Parametric Equations and Curves
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Tangents with Parametric Equations
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Area with Parametric Equations
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[{"Name":"Parametric Equations and Curves","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Parametric Equations and Curves","Duration":"17m 10s","ChapterTopicVideoID":5985,"CourseChapterTopicPlaylistID":4001,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this clip, we\u0027re starting a new topic."},{"Start":"00:02.610 ","End":"00:05.040","Text":"I\u0027m calling it parametric equations and curves,"},{"Start":"00:05.040 ","End":"00:08.865","Text":"and we\u0027ll see later where the name comes from."},{"Start":"00:08.865 ","End":"00:13.545","Text":"Let me just remind you about how we express"},{"Start":"00:13.545 ","End":"00:19.215","Text":"a relationship between x and y and how we draw the graph."},{"Start":"00:19.215 ","End":"00:24.055","Text":"Essentially we\u0027ve seen 3 kinds of such relations."},{"Start":"00:24.055 ","End":"00:28.935","Text":"We\u0027ve seen y equals a function of x,"},{"Start":"00:28.935 ","End":"00:31.080","Text":"and then we know how to draw a graph,"},{"Start":"00:31.080 ","End":"00:32.325","Text":"I\u0027ll give an example."},{"Start":"00:32.325 ","End":"00:36.420","Text":"We\u0027ve also sometimes take x as a function of y."},{"Start":"00:36.420 ","End":"00:38.800","Text":"Let me use another letter g,"},{"Start":"00:38.800 ","End":"00:41.495","Text":"x could be a function of y."},{"Start":"00:41.495 ","End":"00:45.695","Text":"We\u0027ve even seen something called an implicit function where we have"},{"Start":"00:45.695 ","End":"00:52.385","Text":"some capital F of 2 variables equal to 0."},{"Start":"00:52.385 ","End":"00:54.500","Text":"If you haven\u0027t seen this,"},{"Start":"00:54.500 ","End":"00:57.870","Text":"never mind, just ignore that."},{"Start":"01:00.460 ","End":"01:04.015","Text":"Well, our favorite example is always the parabola,"},{"Start":"01:04.015 ","End":"01:07.400","Text":"y equals x squared would be an example."},{"Start":"01:07.400 ","End":"01:10.085","Text":"An example of this could also be a parabola,"},{"Start":"01:10.085 ","End":"01:13.265","Text":"x equals y squared."},{"Start":"01:13.265 ","End":"01:24.150","Text":"An example of this might be x squared plus y squared equals 25."},{"Start":"01:24.990 ","End":"01:27.445","Text":"Some of you may have seen this."},{"Start":"01:27.445 ","End":"01:32.385","Text":"This would be the equation of a circle whose radius is 5,"},{"Start":"01:32.385 ","End":"01:34.410","Text":"because 5 squared is 25."},{"Start":"01:34.410 ","End":"01:37.090","Text":"If I give a quick sketch of each of these,"},{"Start":"01:37.090 ","End":"01:39.820","Text":"so you\u0027ll need a y-axis,"},{"Start":"01:39.820 ","End":"01:42.835","Text":"we need an x-axis,"},{"Start":"01:42.835 ","End":"01:46.385","Text":"and you know what, let me just copy this here and here."},{"Start":"01:46.385 ","End":"01:49.540","Text":"We\u0027ve studied various techniques of how to get from"},{"Start":"01:49.540 ","End":"01:53.575","Text":"this equation to a picture called a curve or a graph."},{"Start":"01:53.575 ","End":"01:56.980","Text":"In this case, it looks roughly something like this."},{"Start":"01:56.980 ","End":"01:59.620","Text":"It\u0027s really not the point here to get anything accurate,"},{"Start":"01:59.620 ","End":"02:01.345","Text":"just to get a general idea."},{"Start":"02:01.345 ","End":"02:02.920","Text":"This would be the equation,"},{"Start":"02:02.920 ","End":"02:04.105","Text":"this is the curve,"},{"Start":"02:04.105 ","End":"02:06.100","Text":"and later we\u0027ll come to the parametric."},{"Start":"02:06.100 ","End":"02:07.969","Text":"Here, we also have an equation,"},{"Start":"02:07.969 ","End":"02:09.365","Text":"x equals y squared,"},{"Start":"02:09.365 ","End":"02:10.910","Text":"corresponding to this equation,"},{"Start":"02:10.910 ","End":"02:13.590","Text":"there\u0027s a curve and it looks something like this,"},{"Start":"02:13.590 ","End":"02:16.940","Text":"a sideways parabola opening to the right."},{"Start":"02:16.940 ","End":"02:18.590","Text":"Here it\u0027s opening above."},{"Start":"02:18.590 ","End":"02:21.535","Text":"In this case we have a circle,"},{"Start":"02:21.535 ","End":"02:24.260","Text":"and it looks something like this."},{"Start":"02:24.260 ","End":"02:29.190","Text":"Now I\u0027m going to show you a different way of getting"},{"Start":"02:29.190 ","End":"02:35.165","Text":"an equation and its curve and the word parametric is the keyword here."},{"Start":"02:35.165 ","End":"02:37.130","Text":"Comes from my parameter."},{"Start":"02:37.130 ","End":"02:41.495","Text":"Now, I\u0027ll give you an example and the example is this."},{"Start":"02:41.495 ","End":"02:50.120","Text":"We give x and y an expression in terms of some third variable,"},{"Start":"02:50.120 ","End":"02:58.340","Text":"usually the letter t. I could say x equals t squared plus t,"},{"Start":"02:58.340 ","End":"03:02.820","Text":"y equals 1 minus 2t."},{"Start":"03:02.950 ","End":"03:10.205","Text":"What this means is that we take various values of t and for each value of t,"},{"Start":"03:10.205 ","End":"03:14.270","Text":"we get an x and a y and that forms a relationship"},{"Start":"03:14.270 ","End":"03:18.575","Text":"between x and y indirectly through t. Now,"},{"Start":"03:18.575 ","End":"03:21.650","Text":"typically this comes from physics,"},{"Start":"03:21.650 ","End":"03:28.805","Text":"where t is often the time and x and y are the coordinates of a point."},{"Start":"03:28.805 ","End":"03:31.640","Text":"So x and y are both functions of time."},{"Start":"03:31.640 ","End":"03:32.810","Text":"At any given time,"},{"Start":"03:32.810 ","End":"03:33.920","Text":"we can compute the x,"},{"Start":"03:33.920 ","End":"03:35.300","Text":"y of a point."},{"Start":"03:35.300 ","End":"03:46.145","Text":"The general form is where x is given as some function of t and y"},{"Start":"03:46.145 ","End":"03:52.460","Text":"is another function maybe g of t. 1 of"},{"Start":"03:52.460 ","End":"03:54.560","Text":"the first things we\u0027ll want to learn to do with"},{"Start":"03:54.560 ","End":"03:59.780","Text":"this parametric form is to sketch it and see what its curve is."},{"Start":"03:59.780 ","End":"04:02.600","Text":"The curve is the graph basically."},{"Start":"04:02.600 ","End":"04:09.905","Text":"Now, in the case of y as a function of x or x as a function of y,"},{"Start":"04:09.905 ","End":"04:17.705","Text":"1 way of sketching is just to draw a table of values and that\u0027s 1 way of drawing"},{"Start":"04:17.705 ","End":"04:22.315","Text":"the curve of a parametric equation"},{"Start":"04:22.315 ","End":"04:29.220","Text":"is you would just make a table,"},{"Start":"04:29.350 ","End":"04:36.800","Text":"only this time you would need 3 columns because you would put t and then various values."},{"Start":"04:36.800 ","End":"04:38.885","Text":"I\u0027m just explaining the general idea."},{"Start":"04:38.885 ","End":"04:40.820","Text":"Then you would, for each of these t\u0027s,"},{"Start":"04:40.820 ","End":"04:42.650","Text":"you would compute the x,"},{"Start":"04:42.650 ","End":"04:45.365","Text":"whether it\u0027s a specific case."},{"Start":"04:45.365 ","End":"04:47.240","Text":"I\u0027ll just give you 1 example."},{"Start":"04:47.240 ","End":"04:49.385","Text":"Suppose in this specific case,"},{"Start":"04:49.385 ","End":"04:51.665","Text":"I let t equals 0."},{"Start":"04:51.665 ","End":"04:56.570","Text":"Then I would say, x is 0 squared plus 0 is"},{"Start":"04:56.570 ","End":"05:02.850","Text":"0 and y is 1 minus twice 0 is 1, and so on."},{"Start":"05:02.850 ","End":"05:05.580","Text":"Then for each of these t\u0027s,"},{"Start":"05:05.580 ","End":"05:09.260","Text":"we could plot the pair x,"},{"Start":"05:09.260 ","End":"05:10.790","Text":"y on the curve."},{"Start":"05:10.790 ","End":"05:16.860","Text":"In fact, why don\u0027t I take this example up in more detail?"},{"Start":"05:16.930 ","End":"05:20.610","Text":"I\u0027ll clear some space."},{"Start":"05:23.800 ","End":"05:28.360","Text":"Why don\u0027t I clear this table?"},{"Start":"05:28.360 ","End":"05:30.420","Text":"There. You know what,"},{"Start":"05:30.420 ","End":"05:33.675","Text":"I\u0027ll move it down here. There we are."},{"Start":"05:33.675 ","End":"05:38.545","Text":"Let me just make these lines a bit longer, there we are."},{"Start":"05:38.545 ","End":"05:42.460","Text":"Now start plugging in some values."},{"Start":"05:42.460 ","End":"05:47.470","Text":"The values I\u0027m going to take and don\u0027t ask me at how I know which values to choose,"},{"Start":"05:47.470 ","End":"05:49.510","Text":"I\u0027m going to choose minus 2,"},{"Start":"05:49.510 ","End":"05:56.500","Text":"minus 1,0,1 and have"},{"Start":"05:56.500 ","End":"06:00.645","Text":"an intuition that minus a half might be good here."},{"Start":"06:00.645 ","End":"06:03.340","Text":"1 of them we did before, when t was 0,"},{"Start":"06:03.340 ","End":"06:07.810","Text":"we already said that x is 0 and y is 1."},{"Start":"06:07.810 ","End":"06:09.430","Text":"Let\u0027s just fill in the other."},{"Start":"06:09.430 ","End":"06:11.605","Text":"We could just take it 5 altogether."},{"Start":"06:11.605 ","End":"06:14.030","Text":"When t is minus 2,"},{"Start":"06:14.030 ","End":"06:19.244","Text":"we have 4 minus 2 is 2,"},{"Start":"06:19.244 ","End":"06:23.715","Text":"and y is 1 minus"},{"Start":"06:23.715 ","End":"06:33.940","Text":"minus 4 is plus 5."},{"Start":"06:34.000 ","End":"06:36.650","Text":"Yeah, I think that\u0027s right."},{"Start":"06:36.650 ","End":"06:38.945","Text":"When t is minus 1,"},{"Start":"06:38.945 ","End":"06:41.600","Text":"x is 1 minus 1,"},{"Start":"06:41.600 ","End":"06:52.395","Text":"that\u0027s a 0, and y is 1 minus twice minus 1 plus 2 is 3."},{"Start":"06:52.395 ","End":"06:54.780","Text":"Let\u0027s do the whole ones first."},{"Start":"06:54.780 ","End":"06:59.025","Text":"When x is 1, 1 squared plus 1 is 2,"},{"Start":"06:59.025 ","End":"07:00.990","Text":"and y is 1 minus twice,"},{"Start":"07:00.990 ","End":"07:03.180","Text":"1 is minus 1."},{"Start":"07:03.180 ","End":"07:05.669","Text":"Then minus a 0.5,"},{"Start":"07:05.669 ","End":"07:12.930","Text":"I have minus 0.5 squared is plus 0.25 minus 0.5,"},{"Start":"07:12.930 ","End":"07:22.640","Text":"that gives me minus 0.25 and y is 1 minus twice minus 0.5,"},{"Start":"07:22.640 ","End":"07:28.090","Text":"which is 1 plus 1, which is 2."},{"Start":"07:28.500 ","End":"07:33.010","Text":"Now we can draw the curve for this."},{"Start":"07:33.010 ","End":"07:35.485","Text":"First of all, we need a set of axes,"},{"Start":"07:35.485 ","End":"07:41.125","Text":"a y-axis and an x-axis,"},{"Start":"07:41.125 ","End":"07:44.725","Text":"and we need some scale."},{"Start":"07:44.725 ","End":"07:49.570","Text":"Let\u0027s see, y goes up to 5 and down to minus 1,"},{"Start":"07:49.570 ","End":"07:53.200","Text":"so let\u0027s this is 5."},{"Start":"07:53.200 ","End":"08:00.260","Text":"This would be 0 and this would be minus 1."},{"Start":"08:00.900 ","End":"08:03.400","Text":"Let\u0027s just say that this is 0,"},{"Start":"08:03.400 ","End":"08:08.320","Text":"of course for both x and y and here we have 1,"},{"Start":"08:08.320 ","End":"08:12.040","Text":"2, 3, 4, 5 and minus 1,"},{"Start":"08:12.040 ","End":"08:13.810","Text":"doesn\u0027t have to be exact."},{"Start":"08:13.810 ","End":"08:23.050","Text":"Then in our x, we see that we need to go from 2 down to minus 1/4 and up to 2 again."},{"Start":"08:23.050 ","End":"08:33.505","Text":"Let\u0027s say that this is minus 1 and then we have 1, 2, 3, 4."},{"Start":"08:33.505 ","End":"08:35.470","Text":"We\u0027re not trying to be precise,"},{"Start":"08:35.470 ","End":"08:37.630","Text":"we\u0027re just getting a general idea."},{"Start":"08:37.630 ","End":"08:40.120","Text":"Now the actual points."},{"Start":"08:40.120 ","End":"08:44.110","Text":"The first point I want to sketch is 2,"},{"Start":"08:44.110 ","End":"08:47.829","Text":"5, x is 2, y is 5,"},{"Start":"08:47.829 ","End":"08:51.085","Text":"so that\u0027s somewhere around here,"},{"Start":"08:51.085 ","End":"08:56.020","Text":"and then x equals 0, y equals 3."},{"Start":"08:56.020 ","End":"09:00.430","Text":"That would be, say here,"},{"Start":"09:00.430 ","End":"09:10.255","Text":"and then x is minus 1/4, y is 2."},{"Start":"09:10.255 ","End":"09:13.825","Text":"Minus 1/4, say here and 1,"},{"Start":"09:13.825 ","End":"09:17.274","Text":"2, something around here."},{"Start":"09:17.274 ","End":"09:23.230","Text":"Then 0, 1, so that would be here,"},{"Start":"09:23.230 ","End":"09:29.095","Text":"and then 2, minus 1,"},{"Start":"09:29.095 ","End":"09:32.275","Text":"that may be here."},{"Start":"09:32.275 ","End":"09:39.760","Text":"It turns out if we draw a line through it and we get something like this,"},{"Start":"09:39.760 ","End":"09:42.340","Text":"it\u0027s actually a parabola."},{"Start":"09:42.340 ","End":"09:44.980","Text":"Doesn\u0027t look great, but you get the idea."},{"Start":"09:44.980 ","End":"09:50.100","Text":"Now, the point I want to make is that not only can we draw the points,"},{"Start":"09:50.100 ","End":"09:51.330","Text":"but we can actually,"},{"Start":"09:51.330 ","End":"09:52.589","Text":"for each of the points,"},{"Start":"09:52.589 ","End":"09:57.250","Text":"associate a parameter, the t I didn\u0027t include here."},{"Start":"09:58.260 ","End":"10:02.380","Text":"Here is the point I got when t was minus 2,"},{"Start":"10:02.380 ","End":"10:07.360","Text":"so I can actually say t equals minus 2 here."},{"Start":"10:07.360 ","End":"10:13.000","Text":"This point here was t equals minus 1."},{"Start":"10:13.000 ","End":"10:20.920","Text":"This point here was t equals minus 1/2 and then we had here,"},{"Start":"10:20.920 ","End":"10:28.060","Text":"where t is equal to 0 and here was"},{"Start":"10:28.060 ","End":"10:36.430","Text":"where we had t equals 1."},{"Start":"10:36.430 ","End":"10:40.029","Text":"Corresponding to each point on the curve,"},{"Start":"10:40.029 ","End":"10:46.660","Text":"there is a value of t. In the model of physics where it\u0027s a particle traveling in time,"},{"Start":"10:46.660 ","End":"10:50.500","Text":"it starts out at time negative 2 here,"},{"Start":"10:50.500 ","End":"10:51.700","Text":"or even before that,"},{"Start":"10:51.700 ","End":"10:53.800","Text":"it came from here."},{"Start":"10:53.800 ","End":"10:56.200","Text":"The present, t equals 0,"},{"Start":"10:56.200 ","End":"10:57.760","Text":"it\u0027s here and in the future,"},{"Start":"10:57.760 ","End":"10:59.500","Text":"t equals 1, it will be here."},{"Start":"10:59.500 ","End":"11:03.730","Text":"There\u0027s an actual motion and a parametric curve"},{"Start":"11:03.730 ","End":"11:09.070","Text":"actually has a direction as opposed to the previous curves."},{"Start":"11:09.070 ","End":"11:11.500","Text":"T is increasing in this direction."},{"Start":"11:11.500 ","End":"11:12.520","Text":"I can draw an arrow,"},{"Start":"11:12.520 ","End":"11:14.365","Text":"t is increasing here,"},{"Start":"11:14.365 ","End":"11:17.830","Text":"t is increasing here and it continues to increase and it"},{"Start":"11:17.830 ","End":"11:21.325","Text":"came from here and put another arrow here,"},{"Start":"11:21.325 ","End":"11:24.565","Text":"and there\u0027s a definite direction of flow."},{"Start":"11:24.565 ","End":"11:28.720","Text":"I believe the technical term is direction of motion,"},{"Start":"11:28.720 ","End":"11:32.380","Text":"which is the direction that we go when we increase"},{"Start":"11:32.380 ","End":"11:35.500","Text":"t. As t is increasing in this direction,"},{"Start":"11:35.500 ","End":"11:41.690","Text":"we get this flow direction."},{"Start":"11:44.040 ","End":"11:47.230","Text":"Now, there\u0027s something I want you to note that\u0027s quite"},{"Start":"11:47.230 ","End":"11:49.810","Text":"important when you sketch the curve."},{"Start":"11:49.810 ","End":"11:55.810","Text":"This would be a parametric curve because the equation is in parametric form,"},{"Start":"11:55.810 ","End":"11:57.700","Text":"t is the parameter."},{"Start":"11:57.700 ","End":"11:59.770","Text":"This is the parametric equation,"},{"Start":"11:59.770 ","End":"12:02.305","Text":"parametric curve, that explains the title."},{"Start":"12:02.305 ","End":"12:09.295","Text":"Now, the important thing is that you notice I didn\u0027t stop here at t equals minus 2,"},{"Start":"12:09.295 ","End":"12:14.700","Text":"I showed an extra bit here and here because obviously, or in this case,"},{"Start":"12:14.700 ","End":"12:16.650","Text":"it\u0027s clear that if I started earlier,"},{"Start":"12:16.650 ","End":"12:18.345","Text":"t equals minus 3,"},{"Start":"12:18.345 ","End":"12:20.730","Text":"where I continued, t equals 2,"},{"Start":"12:20.730 ","End":"12:23.400","Text":"that actually it would continue."},{"Start":"12:23.400 ","End":"12:28.030","Text":"So be sure to leave the extra bits."},{"Start":"12:28.030 ","End":"12:36.805","Text":"However, sometimes the equation is given in the form of a limitation on the parameter."},{"Start":"12:36.805 ","End":"12:45.220","Text":"We might also say that t is between minus 2 and 1."},{"Start":"12:45.220 ","End":"12:49.315","Text":"I\u0027m talking about this part here,"},{"Start":"12:49.315 ","End":"12:50.860","Text":"this is just the general."},{"Start":"12:50.860 ","End":"12:53.650","Text":"If I took this together with this,"},{"Start":"12:53.650 ","End":"12:57.100","Text":"in that case, the curve would stop here,"},{"Start":"12:57.100 ","End":"12:58.945","Text":"so I\u0027m just going to erase that."},{"Start":"12:58.945 ","End":"13:01.000","Text":"If I put that restriction,"},{"Start":"13:01.000 ","End":"13:03.325","Text":"then really we do start here,"},{"Start":"13:03.325 ","End":"13:06.760","Text":"and we do end right here,"},{"Start":"13:06.760 ","End":"13:09.040","Text":"and that\u0027s all there is to the curve."},{"Start":"13:09.040 ","End":"13:12.070","Text":"Sometimes you\u0027ll be given a constraint."},{"Start":"13:12.070 ","End":"13:14.350","Text":"Just like when y is a function of x,"},{"Start":"13:14.350 ","End":"13:19.970","Text":"we sometimes say x is between something and something and restricted."},{"Start":"13:20.310 ","End":"13:23.815","Text":"Another point I want to make is you might say,"},{"Start":"13:23.815 ","End":"13:25.390","Text":"this is a special point."},{"Start":"13:25.390 ","End":"13:28.555","Text":"If this is a parabola and this is the apex,"},{"Start":"13:28.555 ","End":"13:31.405","Text":"in choosing the values of t,"},{"Start":"13:31.405 ","End":"13:34.820","Text":"how did I know to choose minus 1/2?"},{"Start":"13:35.070 ","End":"13:37.510","Text":"It\u0027s really just experience,"},{"Start":"13:37.510 ","End":"13:43.615","Text":"but the answer is there\u0027s no good way and this is not the only way of drawing the curve."},{"Start":"13:43.615 ","End":"13:47.274","Text":"In many cases, not in all cases,"},{"Start":"13:47.274 ","End":"13:54.955","Text":"we can extract either x in terms of y or y in terms of x,"},{"Start":"13:54.955 ","End":"13:57.835","Text":"and let\u0027s see if we can do that here."},{"Start":"13:57.835 ","End":"13:59.965","Text":"In this case here,"},{"Start":"13:59.965 ","End":"14:06.070","Text":"it seems to me that if we are going to try and get one in terms of the other,"},{"Start":"14:06.070 ","End":"14:07.840","Text":"it\u0027s best to first of all,"},{"Start":"14:07.840 ","End":"14:12.565","Text":"get t in terms of y because this is linear and this is quadratic,"},{"Start":"14:12.565 ","End":"14:14.860","Text":"and then once we\u0027ve got t in terms of y,"},{"Start":"14:14.860 ","End":"14:16.180","Text":"to substitute it here."},{"Start":"14:16.180 ","End":"14:21.985","Text":"Let\u0027s do that. If y equals 1 minus 2t,"},{"Start":"14:21.985 ","End":"14:27.235","Text":"then 2t is equal to 1 minus y."},{"Start":"14:27.235 ","End":"14:34.300","Text":"Throw this over here, this over here and then we get that t is 1 minus y over 2."},{"Start":"14:34.300 ","End":"14:36.130","Text":"Once we have that,"},{"Start":"14:36.130 ","End":"14:44.620","Text":"we can then substitute t in the equation for x and say that x is equal to t squared,"},{"Start":"14:44.620 ","End":"14:50.215","Text":"which is 1 minus y over 2 squared, plus t,"},{"Start":"14:50.215 ","End":"14:54.670","Text":"which is 1 minus y over 2,"},{"Start":"14:54.670 ","End":"14:59.370","Text":"and this equals and I\u0027ll save you the algebra,"},{"Start":"14:59.370 ","End":"15:01.145","Text":"I\u0027ll just write the answer."},{"Start":"15:01.145 ","End":"15:08.395","Text":"We get 1/4 y squared minus y plus 3/4,"},{"Start":"15:08.395 ","End":"15:10.540","Text":"if I didn\u0027t make a mistake."},{"Start":"15:10.540 ","End":"15:12.775","Text":"Just copy the x here,"},{"Start":"15:12.775 ","End":"15:16.100","Text":"and we could use this to draw the graph."},{"Start":"15:16.100 ","End":"15:19.610","Text":"For example, I would often try to find the apex first,"},{"Start":"15:19.610 ","End":"15:22.525","Text":"which is what we call minus b over 2a."},{"Start":"15:22.525 ","End":"15:26.860","Text":"In this case, minus b over 2a is 1 over twice 1/4,"},{"Start":"15:26.860 ","End":"15:28.915","Text":"1 over 1/2, which is 2,"},{"Start":"15:28.915 ","End":"15:31.675","Text":"so y would equal 2."},{"Start":"15:31.675 ","End":"15:36.400","Text":"When y is 2, I would say 2 squared over 4 is 1,"},{"Start":"15:36.400 ","End":"15:38.380","Text":"minus 2 is minus 1,"},{"Start":"15:38.380 ","End":"15:40.540","Text":"plus 3/4 is minus a 1/4."},{"Start":"15:40.540 ","End":"15:45.500","Text":"Get the 2 minus 1/4 and so on,"},{"Start":"15:45.500 ","End":"15:46.520","Text":"I plug in values."},{"Start":"15:46.520 ","End":"15:50.030","Text":"However, if we sketch it from this,"},{"Start":"15:50.030 ","End":"15:52.750","Text":"we lose the direction,"},{"Start":"15:52.750 ","End":"15:56.015","Text":"we don\u0027t have a direction which we have in a parameter."},{"Start":"15:56.015 ","End":"16:03.840","Text":"It\u0027s just a curve without any particular flow or direction."},{"Start":"16:05.040 ","End":"16:09.425","Text":"In the case where we have a restriction on t,"},{"Start":"16:09.425 ","End":"16:11.360","Text":"we would also get a restriction on x."},{"Start":"16:11.360 ","End":"16:13.955","Text":"In this case, you can see that x"},{"Start":"16:13.955 ","End":"16:22.345","Text":"is a function of y."},{"Start":"16:22.345 ","End":"16:30.610","Text":"The restriction on y is that y would be between minus 1 and 5."},{"Start":"16:30.610 ","End":"16:38.320","Text":"So I would write y between minus 1 and 5."},{"Start":"16:38.320 ","End":"16:41.350","Text":"If I substituted y equals minus 1, I\u0027d get this,"},{"Start":"16:41.350 ","End":"16:42.700","Text":"I substitute y equals 5,"},{"Start":"16:42.700 ","End":"16:45.580","Text":"I get this, and so on."},{"Start":"16:45.580 ","End":"16:50.160","Text":"That\u0027s 1 way of drawing curves is if you\u0027re lucky enough,"},{"Start":"16:50.160 ","End":"16:52.240","Text":"and it will sometimes happen that you can write"},{"Start":"16:52.240 ","End":"16:56.260","Text":"1 variable in terms of the other, but not always."},{"Start":"16:56.260 ","End":"17:00.730","Text":"Sometimes you\u0027re just going to have to make a table and where you\u0027re not sure,"},{"Start":"17:00.730 ","End":"17:04.340","Text":"add extra points and so on."},{"Start":"17:04.680 ","End":"17:10.470","Text":"I think it\u0027s time overdue for a break."}],"ID":5999},{"Watched":false,"Name":"Parametric Equations and Curves (continued)","Duration":"20m 41s","ChapterTopicVideoID":5984,"CourseChapterTopicPlaylistID":4001,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:04.440","Text":"Continuing with parametric equations and curves,"},{"Start":"00:04.440 ","End":"00:10.110","Text":"I left up the bulk of what we did in the previous clip."},{"Start":"00:10.110 ","End":"00:18.060","Text":"We mentioned that we have x and y both as functions of a parameter called t. In physics,"},{"Start":"00:18.060 ","End":"00:20.145","Text":"t is usually time."},{"Start":"00:20.145 ","End":"00:25.700","Text":"A typical parametric equation would be a pair of things,"},{"Start":"00:25.700 ","End":"00:30.020","Text":"x equals something in terms of t and y equals something in terms of"},{"Start":"00:30.020 ","End":"00:34.840","Text":"t. This translates into a sketch."},{"Start":"00:34.840 ","End":"00:42.950","Text":"Sometimes, we\u0027re given a restriction on the value of the parameter."},{"Start":"00:42.950 ","End":"00:46.580","Text":"That means we only get part of the curve."},{"Start":"00:46.580 ","End":"00:53.615","Text":"It\u0027s not that easy to know what values to choose when we\u0027re making a table."},{"Start":"00:53.615 ","End":"00:56.810","Text":"It\u0027s somehow easier to sketch if we can get it"},{"Start":"00:56.810 ","End":"01:01.955","Text":"into the form of y as a function of x or x as a function of y."},{"Start":"01:01.955 ","End":"01:03.590","Text":"Just like we did here,"},{"Start":"01:03.590 ","End":"01:06.545","Text":"we got x in terms of y."},{"Start":"01:06.545 ","End":"01:13.110","Text":"Then we also found that the restriction here corresponded to a restriction here."},{"Start":"01:15.350 ","End":"01:19.650","Text":"But things don\u0027t always work out that neatly,"},{"Start":"01:19.650 ","End":"01:23.330","Text":"we\u0027ll see some strange things happening in the continuation."},{"Start":"01:23.330 ","End":"01:29.060","Text":"The other thing is that the main thing is that when we have it in closed form,"},{"Start":"01:29.060 ","End":"01:36.290","Text":"we don\u0027t get the direction of motion like we did with the parametric."},{"Start":"01:36.290 ","End":"01:39.950","Text":"We know with parametric form, from the values of t,"},{"Start":"01:39.950 ","End":"01:43.820","Text":"that we go like this,"},{"Start":"01:43.820 ","End":"01:45.890","Text":"we traverse the curve this way."},{"Start":"01:45.890 ","End":"01:48.440","Text":"But from here, it\u0027s just static."},{"Start":"01:48.440 ","End":"01:50.315","Text":"There are no directions."},{"Start":"01:50.315 ","End":"01:54.890","Text":"Still, you could draw it like this and then take a few values of t,"},{"Start":"01:54.890 ","End":"01:58.220","Text":"plug them in and see which way the thing is going."},{"Start":"01:58.220 ","End":"02:01.850","Text":"But we\u0027ll soon see some unusual things happening"},{"Start":"02:01.850 ","End":"02:07.390","Text":"that makes the parametric form quite different from the closed-form."},{"Start":"02:07.390 ","End":"02:10.080","Text":"As I said, you can\u0027t always get"},{"Start":"02:10.080 ","End":"02:14.370","Text":"a closed-form from a parametric form just if you\u0027re lucky."},{"Start":"02:14.660 ","End":"02:17.740","Text":"Let me clear the board."},{"Start":"02:17.740 ","End":"02:22.690","Text":"I want to take another example of a parametric equation and its curve"},{"Start":"02:22.690 ","End":"02:28.095","Text":"and show you what I meant by unusual things that can happen."},{"Start":"02:28.095 ","End":"02:34.480","Text":"Let\u0027s take x equals sine squared t"},{"Start":"02:34.480 ","End":"02:42.720","Text":"and y equals 2 cosine t. You know what?"},{"Start":"02:42.720 ","End":"02:47.334","Text":"I\u0027ll throw in a restriction on t. I\u0027ll let t be,"},{"Start":"02:47.334 ","End":"02:48.910","Text":"because it\u0027s the sine, the cosine,"},{"Start":"02:48.910 ","End":"02:50.050","Text":"and it\u0027s like an angle."},{"Start":"02:50.050 ","End":"02:54.580","Text":"Let\u0027s see what happens if I restrict it between 0 and 2Pi"},{"Start":"02:54.580 ","End":"03:00.380","Text":"which is like 360 degrees if you\u0027re working in degrees."},{"Start":"03:00.770 ","End":"03:07.140","Text":"Let\u0027s see then. We want to draw the sketch so I"},{"Start":"03:07.140 ","End":"03:13.860","Text":"brought a table and coordinate axes."},{"Start":"03:13.860 ","End":"03:18.885","Text":"Let\u0027s start. I see I go from 0 to 2Pi,"},{"Start":"03:18.885 ","End":"03:24.300","Text":"so I\u0027ll just choose some important values in the middle."},{"Start":"03:24.300 ","End":"03:29.765","Text":"Let\u0027s go every 90 degrees and get up to 360."},{"Start":"03:29.765 ","End":"03:31.805","Text":"Or in terms of radians,"},{"Start":"03:31.805 ","End":"03:34.465","Text":"Pi over 2 is 90,"},{"Start":"03:34.465 ","End":"03:39.705","Text":"Pi is 180, 3Pi over 2 is 270,"},{"Start":"03:39.705 ","End":"03:42.660","Text":"and 2Pi which is 360 degrees."},{"Start":"03:42.660 ","End":"03:46.860","Text":"But you should get used to working in radians. Let\u0027s see."},{"Start":"03:46.860 ","End":"03:50.390","Text":"As for x sine squared t,"},{"Start":"03:50.390 ","End":"03:52.220","Text":"which is sine t all squared,"},{"Start":"03:52.220 ","End":"03:54.245","Text":"the sine of 0 is 0,"},{"Start":"03:54.245 ","End":"03:56.090","Text":"0 squared is 0."},{"Start":"03:56.090 ","End":"04:01.185","Text":"Sine of Pi over 2 is 1, squared is 1."},{"Start":"04:01.185 ","End":"04:05.040","Text":"The sine of Pi is also 0,"},{"Start":"04:05.040 ","End":"04:07.574","Text":"squared is still 0."},{"Start":"04:07.574 ","End":"04:11.430","Text":"Sine of 3Pi over 2,"},{"Start":"04:11.430 ","End":"04:13.860","Text":"270 degrees, is minus 1,"},{"Start":"04:13.860 ","End":"04:16.725","Text":"but it\u0027s squared so it\u0027s 1."},{"Start":"04:16.725 ","End":"04:21.095","Text":"The 2Pi is like 0,"},{"Start":"04:21.095 ","End":"04:23.270","Text":"sine of it is 0,"},{"Start":"04:23.270 ","End":"04:27.020","Text":"and squared is 0. That\u0027s the x."},{"Start":"04:27.020 ","End":"04:28.340","Text":"Now, how about the y?"},{"Start":"04:28.340 ","End":"04:38.090","Text":"Twice cosine t. Cosine 0 is 1,"},{"Start":"04:38.090 ","End":"04:39.745","Text":"we double it, we get 2."},{"Start":"04:39.745 ","End":"04:43.235","Text":"Cosine of 90, sorry,"},{"Start":"04:43.235 ","End":"04:45.440","Text":"Pi over 2, work in radians."},{"Start":"04:45.440 ","End":"04:48.110","Text":"Cosine of Pi over 2 is 0,"},{"Start":"04:48.110 ","End":"04:50.104","Text":"double it is still 0."},{"Start":"04:50.104 ","End":"04:54.840","Text":"Cosine of Pi is minus 1 so this is minus 2."},{"Start":"04:54.840 ","End":"04:57.540","Text":"Cosine of 3Pi over 2"},{"Start":"04:57.540 ","End":"05:06.065","Text":"is 0 so double it is still 0."},{"Start":"05:06.065 ","End":"05:07.765","Text":"Cosine of 2Pi is 1,"},{"Start":"05:07.765 ","End":"05:09.925","Text":"so we\u0027re at 2 again."},{"Start":"05:09.925 ","End":"05:15.980","Text":"Now let\u0027s plot these points and just add some ticks to the graph."},{"Start":"05:15.980 ","End":"05:19.505","Text":"Let\u0027s see. X only goes 0 or 1."},{"Start":"05:19.505 ","End":"05:22.190","Text":"Well, this point is 0,0, I\u0027ll leave it."},{"Start":"05:22.190 ","End":"05:24.640","Text":"Let\u0027s say 1 is here,"},{"Start":"05:24.640 ","End":"05:30.075","Text":"so that\u0027s 1, and then y goes 2 or minus 2."},{"Start":"05:30.075 ","End":"05:32.340","Text":"So this is 1,"},{"Start":"05:32.340 ","End":"05:38.235","Text":"this is 2, this is minus 1, minus 2."},{"Start":"05:38.235 ","End":"05:40.470","Text":"I\u0027ll just label them minus 1,"},{"Start":"05:40.470 ","End":"05:43.110","Text":"minus 2, 1, 2."},{"Start":"05:43.110 ","End":"05:47.970","Text":"This is 1, this is 0, of course."},{"Start":"05:47.970 ","End":"05:49.665","Text":"I won\u0027t bother marking it."},{"Start":"05:49.665 ","End":"05:52.090","Text":"I have to take these points then."},{"Start":"05:52.490 ","End":"05:55.440","Text":"0,2, that\u0027s here."},{"Start":"05:55.440 ","End":"06:03.810","Text":"Then 1,0, that\u0027s here,"},{"Start":"06:03.810 ","End":"06:09.705","Text":"and then 0, minus 2 is here."},{"Start":"06:09.705 ","End":"06:13.190","Text":"Now, I happen to know that it\u0027s a curve,"},{"Start":"06:13.190 ","End":"06:15.920","Text":"it\u0027s actually part of a parabola."},{"Start":"06:15.920 ","End":"06:20.420","Text":"Actually, we get something like this."},{"Start":"06:20.420 ","End":"06:23.665","Text":"But we\u0027re not done yet, we\u0027re only up to here."},{"Start":"06:23.665 ","End":"06:25.605","Text":"Even before we\u0027re done,"},{"Start":"06:25.605 ","End":"06:27.710","Text":"I would like to put some arrows on."},{"Start":"06:27.710 ","End":"06:29.765","Text":"We started from here,"},{"Start":"06:29.765 ","End":"06:32.330","Text":"we went up to here, we got up to this,"},{"Start":"06:32.330 ","End":"06:35.225","Text":"then we continued up to here."},{"Start":"06:35.225 ","End":"06:39.480","Text":"Now, we put in again the point 1,0."},{"Start":"06:39.730 ","End":"06:43.920","Text":"Now, we already have this point, it already exists."},{"Start":"06:43.920 ","End":"06:47.990","Text":"Then we get the point 0,2 and it also exists."},{"Start":"06:47.990 ","End":"06:50.090","Text":"But it turns out that in the second half,"},{"Start":"06:50.090 ","End":"06:52.160","Text":"we\u0027re doing the same thing only backwards,"},{"Start":"06:52.160 ","End":"06:54.890","Text":"we\u0027re going from here to here and then to here."},{"Start":"06:54.890 ","End":"07:02.940","Text":"Actually, the whole thing has been traced from here to here and back again."},{"Start":"07:03.320 ","End":"07:07.345","Text":"In fact, you can\u0027t get beyond this."},{"Start":"07:07.345 ","End":"07:10.115","Text":"Even if you increase the values of t,"},{"Start":"07:10.115 ","End":"07:12.200","Text":"you\u0027ll just keep going back and forth,"},{"Start":"07:12.200 ","End":"07:15.424","Text":"back and forth although this is the arc of a parabola."},{"Start":"07:15.424 ","End":"07:19.580","Text":"Now, you don\u0027t see what\u0027s strange yet but if I wrote this"},{"Start":"07:19.580 ","End":"07:24.890","Text":"as an expression of x in terms of y or y in terms of x,"},{"Start":"07:24.890 ","End":"07:30.215","Text":"I\u0027ll just show you how I do this."},{"Start":"07:30.215 ","End":"07:33.455","Text":"I want to make use of the fact that in general,"},{"Start":"07:33.455 ","End":"07:35.720","Text":"sine squared of anything,"},{"Start":"07:35.720 ","End":"07:42.275","Text":"say Alpha plus cosine squared Alpha is 1, famous trigonometrical identity."},{"Start":"07:42.275 ","End":"07:51.290","Text":"In this case, let\u0027s say that if I take x squared and I\u0027ve got sine squared,"},{"Start":"07:51.290 ","End":"07:53.630","Text":"and if I take y squared,"},{"Start":"07:53.630 ","End":"07:55.970","Text":"I\u0027ll have 4 cosine squared."},{"Start":"07:55.970 ","End":"07:59.300","Text":"Let\u0027s take y squared over 4."},{"Start":"07:59.300 ","End":"08:03.490","Text":"What we get is that x squared is sine squared."},{"Start":"08:03.490 ","End":"08:07.905","Text":"Sorry, not squared, just x."},{"Start":"08:07.905 ","End":"08:12.510","Text":"We get sine squared t,"},{"Start":"08:12.510 ","End":"08:17.389","Text":"y squared over 4 is 4 cosine squared t over 4 is"},{"Start":"08:17.389 ","End":"08:22.505","Text":"cosine squared t. By this identity with t instead of Alpha,"},{"Start":"08:22.505 ","End":"08:24.790","Text":"this is equal to 1."},{"Start":"08:24.790 ","End":"08:35.090","Text":"Then from here, I can isolate x and say x is equal to 1 minus y squared over 4."},{"Start":"08:35.090 ","End":"08:42.605","Text":"Now, if I was to just look at x equals 1 minus y squared over 4,"},{"Start":"08:42.605 ","End":"08:47.850","Text":"what I would get would be the full parabola,"},{"Start":"08:47.850 ","End":"08:55.750","Text":"I would get something like this and then going on forever."},{"Start":"08:58.070 ","End":"09:05.720","Text":"There\u0027s definitely a limitation in the parametric form that even if I didn\u0027t restrict t,"},{"Start":"09:05.720 ","End":"09:08.735","Text":"I\u0027d still only get part of the parabola."},{"Start":"09:08.735 ","End":"09:11.370","Text":"Let me just erase that."},{"Start":"09:11.370 ","End":"09:14.430","Text":"Even if I didn\u0027t restrict t,"},{"Start":"09:14.430 ","End":"09:23.805","Text":"I would still have to restrict y to be between minus 2 and 2."},{"Start":"09:23.805 ","End":"09:29.130","Text":"You could tell by looking here that the cosine is between 1 and minus 1,"},{"Start":"09:29.130 ","End":"09:31.950","Text":"so y is going to be between 2 and minus 2."},{"Start":"09:31.950 ","End":"09:34.050","Text":"There\u0027s also a restriction on x,"},{"Start":"09:34.050 ","End":"09:36.630","Text":"but since y here is the independent variable,"},{"Start":"09:36.630 ","End":"09:39.240","Text":"I\u0027m putting the restriction on that."},{"Start":"09:39.240 ","End":"09:44.295","Text":"Often a question will ask you to"},{"Start":"09:44.295 ","End":"09:50.130","Text":"restrict the parameter in such a way that the curve is traced only once."},{"Start":"09:50.130 ","End":"09:55.470","Text":"Notice, in fact, that if I stopped at Pi from 0,"},{"Start":"09:55.470 ","End":"09:58.830","Text":"I didn\u0027t indicate the values of t,"},{"Start":"09:58.830 ","End":"10:01.290","Text":"t equals 0 here,"},{"Start":"10:01.290 ","End":"10:05.205","Text":"t equals Pi over 2,"},{"Start":"10:05.205 ","End":"10:08.160","Text":"here t equals Pi."},{"Start":"10:08.160 ","End":"10:10.920","Text":"That already covers the whole curve,"},{"Start":"10:10.920 ","End":"10:14.340","Text":"which answers the question I was going to ask."},{"Start":"10:14.340 ","End":"10:17.190","Text":"But we\u0027ll get this in a moment."},{"Start":"10:17.190 ","End":"10:20.790","Text":"Then when t is 3Pi over 2,"},{"Start":"10:20.790 ","End":"10:22.440","Text":"I\u0027m back here again,"},{"Start":"10:22.440 ","End":"10:25.185","Text":"t is 3Pi over 2."},{"Start":"10:25.185 ","End":"10:31.635","Text":"Here, once again, t can also be equal to 2Pi,"},{"Start":"10:31.635 ","End":"10:33.105","Text":"and I\u0027m back here."},{"Start":"10:33.105 ","End":"10:36.240","Text":"Now if I wanted to just traverse the curve once,"},{"Start":"10:36.240 ","End":"10:42.360","Text":"I could restrict 0 less than or equal to t,"},{"Start":"10:42.360 ","End":"10:44.025","Text":"less than or equal to Pi."},{"Start":"10:44.025 ","End":"10:49.080","Text":"That would be the smallest range I could get in order to cover everything."},{"Start":"10:49.080 ","End":"10:51.315","Text":"But whatever I do,"},{"Start":"10:51.315 ","End":"10:57.240","Text":"I can never get more of the parabola than I would from this equation without restriction."},{"Start":"10:57.240 ","End":"10:59.505","Text":"So that\u0027s 1 anomaly."},{"Start":"10:59.505 ","End":"11:08.530","Text":"Let me give another example of things that can happen in parametric form."},{"Start":"11:08.810 ","End":"11:11.220","Text":"This time as an example,"},{"Start":"11:11.220 ","End":"11:14.895","Text":"I\u0027ll take x equals 5 cosine t,"},{"Start":"11:14.895 ","End":"11:20.655","Text":"y equals 2 sine t. Let\u0027s keep the same range,"},{"Start":"11:20.655 ","End":"11:26.475","Text":"t from 0-2Pi, and we\u0027ll make a table and sketch it."},{"Start":"11:26.475 ","End":"11:28.770","Text":"I think once again,"},{"Start":"11:28.770 ","End":"11:32.220","Text":"I\u0027ll take the obvious values 0,"},{"Start":"11:32.220 ","End":"11:35.325","Text":"Pi over 2, Pi,"},{"Start":"11:35.325 ","End":"11:38.400","Text":"3Pi over 2, and 2Pi."},{"Start":"11:38.400 ","End":"11:48.900","Text":"Let\u0027s see what we get for x cosine of t. Cosine of 0 is 1."},{"Start":"11:48.900 ","End":"11:52.710","Text":"Remember to multiply everything by 5, so that\u0027s 5."},{"Start":"11:52.710 ","End":"11:55.050","Text":"Here, it\u0027s 5 times 0,"},{"Start":"11:55.050 ","End":"11:57.525","Text":"5 times minus 1,"},{"Start":"11:57.525 ","End":"12:02.490","Text":"5 times 0, and back to 5 times 1."},{"Start":"12:02.490 ","End":"12:04.815","Text":"Here we double everything."},{"Start":"12:04.815 ","End":"12:09.450","Text":"Twice sine 0, twice 0 is 0,"},{"Start":"12:09.450 ","End":"12:12.029","Text":"twice 1 is 2,"},{"Start":"12:12.029 ","End":"12:14.460","Text":"twice 0 is 0,"},{"Start":"12:14.460 ","End":"12:17.100","Text":"twice minus 1 minus 2,"},{"Start":"12:17.100 ","End":"12:20.565","Text":"and back to 0."},{"Start":"12:20.565 ","End":"12:27.300","Text":"I see that x goes from 5 to minus 5 and back to 5."},{"Start":"12:27.300 ","End":"12:29.850","Text":"Just straighten the graph out a bit,"},{"Start":"12:29.850 ","End":"12:32.265","Text":"so we get more symmetrical."},{"Start":"12:32.265 ","End":"12:34.485","Text":"That\u0027s better."},{"Start":"12:34.485 ","End":"12:37.500","Text":"Now, we can plot the points."},{"Start":"12:37.500 ","End":"12:40.395","Text":"I won\u0027t bother with putting ticks in, let\u0027s just guess."},{"Start":"12:40.395 ","End":"12:43.330","Text":"Let\u0027s say 5,0 is here,"},{"Start":"12:43.460 ","End":"12:49.200","Text":"and the minus 5,0 is equidistant on the other side."},{"Start":"12:49.200 ","End":"12:57.790","Text":"Let\u0027s see what else do we get 0,2 and we also have 0, minus 2."},{"Start":"12:58.220 ","End":"13:02.235","Text":"The last point is the same as the first point."},{"Start":"13:02.235 ","End":"13:05.340","Text":"It turns out, and I\u0027ll show you in"},{"Start":"13:05.340 ","End":"13:08.760","Text":"a moment that this actually comes out to be an ellipse."},{"Start":"13:08.760 ","End":"13:11.805","Text":"Let me just roughly sketch an ellipse."},{"Start":"13:11.805 ","End":"13:16.965","Text":"Something like this through here."},{"Start":"13:16.965 ","End":"13:20.380","Text":"That\u0027s not bad for a free-hand."},{"Start":"13:20.420 ","End":"13:22.455","Text":"Straighten it out a bit."},{"Start":"13:22.455 ","End":"13:25.095","Text":"Let\u0027s see, this is 2, this 5,"},{"Start":"13:25.095 ","End":"13:27.360","Text":"minus 2, minus 5."},{"Start":"13:27.360 ","End":"13:29.505","Text":"Why is this an ellipse?"},{"Start":"13:29.505 ","End":"13:31.395","Text":"How could I better draw this?"},{"Start":"13:31.395 ","End":"13:34.590","Text":"Remember, I said that it\u0027s actually more accurate if"},{"Start":"13:34.590 ","End":"13:38.190","Text":"you can get 1 of the variables in terms of the other."},{"Start":"13:38.190 ","End":"13:41.670","Text":"Let\u0027s see if we can get this in an alternate form."},{"Start":"13:41.670 ","End":"13:48.270","Text":"Once again, we could make use of the formula that sine"},{"Start":"13:48.270 ","End":"13:54.315","Text":"squared Alpha plus cosine squared Alpha is always 1 for any Alpha."},{"Start":"13:54.315 ","End":"13:59.655","Text":"Here, if I take x squared,"},{"Start":"13:59.655 ","End":"14:05.325","Text":"I\u0027ve got 5 cosine squared t. But if I divide it by 25,"},{"Start":"14:05.325 ","End":"14:13.995","Text":"then I\u0027ve got just cosine squared t. Now if I add y squared,"},{"Start":"14:13.995 ","End":"14:20.175","Text":"that\u0027ll be 4 sine squared t. But if I put y squared over 4,"},{"Start":"14:20.175 ","End":"14:24.405","Text":"that will be plus sine squared t,"},{"Start":"14:24.405 ","End":"14:28.150","Text":"that\u0027s equal to 1."},{"Start":"14:28.870 ","End":"14:38.280","Text":"What I get is basically that this equals 1."},{"Start":"14:38.280 ","End":"14:43.905","Text":"I\u0027ll just cut out the middleman and write that this equals 1."},{"Start":"14:43.905 ","End":"14:46.710","Text":"Depending on how much analytic geometry you\u0027ve learned,"},{"Start":"14:46.710 ","End":"14:49.950","Text":"you may or may not recognize this as the equation of an ellipse,"},{"Start":"14:49.950 ","End":"14:52.575","Text":"but it\u0027s in implicit form."},{"Start":"14:52.575 ","End":"14:55.335","Text":"If you wanted to actually sketch it,"},{"Start":"14:55.335 ","End":"14:59.655","Text":"what you might do would be say something like,"},{"Start":"14:59.655 ","End":"15:03.270","Text":"y squared is equal to,"},{"Start":"15:03.270 ","End":"15:08.040","Text":"let\u0027s see, 1 minus x squared over 25."},{"Start":"15:08.040 ","End":"15:10.455","Text":"If I bring it over to the other side,"},{"Start":"15:10.455 ","End":"15:18.255","Text":"and then multiply all this by 4."},{"Start":"15:18.255 ","End":"15:21.945","Text":"Then I could say that y,"},{"Start":"15:21.945 ","End":"15:26.040","Text":"see, I can\u0027t get it exactly in explicit form,"},{"Start":"15:26.040 ","End":"15:29.250","Text":"but I could say plus or minus the square root of"},{"Start":"15:29.250 ","End":"15:37.560","Text":"this twice the square root of 1 minus x squared over 25."},{"Start":"15:37.560 ","End":"15:40.290","Text":"The plus being the top part,"},{"Start":"15:40.290 ","End":"15:44.145","Text":"and I could sketch that as y as a function of x,"},{"Start":"15:44.145 ","End":"15:48.675","Text":"and the bottom part with the minus."},{"Start":"15:48.675 ","End":"15:53.175","Text":"With the parameters, I\u0027ll write the parameter in."},{"Start":"15:53.175 ","End":"15:56.820","Text":"Here we started off with t equals 0."},{"Start":"15:56.820 ","End":"16:01.725","Text":"Here we had t is Pi over 2."},{"Start":"16:01.725 ","End":"16:03.450","Text":"Forget the t equals."},{"Start":"16:03.450 ","End":"16:06.360","Text":"Here I had t equals Pi,"},{"Start":"16:06.360 ","End":"16:09.960","Text":"here t equals 3Pi over 2,"},{"Start":"16:09.960 ","End":"16:13.905","Text":"and here we were back again to 2Pi."},{"Start":"16:13.905 ","End":"16:18.165","Text":"The direction we went was this way."},{"Start":"16:18.165 ","End":"16:23.655","Text":"If we increase the range of t,"},{"Start":"16:23.655 ","End":"16:28.780","Text":"we will actually keep going round and round and round,"},{"Start":"16:29.540 ","End":"16:34.390","Text":"because every 2Pi we will return here."},{"Start":"16:34.880 ","End":"16:40.920","Text":"We trace out the ellipse counterclockwise and go round several"},{"Start":"16:40.920 ","End":"16:46.320","Text":"times if you write it in parametric form if we didn\u0027t restrict t. We do restrict t,"},{"Start":"16:46.320 ","End":"16:49.860","Text":"it\u0027s still, we start from here and go around this way and up here."},{"Start":"16:49.860 ","End":"16:52.529","Text":"But this is just a static picture."},{"Start":"16:52.529 ","End":"16:55.275","Text":"It doesn\u0027t tell us anything about the direction of motion,"},{"Start":"16:55.275 ","End":"16:58.755","Text":"or how many times we go around or anything."},{"Start":"16:58.755 ","End":"17:02.400","Text":"I\u0027d like to now take a variation of this,"},{"Start":"17:02.400 ","End":"17:06.225","Text":"where instead of t I\u0027m going to put 3t."},{"Start":"17:06.225 ","End":"17:09.435","Text":"I\u0027m going to erase this, and erase this,"},{"Start":"17:09.435 ","End":"17:18.250","Text":"and erase all this and I\u0027m going to write 3t where I had t before."},{"Start":"17:18.590 ","End":"17:22.455","Text":"The restriction, I\u0027m going to recalculate."},{"Start":"17:22.455 ","End":"17:28.410","Text":"Then I\u0027m also going to clear the, oops."},{"Start":"17:28.410 ","End":"17:33.510","Text":"Yeah, there we are. Now it turns out that the sine formula,"},{"Start":"17:33.510 ","End":"17:37.050","Text":"sine squared Alpha plus cosine squared Alpha is 1,"},{"Start":"17:37.050 ","End":"17:39.810","Text":"will give us exactly the same equation."},{"Start":"17:39.810 ","End":"17:43.050","Text":"If you look at what we did before,"},{"Start":"17:43.050 ","End":"17:47.070","Text":"we also got cosine squared plus sine squared is 1."},{"Start":"17:47.070 ","End":"17:49.740","Text":"The same thing works as before,"},{"Start":"17:49.740 ","End":"17:53.730","Text":"except that instead of Alpha being t, in this case,"},{"Start":"17:53.730 ","End":"17:55.305","Text":"Alpha will be 3t,"},{"Start":"17:55.305 ","End":"17:57.795","Text":"but we\u0027ll get exactly the same equation,"},{"Start":"17:57.795 ","End":"18:00.915","Text":"even though we changed t to a 3t."},{"Start":"18:00.915 ","End":"18:02.850","Text":"What will change however,"},{"Start":"18:02.850 ","End":"18:07.320","Text":"and the reason I erased all the markings for t and the values,"},{"Start":"18:07.320 ","End":"18:11.095","Text":"is that it will happen 3 times as fast."},{"Start":"18:11.095 ","End":"18:16.070","Text":"What do I mean? Well, every time this parameter reaches 2Pi,"},{"Start":"18:16.070 ","End":"18:18.200","Text":"we go around in a full circle."},{"Start":"18:18.200 ","End":"18:21.765","Text":"But for 3t to reach 2Pi,"},{"Start":"18:21.765 ","End":"18:25.950","Text":"it means that t only has to reach 2Pi over 3."},{"Start":"18:25.950 ","End":"18:28.950","Text":"If I go from 0, not to 2 Pi,"},{"Start":"18:28.950 ","End":"18:31.425","Text":"but to 2Pi over 3,"},{"Start":"18:31.425 ","End":"18:33.285","Text":"you\u0027ll see that we get the same thing."},{"Start":"18:33.285 ","End":"18:35.055","Text":"Let me just fill it in."},{"Start":"18:35.055 ","End":"18:40.080","Text":"Here, we have Pi over 6."},{"Start":"18:40.080 ","End":"18:44.880","Text":"Basically, whatever we had in the column before divided by 3."},{"Start":"18:44.880 ","End":"18:47.310","Text":"Here we had Pi,"},{"Start":"18:47.310 ","End":"18:49.230","Text":"so it\u0027s Pi over 3."},{"Start":"18:49.230 ","End":"18:53.235","Text":"Here we had 3Pi over 2,"},{"Start":"18:53.235 ","End":"18:58.060","Text":"so it\u0027s just Pi over 2."},{"Start":"18:59.540 ","End":"19:02.550","Text":"Here we have, 2Pi over 3."},{"Start":"19:02.550 ","End":"19:06.295","Text":"Now look, if I look at 3t,"},{"Start":"19:06.295 ","End":"19:10.695","Text":"suppose I made a column which was,"},{"Start":"19:10.695 ","End":"19:13.035","Text":"let\u0027s say just here, 3t."},{"Start":"19:13.035 ","End":"19:15.905","Text":"You see that 3t would be the same as what we had before."},{"Start":"19:15.905 ","End":"19:18.634","Text":"0, Pi over 2,"},{"Start":"19:18.634 ","End":"19:25.325","Text":"Pi, 3Pi over 2, and then 2Pi."},{"Start":"19:25.325 ","End":"19:30.595","Text":"All that would change is that we go around with a different speed."},{"Start":"19:30.595 ","End":"19:36.420","Text":"Here, we\u0027d have t equals 0."},{"Start":"19:36.420 ","End":"19:38.745","Text":"Looking at this column,"},{"Start":"19:38.745 ","End":"19:44.505","Text":"Pi over 6, Pi over 3,"},{"Start":"19:44.505 ","End":"19:52.920","Text":"Pi over 2, and then 2Pi over 3, so 1/3 of 2 Pi."},{"Start":"19:52.920 ","End":"19:57.960","Text":"But if we kept going up to 2Pi and lengthen the table,"},{"Start":"19:57.960 ","End":"20:01.075","Text":"we\u0027d actually go around 3 times."},{"Start":"20:01.075 ","End":"20:03.785","Text":"To go round just once,"},{"Start":"20:03.785 ","End":"20:12.620","Text":"all we need is for t to go between 0 and 2 Pi over 3."},{"Start":"20:12.620 ","End":"20:17.110","Text":"If we just kept the 0 less than or equal to t,"},{"Start":"20:17.110 ","End":"20:18.575","Text":"less than or equal to 2Pi,"},{"Start":"20:18.575 ","End":"20:21.680","Text":"we would go around the circle 3 times."},{"Start":"20:21.680 ","End":"20:26.960","Text":"There\u0027s quite a difference between the parametric version and the non-parametric."},{"Start":"20:26.960 ","End":"20:30.320","Text":"Non-parametric doesn\u0027t take into account all the motion,"},{"Start":"20:30.320 ","End":"20:34.200","Text":"which direction, and at what speed and all that."},{"Start":"20:35.570 ","End":"20:40.770","Text":"I think it\u0027s time to take a break now and we\u0027ll continue afterwards."}],"ID":6000},{"Watched":false,"Name":"Parametric Equations and Curves (continued2)","Duration":"5m 3s","ChapterTopicVideoID":5987,"CourseChapterTopicPlaylistID":4001,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.537","Text":"We\u0027re almost finished with this section"},{"Start":"00:02.537 ","End":"00:04.210","Text":"on parametric equations and curves."},{"Start":"00:04.210 ","End":"00:06.030","Text":"There\u0027s a lot more than I could say,"},{"Start":"00:06.030 ","End":"00:09.960","Text":"but I just want to say a few important things."},{"Start":"00:09.960 ","End":"00:13.830","Text":"What we saw was that if I changed,"},{"Start":"00:13.830 ","End":"00:16.605","Text":"in this example, instead of t, I put 3t,"},{"Start":"00:16.605 ","End":"00:20.249","Text":"I got exactly the same curve as far as how it looks,"},{"Start":"00:20.249 ","End":"00:24.300","Text":"but the 3 just made it go round 3 times as fast,"},{"Start":"00:24.300 ","End":"00:26.865","Text":"and in general if you put any number here,"},{"Start":"00:26.865 ","End":"00:29.565","Text":"it\u0027ll just change the speed."},{"Start":"00:29.565 ","End":"00:37.365","Text":"If I just remove the 3 and use the Greek letter Omega,"},{"Start":"00:37.365 ","End":"00:40.290","Text":"Omega will be the speed,"},{"Start":"00:40.290 ","End":"00:45.885","Text":"if Omega is 3 then it goes round 3 times when I go from 0 to 2 Pi."},{"Start":"00:45.885 ","End":"00:55.710","Text":"Of course, this stuff\u0027s no longer valid and we\u0027ll have to change the values of t,"},{"Start":"00:55.710 ","End":"00:58.275","Text":"but these same points will exist."},{"Start":"00:58.275 ","End":"01:01.095","Text":"In fact, this is generally an ellipse,"},{"Start":"01:01.095 ","End":"01:06.739","Text":"in more general form, would be instead of 5 and 2,"},{"Start":"01:06.739 ","End":"01:11.705","Text":"5 would be the large radius,"},{"Start":"01:11.705 ","End":"01:14.580","Text":"let\u0027s call that a,"},{"Start":"01:14.830 ","End":"01:17.315","Text":"I\u0027ll rewrite that in a minute,"},{"Start":"01:17.315 ","End":"01:23.585","Text":"and here b which means this is no longer 5 and minus 5,"},{"Start":"01:23.585 ","End":"01:25.625","Text":"and 2 and minus 2."},{"Start":"01:25.625 ","End":"01:31.655","Text":"In general, this will be a and minus"},{"Start":"01:31.655 ","End":"01:38.760","Text":"a and b and minus b. I\u0027m just introducing you to this concept,"},{"Start":"01:38.760 ","End":"01:41.525","Text":"you don\u0027t really have to study the ellipse."},{"Start":"01:41.525 ","End":"01:49.760","Text":"More important is the circle where a equals b and then we usually call it r."},{"Start":"01:49.760 ","End":"01:58.325","Text":"If I erase the a and the b and replace them with r here and here,"},{"Start":"01:58.325 ","End":"02:04.010","Text":"and here and here, it doesn\u0027t look like a circle,"},{"Start":"02:04.010 ","End":"02:06.205","Text":"let me squash it a bit."},{"Start":"02:06.205 ","End":"02:09.330","Text":"There, that looks more like a circle,"},{"Start":"02:09.330 ","End":"02:11.460","Text":"and the Omega is the speed."},{"Start":"02:11.460 ","End":"02:15.640","Text":"But if we want to get rid of the Omega,"},{"Start":"02:15.640 ","End":"02:19.930","Text":"then what will happen is that as t goes from"},{"Start":"02:19.930 ","End":"02:24.850","Text":"0 to 2 Pi with all the values in the middle, Pi over 3,"},{"Start":"02:24.850 ","End":"02:26.420","Text":"and 3Pi over 2,"},{"Start":"02:26.420 ","End":"02:32.275","Text":"then it will go round once with t being 0 here,"},{"Start":"02:32.275 ","End":"02:34.809","Text":"Pi over 2 here, Pi,"},{"Start":"02:34.809 ","End":"02:39.100","Text":"3Pi over 2, and back to 2Pi."},{"Start":"02:42.190 ","End":"02:46.400","Text":"It goes around counterclockwise."},{"Start":"02:46.400 ","End":"02:50.575","Text":"There are other funny things 1 could do, for example,"},{"Start":"02:50.575 ","End":"02:53.090","Text":"and I\u0027m not going to get to too much detail,"},{"Start":"02:53.090 ","End":"02:55.775","Text":"if I change the cosine and the sine,"},{"Start":"02:55.775 ","End":"02:58.100","Text":"if I wrote here instead of cosine,"},{"Start":"02:58.100 ","End":"03:01.715","Text":"sine and instead of sine I wrote cosine,"},{"Start":"03:01.715 ","End":"03:06.995","Text":"what would happen is instead of starting from here and going around counterclockwise,"},{"Start":"03:06.995 ","End":"03:11.450","Text":"I would start here and go around clockwise,"},{"Start":"03:11.450 ","End":"03:13.584","Text":"which all goes to show,"},{"Start":"03:13.584 ","End":"03:15.840","Text":"and let me just undo that,"},{"Start":"03:15.840 ","End":"03:22.490","Text":"that the same curve could be parametrized in different ways,"},{"Start":"03:22.490 ","End":"03:24.230","Text":"could go around at different speeds in"},{"Start":"03:24.230 ","End":"03:27.440","Text":"different directions from different starting points and so on,"},{"Start":"03:27.440 ","End":"03:31.920","Text":"and it would still have the same equation."},{"Start":"03:31.920 ","End":"03:35.405","Text":"In this case this no longer applies."},{"Start":"03:35.405 ","End":"03:39.620","Text":"The previous equations that we had will simplify to x"},{"Start":"03:39.620 ","End":"03:43.265","Text":"squared plus y squared equals r squared,"},{"Start":"03:43.265 ","End":"03:44.690","Text":"and if you want to put, say,"},{"Start":"03:44.690 ","End":"03:45.920","Text":"y in terms of x,"},{"Start":"03:45.920 ","End":"03:48.620","Text":"you can only do it with a plus or minus,"},{"Start":"03:48.620 ","End":"03:53.420","Text":"just the square root of r squared minus x squared,"},{"Start":"03:53.420 ","End":"03:58.100","Text":"where the plus part gives us the top semicircle and the minus gives us the bottom,"},{"Start":"03:58.100 ","End":"03:59.840","Text":"but in the parametrized form,"},{"Start":"03:59.840 ","End":"04:01.630","Text":"we get the whole thing."},{"Start":"04:01.630 ","End":"04:07.940","Text":"There is 1 last point I\u0027d like to make before I conclude the tutorial."},{"Start":"04:07.940 ","End":"04:10.945","Text":"Let me just clear the board."},{"Start":"04:10.945 ","End":"04:15.380","Text":"The point is that if I have a function in explicit form,"},{"Start":"04:15.380 ","End":"04:18.005","Text":"y equals f of x,"},{"Start":"04:18.005 ","End":"04:22.715","Text":"I could always write this in a parametric form."},{"Start":"04:22.715 ","End":"04:30.980","Text":"If I just let x equal t and y equals f of t, this is really the same thing,"},{"Start":"04:30.980 ","End":"04:35.600","Text":"I\u0027m just replacing x by t. But this is a parametric form of the same thing,"},{"Start":"04:35.600 ","End":"04:37.460","Text":"and sometimes this is useful,"},{"Start":"04:37.460 ","End":"04:39.740","Text":"in future you might come across this."},{"Start":"04:39.740 ","End":"04:42.894","Text":"Similarly, if I have,"},{"Start":"04:42.894 ","End":"04:48.890","Text":"let\u0027s say, x equals some function of y, same thing applies."},{"Start":"04:48.890 ","End":"04:57.446","Text":"I could write it in parametric form by saying y equals t and x equals g of t."},{"Start":"04:57.446 ","End":"05:03.780","Text":"Now we\u0027re done with parametric equations and curves."}],"ID":6001}],"Thumbnail":null,"ID":4001},{"Name":"Tangents with Parametric Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tangents with Parametic Equations","Duration":"6m 32s","ChapterTopicVideoID":5988,"CourseChapterTopicPlaylistID":4002,"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 going to talk about tangents with parametric equations."},{"Start":"00:05.190 ","End":"00:11.595","Text":"This is a continuation clip from the parametric equations and curves."},{"Start":"00:11.595 ","End":"00:16.530","Text":"In fact, I\u0027d like to borrow an example that we did in the previous topic."},{"Start":"00:16.530 ","End":"00:21.870","Text":"I copied and pasted the example from the previous section."},{"Start":"00:21.870 ","End":"00:26.120","Text":"It\u0027s also good to have a reminder that parametric equation is when we"},{"Start":"00:26.120 ","End":"00:31.055","Text":"give x and y both as a function of a third variable t,"},{"Start":"00:31.055 ","End":"00:34.390","Text":"which often has a physical meaning of time."},{"Start":"00:34.390 ","End":"00:36.230","Text":"In our particular example,"},{"Start":"00:36.230 ","End":"00:38.695","Text":"this was x and this was y."},{"Start":"00:38.695 ","End":"00:42.305","Text":"We plotted a curve by making a table."},{"Start":"00:42.305 ","End":"00:46.565","Text":"We mentioned that the curve actually has a direction"},{"Start":"00:46.565 ","End":"00:52.190","Text":"where t is increasing and to each point on the curve there is a value of t,"},{"Start":"00:52.190 ","End":"00:55.320","Text":"maybe more than 1, we shall see."},{"Start":"00:55.870 ","End":"01:00.330","Text":"I would like to just continue this topic."},{"Start":"01:00.350 ","End":"01:07.805","Text":"In calculus, we discussed already the concept of a tangent."},{"Start":"01:07.805 ","End":"01:13.745","Text":"But tangent we usually do when we have y as a function of x."},{"Start":"01:13.745 ","End":"01:18.710","Text":"What I\u0027d like to do for this example is to find the equation of the tangent line."},{"Start":"01:18.710 ","End":"01:20.420","Text":"Let\u0027s choose this point."},{"Start":"01:20.420 ","End":"01:24.695","Text":"This is the point where t is minus 1."},{"Start":"01:24.695 ","End":"01:30.770","Text":"In fact, I could just describe the point as this row here where t is minus 1,"},{"Start":"01:30.770 ","End":"01:35.185","Text":"but the point is 0,3 in the plane."},{"Start":"01:35.185 ","End":"01:38.490","Text":"This is the tangent line."},{"Start":"01:38.490 ","End":"01:41.120","Text":"To find its equation,"},{"Start":"01:41.120 ","End":"01:42.590","Text":"I need the slope."},{"Start":"01:42.590 ","End":"01:46.520","Text":"I already have a point, it goes through 0,3."},{"Start":"01:46.520 ","End":"01:54.665","Text":"What I would like to do is find out what m is equal to for this tangent line."},{"Start":"01:54.665 ","End":"02:01.429","Text":"Now, m is equal to the derivative of y with respect to x."},{"Start":"02:01.429 ","End":"02:06.110","Text":"We\u0027re going to use the Leibniz notation rather than saying y prime,"},{"Start":"02:06.110 ","End":"02:08.105","Text":"because y is a function of t,"},{"Start":"02:08.105 ","End":"02:15.710","Text":"This emphasizes that I want the derivative of y with respect to x. I could say,"},{"Start":"02:15.710 ","End":"02:18.500","Text":"in that previous lesson,"},{"Start":"02:18.500 ","End":"02:21.680","Text":"we managed to isolate x in terms of y,"},{"Start":"02:21.680 ","End":"02:23.435","Text":"and I copy pasted it here."},{"Start":"02:23.435 ","End":"02:25.955","Text":"We couldn\u0027t get y in terms of x,"},{"Start":"02:25.955 ","End":"02:27.620","Text":"and you can see that it\u0027s not a function,"},{"Start":"02:27.620 ","End":"02:30.215","Text":"a vertical line could hit the curve twice,"},{"Start":"02:30.215 ","End":"02:32.240","Text":"but we\u0027ve got x in terms of y."},{"Start":"02:32.240 ","End":"02:34.100","Text":"We could say, okay,"},{"Start":"02:34.100 ","End":"02:37.409","Text":"the derivative of x with respect to y,"},{"Start":"02:37.409 ","End":"02:44.950","Text":"dx/dy, is equal to 2 times 1/4 is 1/2y minus 1."},{"Start":"02:44.950 ","End":"02:53.420","Text":"Which means that dx/dy at"},{"Start":"02:53.420 ","End":"03:04.115","Text":"the point where y equals 3 is 1/2 of 3 minus 1,"},{"Start":"03:04.115 ","End":"03:09.900","Text":"1/2 minus 1, which is 1/2."},{"Start":"03:09.900 ","End":"03:11.755","Text":"Now that I have dx/dy,"},{"Start":"03:11.755 ","End":"03:14.820","Text":"I can use the formula that"},{"Start":"03:16.270 ","End":"03:23.450","Text":"dy/dx is 1 over dx/dy."},{"Start":"03:23.450 ","End":"03:27.440","Text":"These ds behave very much like fractions."},{"Start":"03:27.440 ","End":"03:34.520","Text":"The Leibniz notation is very useful that way and I\u0027ll be using that property soon again."},{"Start":"03:34.520 ","End":"03:38.610","Text":"But meanwhile, let\u0027s complete this here."},{"Start":"03:39.340 ","End":"03:45.645","Text":"Dy over dx is the reciprocal of dx/dy,"},{"Start":"03:45.645 ","End":"03:49.510","Text":"so it\u0027s 1/2, which brings it back to 2."},{"Start":"03:49.510 ","End":"03:55.615","Text":"Now I can replace this question mark with its value which is 2."},{"Start":"03:55.615 ","End":"04:00.695","Text":"But suppose we didn\u0027t have x in terms of y or y in terms of x,"},{"Start":"04:00.695 ","End":"04:02.240","Text":"how would I proceed?"},{"Start":"04:02.240 ","End":"04:05.390","Text":"Well, like I said, this Leibniz notation for"},{"Start":"04:05.390 ","End":"04:09.275","Text":"derivative really does behave like a fraction."},{"Start":"04:09.275 ","End":"04:18.665","Text":"In this case, dy/dx if I treat it like a fraction and divide top and bottom by dt,"},{"Start":"04:18.665 ","End":"04:29.060","Text":"I\u0027ll get that this is dy/dt divided by dx/dt."},{"Start":"04:29.060 ","End":"04:31.324","Text":"Let\u0027s see what happens in our case."},{"Start":"04:31.324 ","End":"04:39.410","Text":"In our case, we would get this rule formula dy/dx will equal."},{"Start":"04:39.410 ","End":"04:42.020","Text":"Now dy/dt, I look here,"},{"Start":"04:42.020 ","End":"04:43.445","Text":"y is a function of t,"},{"Start":"04:43.445 ","End":"04:48.420","Text":"so that\u0027s minus 2 over,"},{"Start":"04:49.510 ","End":"04:54.640","Text":"dx/dt is 2t plus 1."},{"Start":"04:54.640 ","End":"05:00.800","Text":"First of all, notice that dy/dx comes out to be a function of t in the parametric form."},{"Start":"05:00.800 ","End":"05:04.430","Text":"Not only is x a function of t and y a function of t,"},{"Start":"05:04.430 ","End":"05:12.140","Text":"but the derivative dy/dx is also given in terms of t. If we want to find our slope,"},{"Start":"05:12.140 ","End":"05:15.845","Text":"which is dy/dx, what we would do"},{"Start":"05:15.845 ","End":"05:20.000","Text":"would be to substitute the value of t corresponding to the point."},{"Start":"05:20.000 ","End":"05:21.845","Text":"In this case, we need the t,"},{"Start":"05:21.845 ","End":"05:23.785","Text":"which is minus 1,"},{"Start":"05:23.785 ","End":"05:26.945","Text":"and then we would get what m equals, m is dy/dx."},{"Start":"05:26.945 ","End":"05:29.015","Text":"Let\u0027s see if t is minus 1,"},{"Start":"05:29.015 ","End":"05:31.610","Text":"you\u0027ve got minus 2 over minus 2 plus 1,"},{"Start":"05:31.610 ","End":"05:34.360","Text":"minus 2 over minus 1 is 2."},{"Start":"05:34.360 ","End":"05:40.080","Text":"That\u0027s good because it confirms what we already got the other way."},{"Start":"05:40.210 ","End":"05:45.395","Text":"Well, to finish it off, we were just to actually find the equation of the tangent."},{"Start":"05:45.395 ","End":"05:51.020","Text":"We would use the formula y minus"},{"Start":"05:51.020 ","End":"05:57.410","Text":"the y of the point is equal to the slope times the x minus the x of the point."},{"Start":"05:57.410 ","End":"06:01.190","Text":"In our case, we would get y minus,"},{"Start":"06:01.190 ","End":"06:03.529","Text":"the y of the point is 3,"},{"Start":"06:03.529 ","End":"06:05.945","Text":"the slope is 2,"},{"Start":"06:05.945 ","End":"06:08.795","Text":"the x of the point is 0."},{"Start":"06:08.795 ","End":"06:14.270","Text":"Or in short, we would get that y equals 2x plus"},{"Start":"06:14.270 ","End":"06:21.030","Text":"3 and that would be the equation of the tangent."},{"Start":"06:21.440 ","End":"06:26.030","Text":"We could do it just using the parametric form."},{"Start":"06:26.030 ","End":"06:29.435","Text":"Let\u0027s take a short break now,"},{"Start":"06:29.435 ","End":"06:32.370","Text":"or a long break, whatever you like."}],"ID":6002},{"Watched":false,"Name":"Tangents with Parametic Equations (continued)","Duration":"18m 27s","ChapterTopicVideoID":5989,"CourseChapterTopicPlaylistID":4002,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.455","Text":"Continuing with the previous clip on tangents with parametric equations,"},{"Start":"00:07.455 ","End":"00:09.615","Text":"let me erase what we don\u0027t need,"},{"Start":"00:09.615 ","End":"00:11.300","Text":"which is pretty much everything."},{"Start":"00:11.300 ","End":"00:13.290","Text":"Let me just rearrange this."},{"Start":"00:13.290 ","End":"00:16.380","Text":"We talked about parametric functions where x and y are"},{"Start":"00:16.380 ","End":"00:20.295","Text":"both functions of t. We talked about dy over dx,"},{"Start":"00:20.295 ","End":"00:21.960","Text":"which was useful for example,"},{"Start":"00:21.960 ","End":"00:25.695","Text":"in calculating the slope of a tangent line."},{"Start":"00:25.695 ","End":"00:30.340","Text":"If we look at y as a function of x was equal to this."},{"Start":"00:30.680 ","End":"00:34.220","Text":"I should have mentioned that this is true"},{"Start":"00:34.220 ","End":"00:37.370","Text":"provided that the denominator is not 0 of course."},{"Start":"00:37.370 ","End":"00:44.995","Text":"So I\u0027ll add that dx over dt is not equal to 0."},{"Start":"00:44.995 ","End":"00:51.485","Text":"You could also write it in this particular case when we have f and g as functions."},{"Start":"00:51.485 ","End":"00:59.495","Text":"You could also write it as f prime of t over g prime of t using the prime notation."},{"Start":"00:59.495 ","End":"01:05.780","Text":"Again, it has to be assumed that the denominator is not 0 otherwise,"},{"Start":"01:05.780 ","End":"01:08.970","Text":"it\u0027s not going to make sense."},{"Start":"01:09.320 ","End":"01:16.860","Text":"I want to complete this with also giving a formula for dx over dy."},{"Start":"01:18.830 ","End":"01:21.850","Text":"We might need dx over dy."},{"Start":"01:21.850 ","End":"01:24.290","Text":"I seem to remember that in the arc length,"},{"Start":"01:24.290 ","End":"01:27.185","Text":"1 of the formulas used dx over dy."},{"Start":"01:27.185 ","End":"01:31.925","Text":"Sometimes we want to see x as a function of y rather than y or a function of x."},{"Start":"01:31.925 ","End":"01:37.955","Text":"In this case, also the same thing applies that it\u0027s equal to dx over dt"},{"Start":"01:37.955 ","End":"01:45.620","Text":"over dy over dt in case we need it and of course,"},{"Start":"01:45.620 ","End":"01:51.605","Text":"the same restriction applies that the denominator must not be 0."},{"Start":"01:51.605 ","End":"01:54.230","Text":"If we have f of t and g of t,"},{"Start":"01:54.230 ","End":"01:59.720","Text":"we could also write this alternately as f prime."},{"Start":"01:59.720 ","End":"02:05.360","Text":"Sorry. Another formula is for dx over dy."},{"Start":"02:05.360 ","End":"02:07.340","Text":"Sometimes we need this."},{"Start":"02:07.340 ","End":"02:09.425","Text":"Not as often as dy over dx."},{"Start":"02:09.425 ","End":"02:12.440","Text":"I seem to remember that in the arc length, for example,"},{"Start":"02:12.440 ","End":"02:14.660","Text":"we had a formula with this,"},{"Start":"02:14.660 ","End":"02:16.520","Text":"same thing you would expect."},{"Start":"02:16.520 ","End":"02:22.565","Text":"It\u0027s dx over dt as if dt could cancel,"},{"Start":"02:22.565 ","End":"02:24.845","Text":"dividing top and bottom by it,"},{"Start":"02:24.845 ","End":"02:31.620","Text":"dy over dt and the same condition applies of course that the denominator,"},{"Start":"02:31.620 ","End":"02:37.060","Text":"in this case, dy over dt must not be 0."},{"Start":"02:38.330 ","End":"02:44.495","Text":"In our case, if we have x and y explicitly as functions f and g of t,"},{"Start":"02:44.495 ","End":"02:47.200","Text":"we could also write this, if we like,"},{"Start":"02:47.200 ","End":"02:55.505","Text":"as g prime of t over f prime of t,"},{"Start":"02:55.505 ","End":"02:59.270","Text":"providing that this is not equal to 0."},{"Start":"02:59.270 ","End":"03:01.210","Text":"For the other case,"},{"Start":"03:01.210 ","End":"03:11.340","Text":"dx over dy is"},{"Start":"03:11.340 ","End":"03:14.805","Text":"going to equal dx over dt is f prime of"},{"Start":"03:14.805 ","End":"03:22.810","Text":"t and dy over dt is g prime of t. Again,"},{"Start":"03:22.810 ","End":"03:27.515","Text":"provided that the denominator is not 0, this will hold."},{"Start":"03:27.515 ","End":"03:31.490","Text":"We\u0027re not going to use this fact in today\u0027s lesson,"},{"Start":"03:31.490 ","End":"03:34.650","Text":"but you should have it for future reference."},{"Start":"03:34.780 ","End":"03:42.775","Text":"Now I\u0027d like to solve another exercise involving tangents."},{"Start":"03:42.775 ","End":"03:44.315","Text":"But in this exercise,"},{"Start":"03:44.315 ","End":"03:46.190","Text":"something unusual is going to happen."},{"Start":"03:46.190 ","End":"03:50.850","Text":"Let\u0027s say we have a set of parametric equations."},{"Start":"03:50.860 ","End":"03:54.790","Text":"We have x equals,"},{"Start":"03:54.790 ","End":"03:58.665","Text":"say, t cubed minus 3t,"},{"Start":"03:58.665 ","End":"04:08.640","Text":"and y is equal to 3t squared minus 9."},{"Start":"04:08.640 ","End":"04:12.285","Text":"The task is to find"},{"Start":"04:12.285 ","End":"04:22.080","Text":"the tangent line at the point where x is 0 and y is 0."},{"Start":"04:22.080 ","End":"04:29.225","Text":"Now, the first thing is who says that 0,0 is even on this curve."},{"Start":"04:29.225 ","End":"04:33.050","Text":"We have to find a value of t such that when we substitute it,"},{"Start":"04:33.050 ","End":"04:38.130","Text":"we get 0, 0, so we get a set of 2 equations in 2 unknowns."},{"Start":"04:38.290 ","End":"04:44.760","Text":"We have that 2 equations in 1 unknown should I say."},{"Start":"04:44.760 ","End":"04:47.130","Text":"We have that x is 0,"},{"Start":"04:47.130 ","End":"04:53.285","Text":"so I have to have that 0 is equal to t cubed minus 3t."},{"Start":"04:53.285 ","End":"05:03.040","Text":"But I also have to have y is 0 so 0 has to be equal to 3t squared minus 9."},{"Start":"05:03.040 ","End":"05:07.460","Text":"In general, there might not be a solution to 2 equations in 1"},{"Start":"05:07.460 ","End":"05:12.965","Text":"unknown but here we\u0027ve made sure that there is a value of t which gives 0,0."},{"Start":"05:12.965 ","End":"05:15.680","Text":"Actually, we have an opposite problem that"},{"Start":"05:15.680 ","End":"05:18.845","Text":"might be more than 1 value of t as we shall see."},{"Start":"05:18.845 ","End":"05:25.850","Text":"The first equation gives us that t cubed minus 3t equals 0,"},{"Start":"05:25.850 ","End":"05:34.075","Text":"which gives us that t times t squared minus 3 is 0."},{"Start":"05:34.075 ","End":"05:36.930","Text":"Then if a product is 0,"},{"Start":"05:36.930 ","End":"05:39.465","Text":"then 1 of them must be 0."},{"Start":"05:39.465 ","End":"05:42.950","Text":"Either t is 0 or t squared minus 3 is 0."},{"Start":"05:42.950 ","End":"05:46.490","Text":"In short, t equals 0 or if this is 0,"},{"Start":"05:46.490 ","End":"05:51.575","Text":"t can equal square root of 3 plus or minus."},{"Start":"05:51.575 ","End":"05:57.905","Text":"It could be the square root of 3 or minus the square root of 3."},{"Start":"05:57.905 ","End":"06:00.500","Text":"How about the other equation?"},{"Start":"06:00.500 ","End":"06:04.100","Text":"This means that we can take 3 out,"},{"Start":"06:04.100 ","End":"06:09.800","Text":"that 3 times t squared minus 3 equals 0."},{"Start":"06:09.800 ","End":"06:13.685","Text":"Here we can divide both sides by 3 because it\u0027s not 0."},{"Start":"06:13.685 ","End":"06:18.760","Text":"The only possibilities for t squared minus 3 equals 0,"},{"Start":"06:18.760 ","End":"06:22.025","Text":"I have t squared minus 3 is 0."},{"Start":"06:22.025 ","End":"06:26.570","Text":"We get just that t could be square root of 3,"},{"Start":"06:26.570 ","End":"06:31.385","Text":"or t equals minus the square root of 3."},{"Start":"06:31.385 ","End":"06:37.040","Text":"Both of these have to hold but we still have 2 possible solutions."},{"Start":"06:37.040 ","End":"06:41.765","Text":"This 1 has been ruled out because it\u0027s not in both but"},{"Start":"06:41.765 ","End":"06:45.140","Text":"the square root of 3 is okay and"},{"Start":"06:45.140 ","End":"06:49.820","Text":"minus the square root of 3 is okay because it appeals here and here."},{"Start":"06:49.820 ","End":"06:58.740","Text":"What to do? The answer is that there are actually 2 tangents at that point."},{"Start":"06:59.900 ","End":"07:03.620","Text":"This is unusual and it didn\u0027t happen in the case of"},{"Start":"07:03.620 ","End":"07:07.580","Text":"non-parametric that the same point had more than 1 tangent."},{"Start":"07:07.580 ","End":"07:09.410","Text":"Let\u0027s compute the tangents and afterwards,"},{"Start":"07:09.410 ","End":"07:13.025","Text":"we\u0027ll get to the bottom of this mystery of why this is happening."},{"Start":"07:13.025 ","End":"07:19.800","Text":"Let\u0027s take the first case where t equals the square root of 3 and then we get"},{"Start":"07:19.800 ","End":"07:24.600","Text":"that m which is dy over"},{"Start":"07:24.600 ","End":"07:30.940","Text":"dx is dy over dt."},{"Start":"07:31.370 ","End":"07:36.330","Text":"Where are we? Is dy over dt,"},{"Start":"07:36.330 ","End":"07:44.850","Text":"which is 6t"},{"Start":"07:44.850 ","End":"07:47.865","Text":"over dx over dt,"},{"Start":"07:47.865 ","End":"07:54.640","Text":"which is 3t squared minus 3."},{"Start":"07:58.230 ","End":"08:06.500","Text":"This if we put in the value t equals square root of 3,"},{"Start":"08:07.650 ","End":"08:17.240","Text":"then we get that m at our point is equal to, let\u0027s see."},{"Start":"08:19.410 ","End":"08:21.820","Text":"We will cancel at the end."},{"Start":"08:21.820 ","End":"08:28.120","Text":"We got 6 times t is the square root of 3 over,"},{"Start":"08:28.120 ","End":"08:31.990","Text":"the square root of 3 squared is just 3,"},{"Start":"08:31.990 ","End":"08:37.235","Text":"3 times 3 is 9, 9 minus 3 is 6."},{"Start":"08:37.235 ","End":"08:41.100","Text":"This is equal to square root of 3."},{"Start":"08:41.100 ","End":"08:48.715","Text":"Once we have the m and we have the point,"},{"Start":"08:48.715 ","End":"08:52.480","Text":"we just use the point-slope form of a tangent,"},{"Start":"08:52.480 ","End":"08:54.650","Text":"so we get that."},{"Start":"08:55.080 ","End":"08:57.430","Text":"I\u0027ll use a different color."},{"Start":"08:57.430 ","End":"09:02.080","Text":"Let\u0027s a y minus the y of the point equals"},{"Start":"09:02.080 ","End":"09:09.250","Text":"the slope times x minus the x of the point."},{"Start":"09:09.250 ","End":"09:18.735","Text":"That just gives us the y equals square root of 3 times x."},{"Start":"09:18.735 ","End":"09:21.315","Text":"That\u0027s only one tangent."},{"Start":"09:21.315 ","End":"09:24.104","Text":"What about the other tangent?"},{"Start":"09:24.104 ","End":"09:26.270","Text":"Let\u0027s take the other case."},{"Start":"09:26.270 ","End":"09:30.370","Text":"This time we take t equals minus the square root of"},{"Start":"09:30.370 ","End":"09:34.465","Text":"3 and we get that m for this not the same m,"},{"Start":"09:34.465 ","End":"09:37.750","Text":"maybe m_1 and m_2, don\u0027t confuse them."},{"Start":"09:37.750 ","End":"09:45.400","Text":"Different m would be using the same formula. I just copy it."},{"Start":"09:45.400 ","End":"09:49.075","Text":"6t over 3t squared minus 3,"},{"Start":"09:49.075 ","End":"09:54.400","Text":"but this time we substitute a different value minus the square root of 3 and so we"},{"Start":"09:54.400 ","End":"09:59.965","Text":"get m for the other case is 6t,"},{"Start":"09:59.965 ","End":"10:03.490","Text":"6 times minus square root of 3,"},{"Start":"10:03.490 ","End":"10:06.265","Text":"so it\u0027s minus 6 square root of 3."},{"Start":"10:06.265 ","End":"10:10.735","Text":"Denominator comes out the same when it\u0027s squared, so it\u0027s 6."},{"Start":"10:10.735 ","End":"10:14.750","Text":"We get minus the square root of 3."},{"Start":"10:15.090 ","End":"10:19.120","Text":"If we do the same thing here,"},{"Start":"10:19.120 ","End":"10:28.720","Text":"we will just get the y is equal to minus the square root of 3x."},{"Start":"10:28.720 ","End":"10:38.955","Text":"This is like tangent number 1 and this is tangent number 2,"},{"Start":"10:38.955 ","End":"10:41.085","Text":"and how can such a thing happen?"},{"Start":"10:41.085 ","End":"10:48.015","Text":"Well, I\u0027ll produce a diagram and that might make things clearer."},{"Start":"10:48.015 ","End":"10:52.255","Text":"I just scroll up a bit."},{"Start":"10:52.255 ","End":"10:56.740","Text":"Here\u0027s the picture and now all is clear."},{"Start":"10:56.740 ","End":"11:00.940","Text":"This goes in the direction of increasing t. When t"},{"Start":"11:00.940 ","End":"11:05.965","Text":"gets to be the minus the square root of 3,"},{"Start":"11:05.965 ","End":"11:09.860","Text":"we get here for the first time."},{"Start":"11:10.650 ","End":"11:13.315","Text":"I can mark this point,"},{"Start":"11:13.315 ","End":"11:16.765","Text":"t equals minus the square root of 3,"},{"Start":"11:16.765 ","End":"11:19.465","Text":"and then it continues."},{"Start":"11:19.465 ","End":"11:25.285","Text":"Actually it turns out this is the point where t is 0,"},{"Start":"11:25.285 ","End":"11:28.390","Text":"because when t is 0, x is 0,"},{"Start":"11:28.390 ","End":"11:30.355","Text":"y is minus 9,"},{"Start":"11:30.355 ","End":"11:33.610","Text":"so we get to t equals 0."},{"Start":"11:33.610 ","End":"11:41.395","Text":"Then we go around and then we get to t equals the square root of 3 and so on."},{"Start":"11:41.395 ","End":"11:46.750","Text":"At this point, we actually have 2 tangents."},{"Start":"11:46.750 ","End":"11:49.630","Text":"Here are the 2 tangent lines,"},{"Start":"11:49.630 ","End":"11:51.535","Text":"so I have drawn them in faintly."},{"Start":"11:51.535 ","End":"11:55.389","Text":"Obviously this one is the one where y equals root 3x,"},{"Start":"11:55.389 ","End":"12:01.315","Text":"it\u0027s a positive slope and the other one is y equals minus root 3x."},{"Start":"12:01.315 ","End":"12:04.285","Text":"That answers that."},{"Start":"12:04.285 ","End":"12:08.560","Text":"I\u0027d like to continue with the theory a little bit."},{"Start":"12:08.560 ","End":"12:12.040","Text":"I want to still talk about vertical lines and horizontal lines,"},{"Start":"12:12.040 ","End":"12:14.470","Text":"but I want to stick with this example."},{"Start":"12:14.470 ","End":"12:17.050","Text":"What I\u0027ll do is I\u0027ll keep the first line."},{"Start":"12:17.050 ","End":"12:19.325","Text":"I\u0027ll erase this stuff."},{"Start":"12:19.325 ","End":"12:23.520","Text":"That\u0027s better and now I\u0027ll add the extra exercise."},{"Start":"12:23.520 ","End":"12:28.140","Text":"This will be part 1 and part 2 will be find"},{"Start":"12:28.140 ","End":"12:37.900","Text":"the points where the tangent is and I\u0027m subdividing this into 2 questions."},{"Start":"12:37.900 ","End":"12:41.110","Text":"Where the tangent is horizontal."},{"Start":"12:41.110 ","End":"12:45.355","Text":"That\u0027s one part and where the tangent is vertical."},{"Start":"12:45.355 ","End":"12:48.355","Text":"We\u0027re missing a little bit of theory for this."},{"Start":"12:48.355 ","End":"12:56.950","Text":"Let\u0027s go back and remember the equation that dy over dx is equal"},{"Start":"12:56.950 ","End":"13:02.185","Text":"to the dy over"},{"Start":"13:02.185 ","End":"13:08.740","Text":"dt divided by dx over dt,"},{"Start":"13:08.740 ","End":"13:12.740","Text":"and this was like m, the slope."},{"Start":"13:12.740 ","End":"13:16.435","Text":"Now, previously in calculus,"},{"Start":"13:16.435 ","End":"13:20.800","Text":"whenever m was 0 then we had a horizontal line."},{"Start":"13:20.800 ","End":"13:26.815","Text":"For the condition for horizontal is as follows."},{"Start":"13:26.815 ","End":"13:31.030","Text":"We want the numerator to be 0,"},{"Start":"13:31.030 ","End":"13:34.510","Text":"dy over dt equals 0,"},{"Start":"13:34.510 ","End":"13:36.940","Text":"but there is an exception."},{"Start":"13:36.940 ","End":"13:38.470","Text":"We can\u0027t have 0 over 0."},{"Start":"13:38.470 ","End":"13:41.259","Text":"0 over 0 makes absolutely no sense,"},{"Start":"13:41.259 ","End":"13:49.030","Text":"so we must have that to check afterwards that dx over dt is not 0 at that point."},{"Start":"13:49.030 ","End":"13:53.650","Text":"Vertical is pretty much the opposite."},{"Start":"13:53.650 ","End":"14:03.070","Text":"In vertical usually it happens when we have y in terms of x that the denominator is 0."},{"Start":"14:03.070 ","End":"14:09.100","Text":"For vertical we want dx over dt to equal 0,"},{"Start":"14:09.100 ","End":"14:12.700","Text":"but once again we don\u0027t want 0 over 0,"},{"Start":"14:12.700 ","End":"14:14.650","Text":"1 over 0 is fine."},{"Start":"14:14.650 ","End":"14:18.010","Text":"I mean, it\u0027s not defined but it could fine for vertical,"},{"Start":"14:18.010 ","End":"14:20.950","Text":"but 0 over 0 could be just anything."},{"Start":"14:20.950 ","End":"14:27.250","Text":"It\u0027s more nonsensical, so dy over dt not equal to 0."},{"Start":"14:27.250 ","End":"14:32.350","Text":"Anyway, you can just accept it that these are the conditions for horizontal and vertical."},{"Start":"14:32.350 ","End":"14:37.885","Text":"That completes the theory but we still want to solve the exercise."},{"Start":"14:37.885 ","End":"14:40.900","Text":"I mean I can see where the tangent is going to be horizontal."},{"Start":"14:40.900 ","End":"14:42.535","Text":"I can see that it\u0027s going to be here."},{"Start":"14:42.535 ","End":"14:44.950","Text":"There\u0027s a couple of points here where it\u0027s going to be vertical."},{"Start":"14:44.950 ","End":"14:47.455","Text":"But suppose we don\u0027t have the picture,"},{"Start":"14:47.455 ","End":"14:50.425","Text":"so here\u0027s what we do."},{"Start":"14:50.425 ","End":"14:52.420","Text":"What should we go for first?"},{"Start":"14:52.420 ","End":"14:54.775","Text":"Let\u0027s go for the horizontal first."},{"Start":"14:54.775 ","End":"15:04.675","Text":"For horizontal, we want dy over dt which is 3t."},{"Start":"15:04.675 ","End":"15:07.075","Text":"Well, we had it before but let\u0027s do it again."},{"Start":"15:07.075 ","End":"15:15.880","Text":"Derivative of y with respect to t was 6t is equal to 0,"},{"Start":"15:15.880 ","End":"15:20.665","Text":"which gives us that t equals 0."},{"Start":"15:20.665 ","End":"15:27.355","Text":"Just have to make sure that the other one is not 0."},{"Start":"15:27.355 ","End":"15:31.165","Text":"When I put it into dx over dt."},{"Start":"15:31.165 ","End":"15:35.930","Text":"Well, I\u0027ll need the dx over dt anyway, dx over dt."},{"Start":"15:37.260 ","End":"15:39.640","Text":"I\u0027ll start the second part,"},{"Start":"15:39.640 ","End":"15:46.120","Text":"for vertical we want dx over dt which is 3t squared minus 3."},{"Start":"15:46.120 ","End":"15:48.640","Text":"Before I continue with the vertical,"},{"Start":"15:48.640 ","End":"15:54.940","Text":"let\u0027s just make sure that when I put in 0 it\u0027s not 0. Clearly it isn\u0027t."},{"Start":"15:54.940 ","End":"15:57.175","Text":"If I put in 0 I get minus 3,"},{"Start":"15:57.175 ","End":"15:59.260","Text":"so that\u0027s not 0."},{"Start":"15:59.260 ","End":"16:03.235","Text":"But for vertical I want it to be 0."},{"Start":"16:03.235 ","End":"16:04.910","Text":"After I get the answer,"},{"Start":"16:04.910 ","End":"16:08.285","Text":"this mustn\u0027t be 0 time the other way around."},{"Start":"16:08.285 ","End":"16:12.400","Text":"3t squared minus 3 equals 0."},{"Start":"16:12.400 ","End":"16:15.325","Text":"I get that t squared equals 1,"},{"Start":"16:15.325 ","End":"16:19.705","Text":"so t is equal to plus or minus 1."},{"Start":"16:19.705 ","End":"16:22.970","Text":"Now I can get the 3 points."},{"Start":"16:22.970 ","End":"16:25.655","Text":"If t equals 0,"},{"Start":"16:25.655 ","End":"16:28.220","Text":"that gives me that the point x,"},{"Start":"16:28.220 ","End":"16:38.305","Text":"y, x is 0 and y is minus 9."},{"Start":"16:38.305 ","End":"16:42.400","Text":"This is the point where again we\u0027ve already mentioned it before that where"},{"Start":"16:42.400 ","End":"16:51.490","Text":"t is 0 and so we get a horizontal tangent."},{"Start":"16:51.490 ","End":"16:53.140","Text":"Let me sketch it."},{"Start":"16:53.140 ","End":"16:59.410","Text":"There we are and for the vertical we have 2 possibilities. Let\u0027s try."},{"Start":"16:59.410 ","End":"17:02.380","Text":"First of all, t equals minus 1."},{"Start":"17:02.380 ","End":"17:05.050","Text":"What do we get for x, y,"},{"Start":"17:05.050 ","End":"17:16.410","Text":"x would be minus 1 cubed plus 3 it gives me 2,"},{"Start":"17:16.410 ","End":"17:19.980","Text":"and y comes out to be,"},{"Start":"17:19.980 ","End":"17:22.700","Text":"when t is minus 1,"},{"Start":"17:22.700 ","End":"17:27.610","Text":"3 minus 9 which is minus 6,"},{"Start":"17:27.610 ","End":"17:31.580","Text":"so I\u0027ll compute them both and we\u0027ll look go to the graph."},{"Start":"17:31.580 ","End":"17:37.610","Text":"When t is 1, we get 1"},{"Start":"17:37.610 ","End":"17:44.390","Text":"minus 3 is minus 2 for x and y comes out the same because it was t squared,"},{"Start":"17:44.390 ","End":"17:46.930","Text":"so it\u0027s also minus 6."},{"Start":"17:46.930 ","End":"17:48.990","Text":"The t is 0 we did already."},{"Start":"17:48.990 ","End":"17:51.390","Text":"At t equals minus 1,"},{"Start":"17:51.390 ","End":"17:56.945","Text":"2 minus 6 must be this point here"},{"Start":"17:56.945 ","End":"18:04.325","Text":"and that\u0027s the t equals minus 1 and if t increasing as we go with the arrows."},{"Start":"18:04.325 ","End":"18:07.335","Text":"Then t equals 1 must be the other one,"},{"Start":"18:07.335 ","End":"18:12.005","Text":"minus 2 minus 6 somewhere like here."},{"Start":"18:12.005 ","End":"18:15.680","Text":"That\u0027s where t is equal to 1."},{"Start":"18:15.680 ","End":"18:19.990","Text":"I\u0027ll just draw in the tangent lines and there we are,"},{"Start":"18:19.990 ","End":"18:25.295","Text":"two vertical tangents and we\u0027re done for now."},{"Start":"18:25.295 ","End":"18:27.450","Text":"We\u0027ll take a break."}],"ID":6003},{"Watched":false,"Name":"Worked Example","Duration":"7m 47s","ChapterTopicVideoID":5990,"CourseChapterTopicPlaylistID":4002,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"Here we are back after the break,"},{"Start":"00:01.920 ","End":"00:05.265","Text":"we\u0027re on tangents with parametric equations."},{"Start":"00:05.265 ","End":"00:09.210","Text":"There\u0027s just 1 more topic here that I want to cover,"},{"Start":"00:09.210 ","End":"00:11.355","Text":"and that\u0027s the second derivative."},{"Start":"00:11.355 ","End":"00:14.640","Text":"But I\u0027d like to stay with the same example,"},{"Start":"00:14.640 ","End":"00:17.280","Text":"and I\u0027m going to erase what I don\u0027t need."},{"Start":"00:17.280 ","End":"00:22.725","Text":"I\u0027m going to add a question part 3, and here it is."},{"Start":"00:22.725 ","End":"00:25.830","Text":"I want to find for which values of t is"},{"Start":"00:25.830 ","End":"00:31.155","Text":"the curve convex and for which values of t is it concave?"},{"Start":"00:31.155 ","End":"00:34.335","Text":"This will lead us into the second derivative."},{"Start":"00:34.335 ","End":"00:39.630","Text":"Because as you remember when we had just y as a function of x,"},{"Start":"00:39.630 ","End":"00:45.210","Text":"we had that convex was y"},{"Start":"00:45.210 ","End":"00:51.700","Text":"double-prime negative and concave,"},{"Start":"00:51.700 ","End":"00:58.115","Text":"or concave up, it was y double-prime positive."},{"Start":"00:58.115 ","End":"01:02.494","Text":"Sometimes this is called concave up and this is concave down."},{"Start":"01:02.494 ","End":"01:06.620","Text":"Sometimes it\u0027s called convex and some books even have it the other way around,"},{"Start":"01:06.620 ","End":"01:09.180","Text":"okay, let\u0027s mark into all that."},{"Start":"01:09.190 ","End":"01:17.059","Text":"I need to tell you how to do second derivative for parametric curves."},{"Start":"01:17.059 ","End":"01:19.355","Text":"It\u0027s not what you might think."},{"Start":"01:19.355 ","End":"01:21.350","Text":"In the case of the first derivative,"},{"Start":"01:21.350 ","End":"01:26.225","Text":"we just take the first derivative of y over the first derivative of x."},{"Start":"01:26.225 ","End":"01:29.150","Text":"But the second derivative of this over"},{"Start":"01:29.150 ","End":"01:32.810","Text":"the second derivative of this is not how we do the second derivative."},{"Start":"01:32.810 ","End":"01:34.729","Text":"The formula is as follows,"},{"Start":"01:34.729 ","End":"01:43.115","Text":"that if we want to take the derivative of the first derivative,"},{"Start":"01:43.115 ","End":"01:46.015","Text":"we want it with respect to x."},{"Start":"01:46.015 ","End":"01:47.960","Text":"This is what I want,"},{"Start":"01:47.960 ","End":"01:51.620","Text":"the derivative of dy over dx with respect to x,"},{"Start":"01:51.620 ","End":"01:54.720","Text":"and this turns out to be,"},{"Start":"01:55.960 ","End":"02:01.670","Text":"assuming that we\u0027ve already computed dy over dx and it\u0027s a function of t remember."},{"Start":"02:01.670 ","End":"02:11.150","Text":"Then this is the derivative of dy over dx with respect to t,"},{"Start":"02:11.150 ","End":"02:15.395","Text":"and then divide it by the derivative of"},{"Start":"02:15.395 ","End":"02:20.090","Text":"x with respect to t. Now some of these we computed already,"},{"Start":"02:20.090 ","End":"02:22.960","Text":"so let me just quote the results."},{"Start":"02:22.960 ","End":"02:25.665","Text":"I copied the result,"},{"Start":"02:25.665 ","End":"02:28.915","Text":"the result was that dy over dx,"},{"Start":"02:28.915 ","End":"02:30.905","Text":"which I also called m,"},{"Start":"02:30.905 ","End":"02:40.310","Text":"was equal to 6t over 3t squared minus 3."},{"Start":"02:40.310 ","End":"02:42.500","Text":"I don\u0027t know why I squashed it in there."},{"Start":"02:42.500 ","End":"02:45.560","Text":"Let me write it over here, dy over dx,"},{"Start":"02:45.560 ","End":"02:48.005","Text":"but at the same time I\u0027ll cancel by 3,"},{"Start":"02:48.005 ","End":"02:53.445","Text":"so it\u0027s 2t over t squared minus 1."},{"Start":"02:53.445 ","End":"02:56.115","Text":"We also computed dx over dt."},{"Start":"02:56.115 ","End":"02:59.175","Text":"Of course we can recompute it because it\u0027s this,"},{"Start":"02:59.175 ","End":"03:02.590","Text":"it\u0027s just 3t squared minus 3."},{"Start":"03:03.050 ","End":"03:07.625","Text":"Then I write that as 3 times t squared minus 1,"},{"Start":"03:07.625 ","End":"03:09.215","Text":"so that will help us."},{"Start":"03:09.215 ","End":"03:12.295","Text":"Now let\u0027s get to do this computation."},{"Start":"03:12.295 ","End":"03:14.820","Text":"I might erase this."},{"Start":"03:14.820 ","End":"03:22.805","Text":"Here we go. We first of all differentiate dy over dx with respect to t. On the numerator,"},{"Start":"03:22.805 ","End":"03:26.600","Text":"I\u0027m going to use the quotient rule on this."},{"Start":"03:26.600 ","End":"03:31.670","Text":"The quotient rule on this differentiated with respect to t,"},{"Start":"03:31.730 ","End":"03:37.795","Text":"we have a denominator which is this thing squared."},{"Start":"03:37.795 ","End":"03:42.380","Text":"We have t squared minus 1 squared,"},{"Start":"03:42.380 ","End":"03:45.055","Text":"and on the numerator,"},{"Start":"03:45.055 ","End":"03:47.830","Text":"we have derivative of the numerator,"},{"Start":"03:47.830 ","End":"03:53.590","Text":"which is 2 times denominator as is t squared minus 1,"},{"Start":"03:53.590 ","End":"03:58.900","Text":"minus the numerator as is times the derivative of the denominator,"},{"Start":"03:58.900 ","End":"04:03.700","Text":"which is another 2t all over this thing squared which we have."},{"Start":"04:03.700 ","End":"04:11.930","Text":"Then here we get the dx over dt which is 3 times t squared minus 1."},{"Start":"04:11.930 ","End":"04:21.705","Text":"Now, this equals, what I can do let me simplify this a bit."},{"Start":"04:21.705 ","End":"04:23.070","Text":"Let me take the numbers out."},{"Start":"04:23.070 ","End":"04:29.140","Text":"I\u0027m going to take 2 out of here and I\u0027m going to take 3 out of here."},{"Start":"04:29.140 ","End":"04:33.660","Text":"I\u0027m going to combine t squared minus 1 squared with t squared minus 1,"},{"Start":"04:33.660 ","End":"04:36.800","Text":"so I\u0027ll get t squared minus 1 cubed."},{"Start":"04:36.800 ","End":"04:43.260","Text":"Now all I have to see is what I have left from here after I\u0027ve taken a 2 out,"},{"Start":"04:43.260 ","End":"04:44.625","Text":"so I can erase this one."},{"Start":"04:44.625 ","End":"04:46.455","Text":"Let\u0027s say I can erase this one,"},{"Start":"04:46.455 ","End":"04:51.705","Text":"so I\u0027ve got t squared minus 1 minus 2t squared,"},{"Start":"04:51.705 ","End":"04:56.520","Text":"so it\u0027s minus t squared minus 1."},{"Start":"04:56.520 ","End":"05:01.260","Text":"I have taken the minus in front and put t squared plus 1"},{"Start":"05:01.260 ","End":"05:09.775","Text":"and that\u0027s our second derivative,"},{"Start":"05:09.775 ","End":"05:17.690","Text":"which is often written as d squared y over dx squared in the Leibnitz notation,"},{"Start":"05:17.690 ","End":"05:22.410","Text":"or if you\u0027re using the Newtonian notation, y double-prime."},{"Start":"05:25.790 ","End":"05:29.810","Text":"The way we decide where this is positive or negative"},{"Start":"05:29.810 ","End":"05:33.500","Text":"as we find that where the numerator or denominator is 0,"},{"Start":"05:33.500 ","End":"05:37.460","Text":"the numerator is never 0 because t squared plus 1 is always positive."},{"Start":"05:37.460 ","End":"05:40.525","Text":"The only 0s, are 0s of the denominator,"},{"Start":"05:40.525 ","End":"05:43.160","Text":"and you can easily see that plus or minus 1,"},{"Start":"05:43.160 ","End":"05:45.485","Text":"that\u0027s where t squared minus 1 is 0."},{"Start":"05:45.485 ","End":"05:53.629","Text":"If we take the number line and we mark on minus 1 and 1,"},{"Start":"05:53.629 ","End":"05:57.020","Text":"the function is not defined here and I\u0027ll indicate that"},{"Start":"05:57.020 ","End":"05:59.780","Text":"by hollow dots because of 0s of the denominator."},{"Start":"05:59.780 ","End":"06:02.525","Text":"But if I take sample points here,"},{"Start":"06:02.525 ","End":"06:07.695","Text":"like let\u0027s say I put 0, 2 and minus 2,"},{"Start":"06:07.695 ","End":"06:09.435","Text":"and then I substitute."},{"Start":"06:09.435 ","End":"06:11.720","Text":"If I put t equals 0,"},{"Start":"06:11.720 ","End":"06:14.570","Text":"this is negative and this is negative."},{"Start":"06:14.570 ","End":"06:18.230","Text":"Let me just put some dotted lines here for the ranges."},{"Start":"06:18.230 ","End":"06:20.530","Text":"At 0 I\u0027m positive,"},{"Start":"06:20.530 ","End":"06:23.960","Text":"and if I put in 2 this thing,"},{"Start":"06:23.960 ","End":"06:26.780","Text":"t squared minus 1 will be positive,"},{"Start":"06:26.780 ","End":"06:29.225","Text":"positive, positive, everything is negative."},{"Start":"06:29.225 ","End":"06:32.000","Text":"Here the same thing, negative."},{"Start":"06:32.000 ","End":"06:39.420","Text":"The second derivative is negative between t equals minus 1 and 1,"},{"Start":"06:39.420 ","End":"06:45.110","Text":"so what I say is that between minus 1 and 1,"},{"Start":"06:45.110 ","End":"06:49.429","Text":"not including because it\u0027s not defined here, second derivative,"},{"Start":"06:49.429 ","End":"06:57.300","Text":"then we know that we have positive meant concave"},{"Start":"06:58.240 ","End":"07:04.970","Text":"and everything else meaning that"},{"Start":"07:04.970 ","End":"07:11.700","Text":"t less than minus 1 or t bigger than 1, we have convex."},{"Start":"07:11.700 ","End":"07:13.730","Text":"It makes sense here because look,"},{"Start":"07:13.730 ","End":"07:16.445","Text":"here\u0027s the minus 1 to the 1,"},{"Start":"07:16.445 ","End":"07:19.805","Text":"we\u0027re coming from minus infinity up to minus 1."},{"Start":"07:19.805 ","End":"07:21.870","Text":"This shape is in"},{"Start":"07:21.870 ","End":"07:30.795","Text":"fact convex or concave down."},{"Start":"07:30.795 ","End":"07:33.090","Text":"From here to here, it holds water,"},{"Start":"07:33.090 ","End":"07:36.139","Text":"it\u0027s concave up from t equals 1 onwards,"},{"Start":"07:36.139 ","End":"07:38.280","Text":"then it\u0027s convex again."},{"Start":"07:38.280 ","End":"07:43.280","Text":"That\u0027s the second derivative and an application to concavity convexity."},{"Start":"07:43.280 ","End":"07:47.340","Text":"Now we\u0027re done on this particular topic."}],"ID":6004}],"Thumbnail":null,"ID":4002},{"Name":"Area with Parametric Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Area with Parametric Equations","Duration":"17m 52s","ChapterTopicVideoID":5991,"CourseChapterTopicPlaylistID":4003,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/5991.jpeg","UploadDate":"2019-12-11T21:21:31.0900000","DurationForVideoObject":"PT17M52S","Description":null,"MetaTitle":"Area with Parametric Equations: Video + Workbook | Proprep","MetaDescription":"Parametric Equations - Area with Parametric Equations. 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/parametric-equations/area-with-parametric-equations/vid6005","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Area under a curve,"},{"Start":"00:02.580 ","End":"00:05.475","Text":"but with parametric equations,"},{"Start":"00:05.475 ","End":"00:10.200","Text":"I just like to give you a brief reminder of how to find the area under"},{"Start":"00:10.200 ","End":"00:16.335","Text":"a curve with regular non-parametric equations where y is a function of x."},{"Start":"00:16.335 ","End":"00:19.530","Text":"Here\u0027s a typical picture, in this case,"},{"Start":"00:19.530 ","End":"00:23.310","Text":"we\u0027ll assume that y is a function of x. I don\u0027t know why is,"},{"Start":"00:23.310 ","End":"00:26.700","Text":"I\u0027ll use capital F because I\u0027m going to use little f later on."},{"Start":"00:26.700 ","End":"00:31.395","Text":"If this area is the area s,"},{"Start":"00:31.395 ","End":"00:37.155","Text":"then the area is given by the integral"},{"Start":"00:37.155 ","End":"00:44.685","Text":"from a to b of f of x dx."},{"Start":"00:44.685 ","End":"00:51.830","Text":"If instead of f we put y we could say that s is equal to in general,"},{"Start":"00:51.830 ","End":"00:57.420","Text":"the integral of ydx from a to b,"},{"Start":"00:57.420 ","End":"00:59.580","Text":"that\u0027s where x goes from."},{"Start":"00:59.580 ","End":"01:06.785","Text":"Now, if we don\u0027t have it in closed form or y is the function of x."},{"Start":"01:06.785 ","End":"01:08.960","Text":"Suppose we haven\u0027t in parametric form,"},{"Start":"01:08.960 ","End":"01:13.970","Text":"then we would have something like x is a function of t,"},{"Start":"01:13.970 ","End":"01:15.710","Text":"y is another function of t,"},{"Start":"01:15.710 ","End":"01:21.305","Text":"I\u0027ll use little f of t and y equals g of"},{"Start":"01:21.305 ","End":"01:29.140","Text":"t. Let\u0027s suppose that t goes from,"},{"Start":"01:29.510 ","End":"01:31.710","Text":"I\u0027ve used up a and b,"},{"Start":"01:31.710 ","End":"01:33.585","Text":"so I\u0027ll use Alpha and Beta."},{"Start":"01:33.585 ","End":"01:36.660","Text":"Let\u0027s say t goes from Alpha to Beta."},{"Start":"01:36.660 ","End":"01:40.930","Text":"Suppose that when t is Alpha,"},{"Start":"01:40.930 ","End":"01:43.505","Text":"that corresponds to x being a."},{"Start":"01:43.505 ","End":"01:48.440","Text":"In other words, we will assume that f of Alpha is a"},{"Start":"01:48.440 ","End":"01:56.940","Text":"and f of Beta equals b."},{"Start":"01:56.940 ","End":"02:04.790","Text":"Then actually, we can use this formula to get the parametric form."},{"Start":"02:04.790 ","End":"02:11.419","Text":"Because if we just replace substitute x and y,"},{"Start":"02:11.419 ","End":"02:16.080","Text":"we get that the area is the integral,"},{"Start":"02:16.310 ","End":"02:19.490","Text":"won\u0027t be from a to b,"},{"Start":"02:19.490 ","End":"02:23.510","Text":"it will be from Alpha to Beta because we\u0027re going to take it in terms of t"},{"Start":"02:23.510 ","End":"02:32.870","Text":"and y is just g of t. Dx is equal to,"},{"Start":"02:32.870 ","End":"02:34.850","Text":"I\u0027ll do this at the side."},{"Start":"02:34.850 ","End":"02:38.390","Text":"Note that if x is f of t,"},{"Start":"02:38.390 ","End":"02:43.420","Text":"then dx is f prime of t, dt."},{"Start":"02:43.420 ","End":"02:45.405","Text":"If I put that here,"},{"Start":"02:45.405 ","End":"02:50.340","Text":"then I get the formula f prime of t dt."},{"Start":"02:50.340 ","End":"02:59.480","Text":"This is the formula for the area under the curve which is given in parametric form."},{"Start":"02:59.480 ","End":"03:03.125","Text":"I think it deserves highlighting,"},{"Start":"03:03.125 ","End":"03:07.315","Text":"not quite covered, there we are."},{"Start":"03:07.315 ","End":"03:10.190","Text":"But there are a couple of remarks,"},{"Start":"03:10.190 ","End":"03:13.100","Text":"it\u0027s not quite so simple."},{"Start":"03:13.100 ","End":"03:15.890","Text":"Remember that with parametric equations,"},{"Start":"03:15.890 ","End":"03:18.319","Text":"there is a direction of motion,"},{"Start":"03:18.319 ","End":"03:21.380","Text":"so to speak, the direction where t increases."},{"Start":"03:21.380 ","End":"03:27.635","Text":"I\u0027m assuming that we start off when t is Alpha and end up in t is Beta,"},{"Start":"03:27.635 ","End":"03:31.175","Text":"which means that we\u0027re essentially going in this direction,"},{"Start":"03:31.175 ","End":"03:35.675","Text":"and then this formula would hold."},{"Start":"03:35.675 ","End":"03:38.990","Text":"But it could be that when t is Alpha,"},{"Start":"03:38.990 ","End":"03:42.845","Text":"we\u0027re here and when t is Beta that we\u0027re here."},{"Start":"03:42.845 ","End":"03:45.520","Text":"Otherwise, we could be going backwards."},{"Start":"03:45.520 ","End":"03:48.590","Text":"If we\u0027re going backward instead of this,"},{"Start":"03:48.590 ","End":"03:56.840","Text":"we\u0027ll have that f of Alpha equals b and f of Beta equals a."},{"Start":"03:56.840 ","End":"04:00.500","Text":"As I go along from Alpha to Beta,"},{"Start":"04:00.500 ","End":"04:02.285","Text":"I\u0027ll be going backwards."},{"Start":"04:02.285 ","End":"04:05.540","Text":"In this case, the formula is different."},{"Start":"04:05.540 ","End":"04:06.950","Text":"You just put a minus in front,"},{"Start":"04:06.950 ","End":"04:08.870","Text":"well, there\u0027s 2 possibilities."},{"Start":"04:08.870 ","End":"04:11.675","Text":"We can either say that s"},{"Start":"04:11.675 ","End":"04:21.575","Text":"equals minus the integral from Beta to Alpha of g of t,"},{"Start":"04:21.575 ","End":"04:23.420","Text":"f prime of t,"},{"Start":"04:23.420 ","End":"04:26.900","Text":"dt, that\u0027s 1 possibility."},{"Start":"04:26.900 ","End":"04:32.150","Text":"There\u0027s another possibility of throwing out the minus and reversing the order,"},{"Start":"04:32.150 ","End":"04:34.070","Text":"I have to choose 1."},{"Start":"04:34.070 ","End":"04:39.440","Text":"I\u0027ll choose with the 1 where I\u0027ll get rid of the minus and reverse the order."},{"Start":"04:39.440 ","End":"04:43.625","Text":"This is the alternative formula for when we\u0027re going backwards."},{"Start":"04:43.625 ","End":"04:47.870","Text":"As I said, the only difference is that I switched alpha and beta,"},{"Start":"04:47.870 ","End":"04:52.065","Text":"or you could keep alpha and beta and put a minus in front, the same thing."},{"Start":"04:52.065 ","End":"04:55.190","Text":"That\u0027s not the only snag though."},{"Start":"04:55.190 ","End":"04:58.235","Text":"Besides going forward or backwards,"},{"Start":"04:58.235 ","End":"05:02.960","Text":"we had an example in a previous clip."},{"Start":"05:02.960 ","End":"05:07.970","Text":"We just went back to that page which I kept, which was,"},{"Start":"05:07.970 ","End":"05:14.300","Text":"we had an example where when t went from 1 end to the other,"},{"Start":"05:14.300 ","End":"05:18.230","Text":"we actually traverse this portion twice."},{"Start":"05:18.230 ","End":"05:22.065","Text":"We went from here to here and all the way back again."},{"Start":"05:22.065 ","End":"05:26.150","Text":"That\u0027s a snag if that happens."},{"Start":"05:26.150 ","End":"05:28.310","Text":"I\u0027ll make a note of that,"},{"Start":"05:28.310 ","End":"05:30.680","Text":"that when t goes from,"},{"Start":"05:30.680 ","End":"05:39.350","Text":"let\u0027s say this was t equals Alpha and t equals Beta or if we had the opposite arrow,"},{"Start":"05:39.350 ","End":"05:43.625","Text":"we would have here that t is Alpha and here t is Beta."},{"Start":"05:43.625 ","End":"05:45.379","Text":"Both of these are okay,"},{"Start":"05:45.379 ","End":"05:48.860","Text":"then it would be going in the opposite direction."},{"Start":"05:48.860 ","End":"05:51.575","Text":"I mean, if we took the Alpha and Beta,"},{"Start":"05:51.575 ","End":"05:52.745","Text":"will be going in the opposite."},{"Start":"05:52.745 ","End":"05:54.755","Text":"Either direction is okay,"},{"Start":"05:54.755 ","End":"05:58.160","Text":"but we have to make sure that we only do it once."},{"Start":"05:58.160 ","End":"06:05.610","Text":"I make another 1, as t goes from Alpha to Beta or it could be Beta to Alpha,"},{"Start":"06:05.610 ","End":"06:10.260","Text":"if it\u0027s in reverse, briefly write that."},{"Start":"06:10.260 ","End":"06:12.500","Text":"If we\u0027re going backwards,"},{"Start":"06:12.500 ","End":"06:19.560","Text":"we have to assume that x goes from a to b only once."},{"Start":"06:24.650 ","End":"06:29.930","Text":"X will increase, but it has to only go 1 time either forward or backwards."},{"Start":"06:29.930 ","End":"06:31.940","Text":"If it\u0027s going forwards,"},{"Start":"06:31.940 ","End":"06:35.630","Text":"if it\u0027s from Alpha to Beta we\u0027ll be using this formula."},{"Start":"06:35.630 ","End":"06:37.040","Text":"If it\u0027s in reverse,"},{"Start":"06:37.040 ","End":"06:38.630","Text":"we\u0027ll be using this formula,"},{"Start":"06:38.630 ","End":"06:42.365","Text":"either switching Alpha and Beta or putting a minus,"},{"Start":"06:42.365 ","End":"06:45.460","Text":"that\u0027s an important note."},{"Start":"06:45.460 ","End":"06:50.910","Text":"This is what I mean by the reverse case,"},{"Start":"06:50.910 ","End":"06:53.810","Text":"but however it is, we must make sure it\u0027s only once."},{"Start":"06:53.810 ","End":"06:58.040","Text":"1 way to do that would be to check, for example,"},{"Start":"06:58.040 ","End":"07:02.990","Text":"that f of t is monotonous,"},{"Start":"07:02.990 ","End":"07:05.450","Text":"meaning, either increasing or"},{"Start":"07:05.450 ","End":"07:14.280","Text":"decreasing when t goes from Alpha to Beta,"},{"Start":"07:14.280 ","End":"07:16.940","Text":"we can check the derivative and make sure it\u0027s"},{"Start":"07:16.940 ","End":"07:20.765","Text":"always non-positive or non-negative, for example."},{"Start":"07:20.765 ","End":"07:22.910","Text":"But it\u0027s important, otherwise,"},{"Start":"07:22.910 ","End":"07:26.420","Text":"we could be ending up with 0 if we went forwards and backwards,"},{"Start":"07:26.420 ","End":"07:28.370","Text":"like in the previous example,"},{"Start":"07:28.370 ","End":"07:30.730","Text":"stuff might cancel out."},{"Start":"07:30.730 ","End":"07:33.880","Text":"Let\u0027s start with an example."},{"Start":"07:35.030 ","End":"07:40.910","Text":"The example I\u0027m going to take is of a curve,"},{"Start":"07:40.910 ","End":"07:44.315","Text":"something called a cycloid."},{"Start":"07:44.315 ","End":"07:46.945","Text":"I\u0027ll sketch and I\u0027ll explain,"},{"Start":"07:46.945 ","End":"07:50.980","Text":"but let\u0027s get some more room here."},{"Start":"07:52.560 ","End":"07:59.710","Text":"There we are, and I brought the formula along with me together with the setup,"},{"Start":"07:59.710 ","End":"08:02.140","Text":"so we\u0027ll have it handy."},{"Start":"08:02.140 ","End":"08:03.940","Text":"Now, what\u0027s a cycloid?"},{"Start":"08:03.940 ","End":"08:10.780","Text":"I borrowed a sketch from the Internet that demonstrates it."},{"Start":"08:10.780 ","End":"08:13.870","Text":"Basically, you imagine a wheel,"},{"Start":"08:13.870 ","End":"08:17.560","Text":"maybe the wheel of a bicycle or a hoop or something,"},{"Start":"08:17.560 ","End":"08:21.160","Text":"that\u0027s traveling along in this direction,"},{"Start":"08:21.160 ","End":"08:24.520","Text":"rolling along, and you mark a certain point,"},{"Start":"08:24.520 ","End":"08:26.860","Text":"initially it\u0027s the point on the bottom."},{"Start":"08:26.860 ","End":"08:28.645","Text":"But as it rolls along,"},{"Start":"08:28.645 ","End":"08:34.450","Text":"that point traces out a curve called a cycloid,"},{"Start":"08:34.450 ","End":"08:39.730","Text":"and eventually, after a complete revolution, a rotation,"},{"Start":"08:39.730 ","End":"08:41.890","Text":"or whatever, it comes back down again,"},{"Start":"08:41.890 ","End":"08:45.830","Text":"and then it goes off again and it goes on to infinity."},{"Start":"08:46.260 ","End":"08:52.825","Text":"It turns out that this is very well-described in parametric form."},{"Start":"08:52.825 ","End":"08:55.270","Text":"Here I have another diagram,"},{"Start":"08:55.270 ","End":"08:57.715","Text":"I also borrowed it from the Internet,"},{"Start":"08:57.715 ","End":"08:59.965","Text":"you can find a lot of stuff there,"},{"Start":"08:59.965 ","End":"09:04.435","Text":"which gives the parametric equation of a cycloid,"},{"Start":"09:04.435 ","End":"09:10.285","Text":"but they use the variable Theta instead of t. You know what?"},{"Start":"09:10.285 ","End":"09:14.020","Text":"Let\u0027s get used to other letters besides t. You\u0027ll"},{"Start":"09:14.020 ","End":"09:17.515","Text":"see a lot of Theta when you study polar coordinates."},{"Start":"09:17.515 ","End":"09:22.220","Text":"So not to be frightened of the Greek letter."},{"Start":"09:22.740 ","End":"09:26.935","Text":"We\u0027ll change our t\u0027s to Theta."},{"Start":"09:26.935 ","End":"09:29.440","Text":"Do it like a circle with a little loop there,"},{"Start":"09:29.440 ","End":"09:32.980","Text":"and yeah, that\u0027s pronounced Theta."},{"Start":"09:32.980 ","End":"09:34.930","Text":"Very common Greek letter,"},{"Start":"09:34.930 ","End":"09:37.405","Text":"often used for angles like Alpha,"},{"Start":"09:37.405 ","End":"09:40.075","Text":"and actually, it is an angle."},{"Start":"09:40.075 ","End":"09:43.795","Text":"What we say is that we\u0027re going to let the angle be the parameter."},{"Start":"09:43.795 ","End":"09:47.440","Text":"As we\u0027re going along,"},{"Start":"09:47.440 ","End":"09:50.635","Text":"imagine 1 of the spokes may be is highlighted."},{"Start":"09:50.635 ","End":"09:54.474","Text":"It\u0027s going to be making an angle,"},{"Start":"09:54.474 ","End":"09:58.885","Text":"and actually at 360 degrees or 2Pi,"},{"Start":"09:58.885 ","End":"10:01.465","Text":"it makes a complete turn."},{"Start":"10:01.465 ","End":"10:05.515","Text":"The point that was on the bottom comes back to the bottom again."},{"Start":"10:05.515 ","End":"10:10.010","Text":"This is the formula and I\u0027m going to write it down here,"},{"Start":"10:10.020 ","End":"10:13.855","Text":"but it gives a general formula when the radius is a."},{"Start":"10:13.855 ","End":"10:22.035","Text":"Let\u0027s take in our case that our radius a is equal to 10, and what\u0027s more,"},{"Start":"10:22.035 ","End":"10:26.940","Text":"we\u0027ll take our parameter to exactly 1 cycle,"},{"Start":"10:26.940 ","End":"10:29.715","Text":"and in the proper direction,"},{"Start":"10:29.715 ","End":"10:37.050","Text":"turns out if I take a Theta between 0 and 360 degrees,"},{"Start":"10:37.050 ","End":"10:38.340","Text":"but we work in radian,"},{"Start":"10:38.340 ","End":"10:43.630","Text":"so that\u0027s 2Pi, then when Theta is 0 is here,"},{"Start":"10:43.630 ","End":"10:45.175","Text":"when Theta is Pi,"},{"Start":"10:45.175 ","End":"10:46.960","Text":"1/2 the way it\u0027s at the top,"},{"Start":"10:46.960 ","End":"10:49.360","Text":"and then it goes down to the bottom again."},{"Start":"10:49.360 ","End":"10:52.165","Text":"We get x equals,"},{"Start":"10:52.165 ","End":"10:55.390","Text":"and we\u0027ll take 10 to be concrete,"},{"Start":"10:55.390 ","End":"11:00.080","Text":"10 inches is the radius of the bicycle wheel."},{"Start":"11:00.420 ","End":"11:07.285","Text":"10 times Theta minus sine Theta."},{"Start":"11:07.285 ","End":"11:09.880","Text":"Doesn\u0027t matter how we get to this formula."},{"Start":"11:09.880 ","End":"11:13.944","Text":"Take it on trust, it\u0027s not the purpose here to derive."},{"Start":"11:13.944 ","End":"11:15.535","Text":"We just take this as a given."},{"Start":"11:15.535 ","End":"11:17.680","Text":"The cycloid is given by,"},{"Start":"11:17.680 ","End":"11:24.355","Text":"and y equals 10 times 1 minus cosine Theta,"},{"Start":"11:24.355 ","End":"11:30.970","Text":"and Theta goes between 0 and 2Pi,"},{"Start":"11:30.970 ","End":"11:33.325","Text":"and because Theta is the angle,"},{"Start":"11:33.325 ","End":"11:38.005","Text":"we know that as we go from 0-2Pi or from 0-360 degrees,"},{"Start":"11:38.005 ","End":"11:41.620","Text":"we\u0027re going exactly once and in the correct direction."},{"Start":"11:41.620 ","End":"11:44.770","Text":"Like I said, we have to make sure it covers exactly once,"},{"Start":"11:44.770 ","End":"11:46.315","Text":"and if it\u0027s in reverse direction,"},{"Start":"11:46.315 ","End":"11:52.540","Text":"we need to reverse this formula by switching the upper and lower limits."},{"Start":"11:52.540 ","End":"11:54.805","Text":"Let\u0027s just compute this."},{"Start":"11:54.805 ","End":"11:57.595","Text":"We have the formula here that\u0027s highlighted."},{"Start":"11:57.595 ","End":"12:00.910","Text":"Our area, I\u0027m not going to shade it,"},{"Start":"12:00.910 ","End":"12:05.230","Text":"it\u0027ll look a mess, but the area or perhaps a little bit."},{"Start":"12:05.230 ","End":"12:15.960","Text":"I\u0027m talking about the area that\u0027s under the cycloid during 1 cycle, so to speak,"},{"Start":"12:15.960 ","End":"12:24.565","Text":"and this is the area S. We get that S is equal to, by the formula,"},{"Start":"12:24.565 ","End":"12:30.685","Text":"the integral from 0-2Pi or 0-360, you can think of it,"},{"Start":"12:30.685 ","End":"12:34.330","Text":"of g of t. Now,"},{"Start":"12:34.330 ","End":"12:37.990","Text":"this is the g and this is the f. I will just make a note of that."},{"Start":"12:37.990 ","End":"12:40.855","Text":"This is our g, this is our f,"},{"Start":"12:40.855 ","End":"12:49.810","Text":"so g of t is 10 times 1 minus cosine Theta,"},{"Start":"12:49.810 ","End":"12:54.490","Text":"and we\u0027re replacing t by Theta in"},{"Start":"12:54.490 ","End":"13:00.595","Text":"the formula everywhere so as not to get too used to t, like I said,"},{"Start":"13:00.595 ","End":"13:04.885","Text":"and then we have times f prime of t,"},{"Start":"13:04.885 ","End":"13:11.755","Text":"so f prime is the derivative of this,"},{"Start":"13:11.755 ","End":"13:14.140","Text":"so it\u0027s 10 times."},{"Start":"13:14.140 ","End":"13:18.780","Text":"But the derivative with respect to Theta is,"},{"Start":"13:18.780 ","End":"13:21.015","Text":"10 is a constant, so it stays,"},{"Start":"13:21.015 ","End":"13:25.470","Text":"derivative of this is 1, derivative of sine is cosine."},{"Start":"13:25.470 ","End":"13:30.850","Text":"We get cosine Theta d Theta."},{"Start":"13:31.230 ","End":"13:33.880","Text":"If I rewrite this,"},{"Start":"13:33.880 ","End":"13:36.625","Text":"I take 10 times 10 is 100,"},{"Start":"13:36.625 ","End":"13:40.870","Text":"outside the brackets times the integral from"},{"Start":"13:40.870 ","End":"13:48.505","Text":"0-2Pi of 1 minus cosine Theta squared d Theta."},{"Start":"13:48.505 ","End":"13:52.525","Text":"Now, all we have to do is compute an integral and we get the answer."},{"Start":"13:52.525 ","End":"13:55.255","Text":"Now I\u0027ll show you how to do this integral."},{"Start":"13:55.255 ","End":"13:57.760","Text":"Since that\u0027s all we have left to compute,"},{"Start":"13:57.760 ","End":"13:59.410","Text":"I don\u0027t need anything else,"},{"Start":"13:59.410 ","End":"14:02.050","Text":"and we\u0027ll just do the integral."},{"Start":"14:02.050 ","End":"14:07.570","Text":"I\u0027ll square this using the standard formula that"},{"Start":"14:07.570 ","End":"14:13.900","Text":"a minus b squared is a squared minus 2ab plus b squared."},{"Start":"14:13.900 ","End":"14:14.980","Text":"You should know this by heart,"},{"Start":"14:14.980 ","End":"14:16.435","Text":"but I\u0027m just reminding you,"},{"Start":"14:16.435 ","End":"14:25.210","Text":"so this is equal to 100 times the integral from"},{"Start":"14:25.210 ","End":"14:32.485","Text":"0-2Pi of 1 minus 2 cosine Theta"},{"Start":"14:32.485 ","End":"14:40.644","Text":"plus cosine squared Theta d Theta."},{"Start":"14:40.644 ","End":"14:48.370","Text":"Then we need a trigonometric formula that cosine squared Theta is equal"},{"Start":"14:48.370 ","End":"14:58.760","Text":"to a 1/2 plus 1/2 cosine 2 Theta, I believe."},{"Start":"14:59.580 ","End":"15:09.820","Text":"We get 100 times the integral from 0-2Pi of 1 plus the 1/2,"},{"Start":"15:09.820 ","End":"15:13.375","Text":"I don\u0027t need the 1, 3/2,"},{"Start":"15:13.375 ","End":"15:20.590","Text":"that\u0027s the 1 plus the 1/2 from here minus 2 cosine Theta"},{"Start":"15:20.590 ","End":"15:29.065","Text":"plus 1/2 cosine of 2 Theta d Theta."},{"Start":"15:29.065 ","End":"15:31.255","Text":"I think we can finally do the integral."},{"Start":"15:31.255 ","End":"15:33.549","Text":"This is equal to 100."},{"Start":"15:33.549 ","End":"15:38.110","Text":"Now let\u0027s see, the integral of 3 over 2,"},{"Start":"15:38.110 ","End":"15:41.935","Text":"it\u0027s a constant, so it\u0027s 3 over 2 Theta."},{"Start":"15:41.935 ","End":"15:46.060","Text":"The integral of cosine is sine and that\u0027s a constant,"},{"Start":"15:46.060 ","End":"15:49.990","Text":"so it\u0027s minus 2 sine Theta."},{"Start":"15:49.990 ","End":"15:57.150","Text":"The integral of cosine 2 Theta would be sine 2 Theta."},{"Start":"15:57.150 ","End":"15:59.810","Text":"But this is a linear function of Theta it\u0027s twice,"},{"Start":"15:59.810 ","End":"16:02.045","Text":"so we have to divide by the 2."},{"Start":"16:02.045 ","End":"16:04.820","Text":"It\u0027s actually 1/4."},{"Start":"16:04.820 ","End":"16:06.680","Text":"If you differentiate this,"},{"Start":"16:06.680 ","End":"16:09.800","Text":"the derivative of sine 2 Theta is not cosine 2 Theta,"},{"Start":"16:09.800 ","End":"16:11.885","Text":"it\u0027s that times 2."},{"Start":"16:11.885 ","End":"16:18.995","Text":"This is okay, and we need to evaluate it between 0 and 2Pi."},{"Start":"16:18.995 ","End":"16:24.200","Text":"What we get is 100."},{"Start":"16:24.200 ","End":"16:29.490","Text":"Now if I put in 2Pi, I get."},{"Start":"16:29.640 ","End":"16:34.480","Text":"First of all, I\u0027ll put in the 2Pi and then I\u0027ll put in the 0, and then we\u0027ll subtract."},{"Start":"16:34.480 ","End":"16:41.540","Text":"If I put in 2Pi, 3 over 2 times 2Pi is just 3Pi."},{"Start":"16:45.930 ","End":"16:50.950","Text":"Sine of 2Pi is the same as sine of 0,"},{"Start":"16:50.950 ","End":"16:56.855","Text":"it\u0027s 0, so I\u0027ll just put minus 0 just to show that I haven\u0027t forgotten."},{"Start":"16:56.855 ","End":"17:00.330","Text":"In fact, I\u0027ll write twice 0."},{"Start":"17:01.530 ","End":"17:07.670","Text":"Now, sine of 4Pi is also 0."},{"Start":"17:07.670 ","End":"17:11.360","Text":"The sine of any multiple of Pi is 0,"},{"Start":"17:11.360 ","End":"17:15.125","Text":"so it\u0027s a 1/4 of 0 minus,"},{"Start":"17:15.125 ","End":"17:17.165","Text":"and now I plug in the 0."},{"Start":"17:17.165 ","End":"17:26.945","Text":"Here I get 0. Sine of 0 is also 0 minus twice 0."},{"Start":"17:26.945 ","End":"17:32.315","Text":"Sine of twice 0 is also 0, and so altogether,"},{"Start":"17:32.315 ","End":"17:38.980","Text":"what we\u0027re left with is just everything 0 except the 3Pi."},{"Start":"17:38.980 ","End":"17:43.545","Text":"We have 300 times Pi,"},{"Start":"17:43.545 ","End":"17:46.175","Text":"which is 900 and something."},{"Start":"17:46.175 ","End":"17:48.125","Text":"I\u0027ll leave it like that,"},{"Start":"17:48.125 ","End":"17:52.110","Text":"and we are done."}],"ID":6005}],"Thumbnail":null,"ID":4003},{"Name":"Surface Area with Parametric Equations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Surface Area with Parametric Equations","Duration":"20m 3s","ChapterTopicVideoID":5992,"CourseChapterTopicPlaylistID":4004,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.870","Text":"In this clip, we\u0027re going to talk about surface area,"},{"Start":"00:05.180 ","End":"00:09.915","Text":"which is going to be a surface area of revolution or rotation,"},{"Start":"00:09.915 ","End":"00:12.690","Text":"but using parametric equations."},{"Start":"00:12.690 ","End":"00:16.140","Text":"Now, we\u0027ve already done this topic,"},{"Start":"00:16.140 ","End":"00:18.030","Text":"but not with parametric equations,"},{"Start":"00:18.030 ","End":"00:20.865","Text":"and I\u0027d like to do a bit of recycling here."},{"Start":"00:20.865 ","End":"00:24.900","Text":"I went and copy pasted the diagrams from"},{"Start":"00:24.900 ","End":"00:33.565","Text":"the explicit forms when y is a function of x or x is a function of y,"},{"Start":"00:33.565 ","End":"00:35.855","Text":"and in one case we rotated about"},{"Start":"00:35.855 ","End":"00:39.740","Text":"the x-axis and then the other case we rotated about the y-axis."},{"Start":"00:39.740 ","End":"00:45.360","Text":"I\u0027ll keep the diagrams and we\u0027ll just modify the formulas for the parametric form."},{"Start":"00:45.790 ","End":"00:54.340","Text":"We\u0027re going to have x equals some function of t and y equals some function of t,"},{"Start":"00:54.340 ","End":"00:56.630","Text":"and not to confuse the letters,"},{"Start":"00:56.630 ","End":"01:01.745","Text":"I\u0027m going to reuse f and g. I\u0027ll just erase this,"},{"Start":"01:01.745 ","End":"01:03.755","Text":"and I\u0027ll erase this."},{"Start":"01:03.755 ","End":"01:06.050","Text":"In both these cases,"},{"Start":"01:06.050 ","End":"01:11.190","Text":"we\u0027ll have x equals f of t,"},{"Start":"01:11.190 ","End":"01:19.480","Text":"y is going to equal g of t. We\u0027re going to assume that we traverse,"},{"Start":"01:19.480 ","End":"01:26.200","Text":"we go from a to b and we go from c to d in the other case when we"},{"Start":"01:26.200 ","End":"01:34.450","Text":"substitute Alpha or Beta for t. In this case,"},{"Start":"01:34.450 ","End":"01:40.560","Text":"we\u0027re going to have that f of Alpha is"},{"Start":"01:40.560 ","End":"01:47.050","Text":"a and f of Beta is b so we go in this direction."},{"Start":"01:47.050 ","End":"01:54.135","Text":"In this case, when we revolve around the y-axis,"},{"Start":"01:54.135 ","End":"02:00.785","Text":"y is going to move so that we are going to have that y of Alpha is c and"},{"Start":"02:00.785 ","End":"02:09.000","Text":"y of Beta is d. Now we need to modify these formulas."},{"Start":"02:09.730 ","End":"02:13.010","Text":"They\u0027re going to have the general same shape,"},{"Start":"02:13.010 ","End":"02:17.870","Text":"it\u0027s still going to be 2Pi times some integral of something"},{"Start":"02:17.870 ","End":"02:24.070","Text":"times the square root of d something."},{"Start":"02:24.070 ","End":"02:27.440","Text":"But even these limits are going to change."},{"Start":"02:27.440 ","End":"02:33.625","Text":"What we will get is everything\u0027s going to be a function of t,"},{"Start":"02:33.625 ","End":"02:35.700","Text":"and instead of from a to b,"},{"Start":"02:35.700 ","End":"02:38.695","Text":"we\u0027ll go from Alpha to Beta."},{"Start":"02:38.695 ","End":"02:43.250","Text":"Here, the function is going to be the y,"},{"Start":"02:43.250 ","End":"02:48.680","Text":"which is g of t. Inside in both cases, well,"},{"Start":"02:48.680 ","End":"02:50.030","Text":"you\u0027ll see that in a moment,"},{"Start":"02:50.030 ","End":"03:00.725","Text":"we\u0027re going to get f prime of t squared plus g prime of t also squared."},{"Start":"03:00.725 ","End":"03:05.120","Text":"I suppose I should really put some extra brackets here,"},{"Start":"03:05.120 ","End":"03:09.440","Text":"because this whole thing is squared and around here too,"},{"Start":"03:09.440 ","End":"03:11.585","Text":"just to be precise."},{"Start":"03:11.585 ","End":"03:13.970","Text":"Now the second formula,"},{"Start":"03:13.970 ","End":"03:16.900","Text":"and let\u0027s erase the old."},{"Start":"03:16.900 ","End":"03:21.180","Text":"Don\u0027t need this, don\u0027t need this, nor this,"},{"Start":"03:21.180 ","End":"03:26.520","Text":"nor this, we still need the 2Pi something times the square root of something."},{"Start":"03:26.860 ","End":"03:31.010","Text":"What we get this time is as before,"},{"Start":"03:31.010 ","End":"03:32.795","Text":"we\u0027re going to have a dt here,"},{"Start":"03:32.795 ","End":"03:36.305","Text":"and t is still going to go from Alpha to Beta."},{"Start":"03:36.305 ","End":"03:42.530","Text":"But this time we\u0027re going to have f of t here."},{"Start":"03:42.530 ","End":"03:44.780","Text":"The 2Pi is still there, yeah."},{"Start":"03:44.780 ","End":"03:48.560","Text":"Here we\u0027re going to get the same thing as here."},{"Start":"03:48.560 ","End":"03:53.480","Text":"We\u0027re going to get f prime of t"},{"Start":"03:53.480 ","End":"04:00.325","Text":"squared plus g prime of t squared."},{"Start":"04:00.325 ","End":"04:03.050","Text":"As before, I\u0027d better put the extra brackets"},{"Start":"04:03.050 ","End":"04:06.410","Text":"in to avoid confusion as to what is being squared,"},{"Start":"04:06.410 ","End":"04:10.235","Text":"the whole thing. That\u0027s basically it."},{"Start":"04:10.235 ","End":"04:12.439","Text":"Let\u0027s just do an example."},{"Start":"04:12.439 ","End":"04:14.380","Text":"These are just dry formulas,"},{"Start":"04:14.380 ","End":"04:18.470","Text":"we\u0027ll just take an example of each and solve it."},{"Start":"04:18.470 ","End":"04:20.810","Text":"For my first example,"},{"Start":"04:20.810 ","End":"04:23.330","Text":"we\u0027ll do a rotation about the x-axis."},{"Start":"04:23.330 ","End":"04:29.210","Text":"Let\u0027s compute the general formula for the surface area of a sphere."},{"Start":"04:29.210 ","End":"04:31.855","Text":"Sounds very challenging."},{"Start":"04:31.855 ","End":"04:34.595","Text":"Now, I\u0027ve drawn a semicircle here,"},{"Start":"04:34.595 ","End":"04:35.630","Text":"because if you think about it,"},{"Start":"04:35.630 ","End":"04:45.470","Text":"a sphere is just what you get if you rotate a semicircle around the x-axis."},{"Start":"04:45.470 ","End":"04:48.245","Text":"I mean, you get the mirror image of it here,"},{"Start":"04:48.245 ","End":"04:51.080","Text":"and you can imagine it."},{"Start":"04:51.080 ","End":"04:54.770","Text":"A parametric equation for a circle."},{"Start":"04:54.770 ","End":"04:58.085","Text":"Well, let\u0027s say the radius is r so this is r here."},{"Start":"04:58.085 ","End":"05:05.045","Text":"This is the point minus r and if we take the angle as the parameter t,"},{"Start":"05:05.045 ","End":"05:08.185","Text":"then we get the parametric equation"},{"Start":"05:08.185 ","End":"05:17.185","Text":"that x equals r cosine t,"},{"Start":"05:17.185 ","End":"05:22.350","Text":"and y equals r sine t,"},{"Start":"05:22.350 ","End":"05:25.955","Text":"and t, the angle goes from 0 to Pi."},{"Start":"05:25.955 ","End":"05:29.125","Text":"So 0 less than or equal to t,"},{"Start":"05:29.125 ","End":"05:31.120","Text":"less than or equal to Pi,"},{"Start":"05:31.120 ","End":"05:34.760","Text":"this is our Alpha and this is our Beta."},{"Start":"05:35.900 ","End":"05:42.100","Text":"Let\u0027s just still use the formula here and see what we get."},{"Start":"05:42.100 ","End":"05:46.680","Text":"We get that, we\u0027ll do it here,"},{"Start":"05:46.680 ","End":"05:53.190","Text":"s is equal to 2Pi times the integral,"},{"Start":"05:53.190 ","End":"05:56.679","Text":"Alpha and Beta are 0 and Pi,"},{"Start":"05:57.050 ","End":"06:05.160","Text":"of g is the function that y is of t so g"},{"Start":"06:05.160 ","End":"06:13.695","Text":"is r sine t. Now we have a square root."},{"Start":"06:13.695 ","End":"06:20.710","Text":"Now, f prime of t is this derivative,"},{"Start":"06:20.710 ","End":"06:23.920","Text":"so that\u0027s minus r"},{"Start":"06:23.920 ","End":"06:32.050","Text":"sine t squared plus the g prime of t,"},{"Start":"06:32.050 ","End":"06:39.860","Text":"which is r cosine t squared and then dt."},{"Start":"06:40.890 ","End":"06:43.660","Text":"What we get is,"},{"Start":"06:43.660 ","End":"06:46.750","Text":"look, when we square this,"},{"Start":"06:46.750 ","End":"06:50.290","Text":"we\u0027re going to get plus r squared sine squared t,"},{"Start":"06:50.290 ","End":"06:54.700","Text":"and here r squared cosine squared t. You know what?"},{"Start":"06:54.700 ","End":"06:57.380","Text":"I\u0027m just going to work on this part of the slide."},{"Start":"06:57.380 ","End":"06:59.905","Text":"Like I said, under the square root sign,"},{"Start":"06:59.905 ","End":"07:09.730","Text":"we get r squared sine squared t plus r squared cosine squared t,"},{"Start":"07:09.730 ","End":"07:15.955","Text":"which is equal to r squared times sine"},{"Start":"07:15.955 ","End":"07:22.650","Text":"squared t plus cosine squared t. But this is equal to 1 by a trigonometric identity,"},{"Start":"07:22.650 ","End":"07:24.325","Text":"which is r squared."},{"Start":"07:24.325 ","End":"07:25.820","Text":"When I take the square root,"},{"Start":"07:25.820 ","End":"07:26.870","Text":"since r is positive,"},{"Start":"07:26.870 ","End":"07:33.030","Text":"it\u0027s just r. We get 2Pi,"},{"Start":"07:33.030 ","End":"07:35.059","Text":"now this r is a constant,"},{"Start":"07:35.059 ","End":"07:36.815","Text":"it\u0027s not dependent on t,"},{"Start":"07:36.815 ","End":"07:44.560","Text":"r. Then we get the square root of r-squared is also r. We get another r here,"},{"Start":"07:44.560 ","End":"07:54.450","Text":"so it\u0027s r squared and all we\u0027re left with is the integral from 0 to Pi of sine t dt."},{"Start":"07:54.730 ","End":"08:02.180","Text":"Maybe I should have mentioned here that the square root of r-squared is r. In general,"},{"Start":"08:02.180 ","End":"08:05.510","Text":"the square root of something squared is the absolute value, but it\u0027s positive,"},{"Start":"08:05.510 ","End":"08:11.040","Text":"so it\u0027s r. Continuing,"},{"Start":"08:11.040 ","End":"08:16.740","Text":"so we get that this is 2Pi r squared and"},{"Start":"08:16.740 ","End":"08:23.924","Text":"now the integral of sine t is minus cosine t,"},{"Start":"08:23.924 ","End":"08:31.500","Text":"and we have to take this between 0 and Pi."},{"Start":"08:32.250 ","End":"08:37.300","Text":"I don\u0027t need this anymore."},{"Start":"08:37.300 ","End":"08:40.000","Text":"We can continue."},{"Start":"08:40.000 ","End":"08:43.735","Text":"I get 2Pir squared."},{"Start":"08:43.735 ","End":"08:45.970","Text":"When I put in Pi,"},{"Start":"08:45.970 ","End":"08:50.695","Text":"I get minus cosine of Pi."},{"Start":"08:50.695 ","End":"08:52.390","Text":"Then when I put in 0,"},{"Start":"08:52.390 ","End":"08:55.360","Text":"I get minus cosine 0 because I subtract,"},{"Start":"08:55.360 ","End":"08:59.750","Text":"it becomes plus cosine 0."},{"Start":"09:00.510 ","End":"09:06.910","Text":"Cosine of Pi, this is minus 1,"},{"Start":"09:06.910 ","End":"09:09.070","Text":"and cosine of 0 is 1."},{"Start":"09:09.070 ","End":"09:11.215","Text":"So it\u0027s minus, minus 1,"},{"Start":"09:11.215 ","End":"09:13.330","Text":"plus 1, which is 2."},{"Start":"09:13.330 ","End":"09:16.810","Text":"It comes out because 1 plus 1 is 2,"},{"Start":"09:16.810 ","End":"09:22.705","Text":"that this is 4Pir squared."},{"Start":"09:22.705 ","End":"09:25.210","Text":"I\u0027ll highlight this."},{"Start":"09:25.210 ","End":"09:29.860","Text":"Actually, this is the famous formula for the surface of"},{"Start":"09:29.860 ","End":"09:36.205","Text":"a sphere with radius r and is the correct answer, 4Pir squared."},{"Start":"09:36.205 ","End":"09:40.660","Text":"I\u0027m going to erase this example. All clear."},{"Start":"09:40.660 ","End":"09:45.775","Text":"Now I\u0027ll do an example of 1 of these where we rotate around the y-axis."},{"Start":"09:45.775 ","End":"09:48.970","Text":"Here\u0027s some axes in case I need a sketch."},{"Start":"09:48.970 ","End":"09:51.790","Text":"Let\u0027s take a more abstract example."},{"Start":"09:51.790 ","End":"09:56.695","Text":"This time, x is equal to,"},{"Start":"09:56.695 ","End":"10:01.675","Text":"let\u0027s see, cosine cubed."},{"Start":"10:01.675 ","End":"10:04.825","Text":"You know what? I\u0027ll make the angle Theta instead of"},{"Start":"10:04.825 ","End":"10:11.320","Text":"t. We\u0027ll use the parameter Theta just to get you used to it."},{"Start":"10:11.320 ","End":"10:16.615","Text":"Y is going to equal sine cubed Theta,"},{"Start":"10:16.615 ","End":"10:24.910","Text":"and Theta will go from 0-90 degrees,"},{"Start":"10:24.910 ","End":"10:27.950","Text":"which is Pi over 2."},{"Start":"10:28.110 ","End":"10:30.955","Text":"This does have a shape."},{"Start":"10:30.955 ","End":"10:34.300","Text":"Well, for 1 thing, I can compute the start and end."},{"Start":"10:34.300 ","End":"10:40.540","Text":"When Theta is 0, then cosine Theta is 1, and sine Theta 0."},{"Start":"10:40.540 ","End":"10:43.555","Text":"I get the point 1,0."},{"Start":"10:43.555 ","End":"10:46.150","Text":"I know it starts here."},{"Start":"10:46.150 ","End":"10:48.280","Text":"When it\u0027s Pi over 2,"},{"Start":"10:48.280 ","End":"10:54.470","Text":"x is, let\u0027s see."},{"Start":"10:54.470 ","End":"10:56.970","Text":"Cosine of Pi over 2 is 0,"},{"Start":"10:56.970 ","End":"10:59.715","Text":"sine of Pi over 2 is 1."},{"Start":"10:59.715 ","End":"11:02.550","Text":"When I cube them, it\u0027s still 0,1,"},{"Start":"11:02.550 ","End":"11:05.590","Text":"so it ends up here."},{"Start":"11:05.590 ","End":"11:08.320","Text":"I think I\u0027ve encountered this somewhere."},{"Start":"11:08.320 ","End":"11:13.045","Text":"It looks something like this."},{"Start":"11:13.045 ","End":"11:14.680","Text":"Not exactly sure."},{"Start":"11:14.680 ","End":"11:20.860","Text":"Anyway, we\u0027re going to rotate or revolve around the y-axis,"},{"Start":"11:20.860 ","End":"11:27.865","Text":"so we get a surface because it\u0027s got rings."},{"Start":"11:27.865 ","End":"11:30.130","Text":"Those rings we didn\u0027t hope."},{"Start":"11:30.130 ","End":"11:31.360","Text":"Anyway, get the idea."},{"Start":"11:31.360 ","End":"11:37.930","Text":"Take this and get a surface by revolving this line around the y-axis."},{"Start":"11:37.930 ","End":"11:41.455","Text":"Let\u0027s just do it using the formula."},{"Start":"11:41.455 ","End":"11:45.040","Text":"This is my f, well,"},{"Start":"11:45.040 ","End":"11:47.110","Text":"it\u0027s f of not t,"},{"Start":"11:47.110 ","End":"11:48.625","Text":"but f of Theta,"},{"Start":"11:48.625 ","End":"11:51.920","Text":"and this is my g of Theta."},{"Start":"11:52.170 ","End":"11:56.140","Text":"Instead of t, I\u0027ll take Theta everywhere in the formula,"},{"Start":"11:56.140 ","End":"11:58.570","Text":"but other than that same idea."},{"Start":"11:58.570 ","End":"11:59.740","Text":"I\u0027m using this formula,"},{"Start":"11:59.740 ","End":"12:02.200","Text":"but I\u0027m just starting over here."},{"Start":"12:02.200 ","End":"12:05.395","Text":"This is the formula I\u0027m using."},{"Start":"12:05.395 ","End":"12:12.565","Text":"S is equal to 2Pi times the integral,"},{"Start":"12:12.565 ","End":"12:16.850","Text":"Alpha and Beta are 0 and Pi over 2."},{"Start":"12:17.070 ","End":"12:19.960","Text":"By the way, the arrow does go this way,"},{"Start":"12:19.960 ","End":"12:23.515","Text":"but that doesn\u0027t really matter."},{"Start":"12:23.515 ","End":"12:29.965","Text":"From 0 to Pi over 2 of f of t,"},{"Start":"12:29.965 ","End":"12:32.200","Text":"which is f of Theta really."},{"Start":"12:32.200 ","End":"12:36.649","Text":"That\u0027s cosine cube Theta"},{"Start":"12:37.440 ","End":"12:42.655","Text":"times the square root of,"},{"Start":"12:42.655 ","End":"12:46.450","Text":"let\u0027s see, what is f prime?"},{"Start":"12:46.450 ","End":"12:49.480","Text":"The derivative of cosine cubed,"},{"Start":"12:49.480 ","End":"12:51.235","Text":"it\u0027s using the chain rule,"},{"Start":"12:51.235 ","End":"12:56.710","Text":"is 3 cosine squared Theta."},{"Start":"12:56.710 ","End":"13:03.550","Text":"But the inner derivative of cosine Theta is minus sine Theta,"},{"Start":"13:03.550 ","End":"13:06.880","Text":"and all this is squared."},{"Start":"13:06.880 ","End":"13:09.860","Text":"I\u0027m going to need a bit more room."},{"Start":"13:09.900 ","End":"13:14.905","Text":"Now g prime of Theta squared,"},{"Start":"13:14.905 ","End":"13:18.260","Text":"sorry, from here, it\u0027s the same thing."},{"Start":"13:18.480 ","End":"13:25.990","Text":"G prime of Theta is, let\u0027s see,"},{"Start":"13:25.990 ","End":"13:36.175","Text":"3 sine squared Theta times inner derivative is just cosine Theta,"},{"Start":"13:36.175 ","End":"13:37.930","Text":"and this is squared."},{"Start":"13:37.930 ","End":"13:42.905","Text":"Well, it\u0027s a long expression and not dt but d Theta."},{"Start":"13:42.905 ","End":"13:46.000","Text":"Let\u0027s see what we get."},{"Start":"13:47.580 ","End":"13:51.440","Text":"Need to do some trigonometry here."},{"Start":"13:52.590 ","End":"14:02.245","Text":"We get 2Pi times the integral from 0 to Pi over 2, cosine cubed Theta."},{"Start":"14:02.245 ","End":"14:04.390","Text":"What do we get inside here?"},{"Start":"14:04.390 ","End":"14:07.930","Text":"We get the square root of,"},{"Start":"14:07.930 ","End":"14:13.165","Text":"now, 3 squared is 9."},{"Start":"14:13.165 ","End":"14:16.840","Text":"We get 9, and here we\u0027re also going to get 9."},{"Start":"14:16.840 ","End":"14:19.165","Text":"Well, I\u0027ll just leave it for the time being."},{"Start":"14:19.165 ","End":"14:21.850","Text":"Then we\u0027re going to get cosine to"},{"Start":"14:21.850 ","End":"14:33.055","Text":"the 4th Theta sine squared Theta."},{"Start":"14:33.055 ","End":"14:38.380","Text":"Here we\u0027re going to get 9, the other way around,"},{"Start":"14:38.380 ","End":"14:44.770","Text":"sine to the 4th Theta cosine"},{"Start":"14:44.770 ","End":"14:49.495","Text":"squared Theta d Theta."},{"Start":"14:49.495 ","End":"14:51.730","Text":"I\u0027m going to scroll down."},{"Start":"14:51.730 ","End":"14:54.220","Text":"I don\u0027t really need any of these formulas anymore."},{"Start":"14:54.220 ","End":"14:57.790","Text":"We just have to finish evaluating this integral and we\u0027re done."},{"Start":"14:57.790 ","End":"15:03.790","Text":"Now, what I can do here is I can take cosine squared sine squared."},{"Start":"15:03.790 ","End":"15:08.215","Text":"In fact, I can take 9 cosine squared sine squared outside the brackets."},{"Start":"15:08.215 ","End":"15:17.350","Text":"What I get is 2Pi integral from 0 to Pi over 2 cosine cubed Theta,"},{"Start":"15:17.350 ","End":"15:18.715","Text":"so far the same."},{"Start":"15:18.715 ","End":"15:22.825","Text":"Then I can take the square root of 2 bits separately."},{"Start":"15:22.825 ","End":"15:32.545","Text":"I can take 9 cosine squared Theta sine squared Theta outside the brackets."},{"Start":"15:32.545 ","End":"15:35.470","Text":"After I take it outside the brackets,"},{"Start":"15:35.470 ","End":"15:37.960","Text":"and I can split it up into a product of 2 things,"},{"Start":"15:37.960 ","End":"15:42.040","Text":"what I\u0027m left with here is cosine squared,"},{"Start":"15:42.040 ","End":"15:49.960","Text":"and here I\u0027m left with sine squared d Theta."},{"Start":"15:49.960 ","End":"15:54.040","Text":"Now look, cosine squared plus sine squared equals 1."},{"Start":"15:54.040 ","End":"15:56.425","Text":"This disappears."},{"Start":"15:56.425 ","End":"16:07.375","Text":"What I get is 2Pi integral from 0 to Pi over 2 cosine cubed Theta."},{"Start":"16:07.375 ","End":"16:09.790","Text":"This thing is 1."},{"Start":"16:09.790 ","End":"16:13.390","Text":"This thing is"},{"Start":"16:13.390 ","End":"16:20.905","Text":"3 cosine Theta sine Theta."},{"Start":"16:20.905 ","End":"16:24.385","Text":"Normally, I should put it in absolute value."},{"Start":"16:24.385 ","End":"16:29.890","Text":"But because Theta is between 0 and Pi over 2 and the sine and the cosine,"},{"Start":"16:29.890 ","End":"16:34.585","Text":"everything\u0027s positive there, I don\u0027t need the absolute value."},{"Start":"16:34.585 ","End":"16:42.445","Text":"Altogether, I can put the 3 here in front, 3 times 2,"},{"Start":"16:42.445 ","End":"16:52.945","Text":"and here I can put another cosine Theta sine Theta d Theta."},{"Start":"16:52.945 ","End":"16:55.060","Text":"What do we end up with?"},{"Start":"16:55.060 ","End":"16:57.520","Text":"Still close to the end now."},{"Start":"16:57.520 ","End":"17:04.075","Text":"We\u0027ve got 6Pi times the integral from 0 to Pi over 2,"},{"Start":"17:04.075 ","End":"17:13.425","Text":"cosine to the 4th Theta times sine Theta d Theta."},{"Start":"17:13.425 ","End":"17:16.340","Text":"This is more an integration problem than anything else."},{"Start":"17:16.340 ","End":"17:19.310","Text":"What we\u0027ll do here is we\u0027ll make a substitution."},{"Start":"17:19.310 ","End":"17:23.600","Text":"I\u0027ll substitute cosine Theta as some letter because I"},{"Start":"17:23.600 ","End":"17:28.140","Text":"already have the derivative. You know what?"},{"Start":"17:28.140 ","End":"17:30.590","Text":"We can do it without substitution."},{"Start":"17:30.590 ","End":"17:39.365","Text":"If we just noticed that the derivative of cosine Theta is minus sine Theta,"},{"Start":"17:39.365 ","End":"17:42.680","Text":"I can put a minus here."},{"Start":"17:42.680 ","End":"17:44.335","Text":"Let me just say this."},{"Start":"17:44.335 ","End":"17:46.685","Text":"It\u0027s minus 6Pi."},{"Start":"17:46.685 ","End":"17:52.120","Text":"The integral from 0 to Pi over 2 of cosine to"},{"Start":"17:52.120 ","End":"17:58.165","Text":"the 4th Theta times minus sine Theta d Theta."},{"Start":"17:58.165 ","End":"18:01.640","Text":"Because I have the derivative of cosine outside,"},{"Start":"18:01.640 ","End":"18:04.790","Text":"you could do it with a substitution."},{"Start":"18:04.790 ","End":"18:06.880","Text":"You could do it in 1 step."},{"Start":"18:06.880 ","End":"18:08.110","Text":"Let me think. You know what?"},{"Start":"18:08.110 ","End":"18:09.820","Text":"I\u0027ll do it the longer way."},{"Start":"18:09.820 ","End":"18:12.415","Text":"We\u0027ll do it with a substitution."},{"Start":"18:12.415 ","End":"18:14.915","Text":"I\u0027m going to substitute in here,"},{"Start":"18:14.915 ","End":"18:21.150","Text":"let\u0027s say u is equal to cosine Theta"},{"Start":"18:21.150 ","End":"18:27.890","Text":"and then du is equal to the derivative of this,"},{"Start":"18:27.890 ","End":"18:33.190","Text":"which is minus sine Theta d Theta."},{"Start":"18:33.190 ","End":"18:35.960","Text":"After I substitute this,"},{"Start":"18:35.960 ","End":"18:42.585","Text":"I get minus 6Pi integral"},{"Start":"18:42.585 ","End":"18:52.185","Text":"of cosine to the 4th Theta is u to the 4th minus sine Theta d Theta is du."},{"Start":"18:52.185 ","End":"18:57.320","Text":"But we also have to substitute the limits."},{"Start":"18:57.320 ","End":"19:04.155","Text":"When Theta is 0, cosine Theta is 1,"},{"Start":"19:04.155 ","End":"19:07.550","Text":"and when Theta is 90 degrees or Pi over 2,"},{"Start":"19:07.550 ","End":"19:10.760","Text":"the cosine is 0."},{"Start":"19:10.760 ","End":"19:13.250","Text":"What we do, in this case,"},{"Start":"19:13.250 ","End":"19:16.770","Text":"I can get rid of the minus if I reverse top and bottom."},{"Start":"19:16.770 ","End":"19:24.555","Text":"It\u0027s 6Pi times the integral from 0-1 of u to the 4th du."},{"Start":"19:24.555 ","End":"19:27.760","Text":"This is equal to"},{"Start":"19:29.480 ","End":"19:37.870","Text":"6Pi times u to the 5th over 5."},{"Start":"19:37.870 ","End":"19:44.450","Text":"All this taken between 0 and 1."},{"Start":"19:45.000 ","End":"19:48.370","Text":"When u is 0, I just get 0."},{"Start":"19:48.370 ","End":"19:51.280","Text":"When u is 1, I get a 5th."},{"Start":"19:51.280 ","End":"19:59.245","Text":"Altogether I just get 6Pi over 5,"},{"Start":"19:59.245 ","End":"20:03.920","Text":"and that is the answer. We are done."}],"ID":6006}],"Thumbnail":null,"ID":4004},{"Name":"Curve Length","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Curve Length","Duration":"3m 29s","ChapterTopicVideoID":8449,"CourseChapterTopicPlaylistID":4907,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.605","Text":"Okay, now we come to part 3 of length of a curve."},{"Start":"00:04.605 ","End":"00:08.430","Text":"This time, the curve is given in parametric form."},{"Start":"00:08.430 ","End":"00:13.850","Text":"Both x and y are given as functions of a parameter t. Often but not always,"},{"Start":"00:13.850 ","End":"00:18.069","Text":"this is expressed as a physical problem where t is time."},{"Start":"00:18.069 ","End":"00:25.040","Text":"The quantities that are important here are a function of t that defines x,"},{"Start":"00:25.040 ","End":"00:27.425","Text":"another function y of t,"},{"Start":"00:27.425 ","End":"00:31.690","Text":"and 2 constants, time a and time b,"},{"Start":"00:31.690 ","End":"00:35.010","Text":"that when we plug them in to x of t,"},{"Start":"00:35.010 ","End":"00:37.110","Text":"they will give me a and b."},{"Start":"00:37.110 ","End":"00:43.535","Text":"So t_a actually corresponds to a and t_b corresponds to b and we have a formula again."},{"Start":"00:43.535 ","End":"00:45.650","Text":"This time it\u0027s a little bit different."},{"Start":"00:45.650 ","End":"00:49.880","Text":"There\u0027s no 1 plus l is l from a to b."},{"Start":"00:49.880 ","End":"00:51.335","Text":"You could write that here too."},{"Start":"00:51.335 ","End":"00:53.900","Text":"We need to know the derivative of x,"},{"Start":"00:53.900 ","End":"00:55.730","Text":"the derivative of y with respect to t,"},{"Start":"00:55.730 ","End":"00:57.155","Text":"plug into the formula,"},{"Start":"00:57.155 ","End":"01:00.065","Text":"substitute the limits and we\u0027ll get the answer."},{"Start":"01:00.065 ","End":"01:02.885","Text":"Best thing to do is to give an example."},{"Start":"01:02.885 ","End":"01:06.635","Text":"We have to find the length of the parameterized curve."},{"Start":"01:06.635 ","End":"01:10.190","Text":"This time I give you the functions x of t,"},{"Start":"01:10.190 ","End":"01:13.880","Text":"which in this case is cosine t. The other function which gives y,"},{"Start":"01:13.880 ","End":"01:21.110","Text":"which is sine t. We have the t_a and the t_b which are 0 and 2 Pi."},{"Start":"01:21.110 ","End":"01:23.270","Text":"Now the formula\u0027s still up here,"},{"Start":"01:23.270 ","End":"01:26.015","Text":"but let\u0027s just write the derivatives."},{"Start":"01:26.015 ","End":"01:32.165","Text":"The x prime with respect to t is minus sine t"},{"Start":"01:32.165 ","End":"01:39.275","Text":"and y prime is cosine t. If I substitute in here,"},{"Start":"01:39.275 ","End":"01:45.950","Text":"I will get that the length of curve is equal to the integral from 0 to"},{"Start":"01:45.950 ","End":"01:53.895","Text":"2 Pi of the derivative with respect to x squared is sine squared t. Yeah,"},{"Start":"01:53.895 ","End":"01:56.235","Text":"there\u0027s a square root also."},{"Start":"01:56.235 ","End":"02:02.090","Text":"Y prime squared will be cosine squared t dt."},{"Start":"02:02.090 ","End":"02:05.990","Text":"If you remember your trigonometrical identities,"},{"Start":"02:05.990 ","End":"02:10.390","Text":"you\u0027ll recall that sine squared t plus cosine squared t is 1."},{"Start":"02:10.390 ","End":"02:16.670","Text":"This comes out to be the integral from 0 to 2 Pi of 1 dt."},{"Start":"02:16.670 ","End":"02:22.410","Text":"The integral of 1 is just t from 0 to 2 Pi."},{"Start":"02:22.410 ","End":"02:25.350","Text":"That makes it 2 Pi minus 0."},{"Start":"02:25.350 ","End":"02:28.395","Text":"The answer is 2 Pi."},{"Start":"02:28.395 ","End":"02:31.120","Text":"We\u0027re done with part 3."},{"Start":"02:31.120 ","End":"02:35.150","Text":"As for part 4, I\u0027ll mention it just very briefly."},{"Start":"02:35.150 ","End":"02:38.810","Text":"I wasn\u0027t really going to talk about implicit differentiation,"},{"Start":"02:38.810 ","End":"02:40.400","Text":"but I\u0027ll just say a few words."},{"Start":"02:40.400 ","End":"02:43.280","Text":"When we have f of x and y equals 0,"},{"Start":"02:43.280 ","End":"02:45.534","Text":"that\u0027s an implicit function."},{"Start":"02:45.534 ","End":"02:50.120","Text":"If we have a length of curve problem involving this,"},{"Start":"02:50.120 ","End":"02:52.490","Text":"there\u0027s really several things you could do."},{"Start":"02:52.490 ","End":"02:53.780","Text":"In the problem itself,"},{"Start":"02:53.780 ","End":"02:59.300","Text":"you\u0027ll see you could isolate y in terms of x if that\u0027s possible or you"},{"Start":"02:59.300 ","End":"03:05.735","Text":"could isolate x in terms of y or you could do implicit differentiation."},{"Start":"03:05.735 ","End":"03:06.920","Text":"I\u0027m not going to go into this."},{"Start":"03:06.920 ","End":"03:09.800","Text":"At least 1 of the exercises contains an implicit function"},{"Start":"03:09.800 ","End":"03:12.860","Text":"and an implicit differentiation and you\u0027ll see it there."},{"Start":"03:12.860 ","End":"03:16.805","Text":"It\u0027s not that frequent that you get asked this kind of question anyway."},{"Start":"03:16.805 ","End":"03:19.175","Text":"Having said a few words about part 4,"},{"Start":"03:19.175 ","End":"03:23.870","Text":"we\u0027re done with the subject of length of curve via integration."},{"Start":"03:23.870 ","End":"03:30.150","Text":"I leave you to solve the exercises that follow this tutorial."}],"ID":8663},{"Watched":false,"Name":"Exercise 1","Duration":"4m 17s","ChapterTopicVideoID":8448,"CourseChapterTopicPlaylistID":4907,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"This exercise talks about the distance traveled by a particle."},{"Start":"00:04.965 ","End":"00:09.450","Text":"We\u0027re given its x and y coordinates as functions of time,"},{"Start":"00:09.450 ","End":"00:12.570","Text":"and also, we have the start time and the end time."},{"Start":"00:12.570 ","End":"00:19.080","Text":"This type of problem is actually a length of curve problem in the parametric dial."},{"Start":"00:19.080 ","End":"00:21.105","Text":"Basically what we need to know,"},{"Start":"00:21.105 ","End":"00:24.270","Text":"the x coordinate as a function of time,"},{"Start":"00:24.270 ","End":"00:27.180","Text":"the y coordinate to the function of time,"},{"Start":"00:27.180 ","End":"00:31.095","Text":"and also the start and end times and we have all these."},{"Start":"00:31.095 ","End":"00:33.930","Text":"This is our x, which is x of t, this is y,"},{"Start":"00:33.930 ","End":"00:37.795","Text":"which is y of t. This is the start time,"},{"Start":"00:37.795 ","End":"00:40.310","Text":"which is called here a t_a."},{"Start":"00:40.310 ","End":"00:43.805","Text":"This is the end time which is called t_b."},{"Start":"00:43.805 ","End":"00:48.215","Text":"All we need to do is apply this formula here in red."},{"Start":"00:48.215 ","End":"00:51.580","Text":"I see that we need both derivatives."},{"Start":"00:51.580 ","End":"00:54.855","Text":"Let\u0027s start and differentiate."},{"Start":"00:54.855 ","End":"01:03.920","Text":"X prime is equal to cosine t has derivative minus sine t."},{"Start":"01:03.920 ","End":"01:07.820","Text":"Here we have a product, this times this,"},{"Start":"01:07.820 ","End":"01:09.815","Text":"So we use the product rule."},{"Start":"01:09.815 ","End":"01:13.190","Text":"It\u0027s the derivative of the first which is 1,"},{"Start":"01:13.190 ","End":"01:17.120","Text":"times the second which is sine t. I\u0027ll write it as 1 times"},{"Start":"01:17.120 ","End":"01:23.690","Text":"sine t. Then we have t as is and the derivative of sine t,"},{"Start":"01:23.690 ","End":"01:31.985","Text":"which is cosine t. Y prime is equal to, derivative of sine t,"},{"Start":"01:31.985 ","End":"01:37.505","Text":"which is cosine t. Now here we have a product minus,"},{"Start":"01:37.505 ","End":"01:41.240","Text":"first of all, differentiate the t which is 1,"},{"Start":"01:41.240 ","End":"01:44.400","Text":"and leave cosine t alone."},{"Start":"01:44.400 ","End":"01:46.020","Text":"In the second 1,"},{"Start":"01:46.020 ","End":"01:49.329","Text":"we leave the t as is and differentiate"},{"Start":"01:49.329 ","End":"01:52.715","Text":"the cosine to get minus sine."},{"Start":"01:52.715 ","End":"01:55.690","Text":"Now, we have stuff that cancels."},{"Start":"01:55.690 ","End":"02:00.236","Text":"The minus sine t cancels with plus sine t."},{"Start":"02:00.236 ","End":"02:01.730","Text":"The 1 makes no difference and"},{"Start":"02:01.730 ","End":"02:04.653","Text":"the cosine t cancels with the cosine t."},{"Start":"02:04.653 ","End":"02:07.595","Text":"These 2 minuses combine to give a plus."},{"Start":"02:07.595 ","End":"02:13.355","Text":"Then we can now compute what it says here that the length l,"},{"Start":"02:13.355 ","End":"02:18.645","Text":"which is equal to the square root from, let\u0027s see,"},{"Start":"02:18.645 ","End":"02:23.910","Text":"the integral from 0 to Pi over 2."},{"Start":"02:23.910 ","End":"02:25.610","Text":"Now, what do we have here?"},{"Start":"02:25.610 ","End":"02:30.830","Text":"We have x prime squared, which is t cosine t squared,"},{"Start":"02:30.830 ","End":"02:35.300","Text":"which is just t squared cosine squared t."},{"Start":"02:35.300 ","End":"02:38.825","Text":"Then y prime squared is just t"},{"Start":"02:38.825 ","End":"02:44.395","Text":"squared sine squared t and all this dt."},{"Start":"02:44.395 ","End":"02:47.465","Text":"Now look, the cosine squared plus"},{"Start":"02:47.465 ","End":"02:51.335","Text":"sine squared is 1 sine squared of Alpha,"},{"Start":"02:51.335 ","End":"02:54.770","Text":"plus cosine squared of Alpha, equals 1."},{"Start":"02:54.770 ","End":"02:58.270","Text":"Here, I can take t squared outside the brackets."},{"Start":"02:58.270 ","End":"03:04.340","Text":"What we get is the integral from 0"},{"Start":"03:04.340 ","End":"03:10.700","Text":"to Pi over 2 of the square root of just t squared."},{"Start":"03:10.700 ","End":"03:13.940","Text":"Because after we\u0027ve taken the cosine squared plus"},{"Start":"03:13.940 ","End":"03:16.265","Text":"sine squared equals 1 and taken t squared"},{"Start":"03:16.265 ","End":"03:18.748","Text":"outside the brackets, that\u0027s what we\u0027re left with."},{"Start":"03:18.748 ","End":"03:27.020","Text":"Dt, which is just equal to the integral from 0 to Pi over 2."},{"Start":"03:27.020 ","End":"03:29.900","Text":"Now, the square root of t squared is absolute value of t,"},{"Start":"03:29.900 ","End":"03:33.900","Text":"but t is positive, so it\u0027s just tdt."},{"Start":"03:34.270 ","End":"03:37.565","Text":"That\u0027s a pretty straightforward integral."},{"Start":"03:37.565 ","End":"03:47.924","Text":"We get t squared over 2 between 0 and Pi over 2 and this equals,"},{"Start":"03:47.924 ","End":"03:56.760","Text":"if we put t is Pi over 2, we get Pi over 2 squared over 2."},{"Start":"03:56.760 ","End":"04:03.470","Text":"If we put 0 in, we get 0 squared over 2."},{"Start":"04:03.470 ","End":"04:10.055","Text":"Now this thing is 0. The numerator is Pi squared over 4 divided by 2."},{"Start":"04:10.055 ","End":"04:18.210","Text":"We\u0027re just left with Pi squared over 8 and that is the answer. We are done."}],"ID":8664},{"Watched":false,"Name":"Exercise 2","Duration":"3m 14s","ChapterTopicVideoID":8450,"CourseChapterTopicPlaylistID":4907,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"This is 1 of those distance problems,"},{"Start":"00:02.400 ","End":"00:07.245","Text":"which is really a length of curve problem in the parametric form."},{"Start":"00:07.245 ","End":"00:11.220","Text":"The things we need to know in order to use the formula here,"},{"Start":"00:11.220 ","End":"00:14.040","Text":"we need to know what the function x of t is,"},{"Start":"00:14.040 ","End":"00:15.465","Text":"and this is what it is,"},{"Start":"00:15.465 ","End":"00:17.850","Text":"y of t is this,"},{"Start":"00:17.850 ","End":"00:23.355","Text":"and the start and endpoints for t are 0 and 4."},{"Start":"00:23.355 ","End":"00:25.440","Text":"Now that we know all these,"},{"Start":"00:25.440 ","End":"00:28.920","Text":"we\u0027re going to start computing l from this formula."},{"Start":"00:28.920 ","End":"00:34.104","Text":"I see I need both x prime and y prime and derivatives with respect to t."},{"Start":"00:34.104 ","End":"00:43.150","Text":"Let\u0027s see, x prime of t is equal to the derivative of t squared over 2 is just t,"},{"Start":"00:43.150 ","End":"00:53.175","Text":"and y prime is equal to, let\u0027s see, 3 over 2 times 1/3 is 1/2."},{"Start":"00:53.175 ","End":"00:56.910","Text":"That\u0027s a simple fraction problem."},{"Start":"00:56.910 ","End":"01:01.280","Text":"Now we have to lower by 1 the exponent,"},{"Start":"01:01.280 ","End":"01:06.575","Text":"so we get 2t plus 1 to the power of 1/2."},{"Start":"01:06.575 ","End":"01:11.380","Text":"We also have to multiply by the inner derivative, which is 2,"},{"Start":"01:11.380 ","End":"01:17.690","Text":"and actually we can cancel the 2 goes with the 1.5."},{"Start":"01:17.690 ","End":"01:20.050","Text":"That makes life simpler."},{"Start":"01:20.050 ","End":"01:22.730","Text":"Now I want to see what this expression is."},{"Start":"01:22.730 ","End":"01:25.580","Text":"X prime squared plus y prime squared,"},{"Start":"01:25.580 ","End":"01:32.825","Text":"so x prime squared plus y prime squared is,"},{"Start":"01:32.825 ","End":"01:39.765","Text":"this part is easy, that\u0027s t squared plus, this is easy too,"},{"Start":"01:39.765 ","End":"01:45.350","Text":"because 1^1/2, when you square it is just to the power of 1,"},{"Start":"01:45.350 ","End":"01:49.360","Text":"so we get plus 2t plus 1."},{"Start":"01:49.360 ","End":"01:54.335","Text":"Notice that this is a perfect square, it\u0027s so familiar."},{"Start":"01:54.335 ","End":"01:55.725","Text":"It should be."},{"Start":"01:55.725 ","End":"02:00.425","Text":"Anyway, you can always check by multiplying this and you\u0027ll get this."},{"Start":"02:00.425 ","End":"02:04.015","Text":"Now we\u0027re ready to tackle the integral."},{"Start":"02:04.015 ","End":"02:12.500","Text":"L is the integral from t_a to t_b, which is 0 to 4 of"},{"Start":"02:13.300 ","End":"02:20.305","Text":"x squared plus y squared is just the square root of t plus 1 squared,"},{"Start":"02:20.305 ","End":"02:22.600","Text":"which is just t plus 1."},{"Start":"02:22.600 ","End":"02:25.840","Text":"Normally, I would need absolute values because"},{"Start":"02:25.840 ","End":"02:29.260","Text":"the square root of a squared is absolute value of a."},{"Start":"02:29.260 ","End":"02:31.970","Text":"But since t is from 0 to 4, I don\u0027t need that,"},{"Start":"02:31.970 ","End":"02:33.699","Text":"so just regular brackets,"},{"Start":"02:33.699 ","End":"02:38.460","Text":"and dt, this is a straight forward integral."},{"Start":"02:38.460 ","End":"02:48.200","Text":"This is equal to t squared over 2 plus t taken between 0 and 4."},{"Start":"02:48.200 ","End":"02:57.990","Text":"If I put in 4, I get 4 squared over 2 plus 4, and if I put in 0,"},{"Start":"02:57.990 ","End":"03:04.290","Text":"everything\u0027s 0, well, I could write minus 0 plus 0,"},{"Start":"03:04.290 ","End":"03:07.890","Text":"and in short 4 squared over 2 is,"},{"Start":"03:07.890 ","End":"03:13.460","Text":"16 over 2 is 8, plus 4 is 12, and 12 is our final answer,"},{"Start":"03:13.460 ","End":"03:15.600","Text":"and we are done."}],"ID":8665},{"Watched":false,"Name":"Exercise 3","Duration":"4m 5s","ChapterTopicVideoID":8447,"CourseChapterTopicPlaylistID":4907,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"Here again, we have 1 of these distance problems."},{"Start":"00:04.050 ","End":"00:06.135","Text":"Distance traveled by a particle,"},{"Start":"00:06.135 ","End":"00:12.029","Text":"which really is just the parametric form of a length of curve problem."},{"Start":"00:12.029 ","End":"00:15.270","Text":"The things we need to know in length of curve,"},{"Start":"00:15.270 ","End":"00:17.189","Text":"as you can see from this formula,"},{"Start":"00:17.189 ","End":"00:18.390","Text":"are those 4 quantities."},{"Start":"00:18.390 ","End":"00:20.520","Text":"We need to know x as a function of t,"},{"Start":"00:20.520 ","End":"00:24.120","Text":"that\u0027s the x of t. We need to know y as a function of"},{"Start":"00:24.120 ","End":"00:28.500","Text":"t and we need to know the limits of integration,"},{"Start":"00:28.500 ","End":"00:33.240","Text":"which is from 0-4 for t."},{"Start":"00:33.240 ","End":"00:35.520","Text":"Let\u0027s go about computing l."},{"Start":"00:35.520 ","End":"00:38.205","Text":"We need each of the derivatives first."},{"Start":"00:38.205 ","End":"00:42.630","Text":"Let\u0027s see what is x prime, what\u0027s y prime."},{"Start":"00:42.630 ","End":"00:49.515","Text":"x prime equals, now here we have 3 over 2 times 1/3,"},{"Start":"00:49.515 ","End":"00:52.075","Text":"which is just 1/2."},{"Start":"00:52.075 ","End":"00:54.935","Text":"Then we reduce the exponent by 1."},{"Start":"00:54.935 ","End":"00:59.975","Text":"So we get 2t plus 1 to the power of 1/2."},{"Start":"00:59.975 ","End":"01:05.495","Text":"But there\u0027s also an inner derivative which is 2,"},{"Start":"01:05.495 ","End":"01:09.515","Text":"and this 2 just cancels with this 1/2."},{"Start":"01:09.515 ","End":"01:13.615","Text":"Now, y prime is equal to,"},{"Start":"01:13.615 ","End":"01:21.110","Text":"derivative of t squared over 2 is just t. Derivative of t is 1."},{"Start":"01:21.110 ","End":"01:23.255","Text":"Now that we have both of these,"},{"Start":"01:23.255 ","End":"01:27.530","Text":"let\u0027s see what is x prime squared plus y prime squared?"},{"Start":"01:27.530 ","End":"01:32.065","Text":"So x prime squared plus y prime squared,"},{"Start":"01:32.065 ","End":"01:35.460","Text":"brackets here, is equal to."},{"Start":"01:35.460 ","End":"01:40.160","Text":"This 1 squared, the power of 2 and the power of 1/2 cancels,"},{"Start":"01:40.160 ","End":"01:42.470","Text":"so it\u0027s just 2t plus 1."},{"Start":"01:42.470 ","End":"01:45.220","Text":"The second bit is, this thing squared,"},{"Start":"01:45.220 ","End":"01:49.115","Text":"is just t plus 1 squared."},{"Start":"01:49.115 ","End":"01:51.050","Text":"What does this give us?"},{"Start":"01:51.050 ","End":"01:55.670","Text":"This is 2 squared plus 2t plus 1."},{"Start":"01:55.670 ","End":"01:58.934","Text":"That makes it t squared,"},{"Start":"01:58.934 ","End":"02:01.620","Text":"plus 2t plus 2t is 4t,"},{"Start":"02:01.620 ","End":"02:05.860","Text":"plus 1 plus 1 is plus 2."},{"Start":"02:05.860 ","End":"02:10.250","Text":"At this point, I was really expecting to get a perfect square,"},{"Start":"02:10.250 ","End":"02:12.905","Text":"which it would have been if there was a 4 at the end,"},{"Start":"02:12.905 ","End":"02:14.480","Text":"I may have miscopied the problem."},{"Start":"02:14.480 ","End":"02:19.220","Text":"Let\u0027s just change this 1 into a 3 and then this will have to become"},{"Start":"02:19.220 ","End":"02:24.980","Text":"a 3 and this will have to become a 3."},{"Start":"02:24.980 ","End":"02:26.565","Text":"So I\u0027ll write here,"},{"Start":"02:26.565 ","End":"02:32.355","Text":"3 and 3 and now this becomes a 4."},{"Start":"02:32.355 ","End":"02:36.690","Text":"Erase the 2 and write a 4."},{"Start":"02:36.690 ","End":"02:40.280","Text":"Now I\u0027m happier, even though it\u0027s fudging a bit."},{"Start":"02:40.280 ","End":"02:42.890","Text":"I just presume I must have miscopied the- Okay."},{"Start":"02:42.890 ","End":"02:50.810","Text":"Let\u0027s continue. What we need now is the integral from 0-4,"},{"Start":"02:50.810 ","End":"02:53.605","Text":"put the 0 and 4 back in view,"},{"Start":"02:53.605 ","End":"02:57.200","Text":"of, now this thing I said is a perfect square."},{"Start":"02:57.200 ","End":"02:59.780","Text":"It is in fact t plus 2 all squared,"},{"Start":"02:59.780 ","End":"03:02.555","Text":"as you can check by multiplying it out."},{"Start":"03:02.555 ","End":"03:10.780","Text":"The square root of x prime squared plus y prime squared is the square root of this,"},{"Start":"03:10.780 ","End":"03:16.975","Text":"which is just t plus 2 and this is a straightforward integral."},{"Start":"03:16.975 ","End":"03:26.955","Text":"This is equal to t. Integral of t is 1/2t squared or t squared over 2 plus 2t,"},{"Start":"03:26.955 ","End":"03:31.575","Text":"all this taken between 0 and 4."},{"Start":"03:31.575 ","End":"03:34.540","Text":"If I put in 4,"},{"Start":"03:34.540 ","End":"03:42.264","Text":"I get 4 squared over 2 is 16 over 2 is 8."},{"Start":"03:42.264 ","End":"03:46.315","Text":"2 times 4 is 8."},{"Start":"03:46.315 ","End":"03:50.295","Text":"When I put in 0, I get 1/2t squared,"},{"Start":"03:50.295 ","End":"03:54.445","Text":"1/2 0 squared is 0 and twice 0 is 0."},{"Start":"03:54.445 ","End":"04:00.610","Text":"According to this, what I get is 16."},{"Start":"04:00.610 ","End":"04:02.795","Text":"We are done."},{"Start":"04:02.795 ","End":"04:06.570","Text":"Forgive me for the miscopying of the problem."}],"ID":8666}],"Thumbnail":null,"ID":4907}]

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