32.920 ","End":"04:35.374","Text":"If we let x equals 9,"},{"Start":"04:35.374 ","End":"04:42.220","Text":"what do we let x_0 be something very close or closest possible to 9,"},{"Start":"04:42.220 ","End":"04:43.985","Text":"and we do know its cube root."},{"Start":"04:43.985 ","End":"04:46.775","Text":"Well, the obvious candidate is 8."},{"Start":"04:46.775 ","End":"04:49.100","Text":"Let\u0027s take these 2 values,"},{"Start":"04:49.100 ","End":"04:51.035","Text":"and then if we put them in here,"},{"Start":"04:51.035 ","End":"04:54.930","Text":"we shall get with these values of x and x_0,"},{"Start":"04:54.930 ","End":"05:04.205","Text":"we get the cube root of 9 is approximately equal to the cube root of x, now,"},{"Start":"05:04.205 ","End":"05:07.505","Text":"which is 8 plus"},{"Start":"05:07.505 ","End":"05:12.560","Text":"1 over 3 times the cube root of"},{"Start":"05:12.560 ","End":"05:18.835","Text":"8 squared times 9 minus 8."},{"Start":"05:18.835 ","End":"05:20.420","Text":"What does this give us?"},{"Start":"05:20.420 ","End":"05:22.535","Text":"Cube root of 8 is 2,"},{"Start":"05:22.535 ","End":"05:25.640","Text":"the cube root of 8 squared is going to be"},{"Start":"05:25.640 ","End":"05:28.775","Text":"4 because I can do first the cube root and then square it."},{"Start":"05:28.775 ","End":"05:33.415","Text":"That\u0027s 4, so that\u0027s 1 over 3 times 4."},{"Start":"05:33.415 ","End":"05:38.070","Text":"Then 9 minus 8 is just 1 times 1,"},{"Start":"05:38.070 ","End":"05:40.650","Text":"which is nothing here."},{"Start":"05:40.650 ","End":"05:42.810","Text":"We get 2 1/12,"},{"Start":"05:42.810 ","End":"05:52.860","Text":"and in decimal I remember that 1/12 is 8 and 1/3 percent."},{"Start":"05:52.860 ","End":"05:59.160","Text":"It would be like 2.083 or something like that."},{"Start":"05:59.160 ","End":"06:02.720","Text":"This is my approximation for the cube root of 9,"},{"Start":"06:02.720 ","End":"06:04.715","Text":"check it on the calculator."},{"Start":"06:04.715 ","End":"06:08.750","Text":"I feel that these 2 examples are enough for getting along with meanwhile"},{"Start":"06:08.750 ","End":"06:13.100","Text":"together with the examples from the previous clip."},{"Start":"06:13.100 ","End":"06:16.500","Text":"I declare that we\u0027re done."}],"ID":1912},{"Watched":false,"Name":"Exercise 1","Duration":"3m 44s","ChapterTopicVideoID":6439,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, I want to get an approximate value for the fifth root of"},{"Start":"00:03.990 ","End":"00:08.175","Text":"33 and we\u0027re going to use the linear approximation formula,"},{"Start":"00:08.175 ","End":"00:09.780","Text":"which I brought with me;"},{"Start":"00:09.780 ","End":"00:12.000","Text":"2 things to identify, first of all,"},{"Start":"00:12.000 ","End":"00:14.190","Text":"are what are the function I\u0027m going to use and"},{"Start":"00:14.190 ","End":"00:18.030","Text":"I think it\u0027s fairly clear that we\u0027re going to use the function of x,"},{"Start":"00:18.030 ","End":"00:20.655","Text":"which is the fifth root of x."},{"Start":"00:20.655 ","End":"00:27.285","Text":"The other thing I\u0027ll need to decide is x naught and x naught is a value of x,"},{"Start":"00:27.285 ","End":"00:32.685","Text":"for which I do know what is the fifth root of or f of that particular value."},{"Start":"00:32.685 ","End":"00:35.190","Text":"Obvious choice is 32,"},{"Start":"00:35.190 ","End":"00:41.460","Text":"the reason I say 32 is that we do know the fifth root of 32 is 2."},{"Start":"00:41.460 ","End":"00:43.295","Text":"If you don\u0027t see this right away,"},{"Start":"00:43.295 ","End":"00:46.230","Text":"then compute 2 to the power of 5,"},{"Start":"00:46.230 ","End":"00:50.640","Text":"2 times 2 times 2 times 2 times 2 and that will give you 32,"},{"Start":"00:50.640 ","End":"00:53.395","Text":"which means that the fifth root of 32 is 2."},{"Start":"00:53.395 ","End":"00:55.910","Text":"Not going to use this value 33 just yet,"},{"Start":"00:55.910 ","End":"01:02.480","Text":"what I do need is the derivative of f. F prime of x is equal to,"},{"Start":"01:02.480 ","End":"01:05.870","Text":"I can think of fifth root of x as just x to"},{"Start":"01:05.870 ","End":"01:09.635","Text":"the power of 1/5 and if you look at it that way,"},{"Start":"01:09.635 ","End":"01:16.790","Text":"then what we get is 1/5 x to the power of minus 4/5."},{"Start":"01:16.790 ","End":"01:24.915","Text":"What I\u0027d like to do next is compute f prime of x naught but x naught is 32,"},{"Start":"01:24.915 ","End":"01:35.175","Text":"this is equal to 1 over 5 times 32 to the minus 4/5."},{"Start":"01:35.175 ","End":"01:37.440","Text":"This bit here I\u0027ll compute at the side,"},{"Start":"01:37.440 ","End":"01:44.340","Text":"it\u0027s 1 over 32 to the power of positive 4/5."},{"Start":"01:44.410 ","End":"01:47.810","Text":"I want to have 4/5, I need to take to the power of"},{"Start":"01:47.810 ","End":"01:50.390","Text":"the 4 and then fifth root or the other way around."},{"Start":"01:50.390 ","End":"01:54.650","Text":"I can take first of all the fifth root and then raise it to the power of 4,"},{"Start":"01:54.650 ","End":"01:59.670","Text":"the fifth root of 32 is 2 raised to the power of 4,"},{"Start":"01:59.670 ","End":"02:03.435","Text":"and that gives me 1/16."},{"Start":"02:03.435 ","End":"02:06.150","Text":"This is equal to 1 over,"},{"Start":"02:06.150 ","End":"02:09.930","Text":"I can put the 1/5 together with the 16,"},{"Start":"02:09.930 ","End":"02:14.590","Text":"the denominator, I end up with 1 over 80."},{"Start":"02:14.590 ","End":"02:18.095","Text":"Now, we\u0027re ready to plug into this formula,"},{"Start":"02:18.095 ","End":"02:24.380","Text":"we get in general that f of x for any x close to 32,"},{"Start":"02:24.380 ","End":"02:26.330","Text":"and we don\u0027t define what is close,"},{"Start":"02:26.330 ","End":"02:30.410","Text":"is going to equal approximately f of x naught,"},{"Start":"02:30.410 ","End":"02:33.565","Text":"which is the fifth root of 32,"},{"Start":"02:33.565 ","End":"02:37.875","Text":"is 2 plus 1 over 80,"},{"Start":"02:37.875 ","End":"02:44.680","Text":"which is this times x minus x naught is 32."},{"Start":"02:44.680 ","End":"02:46.490","Text":"This is true,"},{"Start":"02:46.490 ","End":"02:47.750","Text":"I\u0027ll just write it in brackets,"},{"Start":"02:47.750 ","End":"02:52.655","Text":"for whenever x is approximately equal to 32."},{"Start":"02:52.655 ","End":"02:55.580","Text":"So far I haven\u0027t used the value 33,"},{"Start":"02:55.580 ","End":"02:58.115","Text":"I\u0027ve got a general formula for f of x,"},{"Start":"02:58.115 ","End":"03:01.160","Text":"what it\u0027s approximately equal to when x is near 32,"},{"Start":"03:01.160 ","End":"03:04.225","Text":"let\u0027s say 33 is near 32."},{"Start":"03:04.225 ","End":"03:09.230","Text":"I can now say that f of 33 is approximately"},{"Start":"03:09.230 ","End":"03:15.755","Text":"equal to 2 plus 33 minus 32 over 80."},{"Start":"03:15.755 ","End":"03:18.800","Text":"This is 2 plus 1 over 80."},{"Start":"03:18.800 ","End":"03:24.500","Text":"Actually, I can do this in my head because I know that 1/8 is 0.125."},{"Start":"03:24.500 ","End":"03:29.050","Text":"All I need is an extra 0125,"},{"Start":"03:29.050 ","End":"03:33.980","Text":"this is an approximate value for the fifth root of 33."},{"Start":"03:33.980 ","End":"03:40.050","Text":"Just to check, I did this on the calculator and it comes to be 2.0123,"},{"Start":"03:40.510 ","End":"03:45.059","Text":"we see that we actually were fairly close."}],"ID":6466},{"Watched":false,"Name":"Exercise 2","Duration":"3m 50s","ChapterTopicVideoID":6440,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.355","Text":"In this exercise, I want to approximate the value of the 4th root of 15."},{"Start":"00:05.355 ","End":"00:09.855","Text":"I brought with me the formula for linear approximation."},{"Start":"00:09.855 ","End":"00:11.730","Text":"The first things I want to identify,"},{"Start":"00:11.730 ","End":"00:12.960","Text":"what is the function f,"},{"Start":"00:12.960 ","End":"00:15.000","Text":"and what is the point x naught."},{"Start":"00:15.000 ","End":"00:17.100","Text":"F of x, fairly clear,"},{"Start":"00:17.100 ","End":"00:19.640","Text":"we want the function 4th root of x."},{"Start":"00:19.640 ","End":"00:25.550","Text":"X naught is going to be a point for which I do know the value."},{"Start":"00:25.550 ","End":"00:32.945","Text":"I\u0027ll take it as 16 because I know that the 4th root of 16 is 2."},{"Start":"00:32.945 ","End":"00:35.135","Text":"I\u0027m also going to need the derivative,"},{"Start":"00:35.135 ","End":"00:38.795","Text":"f prime of x is equal to,"},{"Start":"00:38.795 ","End":"00:43.765","Text":"if I write this as x^1/4,"},{"Start":"00:43.765 ","End":"00:45.000","Text":"that\u0027s what this is."},{"Start":"00:45.000 ","End":"00:46.275","Text":"It\u0027s a 4th root."},{"Start":"00:46.275 ","End":"00:53.950","Text":"Then I can get that f prime of x is 1/4 x to the power of minus 3/4."},{"Start":"00:53.950 ","End":"00:55.864","Text":"But for the formula,"},{"Start":"00:55.864 ","End":"00:58.340","Text":"I need f prime of x naught."},{"Start":"00:58.340 ","End":"01:02.375","Text":"I need f prime of 16."},{"Start":"01:02.375 ","End":"01:11.085","Text":"This is going to equal 1/4 times 16 to the power of minus 3/4."},{"Start":"01:11.085 ","End":"01:12.700","Text":"I\u0027ll compute this at the side."},{"Start":"01:12.700 ","End":"01:15.190","Text":"This is 1/4 times,"},{"Start":"01:15.190 ","End":"01:16.705","Text":"now instead of minus 3/4,"},{"Start":"01:16.705 ","End":"01:22.465","Text":"I can put 1 over 16^3/4."},{"Start":"01:22.465 ","End":"01:26.045","Text":"This is equal to 1/4."},{"Start":"01:26.045 ","End":"01:28.410","Text":"Now to the power of 3/4,"},{"Start":"01:28.410 ","End":"01:29.730","Text":"I could take first of all,"},{"Start":"01:29.730 ","End":"01:31.785","Text":"the 4th root and then cube it."},{"Start":"01:31.785 ","End":"01:34.980","Text":"The 4th root of 16 is 2,"},{"Start":"01:34.980 ","End":"01:38.190","Text":"so it\u0027s 2 cubed."},{"Start":"01:38.190 ","End":"01:40.035","Text":"2 cubed is 8,"},{"Start":"01:40.035 ","End":"01:42.480","Text":"8 times 4 is 32."},{"Start":"01:42.480 ","End":"01:46.170","Text":"This is just 1 over 32."},{"Start":"01:46.170 ","End":"01:48.120","Text":"What we get is f of x,"},{"Start":"01:48.120 ","End":"01:52.040","Text":"which is the 4th root of x,"},{"Start":"01:52.040 ","End":"01:56.540","Text":"is approximately equal to f of x naught,"},{"Start":"01:56.540 ","End":"01:58.880","Text":"which is 16,"},{"Start":"01:58.880 ","End":"02:01.415","Text":"4th root of 16 is 2,"},{"Start":"02:01.415 ","End":"02:04.470","Text":"plus f prime of x naught,"},{"Start":"02:04.470 ","End":"02:06.630","Text":"that\u0027s f prime of 16,"},{"Start":"02:06.630 ","End":"02:10.010","Text":"that\u0027s 1/32."},{"Start":"02:10.010 ","End":"02:14.255","Text":"This here was our f of x naught."},{"Start":"02:14.255 ","End":"02:18.060","Text":"This here was our f prime of x naught,"},{"Start":"02:18.060 ","End":"02:21.130","Text":"so you might see it more clearly in the formula,"},{"Start":"02:21.130 ","End":"02:23.310","Text":"times x minus,"},{"Start":"02:23.310 ","End":"02:26.675","Text":"and x naught is 16."},{"Start":"02:26.675 ","End":"02:32.779","Text":"This gives us a formula for how to approximate the 4th root of x in general."},{"Start":"02:32.779 ","End":"02:35.405","Text":"But x has to be close to 16."},{"Start":"02:35.405 ","End":"02:38.045","Text":"We can\u0027t really say how close is close."},{"Start":"02:38.045 ","End":"02:41.120","Text":"But let\u0027s assume that 15 is close enough."},{"Start":"02:41.120 ","End":"02:47.015","Text":"What we can get if we put in here x equals 15 is that"},{"Start":"02:47.015 ","End":"02:56.090","Text":"the 4th root of 15 is approximately equal to 2 plus 1/32,"},{"Start":"02:56.090 ","End":"03:01.010","Text":"times 15 minus 16."},{"Start":"03:01.010 ","End":"03:07.040","Text":"This is just equal to 2 minus 1/32."},{"Start":"03:07.040 ","End":"03:09.875","Text":"We could do this even without a calculator."},{"Start":"03:09.875 ","End":"03:12.030","Text":"We could manually compute that,"},{"Start":"03:12.030 ","End":"03:14.425","Text":"or just keep dividing by 2,"},{"Start":"03:14.425 ","End":"03:16.100","Text":"or cheat and use the calculator,"},{"Start":"03:16.100 ","End":"03:18.110","Text":"that\u0027s why we have a calculator for this,"},{"Start":"03:18.110 ","End":"03:19.505","Text":"but not for the 4th root,"},{"Start":"03:19.505 ","End":"03:25.730","Text":"is 2 minus 0.03125,"},{"Start":"03:25.730 ","End":"03:35.285","Text":"which gives us 1.96875."},{"Start":"03:35.285 ","End":"03:40.085","Text":"I checked on the calculator what is the 4th root of 15,"},{"Start":"03:40.085 ","End":"03:47.810","Text":"and it came out to 1.96799 to 5 decimal places."},{"Start":"03:47.810 ","End":"03:49.220","Text":"It\u0027s an approximation."},{"Start":"03:49.220 ","End":"03:51.180","Text":"That\u0027s it."}],"ID":6467},{"Watched":false,"Name":"Exercise 3","Duration":"2m 51s","ChapterTopicVideoID":6441,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"We want to get an approximate value for the sine of 3 degrees,"},{"Start":"00:03.975 ","End":"00:06.240","Text":"and we\u0027re going to use linear approximation."},{"Start":"00:06.240 ","End":"00:08.205","Text":"The formula for it is here."},{"Start":"00:08.205 ","End":"00:11.910","Text":"We have to identify what is the function f we want to use."},{"Start":"00:11.910 ","End":"00:15.570","Text":"Really obvious that we want to use sine x."},{"Start":"00:15.570 ","End":"00:18.570","Text":"We also want a value x naught,"},{"Start":"00:18.570 ","End":"00:21.570","Text":"something that we know what is f of x naught,"},{"Start":"00:21.570 ","End":"00:23.250","Text":"we want to know the sine of it."},{"Start":"00:23.250 ","End":"00:25.170","Text":"We don\u0027t know the sine of very many angles,"},{"Start":"00:25.170 ","End":"00:26.820","Text":"but 0 we do know,"},{"Start":"00:26.820 ","End":"00:30.645","Text":"so this is going to be equal to 0."},{"Start":"00:30.645 ","End":"00:33.570","Text":"I want to point out right away that we\u0027re going to be using"},{"Start":"00:33.570 ","End":"00:37.605","Text":"radians because if we say f prime of x is cosine x,"},{"Start":"00:37.605 ","End":"00:40.290","Text":"this derivative only works in radians."},{"Start":"00:40.290 ","End":"00:43.800","Text":"Might as well already convert 3 degrees to radians,"},{"Start":"00:43.800 ","End":"00:48.380","Text":"I take the 3 and I multiply it by Pi over a 180."},{"Start":"00:48.380 ","End":"00:52.820","Text":"I\u0027ll leave it for the moment as Pi over 60."},{"Start":"00:52.820 ","End":"00:57.490","Text":"What I\u0027m still missing in the formula is f prime of x naught,"},{"Start":"00:57.490 ","End":"01:04.530","Text":"so that\u0027s f prime of 0 is cosine 0, which is 1."},{"Start":"01:04.530 ","End":"01:06.170","Text":"I have most everything,"},{"Start":"01:06.170 ","End":"01:08.480","Text":"I have f of x naught."},{"Start":"01:08.480 ","End":"01:11.720","Text":"I know what is f of x naught,"},{"Start":"01:11.720 ","End":"01:13.925","Text":"because that\u0027s f of 0,"},{"Start":"01:13.925 ","End":"01:17.620","Text":"which is sine 0, which is 0."},{"Start":"01:17.620 ","End":"01:20.940","Text":"I\u0027ve got that this is 0,"},{"Start":"01:20.940 ","End":"01:24.150","Text":"I\u0027ve got that this is 1,"},{"Start":"01:24.150 ","End":"01:31.940","Text":"and f of x is sine x. I can say that sine x is approximately"},{"Start":"01:31.940 ","End":"01:40.710","Text":"equal to 0 plus 1 times x minus x naught is 0."},{"Start":"01:40.710 ","End":"01:45.225","Text":"All we\u0027re left with, is that this is equal to just x."},{"Start":"01:45.225 ","End":"01:51.410","Text":"Sine x is approximately equal to x as long as x is close to x naught."},{"Start":"01:51.410 ","End":"01:56.465","Text":"In other words, it\u0027s got to be that x is close to 0."},{"Start":"01:56.465 ","End":"01:59.435","Text":"Now plugging in to this formula,"},{"Start":"01:59.435 ","End":"02:02.600","Text":"Pi over 60, because remember we\u0027re using radians,"},{"Start":"02:02.600 ","End":"02:07.070","Text":"this is my value of x. I\u0027ve got sine of"},{"Start":"02:07.070 ","End":"02:16.445","Text":"Pi over 60 radians is approximately equal to Pi over 60."},{"Start":"02:16.445 ","End":"02:20.255","Text":"Let\u0027s just say we have a primitive calculator without trigonometric functions,"},{"Start":"02:20.255 ","End":"02:29.795","Text":"so we can compute this to 5 decimal places, 0.05236."},{"Start":"02:29.795 ","End":"02:31.730","Text":"That\u0027s actually in radians,"},{"Start":"02:31.730 ","End":"02:35.265","Text":"but it\u0027s really sine of 3 degrees."},{"Start":"02:35.265 ","End":"02:38.820","Text":"Computing sine of 3 degrees and let\u0027s see what we get,"},{"Start":"02:38.820 ","End":"02:43.750","Text":"I rounded to 4 decimal places and I got 0.05234."},{"Start":"02:47.090 ","End":"02:52.180","Text":"Not a bad approximation and we\u0027re done."}],"ID":6468},{"Watched":false,"Name":"Exercise 4","Duration":"2m 41s","ChapterTopicVideoID":6442,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.895","Text":"In this exercise we want to get an approximate value for arctangent of 0.25."},{"Start":"00:05.895 ","End":"00:08.460","Text":"We\u0027re going to use linear approximation."},{"Start":"00:08.460 ","End":"00:10.725","Text":"I brought the formula with me."},{"Start":"00:10.725 ","End":"00:12.420","Text":"What we want to do is identify,"},{"Start":"00:12.420 ","End":"00:15.135","Text":"first of all, what is f and what is x naught."},{"Start":"00:15.135 ","End":"00:16.965","Text":"The function we want to use,"},{"Start":"00:16.965 ","End":"00:19.905","Text":"nothing much else but arctangent."},{"Start":"00:19.905 ","End":"00:22.590","Text":"We\u0027ll take it as arctangent of x."},{"Start":"00:22.590 ","End":"00:29.340","Text":"X naught is a value which is close to the value that we want,"},{"Start":"00:29.340 ","End":"00:32.125","Text":"but something that we know the arctangent of."},{"Start":"00:32.125 ","End":"00:34.770","Text":"We don\u0027t know the arctangent of very many things."},{"Start":"00:34.770 ","End":"00:39.080","Text":"We know the arctangent of 0, the arctangent of 1."},{"Start":"00:39.080 ","End":"00:42.790","Text":"Maybe we even know the arctangent of 1 over root 3."},{"Start":"00:42.790 ","End":"00:49.220","Text":"The closest of those to 0.25 and the simplest is just to take it as 0."},{"Start":"00:49.220 ","End":"00:52.309","Text":"We know what is f of x naught,"},{"Start":"00:52.309 ","End":"00:55.490","Text":"which is arctangent of 0."},{"Start":"00:55.490 ","End":"00:57.485","Text":"You should know that this is 0."},{"Start":"00:57.485 ","End":"00:59.720","Text":"Reason is that the tangent of 0 is 0,"},{"Start":"00:59.720 ","End":"01:01.880","Text":"so the other way around also."},{"Start":"01:01.880 ","End":"01:03.770","Text":"Now we need the derivative of f."},{"Start":"01:03.770 ","End":"01:05.270","Text":"We always need that."},{"Start":"01:05.270 ","End":"01:06.890","Text":"F prime of x,"},{"Start":"01:06.890 ","End":"01:10.985","Text":"you should remember or look it up in the table of derivatives."},{"Start":"01:10.985 ","End":"01:17.615","Text":"The derivative of the arctangent of x is 1 over 1 plus x squared."},{"Start":"01:17.615 ","End":"01:25.460","Text":"F prime of x naught is 1 over 1 plus 0 squared, which is just 1."},{"Start":"01:25.460 ","End":"01:29.510","Text":"We have f of x naught is naught."},{"Start":"01:29.510 ","End":"01:34.320","Text":"We have f prime of x naught is 1."},{"Start":"01:34.430 ","End":"01:37.500","Text":"We can say that f of x,"},{"Start":"01:37.500 ","End":"01:41.185","Text":"which is arctangent of x,"},{"Start":"01:41.185 ","End":"01:49.935","Text":"is approximately equal to 0 plus 1 times x minus,"},{"Start":"01:49.935 ","End":"01:52.590","Text":"x naught is naught."},{"Start":"01:52.590 ","End":"01:54.090","Text":"I used this other symbol."},{"Start":"01:54.090 ","End":"01:56.675","Text":"This is not the symbol for approximately equal to."},{"Start":"01:56.675 ","End":"01:59.305","Text":"You could also write it as 2 wavy lines."},{"Start":"01:59.305 ","End":"02:04.520","Text":"Arctangent of x is approximately equal to just x."},{"Start":"02:04.520 ","End":"02:10.445","Text":"In our case, arctangent of 0.25,"},{"Start":"02:10.445 ","End":"02:16.820","Text":"we can say, is approximately equal to 0.25."},{"Start":"02:16.820 ","End":"02:19.880","Text":"This, by the way, is in radians."},{"Start":"02:19.880 ","End":"02:21.560","Text":"If you wanted to do it in degrees,"},{"Start":"02:21.560 ","End":"02:26.615","Text":"you\u0027d have to multiply it by 180 over Pi."},{"Start":"02:26.615 ","End":"02:27.900","Text":"We\u0027re basically done,"},{"Start":"02:27.900 ","End":"02:31.430","Text":"but in case you\u0027re curious as to what the exact answer is,"},{"Start":"02:31.430 ","End":"02:33.320","Text":"let me do it on the calculator."},{"Start":"02:33.320 ","End":"02:38.960","Text":"I did it to 3 decimal places and it comes out 0.245."},{"Start":"02:38.960 ","End":"02:41.310","Text":"We\u0027re done."}],"ID":6469},{"Watched":false,"Name":"Exercise 5","Duration":"2m 26s","ChapterTopicVideoID":6443,"CourseChapterTopicPlaylistID":1674,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.086","Text":"In this exercise, we want to get an approximate value for 1/e."},{"Start":"00:04.086 ","End":"00:08.565","Text":"We\u0027re going to use the linear approximation formula, which I brought."},{"Start":"00:08.565 ","End":"00:12.510","Text":"First thing to do is to identify what is our function."},{"Start":"00:12.510 ","End":"00:18.765","Text":"I would say that because 1/e is just e^minus 1,"},{"Start":"00:18.765 ","End":"00:27.290","Text":"it looks like we could go for f of x equals e^x and eventually,"},{"Start":"00:27.290 ","End":"00:29.510","Text":"we\u0027ll want x to be minus 1."},{"Start":"00:29.510 ","End":"00:35.570","Text":"But meanwhile, we choose something close that we do know the value of the function there."},{"Start":"00:35.570 ","End":"00:37.520","Text":"Let\u0027s take x_naught equals naught."},{"Start":"00:37.520 ","End":"00:43.025","Text":"We don\u0027t really know the value of e to the anything except that e^0 is 1."},{"Start":"00:43.025 ","End":"00:44.720","Text":"Let\u0027s go for that."},{"Start":"00:44.720 ","End":"00:53.135","Text":"That means that f of x_naught is e^0, which is 1."},{"Start":"00:53.135 ","End":"00:55.340","Text":"We also need f prime of x_naught."},{"Start":"00:55.340 ","End":"00:57.215","Text":"Get the general derivative first,"},{"Start":"00:57.215 ","End":"01:01.080","Text":"f prime of x is just e^x itself."},{"Start":"01:01.080 ","End":"01:02.820","Text":"It\u0027s 1 of those functions when you differentiate it,"},{"Start":"01:02.820 ","End":"01:06.585","Text":"it stays the same and so f prime of"},{"Start":"01:06.585 ","End":"01:11.935","Text":"x_naught is same thing as f of x_naught is also equal to 1."},{"Start":"01:11.935 ","End":"01:14.630","Text":"Well, we have everything for plugging in here,"},{"Start":"01:14.630 ","End":"01:18.800","Text":"f of x is e^x."},{"Start":"01:18.800 ","End":"01:24.610","Text":"That\u0027s approximately equal to f of x_naught is 1,"},{"Start":"01:24.610 ","End":"01:28.544","Text":"f prime of x_naught is also 1,"},{"Start":"01:28.544 ","End":"01:32.910","Text":"x minus x_naught is 0."},{"Start":"01:32.910 ","End":"01:40.700","Text":"What we get is that e^x is approximately equal to 1 plus x."},{"Start":"01:40.700 ","End":"01:47.780","Text":"This approximation holds for when x is near x_naught, x is near 0."},{"Start":"01:47.780 ","End":"01:52.105","Text":"If we plug in x equals minus 1,"},{"Start":"01:52.105 ","End":"02:01.495","Text":"we\u0027ll get e^minus 1 is approximately equal to 1 plus minus 1."},{"Start":"02:01.495 ","End":"02:04.530","Text":"It came out to be 0."},{"Start":"02:04.530 ","End":"02:08.300","Text":"This is a very bad approximation,"},{"Start":"02:08.300 ","End":"02:11.930","Text":"I guess because minus 1 is too far from 0,"},{"Start":"02:11.930 ","End":"02:14.075","Text":"we don\u0027t always get a good approximation."},{"Start":"02:14.075 ","End":"02:22.045","Text":"The actual value on the calculator came out 0.367 something."},{"Start":"02:22.045 ","End":"02:26.160","Text":"A very bad approximation, but there we are."}],"ID":6470}],"Thumbnail":null,"ID":1674}]

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