Continuity of a Function
0/12 completed

Points of Discontinuity
0/6 completed

The Intermediate Value Theorem
0/7 completed

{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Continuity of a Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Continuity of a Function at a Point","Duration":"9m 43s","ChapterTopicVideoID":8254,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8254.jpeg","UploadDate":"2019-11-14T06:52:03.3770000","DurationForVideoObject":"PT9M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"In this clip, I\u0027ll talk about the concept of continuity of a function at a point."},{"Start":"00:05.700 ","End":"00:09.375","Text":"I\u0027ll start right away with a very informal definition,"},{"Start":"00:09.375 ","End":"00:11.025","Text":"and it goes like this."},{"Start":"00:11.025 ","End":"00:15.330","Text":"A function f is continuous at a point if we can sketch"},{"Start":"00:15.330 ","End":"00:20.160","Text":"its graph without lifting the pen off the paper as we pass through the point."},{"Start":"00:20.160 ","End":"00:22.620","Text":"I\u0027ll give you an example right away."},{"Start":"00:22.620 ","End":"00:25.890","Text":"Here I\u0027ve drawn the graph of a function y equals f of x,"},{"Start":"00:25.890 ","End":"00:30.165","Text":"and I want to concentrate on this blue point where x equals 3."},{"Start":"00:30.165 ","End":"00:32.595","Text":"Now if you look at this point on the graph,"},{"Start":"00:32.595 ","End":"00:35.804","Text":"you see that as I pass through that point,"},{"Start":"00:35.804 ","End":"00:40.960","Text":"I don\u0027t have to lift the pen off the paper in order to get this point drawn."},{"Start":"00:40.960 ","End":"00:44.060","Text":"The function is continuous around x equals 3,"},{"Start":"00:44.060 ","End":"00:46.385","Text":"or more precisely at the point 3, 2."},{"Start":"00:46.385 ","End":"00:48.620","Text":"Before I give a formal definition of what it"},{"Start":"00:48.620 ","End":"00:51.020","Text":"means for a function to be continuous at a point,"},{"Start":"00:51.020 ","End":"00:55.400","Text":"I\u0027d like to look at a couple of points where f is not continuous."},{"Start":"00:55.400 ","End":"00:56.570","Text":"Let\u0027s look, for example,"},{"Start":"00:56.570 ","End":"00:59.930","Text":"at the point where x equals minus 6."},{"Start":"00:59.930 ","End":"01:03.305","Text":"At minus 6 when I draw the function,"},{"Start":"01:03.305 ","End":"01:04.720","Text":"I certainly make a jump,"},{"Start":"01:04.720 ","End":"01:07.160","Text":"I have to take the pen off the paper."},{"Start":"01:07.160 ","End":"01:10.610","Text":"The main reason for this is that the function is not even"},{"Start":"01:10.610 ","End":"01:14.530","Text":"defined at the point where x equals minus 6."},{"Start":"01:14.530 ","End":"01:19.984","Text":"Conclusion. In order for a function to be continuous at a point,"},{"Start":"01:19.984 ","End":"01:22.430","Text":"it must be defined at the point."},{"Start":"01:22.430 ","End":"01:24.470","Text":"As you see if it\u0027s not defined here,"},{"Start":"01:24.470 ","End":"01:28.115","Text":"when I pass through I have to lift my pen off the paper."},{"Start":"01:28.115 ","End":"01:31.205","Text":"However, even if it\u0027s defined at a point,"},{"Start":"01:31.205 ","End":"01:33.290","Text":"that doesn\u0027t mean it\u0027s continuous there."},{"Start":"01:33.290 ","End":"01:35.765","Text":"This time, take a look at x equals 4,"},{"Start":"01:35.765 ","End":"01:38.210","Text":"it\u0027s defined at x equals 4,"},{"Start":"01:38.210 ","End":"01:42.675","Text":"y equals 4, but nevertheless, it jumps."},{"Start":"01:42.675 ","End":"01:43.875","Text":"As we pass through,"},{"Start":"01:43.875 ","End":"01:46.450","Text":"it makes a jump from 4 to 5,"},{"Start":"01:46.450 ","End":"01:48.920","Text":"and we have to pick our pen off the paper."},{"Start":"01:48.920 ","End":"01:53.480","Text":"Here\u0027s an example where a function is defined at a point,"},{"Start":"01:53.480 ","End":"01:55.900","Text":"but it\u0027s still not continuous there."},{"Start":"01:55.900 ","End":"01:59.480","Text":"What happened here to make this function not continuous?"},{"Start":"01:59.480 ","End":"02:01.100","Text":"Let\u0027s look more closely."},{"Start":"02:01.100 ","End":"02:05.180","Text":"Let\u0027s look at the limit as x goes to 4 from the left,"},{"Start":"02:05.180 ","End":"02:09.010","Text":"and then let\u0027s look what happens as we take the limit from the right."},{"Start":"02:09.010 ","End":"02:14.735","Text":"The limit as x goes to 4 from the left is equal to 4,"},{"Start":"02:14.735 ","End":"02:18.805","Text":"but the limit as x goes to 4 from the right,"},{"Start":"02:18.805 ","End":"02:21.300","Text":"which is here, is 5."},{"Start":"02:21.300 ","End":"02:22.715","Text":"These are not equal,"},{"Start":"02:22.715 ","End":"02:27.680","Text":"which implies that f doesn\u0027t have a limit at x equals 4, because I have a limit."},{"Start":"02:27.680 ","End":"02:31.025","Text":"There has to be a left limit and a right limit which are equal."},{"Start":"02:31.025 ","End":"02:33.470","Text":"Our second conclusion is this,"},{"Start":"02:33.470 ","End":"02:37.955","Text":"which is pretty much the same as the first one with the modification, and I\u0027ll read it."},{"Start":"02:37.955 ","End":"02:41.330","Text":"In order for a function to be continuous at a point,"},{"Start":"02:41.330 ","End":"02:43.880","Text":"it must have a limit at the point."},{"Start":"02:43.880 ","End":"02:46.580","Text":"Previously we said it must be defined at the point,"},{"Start":"02:46.580 ","End":"02:48.215","Text":"and now we have an additional requirement,"},{"Start":"02:48.215 ","End":"02:49.570","Text":"must have a limit at the point,"},{"Start":"02:49.570 ","End":"02:51.470","Text":"which brings up another question."},{"Start":"02:51.470 ","End":"02:54.490","Text":"Suppose it\u0027s defined and has a limit,"},{"Start":"02:54.490 ","End":"02:57.410","Text":"now is that enough for it to be continuous?"},{"Start":"02:57.410 ","End":"03:02.140","Text":"Well, unfortunately not, as our last example will show."},{"Start":"03:02.140 ","End":"03:08.935","Text":"What happens here is that I modified the functions of that the value at minus 3,"},{"Start":"03:08.935 ","End":"03:10.680","Text":"whereas it used to be 2,"},{"Start":"03:10.680 ","End":"03:12.675","Text":"I\u0027ve changed it to be 4."},{"Start":"03:12.675 ","End":"03:16.265","Text":"Just pluck it quickly go back and look at the other points."},{"Start":"03:16.265 ","End":"03:18.110","Text":"When x was minus 6,"},{"Start":"03:18.110 ","End":"03:20.170","Text":"the problem was that it wasn\u0027t defined there,"},{"Start":"03:20.170 ","End":"03:22.775","Text":"so f of minus 6 was not defined."},{"Start":"03:22.775 ","End":"03:26.985","Text":"Then we had another type of problem at x equals 4,"},{"Start":"03:26.985 ","End":"03:29.630","Text":"where it was defined but the limit from"},{"Start":"03:29.630 ","End":"03:32.960","Text":"the left and the limit from the right were not equal."},{"Start":"03:32.960 ","End":"03:35.240","Text":"In other words, the limit from the left was 4,"},{"Start":"03:35.240 ","End":"03:38.150","Text":"the limit from the right was 5 and they were not equal,"},{"Start":"03:38.150 ","End":"03:39.445","Text":"so there was no limit."},{"Start":"03:39.445 ","End":"03:42.770","Text":"Now, if we turn our attention to x equals minus 3,"},{"Start":"03:42.770 ","End":"03:50.420","Text":"it is defined there because f of minus 3 is equal to 4 and it does have a limit."},{"Start":"03:50.420 ","End":"03:55.825","Text":"The limit as x goes to minus 3 of f of x,"},{"Start":"03:55.825 ","End":"03:57.640","Text":"I claim it equals 2."},{"Start":"03:57.640 ","End":"04:01.355","Text":"If I take the limit from the right and I go along here,"},{"Start":"04:01.355 ","End":"04:02.990","Text":"I get to 2."},{"Start":"04:02.990 ","End":"04:08.300","Text":"If I go from the left and get closer and closer to minus 3,"},{"Start":"04:08.300 ","End":"04:10.415","Text":"it also gets close to 2."},{"Start":"04:10.415 ","End":"04:14.375","Text":"The limit from the left is 2 and the limit from the right is 2,"},{"Start":"04:14.375 ","End":"04:15.865","Text":"so the limit is 2."},{"Start":"04:15.865 ","End":"04:17.880","Text":"It has a definition,"},{"Start":"04:17.880 ","End":"04:19.800","Text":"it\u0027s defined as 4,"},{"Start":"04:19.800 ","End":"04:21.930","Text":"it has a limit which is 2,"},{"Start":"04:21.930 ","End":"04:25.685","Text":"but the trouble this time is that 4 is not equal to 2,"},{"Start":"04:25.685 ","End":"04:29.360","Text":"by which the value of the function at the point"},{"Start":"04:29.360 ","End":"04:33.305","Text":"is not equal to the limit of the function as extends to the point."},{"Start":"04:33.305 ","End":"04:35.165","Text":"This is the problem here."},{"Start":"04:35.165 ","End":"04:38.570","Text":"Now I think we\u0027re about ready to formalize the definition of"},{"Start":"04:38.570 ","End":"04:41.975","Text":"what it means for a function to be continuous at a point."},{"Start":"04:41.975 ","End":"04:47.540","Text":"Here are the previous conclusions just rewritten as the first 2 parts of the 3 part."},{"Start":"04:47.540 ","End":"04:50.930","Text":"What we learned from this case where x equals minus 3 was"},{"Start":"04:50.930 ","End":"04:54.605","Text":"that it did have a definition or a value if you like,"},{"Start":"04:54.605 ","End":"04:56.175","Text":"it did have a limit,"},{"Start":"04:56.175 ","End":"05:00.140","Text":"what was wrong was that the value and the limit were not equal."},{"Start":"05:00.140 ","End":"05:02.915","Text":"This brings us to point number 3,"},{"Start":"05:02.915 ","End":"05:07.895","Text":"the value at the point must equal the limit."},{"Start":"05:07.895 ","End":"05:10.490","Text":"Now that we have these 3 conditions,"},{"Start":"05:10.490 ","End":"05:14.390","Text":"we\u0027re actually going to make these the definition of continuity."},{"Start":"05:14.390 ","End":"05:17.225","Text":"Let me write this in mathematical language."},{"Start":"05:17.225 ","End":"05:18.740","Text":"I\u0027m just going to translate 1, 2,"},{"Start":"05:18.740 ","End":"05:21.540","Text":"and 3 into mathematics, and here it goes."},{"Start":"05:21.540 ","End":"05:26.585","Text":"A function f is called continuous at a point x equals a if"},{"Start":"05:26.585 ","End":"05:33.050","Text":"the limit on the right as x approaches a equals the limit on the left as x approaches a,"},{"Start":"05:33.050 ","End":"05:36.980","Text":"which is also equal to the value of f at the point a."},{"Start":"05:36.980 ","End":"05:38.660","Text":"How does this work with this?"},{"Start":"05:38.660 ","End":"05:40.670","Text":"Must be defined, means it has a value,"},{"Start":"05:40.670 ","End":"05:42.995","Text":"means that f of a makes sense."},{"Start":"05:42.995 ","End":"05:45.930","Text":"If f of a make sense then f is defined at the point."},{"Start":"05:45.930 ","End":"05:47.450","Text":"It must have a limit,"},{"Start":"05:47.450 ","End":"05:51.200","Text":"to have a limit it has to have a left limit and right limit and they must be equal."},{"Start":"05:51.200 ","End":"05:53.120","Text":"That gives us this equality."},{"Start":"05:53.120 ","End":"05:57.500","Text":"Finally, the value at the point must equal the limit at the point."},{"Start":"05:57.500 ","End":"06:00.410","Text":"Allows us to write this equals in here."},{"Start":"06:00.410 ","End":"06:04.280","Text":"There it is, the formal definition of continuity at a point."},{"Start":"06:04.280 ","End":"06:06.185","Text":"Let\u0027s take a look on the graph."},{"Start":"06:06.185 ","End":"06:08.775","Text":"This point, the blue one was our good one."},{"Start":"06:08.775 ","End":"06:14.400","Text":"There you can see that the limit from the left of the function was 2,"},{"Start":"06:14.400 ","End":"06:16.215","Text":"the limit from the right was 2,"},{"Start":"06:16.215 ","End":"06:19.610","Text":"and the value at the point was also equal to 2."},{"Start":"06:19.610 ","End":"06:21.260","Text":"So all 3 things were equal."},{"Start":"06:21.260 ","End":"06:23.440","Text":"There\u0027s another thing I\u0027d like to mention."},{"Start":"06:23.440 ","End":"06:27.785","Text":"In case these 2 are equal but not equal to the last one,"},{"Start":"06:27.785 ","End":"06:30.800","Text":"I\u0027ll show you what that means in our picture."},{"Start":"06:30.800 ","End":"06:33.155","Text":"In our case, this was minus 3,"},{"Start":"06:33.155 ","End":"06:35.975","Text":"the left limit was equal to 2,"},{"Start":"06:35.975 ","End":"06:38.375","Text":"and the right limit was equal to 2,"},{"Start":"06:38.375 ","End":"06:40.800","Text":"but the value was not equal to 2,"},{"Start":"06:40.800 ","End":"06:42.045","Text":"it was equal to 4."},{"Start":"06:42.045 ","End":"06:43.680","Text":"In this condition,"},{"Start":"06:43.680 ","End":"06:48.410","Text":"this discontinuity is called a removable discontinuity."},{"Start":"06:48.410 ","End":"06:52.700","Text":"Because if you just change the value of this 1 point to make it go into the whole here,"},{"Start":"06:52.700 ","End":"06:54.125","Text":"then everything will be fine."},{"Start":"06:54.125 ","End":"06:57.380","Text":"Let me take these values and go down to the definition."},{"Start":"06:57.380 ","End":"07:00.020","Text":"What we had up there was that,"},{"Start":"07:00.020 ","End":"07:01.960","Text":"when a was minus 3,"},{"Start":"07:01.960 ","End":"07:05.435","Text":"that this limit from the right was equal to 2,"},{"Start":"07:05.435 ","End":"07:07.715","Text":"the limit from the left was equal to 2."},{"Start":"07:07.715 ","End":"07:09.740","Text":"But the value of function was 4."},{"Start":"07:09.740 ","End":"07:14.670","Text":"If only we just changed the value at x equals minus 3 to 2,"},{"Start":"07:14.670 ","End":"07:17.010","Text":"also, we would have continuity."},{"Start":"07:17.010 ","End":"07:18.890","Text":"That\u0027s why this discontinuity is"},{"Start":"07:18.890 ","End":"07:22.460","Text":"only a light discontinuity and it\u0027s considered to be removable."},{"Start":"07:22.460 ","End":"07:24.800","Text":"I\u0027ll just like to leave you with the term removable"},{"Start":"07:24.800 ","End":"07:28.970","Text":"discontinuity and you\u0027ll be encountering it again in another chapter."},{"Start":"07:28.970 ","End":"07:32.390","Text":"I\u0027ll finish this clip with an exercise on continuity."},{"Start":"07:32.390 ","End":"07:35.960","Text":"Typically you will not be getting any graphs or sketches,"},{"Start":"07:35.960 ","End":"07:37.970","Text":"just equations and formulas."},{"Start":"07:37.970 ","End":"07:40.610","Text":"The most typical kind is where you\u0027re"},{"Start":"07:40.610 ","End":"07:43.400","Text":"given a function that\u0027s defined piecewise and you\u0027re"},{"Start":"07:43.400 ","End":"07:48.815","Text":"asked to ascertain whether the function is continuous at the seamline."},{"Start":"07:48.815 ","End":"07:52.040","Text":"In our case, the function is defined as follows,"},{"Start":"07:52.040 ","End":"07:54.335","Text":"one way for x bigger or equal to 3,"},{"Start":"07:54.335 ","End":"07:56.360","Text":"and another way for x less than 3,"},{"Start":"07:56.360 ","End":"07:57.875","Text":"3 is our seamline,"},{"Start":"07:57.875 ","End":"08:02.060","Text":"and we\u0027re asked if f is continuous at x equals 3."},{"Start":"08:02.060 ","End":"08:03.350","Text":"In other words, informally,"},{"Start":"08:03.350 ","End":"08:06.620","Text":"when I joined these 2 bits of function together,"},{"Start":"08:06.620 ","End":"08:09.710","Text":"whether I can sketch the thing around x equals"},{"Start":"08:09.710 ","End":"08:13.280","Text":"3 without lifting the pen off the paper, so to speak."},{"Start":"08:13.280 ","End":"08:16.685","Text":"But notice that the definition of continuity is still up there,"},{"Start":"08:16.685 ","End":"08:19.805","Text":"we just have to check if 3 different things are equal."},{"Start":"08:19.805 ","End":"08:22.580","Text":"What we\u0027re going to do is check whether those 3 things are equal."},{"Start":"08:22.580 ","End":"08:24.080","Text":"In our case, in other words,"},{"Start":"08:24.080 ","End":"08:31.505","Text":"we\u0027ll check the limit as x goes to 3 from the right of f of x and see what that is."},{"Start":"08:31.505 ","End":"08:37.785","Text":"Then we\u0027ll check the limit as x goes to 3 from the left of f of x."},{"Start":"08:37.785 ","End":"08:39.425","Text":"Then the value at the point,"},{"Start":"08:39.425 ","End":"08:42.065","Text":"what is f of 3 equal to?"},{"Start":"08:42.065 ","End":"08:45.215","Text":"If all these 3 quantities are equal,"},{"Start":"08:45.215 ","End":"08:46.790","Text":"then the answer will be yes,"},{"Start":"08:46.790 ","End":"08:48.470","Text":"otherwise the answer will be no."},{"Start":"08:48.470 ","End":"08:50.945","Text":"First one, x goes to 3 from the right."},{"Start":"08:50.945 ","End":"08:53.360","Text":"That means that x is bigger than 3,"},{"Start":"08:53.360 ","End":"08:56.030","Text":"which means that we\u0027re in the first top row."},{"Start":"08:56.030 ","End":"09:02.390","Text":"We just have to substitute x equals 3 here and we get 3 squared plus twice 3,"},{"Start":"09:02.390 ","End":"09:06.680","Text":"that would be 9 plus 6, that would be 15."},{"Start":"09:06.680 ","End":"09:10.020","Text":"Now, from the left x will be less than 3,"},{"Start":"09:10.020 ","End":"09:11.925","Text":"so it\u0027ll be in this bottom row."},{"Start":"09:11.925 ","End":"09:13.860","Text":"We substitute to find the limit,"},{"Start":"09:13.860 ","End":"09:16.700","Text":"so minus 3 plus 18,"},{"Start":"09:16.700 ","End":"09:19.955","Text":"that\u0027s also easy to figure, that\u0027s also 15."},{"Start":"09:19.955 ","End":"09:23.190","Text":"Now, finally, f of 3."},{"Start":"09:23.190 ","End":"09:24.690","Text":"Where does 3 belong?"},{"Start":"09:24.690 ","End":"09:29.630","Text":"Our 3 belongs in this category in the x bigger or equal to 3, includes the equals."},{"Start":"09:29.630 ","End":"09:31.610","Text":"Just put in x equals 3,"},{"Start":"09:31.610 ","End":"09:34.790","Text":"3 squared plus twice 3 is 15."},{"Start":"09:34.790 ","End":"09:36.560","Text":"The answer is yes,"},{"Start":"09:36.560 ","End":"09:37.910","Text":"they are all equal,"},{"Start":"09:37.910 ","End":"09:44.310","Text":"so f is continuous at the point 3. We\u0027re done."}],"ID":8414},{"Watched":false,"Name":"Exercise 1","Duration":"2m 45s","ChapterTopicVideoID":91,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/91.jpeg","UploadDate":"2017-06-15T14:33:15.3630000","DurationForVideoObject":"PT2M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we have to check whether the function"},{"Start":"00:03.000 ","End":"00:07.325","Text":"f is or isn\u0027t continuous at x equals 1,"},{"Start":"00:07.325 ","End":"00:09.720","Text":"and at the end, we need to draw a sketch."},{"Start":"00:09.720 ","End":"00:12.540","Text":"F is defined piecewise as follows,"},{"Start":"00:12.540 ","End":"00:15.510","Text":"and the only interesting point is where"},{"Start":"00:15.510 ","End":"00:19.145","Text":"x is equal to 1 because then there\u0027s a transition, a change."},{"Start":"00:19.145 ","End":"00:21.535","Text":"We first check 3 things,"},{"Start":"00:21.535 ","End":"00:25.245","Text":"the limit as x goes to 1 from the right,"},{"Start":"00:25.245 ","End":"00:28.470","Text":"from the left, and at the point 1 itself."},{"Start":"00:28.470 ","End":"00:30.435","Text":"We\u0027re going to write down 3 things."},{"Start":"00:30.435 ","End":"00:35.100","Text":"Limit x goes to 1 from the right,"},{"Start":"00:35.100 ","End":"00:37.275","Text":"f of x equals,"},{"Start":"00:37.275 ","End":"00:43.905","Text":"then we\u0027ll check the limit as x goes to 1 from the left of x,"},{"Start":"00:43.905 ","End":"00:47.820","Text":"and finally, we check f of 1 itself."},{"Start":"00:47.820 ","End":"00:50.270","Text":"For this first piece,"},{"Start":"00:50.270 ","End":"00:53.925","Text":"we use the formula x equals 1,"},{"Start":"00:53.925 ","End":"00:56.205","Text":"so at 1 it\u0027s just equal to 1."},{"Start":"00:56.205 ","End":"00:58.470","Text":"Similarly x squared at 1,"},{"Start":"00:58.470 ","End":"01:00.405","Text":"it\u0027s also equal 1."},{"Start":"01:00.405 ","End":"01:02.865","Text":"Meanwhile, we\u0027ve got 2 out of 3."},{"Start":"01:02.865 ","End":"01:06.395","Text":"What we want is for all 3 to exist and all to be the same."},{"Start":"01:06.395 ","End":"01:09.260","Text":"Finally, we let x equals 1,"},{"Start":"01:09.260 ","End":"01:11.350","Text":"which comes from this definition,"},{"Start":"01:11.350 ","End":"01:13.890","Text":"and yes, we have 1."},{"Start":"01:13.890 ","End":"01:15.570","Text":"Of these 3 things,"},{"Start":"01:15.570 ","End":"01:18.375","Text":"this, this, and this are the same,"},{"Start":"01:18.375 ","End":"01:25.650","Text":"and therefore f is continuous at x equals 1."},{"Start":"01:25.650 ","End":"01:27.830","Text":"Then we\u0027re done with that part,"},{"Start":"01:27.830 ","End":"01:30.395","Text":"but we still have the sketch."},{"Start":"01:30.395 ","End":"01:33.815","Text":"Let\u0027s draw each of these functions."},{"Start":"01:33.815 ","End":"01:38.540","Text":"Let\u0027s try f of x equals x. I\u0027ll ignore first of"},{"Start":"01:38.540 ","End":"01:46.415","Text":"all the boundaries of the joining points and just draw f of x equals x in general,"},{"Start":"01:46.415 ","End":"01:51.815","Text":"clearly a straight line 45 degrees through the origin, something like this."},{"Start":"01:51.815 ","End":"01:54.995","Text":"The main point is that at 1,"},{"Start":"01:54.995 ","End":"01:56.645","Text":"when x is 1,"},{"Start":"01:56.645 ","End":"01:58.430","Text":"y is equal to 1,"},{"Start":"01:58.430 ","End":"01:59.869","Text":"and the other function,"},{"Start":"01:59.869 ","End":"02:03.665","Text":"x squared also goes through 1, 1."},{"Start":"02:03.665 ","End":"02:06.810","Text":"That\u0027s the parabola upward facing."},{"Start":"02:06.810 ","End":"02:10.020","Text":"Now let\u0027s use some colors to draw"},{"Start":"02:10.020 ","End":"02:14.360","Text":"f. Because what we have to do is not take both of these graphs,"},{"Start":"02:14.360 ","End":"02:17.440","Text":"but just part of this graph and part of this graph,"},{"Start":"02:17.440 ","End":"02:20.780","Text":"and the border is where x is equal to 1."},{"Start":"02:20.780 ","End":"02:24.265","Text":"To the right, it\u0027s equal to x,"},{"Start":"02:24.265 ","End":"02:26.130","Text":"so that would be this part here,"},{"Start":"02:26.130 ","End":"02:28.760","Text":"and from 1 and to the left,"},{"Start":"02:28.760 ","End":"02:32.530","Text":"we take it from here, again,"},{"Start":"02:32.530 ","End":"02:36.305","Text":"not the greatest, but you\u0027ll get the general idea."},{"Start":"02:36.305 ","End":"02:39.970","Text":"We take part of the parabola and part of the straight line,"},{"Start":"02:39.970 ","End":"02:43.730","Text":"so the colored bit is the graph of f of x,"},{"Start":"02:43.730 ","End":"02:46.470","Text":"and that\u0027s it, we\u0027re done."}],"ID":90},{"Watched":false,"Name":"Exercise 2","Duration":"3m 44s","ChapterTopicVideoID":92,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/92.jpeg","UploadDate":"2017-06-15T14:34:23.4970000","DurationForVideoObject":"PT3M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.275","Text":"In this exercise,"},{"Start":"00:01.275 ","End":"00:04.920","Text":"we have to check whether or not f is continuous at x equals 2,"},{"Start":"00:04.920 ","End":"00:07.095","Text":"and at the end we\u0027ll also sketch the graph."},{"Start":"00:07.095 ","End":"00:09.240","Text":"F is defined piecewise,"},{"Start":"00:09.240 ","End":"00:12.420","Text":"defined 1 way here and another way there,"},{"Start":"00:12.420 ","End":"00:14.760","Text":"and 2 is the border point,"},{"Start":"00:14.760 ","End":"00:18.675","Text":"the crossing point between 1 definition and another."},{"Start":"00:18.675 ","End":"00:23.445","Text":"What we have to do as usual is to check 3 things."},{"Start":"00:23.445 ","End":"00:31.920","Text":"First of all, the limit as x goes to 2 on 1 side let\u0027s say the right of f of x."},{"Start":"00:31.920 ","End":"00:36.254","Text":"Secondly, we have to check the limit on the other side."},{"Start":"00:36.254 ","End":"00:39.540","Text":"We also have to check that all these things exist,"},{"Start":"00:39.540 ","End":"00:42.370","Text":"whether the function has a limit, whether it\u0027s defined."},{"Start":"00:42.370 ","End":"00:47.805","Text":"Finally, the third 1 is to check what is f of 2 itself."},{"Start":"00:47.805 ","End":"00:49.920","Text":"Hopefully, will this exist as the same?"},{"Start":"00:49.920 ","End":"00:52.055","Text":"Then yes, the function is continuous."},{"Start":"00:52.055 ","End":"00:54.840","Text":"So when x goes to 2,"},{"Start":"00:54.840 ","End":"00:56.550","Text":"then we have the limit."},{"Start":"00:56.550 ","End":"01:00.555","Text":"We take our definition from the right, from here."},{"Start":"01:00.555 ","End":"01:07.680","Text":"x goes to 2 from the right of x plus 1,"},{"Start":"01:07.680 ","End":"01:10.335","Text":"and that\u0027s just substitution,"},{"Start":"01:10.335 ","End":"01:12.030","Text":"2 plus 1 is 3."},{"Start":"01:12.030 ","End":"01:13.605","Text":"Now let\u0027s try the other side,"},{"Start":"01:13.605 ","End":"01:15.060","Text":"2 from the left,"},{"Start":"01:15.060 ","End":"01:17.835","Text":"this time not x plus 1, 5 minus x,"},{"Start":"01:17.835 ","End":"01:22.215","Text":"so just substitute x equals 2, no problem, 3."},{"Start":"01:22.215 ","End":"01:25.890","Text":"Finally, F of 2 itself,"},{"Start":"01:25.890 ","End":"01:28.905","Text":"to itself actually falls in this part,"},{"Start":"01:28.905 ","End":"01:31.115","Text":"so that\u0027s equal to 2 plus 1,"},{"Start":"01:31.115 ","End":"01:33.100","Text":"and that\u0027s also 3."},{"Start":"01:33.100 ","End":"01:35.550","Text":"So here we have it, 3,"},{"Start":"01:35.550 ","End":"01:39.210","Text":"3 and 3 are all the same,"},{"Start":"01:39.210 ","End":"01:41.930","Text":"so the answer is yes."},{"Start":"01:41.930 ","End":"01:49.220","Text":"In other words, F is continuous at x equals 2."},{"Start":"01:49.220 ","End":"01:50.820","Text":"Let\u0027s do it in colors."},{"Start":"01:50.820 ","End":"01:51.970","Text":"Let\u0027s say the first part,"},{"Start":"01:51.970 ","End":"01:58.040","Text":"we\u0027ll do that in red and we\u0027ll make a little table but only 1 to go up to x equals 2."},{"Start":"01:58.040 ","End":"02:01.455","Text":"Let\u0027s take x is equal to 2,"},{"Start":"02:01.455 ","End":"02:04.245","Text":"then y is equal to 3."},{"Start":"02:04.245 ","End":"02:07.095","Text":"Let\u0027s take x equals 0,"},{"Start":"02:07.095 ","End":"02:10.110","Text":"then y is equal to 1,"},{"Start":"02:10.110 ","End":"02:11.910","Text":"but really should do this."},{"Start":"02:11.910 ","End":"02:14.145","Text":"If we take 0,1,"},{"Start":"02:14.145 ","End":"02:15.765","Text":"it will be somewhere here."},{"Start":"02:15.765 ","End":"02:16.860","Text":"If we take 2,"},{"Start":"02:16.860 ","End":"02:21.350","Text":"3 and 2 points is really all we need for drawing a line,"},{"Start":"02:21.350 ","End":"02:23.620","Text":"but I\u0027m not going to go further right."},{"Start":"02:23.620 ","End":"02:25.115","Text":"I\u0027ll start from here,"},{"Start":"02:25.115 ","End":"02:26.530","Text":"and it goes all the way too."},{"Start":"02:26.530 ","End":"02:29.000","Text":"Next we\u0027ll do the other side,"},{"Start":"02:29.000 ","End":"02:31.159","Text":"which we\u0027ll do in blue."},{"Start":"02:31.159 ","End":"02:33.140","Text":"Again, draw table,"},{"Start":"02:33.140 ","End":"02:38.535","Text":"only this time we start from x is 2 and continue rightward."},{"Start":"02:38.535 ","End":"02:40.965","Text":"So when x is 2 by 3,"},{"Start":"02:40.965 ","End":"02:43.410","Text":"now you might say that yes,"},{"Start":"02:43.410 ","End":"02:45.450","Text":"x is not defined at 2,"},{"Start":"02:45.450 ","End":"02:48.870","Text":"but since we\u0027ve already ascertained that this is continuous,"},{"Start":"02:48.870 ","End":"02:52.820","Text":"we\u0027re going to be able to draw it without taking a hand off the paper, so to speak."},{"Start":"02:52.820 ","End":"02:55.110","Text":"We can actually use the 2,"},{"Start":"02:55.110 ","End":"02:57.560","Text":"3 or if you like you can do"},{"Start":"02:57.560 ","End":"03:06.410","Text":"2.001 and then y will be 2.99 or something."},{"Start":"03:06.410 ","End":"03:09.750","Text":"Anyway, 2,3 is okay."},{"Start":"03:09.750 ","End":"03:17.940","Text":"Let\u0027s also do x equals 5 because then y will be 5 minus 5 is 0."},{"Start":"03:17.940 ","End":"03:25.275","Text":"So 5 is somewhere here,"},{"Start":"03:25.275 ","End":"03:31.865","Text":"so we\u0027ll continue and rightward, that\u0027s the graph."},{"Start":"03:31.865 ","End":"03:38.420","Text":"The main thing to notice about the graph is what happens in this area here,"},{"Start":"03:38.420 ","End":"03:40.140","Text":"where it\u0027s continuous,"},{"Start":"03:40.140 ","End":"03:42.455","Text":"we can draw it without taking our hand off the paper."},{"Start":"03:42.455 ","End":"03:44.970","Text":"So that\u0027s it, the sketch."}],"ID":91},{"Watched":false,"Name":"Exercise 3","Duration":"42s","ChapterTopicVideoID":93,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/93.jpeg","UploadDate":"2017-06-15T14:34:36.5030000","DurationForVideoObject":"PT42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.860","Text":"In this exercise, we have to say whether or"},{"Start":"00:02.860 ","End":"00:06.354","Text":"not the function f is continuous at x equals 0,"},{"Start":"00:06.354 ","End":"00:09.175","Text":"where f is defined piecewise as follows."},{"Start":"00:09.175 ","End":"00:14.155","Text":"Now before you start jumping to compute any limits left or right or whatever,"},{"Start":"00:14.155 ","End":"00:16.660","Text":"just notice something very important."},{"Start":"00:16.660 ","End":"00:20.860","Text":"F is defined for x bigger than 0 and x is smaller than 0,"},{"Start":"00:20.860 ","End":"00:24.000","Text":"but f is not actually defined when x equals 0."},{"Start":"00:24.000 ","End":"00:26.350","Text":"If a function is not defined,"},{"Start":"00:26.350 ","End":"00:28.375","Text":"and it can\u0027t be continuous,"},{"Start":"00:28.375 ","End":"00:32.685","Text":"so the answer is no, it isn\u0027t."},{"Start":"00:32.685 ","End":"00:34.450","Text":"If you want to add a few words,"},{"Start":"00:34.450 ","End":"00:43.420","Text":"you can basically because it\u0027s not defined at x equals 0. That\u0027s all there is to it."}],"ID":92},{"Watched":false,"Name":"Exercise 4","Duration":"2m 49s","ChapterTopicVideoID":2945,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/2945.jpeg","UploadDate":"2014-11-25T21:12:05.3530000","DurationForVideoObject":"PT2M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this exercise, we have to say whether the function"},{"Start":"00:03.990 ","End":"00:08.565","Text":"f is continuous or not at x equals 0."},{"Start":"00:08.565 ","End":"00:11.235","Text":"The function f is defined piece-wise,"},{"Start":"00:11.235 ","End":"00:13.680","Text":"defined one way for x bigger than 0,"},{"Start":"00:13.680 ","End":"00:16.965","Text":"for x equals 0, and for x smaller than 0."},{"Start":"00:16.965 ","End":"00:23.040","Text":"Basically what we have to do is to check 3 things and make sure that they\u0027re all equal."},{"Start":"00:23.040 ","End":"00:24.405","Text":"Here\u0027s what they are."},{"Start":"00:24.405 ","End":"00:32.430","Text":"We have to check the limit as x goes to 0 from the right of f of x,"},{"Start":"00:32.430 ","End":"00:40.535","Text":"then we have to check the limit of x goes to 0 from the left of f of x,"},{"Start":"00:40.535 ","End":"00:46.870","Text":"and thirdly, we have to check the actual value of f at 0 itself."},{"Start":"00:46.870 ","End":"00:51.200","Text":"Now, all these 3 things have to exist and they all have to be equal and then"},{"Start":"00:51.200 ","End":"00:56.960","Text":"we can say that f is continuous at 0. Let\u0027s see."},{"Start":"00:56.960 ","End":"00:59.015","Text":"What\u0027s the first one?"},{"Start":"00:59.015 ","End":"01:02.150","Text":"When x goes to 0 from the right,"},{"Start":"01:02.150 ","End":"01:08.255","Text":"then what we have is the sine x over x."},{"Start":"01:08.255 ","End":"01:18.520","Text":"That\u0027s the limit as x goes to 0 from the right of sine x over x."},{"Start":"01:18.520 ","End":"01:22.460","Text":"The next one is on the left,"},{"Start":"01:22.460 ","End":"01:25.835","Text":"is 1 plus e to the 1 over x."},{"Start":"01:25.835 ","End":"01:31.520","Text":"That\u0027s 1 plus e to the 1 over"},{"Start":"01:31.520 ","End":"01:37.670","Text":"x limit as x goes to 0 from the left."},{"Start":"01:37.670 ","End":"01:40.895","Text":"Finally, f of 0 itself,"},{"Start":"01:40.895 ","End":"01:43.850","Text":"which is, we\u0027ll see what it is."},{"Start":"01:43.850 ","End":"01:46.280","Text":"Let\u0027s start with that one then because it\u0027s easiest."},{"Start":"01:46.280 ","End":"01:51.500","Text":"Just put x equals 0 and we get 2 to constant function."},{"Start":"01:51.500 ","End":"01:55.460","Text":"Now back to the limits on the right."},{"Start":"01:55.460 ","End":"01:58.610","Text":"Well, it\u0027s a famous limit, sine x over x."},{"Start":"01:58.610 ","End":"02:02.015","Text":"When x goes to 0, even if it\u0027s just on the right,"},{"Start":"02:02.015 ","End":"02:05.520","Text":"this is equal to 1, famous limit."},{"Start":"02:05.520 ","End":"02:09.995","Text":"The last one, if X goes to 0 from the left,"},{"Start":"02:09.995 ","End":"02:14.015","Text":"1 over x goes to minus infinity."},{"Start":"02:14.015 ","End":"02:20.430","Text":"We get basically 1 plus e to the minus infinity,"},{"Start":"02:20.430 ","End":"02:22.835","Text":"and e to the minus infinity is 0,"},{"Start":"02:22.835 ","End":"02:25.355","Text":"so this is equal to 1."},{"Start":"02:25.355 ","End":"02:29.150","Text":"I could\u0027ve stopped earlier on once I see that 2 of them are not equal,"},{"Start":"02:29.150 ","End":"02:31.340","Text":"but I did all of them and it\u0027s a pity."},{"Start":"02:31.340 ","End":"02:32.840","Text":"We have 3 things."},{"Start":"02:32.840 ","End":"02:35.495","Text":"We have this, we have this,"},{"Start":"02:35.495 ","End":"02:37.955","Text":"and we have this, but they\u0027re not all the same."},{"Start":"02:37.955 ","End":"02:40.310","Text":"This one is the odd one out."},{"Start":"02:40.310 ","End":"02:42.725","Text":"The answer is no,"},{"Start":"02:42.725 ","End":"02:46.650","Text":"not continuous because they\u0027re not all 3."},{"Start":"02:46.650 ","End":"02:48.285","Text":"2 out of 3 is not good enough."},{"Start":"02:48.285 ","End":"02:50.680","Text":"Then we\u0027re done."}],"ID":2957},{"Watched":false,"Name":"Exercise 5","Duration":"3m 34s","ChapterTopicVideoID":2946,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/2946.jpeg","UploadDate":"2014-11-25T21:12:49.4530000","DurationForVideoObject":"PT3M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this exercise, we have to check whether or not"},{"Start":"00:02.520 ","End":"00:06.435","Text":"the function f is continuous at these 3 points,"},{"Start":"00:06.435 ","End":"00:08.175","Text":"0, 1, and 2."},{"Start":"00:08.175 ","End":"00:11.940","Text":"Notice that these are the points where f,"},{"Start":"00:11.940 ","End":"00:16.020","Text":"which is piece wise defined, changes its formula."},{"Start":"00:16.020 ","End":"00:20.295","Text":"When it crosses 0, we change formula from here to here."},{"Start":"00:20.295 ","End":"00:23.030","Text":"When we cross the 1 from left to right."},{"Start":"00:23.030 ","End":"00:25.025","Text":"We go from x square to 2 minus x."},{"Start":"00:25.025 ","End":"00:26.450","Text":"When we cross to 2,"},{"Start":"00:26.450 ","End":"00:28.130","Text":"we go from 2 minus x,"},{"Start":"00:28.130 ","End":"00:29.950","Text":"to x minus 3."},{"Start":"00:29.950 ","End":"00:34.655","Text":"These are 3 important points which to check continuity."},{"Start":"00:34.655 ","End":"00:37.790","Text":"Now remember when we check continuity at a point,"},{"Start":"00:37.790 ","End":"00:40.950","Text":"we have to check 3 things we have to check,"},{"Start":"00:40.950 ","End":"00:42.890","Text":"the limit from the left,"},{"Start":"00:42.890 ","End":"00:44.210","Text":"the limit from the right,"},{"Start":"00:44.210 ","End":"00:45.755","Text":"and the value at the point."},{"Start":"00:45.755 ","End":"00:47.970","Text":"That\u0027s for 0, and also for 1,"},{"Start":"00:47.970 ","End":"00:49.750","Text":"and also for 2."},{"Start":"00:49.750 ","End":"00:53.690","Text":"All 3 have to be the same for each of these points,"},{"Start":"00:53.690 ","End":"00:56.525","Text":"in order for it to be continuous at these points."},{"Start":"00:56.525 ","End":"00:59.480","Text":"I wrote all this stuff in advance,"},{"Start":"00:59.480 ","End":"01:02.435","Text":"looks like I even did the first 1,"},{"Start":"01:02.435 ","End":"01:05.000","Text":"carried away there. Let\u0027s check."},{"Start":"01:05.000 ","End":"01:07.315","Text":"We start off with x is 0."},{"Start":"01:07.315 ","End":"01:11.290","Text":"As x goes to 0 from the left for using the formula sine x,"},{"Start":"01:11.290 ","End":"01:12.785","Text":"and when x goes to 0,"},{"Start":"01:12.785 ","End":"01:14.870","Text":"sine x goes to 0."},{"Start":"01:14.870 ","End":"01:17.360","Text":"When x goes to 0 from the right,"},{"Start":"01:17.360 ","End":"01:21.220","Text":"however, then we\u0027re going to use the x squared formula."},{"Start":"01:21.220 ","End":"01:26.780","Text":"The limit we just substitute x equals 0 and x squared is also 0."},{"Start":"01:26.780 ","End":"01:30.350","Text":"That holds for here for the limit as well as for the value at the point."},{"Start":"01:30.350 ","End":"01:33.160","Text":"In both cases we\u0027re using the x squared."},{"Start":"01:33.160 ","End":"01:36.315","Text":"Next, at 1,"},{"Start":"01:36.315 ","End":"01:41.130","Text":"we\u0027re changing from x squared to 2 minus x at the right,"},{"Start":"01:41.130 ","End":"01:43.450","Text":"is where when x equals 1,"},{"Start":"01:43.450 ","End":"01:45.140","Text":"we take our value from the right."},{"Start":"01:45.140 ","End":"01:47.290","Text":"In other words, from 2 minus x,"},{"Start":"01:47.290 ","End":"01:51.315","Text":"which is 2 minus 1, which is 1."},{"Start":"01:51.315 ","End":"01:54.485","Text":"That\u0027s also when x goes to 1 from the right."},{"Start":"01:54.485 ","End":"01:56.060","Text":"We\u0027re also using this formula,"},{"Start":"01:56.060 ","End":"01:57.905","Text":"so this is also 1."},{"Start":"01:57.905 ","End":"02:02.040","Text":"From the left we\u0027re using the formula x squared."},{"Start":"02:02.040 ","End":"02:08.300","Text":"To find the limit, we substitute x equals 1 into x squared and that\u0027s also equal to 1."},{"Start":"02:08.300 ","End":"02:09.800","Text":"So far, so good."},{"Start":"02:09.800 ","End":"02:13.170","Text":"Now let\u0027s try the 2,"},{"Start":"02:13.180 ","End":"02:17.765","Text":"here we change from 2 minus x to x minus 3."},{"Start":"02:17.765 ","End":"02:19.760","Text":"But at x equals 2,"},{"Start":"02:19.760 ","End":"02:21.470","Text":"we use this 1,"},{"Start":"02:21.470 ","End":"02:23.485","Text":"the x minus 3."},{"Start":"02:23.485 ","End":"02:25.225","Text":"When x is 2,"},{"Start":"02:25.225 ","End":"02:27.755","Text":"then we have 2 minus 3,"},{"Start":"02:27.755 ","End":"02:29.510","Text":"and that\u0027s minus 1,"},{"Start":"02:29.510 ","End":"02:35.410","Text":"and that\u0027s also what happens when x equals 2."},{"Start":"02:35.410 ","End":"02:36.925","Text":"It\u0027s also minus 1."},{"Start":"02:36.925 ","End":"02:39.695","Text":"Now the only 1 more to check,"},{"Start":"02:39.695 ","End":"02:42.154","Text":"and that\u0027s when x goes to 2 from the left."},{"Start":"02:42.154 ","End":"02:44.915","Text":"When x goes to 2 from the left,"},{"Start":"02:44.915 ","End":"02:48.320","Text":"we have to substitute x equals 2 in this formula,"},{"Start":"02:48.320 ","End":"02:50.600","Text":"2 minus x. Oh dear,"},{"Start":"02:50.600 ","End":"02:52.880","Text":"it doesn\u0027t come out the same."},{"Start":"02:52.880 ","End":"02:54.815","Text":"What can we say?"},{"Start":"02:54.815 ","End":"02:56.420","Text":"Well, as a whole,"},{"Start":"02:56.420 ","End":"02:58.130","Text":"the function is not continuous,"},{"Start":"02:58.130 ","End":"03:01.370","Text":"but at x equals 0, this,"},{"Start":"03:01.370 ","End":"03:03.560","Text":"this, and this are the same,"},{"Start":"03:03.560 ","End":"03:06.380","Text":"so it is continuous at 0."},{"Start":"03:06.380 ","End":"03:08.975","Text":"Similarly here we have 1, 1,"},{"Start":"03:08.975 ","End":"03:12.410","Text":"and 1, so it is continuous."},{"Start":"03:12.410 ","End":"03:16.325","Text":"But here the 2 is the odd 1 out,"},{"Start":"03:16.325 ","End":"03:20.570","Text":"so the answer is that it is not continuous."},{"Start":"03:20.570 ","End":"03:22.780","Text":"2 out of 3,"},{"Start":"03:22.780 ","End":"03:25.230","Text":"that\u0027s not too bad."},{"Start":"03:25.230 ","End":"03:28.160","Text":"As far as I can\u0027t give a yes or no for all 3,"},{"Start":"03:28.160 ","End":"03:30.425","Text":"but here I can say that,"},{"Start":"03:30.425 ","End":"03:33.170","Text":"yes, yes, no."},{"Start":"03:33.170 ","End":"03:35.880","Text":"That\u0027s it. We\u0027re done."}],"ID":2958},{"Watched":false,"Name":"Exercise 6","Duration":"3m 50s","ChapterTopicVideoID":2947,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/2947.jpeg","UploadDate":"2014-11-25T21:13:37.1730000","DurationForVideoObject":"PT3M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we have to check whether or not f is"},{"Start":"00:03.120 ","End":"00:07.875","Text":"continuous at x equals 1 and also at x equals 2."},{"Start":"00:07.875 ","End":"00:12.930","Text":"Function is defined piecewise and the reason that we asked to"},{"Start":"00:12.930 ","End":"00:17.894","Text":"find at 1 and 2 is because here the function changes its formula."},{"Start":"00:17.894 ","End":"00:19.785","Text":"These are transition points."},{"Start":"00:19.785 ","End":"00:22.950","Text":"For example, at x equals 1,"},{"Start":"00:22.950 ","End":"00:29.535","Text":"we change from the left to the right and the same at x crosses 2,"},{"Start":"00:29.535 ","End":"00:32.580","Text":"we actually have 3 different formulae,"},{"Start":"00:32.580 ","End":"00:34.215","Text":"to the left of 2,"},{"Start":"00:34.215 ","End":"00:36.645","Text":"at 2 and to the right of 2."},{"Start":"00:36.645 ","End":"00:40.050","Text":"We have to check several things."},{"Start":"00:40.050 ","End":"00:43.775","Text":"What we have to do is to check of 6 things,"},{"Start":"00:43.775 ","End":"00:51.845","Text":"3 for each point and we have to check the limit"},{"Start":"00:51.845 ","End":"01:01.100","Text":"as x goes to 1 from the left of f of x, what this equals."},{"Start":"01:01.100 ","End":"01:06.900","Text":"Then we have to check the limit as x goes to 1 from"},{"Start":"01:06.900 ","End":"01:14.000","Text":"the right of f of x and also the value at the point itself."},{"Start":"01:14.000 ","End":"01:18.380","Text":"What f equals when x equals 1."},{"Start":"01:18.380 ","End":"01:20.690","Text":"Then after we\u0027ve checked these 3,"},{"Start":"01:20.690 ","End":"01:22.705","Text":"that\u0027s for x equals 1,"},{"Start":"01:22.705 ","End":"01:24.960","Text":"we\u0027ll have to check the limit."},{"Start":"01:24.960 ","End":"01:26.130","Text":"Same thing as these,"},{"Start":"01:26.130 ","End":"01:27.895","Text":"just with 2 instead of 1."},{"Start":"01:27.895 ","End":"01:32.975","Text":"X goes to 2 from the left of f of x."},{"Start":"01:32.975 ","End":"01:39.350","Text":"The limit as x goes to 2 from the right of f of x"},{"Start":"01:39.350 ","End":"01:46.980","Text":"and also the value of the function f at 2 itself and see what that equals."},{"Start":"01:46.980 ","End":"01:49.620","Text":"X equals 1 first."},{"Start":"01:49.620 ","End":"01:51.590","Text":"The limit from the left,"},{"Start":"01:51.590 ","End":"01:55.595","Text":"we use the formula 1 over x and when x goes to 1,"},{"Start":"01:55.595 ","End":"01:59.665","Text":"we just get 1 over 1, which is 1."},{"Start":"01:59.665 ","End":"02:03.170","Text":"Also because we use this formula then at 1 also,"},{"Start":"02:03.170 ","End":"02:07.525","Text":"it\u0027s going to equal 1 because we\u0027re using it from the same formula."},{"Start":"02:07.525 ","End":"02:10.730","Text":"Now the limit as x goes to 1 from the right,"},{"Start":"02:10.730 ","End":"02:16.805","Text":"then we have to use this formula and we just have to substitute x equals 1,"},{"Start":"02:16.805 ","End":"02:20.960","Text":"so it\u0027s 1 minus 2 and the absolute value of that,"},{"Start":"02:20.960 ","End":"02:24.865","Text":"absolute value of minus 1 is just 1."},{"Start":"02:24.865 ","End":"02:26.720","Text":"Here we\u0027re fine."},{"Start":"02:26.720 ","End":"02:28.820","Text":"We have limit from the left,"},{"Start":"02:28.820 ","End":"02:29.855","Text":"limit from the right,"},{"Start":"02:29.855 ","End":"02:32.480","Text":"value at the point or equal."},{"Start":"02:32.480 ","End":"02:38.380","Text":"Here, it is continuous when x is equal to 1."},{"Start":"02:38.380 ","End":"02:41.820","Text":"Now let\u0027s try x equals 2."},{"Start":"02:41.820 ","End":"02:44.410","Text":"We need these 3."},{"Start":"02:44.410 ","End":"02:47.599","Text":"When x goes to 2 from the left,"},{"Start":"02:47.599 ","End":"02:50.570","Text":"we\u0027re using this formula and we just substitute"},{"Start":"02:50.570 ","End":"02:55.805","Text":"absolute value of 2 minus 2 comes out to be 0."},{"Start":"02:55.805 ","End":"02:58.805","Text":"If we check the limit from the right,"},{"Start":"02:58.805 ","End":"03:00.755","Text":"then we\u0027re going to use this formula."},{"Start":"03:00.755 ","End":"03:04.579","Text":"We just have 2 minus 2, which is 0,"},{"Start":"03:04.579 ","End":"03:09.094","Text":"but the value of f at the point x equals 2 itself,"},{"Start":"03:09.094 ","End":"03:10.775","Text":"we take from this formula,"},{"Start":"03:10.775 ","End":"03:12.230","Text":"and it doesn\u0027t matter what x is,"},{"Start":"03:12.230 ","End":"03:15.610","Text":"is a constant function and this equals 1,"},{"Start":"03:15.610 ","End":"03:20.825","Text":"which is a pity because not all 3 are the same."},{"Start":"03:20.825 ","End":"03:23.240","Text":"This is the same and this is the same,"},{"Start":"03:23.240 ","End":"03:25.145","Text":"but this is the odd 1 out."},{"Start":"03:25.145 ","End":"03:29.750","Text":"Because of this, we\u0027re not all the same or because of those 2,"},{"Start":"03:29.750 ","End":"03:31.220","Text":"if you like to look at it that way."},{"Start":"03:31.220 ","End":"03:33.785","Text":"In any event they\u0027re not all 3 the same,"},{"Start":"03:33.785 ","End":"03:39.890","Text":"the answer is that it is not continuous when x is equal to 2."},{"Start":"03:39.890 ","End":"03:41.915","Text":"Here it is, and here it isn\u0027t."},{"Start":"03:41.915 ","End":"03:45.680","Text":"When I\u0027m asked what happens at x equals 1 and 2,"},{"Start":"03:45.680 ","End":"03:51.060","Text":"I say, here it is and here it isn\u0027t and that\u0027s it, we\u0027re done."}],"ID":2959},{"Watched":false,"Name":"Exercise 7","Duration":"2m 55s","ChapterTopicVideoID":2948,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/2948.jpeg","UploadDate":"2014-11-25T21:14:13.5870000","DurationForVideoObject":"PT2M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have to find the value of k,"},{"Start":"00:03.300 ","End":"00:06.015","Text":"which makes f continuous everywhere."},{"Start":"00:06.015 ","End":"00:08.805","Text":"Notice that f is defined piecewise,"},{"Start":"00:08.805 ","End":"00:11.700","Text":"here in 1 place and here in another place,"},{"Start":"00:11.700 ","End":"00:15.105","Text":"k is a number or sometimes called a parameter."},{"Start":"00:15.105 ","End":"00:19.050","Text":"I\u0027ll also note that when x is not equal to 2,"},{"Start":"00:19.050 ","End":"00:23.250","Text":"there is no problem at all with the continuity because this is an elementary function,"},{"Start":"00:23.250 ","End":"00:26.865","Text":"even a polynomial, and it\u0027s going to be continuous everywhere."},{"Start":"00:26.865 ","End":"00:30.810","Text":"The only problem that could go wrong is that a dividing line transition"},{"Start":"00:30.810 ","End":"00:35.720","Text":"from 1 formula to another and this is what happens when x equals 2."},{"Start":"00:35.720 ","End":"00:41.200","Text":"Now I\u0027ve already written something because we would do this thing very often."},{"Start":"00:41.200 ","End":"00:43.235","Text":"Continuity at a point,"},{"Start":"00:43.235 ","End":"00:47.450","Text":"in this case 2 is determined by the equality of 3 things."},{"Start":"00:47.450 ","End":"00:51.455","Text":"The limit as x goes to 2 from the right of the function,"},{"Start":"00:51.455 ","End":"00:54.244","Text":"the limit as x goes to 2 from the left,"},{"Start":"00:54.244 ","End":"00:57.980","Text":"and the value at the point itself at the point 2."},{"Start":"00:57.980 ","End":"01:00.725","Text":"If these 3 things turn out to be equal,"},{"Start":"01:00.725 ","End":"01:03.290","Text":"then the function f is going to be continuous,"},{"Start":"01:03.290 ","End":"01:05.560","Text":"not only at 2 but everywhere."},{"Start":"01:05.560 ","End":"01:08.030","Text":"Let\u0027s do some substitution."},{"Start":"01:08.030 ","End":"01:10.370","Text":"If x goes to 2 from the right,"},{"Start":"01:10.370 ","End":"01:13.040","Text":"2 from the right means that we\u0027re bigger than 2,"},{"Start":"01:13.040 ","End":"01:15.935","Text":"that means that we\u0027re working in this formula here."},{"Start":"01:15.935 ","End":"01:18.640","Text":"Then all we do is put x,"},{"Start":"01:18.640 ","End":"01:23.395","Text":"we can just substitute x equals 2 in this formula and we get"},{"Start":"01:23.395 ","End":"01:30.425","Text":"5 times k times 2 from the x minus 6,"},{"Start":"01:30.425 ","End":"01:34.355","Text":"and this is equal to 10k minus 6."},{"Start":"01:34.355 ","End":"01:36.290","Text":"Next on the left."},{"Start":"01:36.290 ","End":"01:41.465","Text":"Well, same thing except that we are going to be using this formula and put x equals 2 in"},{"Start":"01:41.465 ","End":"01:49.875","Text":"here so we get k times 2 squared plus 2 minus 2,"},{"Start":"01:49.875 ","End":"01:52.755","Text":"and this gives us 4K."},{"Start":"01:52.755 ","End":"01:55.170","Text":"The value at 2, well,"},{"Start":"01:55.170 ","End":"01:58.570","Text":"its just going to be the same as from this formula here."},{"Start":"01:58.570 ","End":"02:01.355","Text":"What we get is basically 4k."},{"Start":"02:01.355 ","End":"02:08.125","Text":"Now we found the 3 quantities and I\u0027ll just underline them."},{"Start":"02:08.125 ","End":"02:10.945","Text":"On the 1 hand, we have the limit from the right,"},{"Start":"02:10.945 ","End":"02:12.460","Text":"on the other hand,"},{"Start":"02:12.460 ","End":"02:14.870","Text":"we have the limit from the left,"},{"Start":"02:14.870 ","End":"02:19.480","Text":"on the third hand we have the value at the point itself of the function."},{"Start":"02:19.480 ","End":"02:22.540","Text":"Well, this is certainly equal regardless of k,"},{"Start":"02:22.540 ","End":"02:25.390","Text":"so what we have left to do now"},{"Start":"02:25.390 ","End":"02:29.210","Text":"is to make sure that these 2 are equal and then we are all set."},{"Start":"02:29.210 ","End":"02:37.730","Text":"What we need is the equation that 10k minus 6 must equal 4k."},{"Start":"02:37.730 ","End":"02:40.610","Text":"Now if we do small bit of algebra,"},{"Start":"02:40.610 ","End":"02:44.975","Text":"just take the 4k to the other side and it becomes 6k,"},{"Start":"02:44.975 ","End":"02:48.245","Text":"bring the 6 over to the other side is 6."},{"Start":"02:48.245 ","End":"02:52.585","Text":"All we get is that k is equal to 1,"},{"Start":"02:52.585 ","End":"02:55.810","Text":"and that\u0027s it. We\u0027re done."}],"ID":2960},{"Watched":false,"Name":"Exercise 8","Duration":"5m 13s","ChapterTopicVideoID":2949,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/2949.jpeg","UploadDate":"2014-11-25T21:15:18.2370000","DurationForVideoObject":"PT5M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.695","Text":"Here we have to find k,"},{"Start":"00:01.695 ","End":"00:06.870","Text":"which makes the function f continuous for every x. K is over here,"},{"Start":"00:06.870 ","End":"00:09.360","Text":"it\u0027s a constant parameter."},{"Start":"00:09.360 ","End":"00:15.960","Text":"F is defined piecewise and notice that f is actually defined for all x."},{"Start":"00:15.960 ","End":"00:21.720","Text":"The domain is all real numbers because for x equals 1, it\u0027s explicitly given."},{"Start":"00:21.720 ","End":"00:24.045","Text":"When x is not equal to 1,"},{"Start":"00:24.045 ","End":"00:26.910","Text":"it\u0027s a polynomial over a polynomial and"},{"Start":"00:26.910 ","End":"00:30.524","Text":"the denominator is not 0, it\u0027s defined everywhere."},{"Start":"00:30.524 ","End":"00:34.440","Text":"Also noticed that when x is not equal to 1, again,"},{"Start":"00:34.440 ","End":"00:36.780","Text":"because it\u0027s a rational function or a polynomial over"},{"Start":"00:36.780 ","End":"00:41.320","Text":"polynomial, it\u0027s continuous everywhere."},{"Start":"00:41.320 ","End":"00:47.105","Text":"It\u0027s defined as an elementary function and the denominator is not 0."},{"Start":"00:47.105 ","End":"00:52.220","Text":"The only thing we have to worry about is the x equals 1."},{"Start":"00:52.220 ","End":"00:54.765","Text":"Everywhere else, regardless of k,"},{"Start":"00:54.765 ","End":"00:56.460","Text":"in fact k doesn\u0027t even appear there,"},{"Start":"00:56.460 ","End":"00:58.485","Text":"it\u0027s going to be continuous."},{"Start":"00:58.485 ","End":"01:01.745","Text":"Now, what makes it continuous when x equals 1?"},{"Start":"01:01.745 ","End":"01:04.805","Text":"This is what we need to check,"},{"Start":"01:04.805 ","End":"01:07.595","Text":"is we check the usual 3 things."},{"Start":"01:07.595 ","End":"01:11.360","Text":"We check the limit from the left,"},{"Start":"01:11.360 ","End":"01:12.380","Text":"limit from the right,"},{"Start":"01:12.380 ","End":"01:15.035","Text":"and the value of the function at the point."},{"Start":"01:15.035 ","End":"01:19.205","Text":"First of all, just compute the following 3 things."},{"Start":"01:19.205 ","End":"01:22.980","Text":"We need the limit as x goes to 1,"},{"Start":"01:22.980 ","End":"01:28.430","Text":"from 1 side that say from the left of f of x."},{"Start":"01:28.430 ","End":"01:32.105","Text":"Second thing we have to compute is the same thing,"},{"Start":"01:32.105 ","End":"01:36.290","Text":"except that goes to 1 on the other side and"},{"Start":"01:36.290 ","End":"01:41.565","Text":"the third thing we have to compute is f of 1 itself."},{"Start":"01:41.565 ","End":"01:43.955","Text":"We would like these 3 things to be equal,"},{"Start":"01:43.955 ","End":"01:47.460","Text":"and then it will be continuous at x equals 1."},{"Start":"01:47.660 ","End":"01:50.660","Text":"Why don\u0027t I go for the easy stuff first?"},{"Start":"01:50.660 ","End":"01:56.510","Text":"When x f of 1 just means that x equals 1 so the answer is k. Well, that was easy."},{"Start":"01:56.510 ","End":"02:01.234","Text":"Now, in both of these formula from the left or the right,"},{"Start":"02:01.234 ","End":"02:03.320","Text":"we\u0027re going to try, first of all,"},{"Start":"02:03.320 ","End":"02:06.545","Text":"substituting x equals 1."},{"Start":"02:06.545 ","End":"02:10.520","Text":"Well, that won\u0027t do because if as soon as we see the denominator,"},{"Start":"02:10.520 ","End":"02:11.725","Text":"when x is 1,"},{"Start":"02:11.725 ","End":"02:14.715","Text":"then that\u0027s here is 0."},{"Start":"02:14.715 ","End":"02:19.765","Text":"In fact in the numerator also 1 squared plus twice 1 minus 3 also 0."},{"Start":"02:19.765 ","End":"02:23.585","Text":"It\u0027s 0 over 0 for both of these,"},{"Start":"02:23.585 ","End":"02:29.480","Text":"just to indicate that both these cases are cases of 0 over 0."},{"Start":"02:29.480 ","End":"02:35.075","Text":"We have to do something else and that is in this case to"},{"Start":"02:35.075 ","End":"02:37.790","Text":"try and factorize the numerator and"},{"Start":"02:37.790 ","End":"02:41.390","Text":"hopefully something cancels and then we will be able to substitute."},{"Start":"02:41.390 ","End":"02:44.855","Text":"I drew a little margin we can do some calculations."},{"Start":"02:44.855 ","End":"02:47.845","Text":"What I want to do is factorize the numerator,"},{"Start":"02:47.845 ","End":"02:51.260","Text":"I\u0027ll just first write what it is that I\u0027m doing."},{"Start":"02:51.260 ","End":"03:01.160","Text":"I\u0027m doing the limit as x goes to 1 minus of x squared plus 2x minus 3."},{"Start":"03:01.160 ","End":"03:04.755","Text":"Just copying over x minus 1."},{"Start":"03:04.755 ","End":"03:09.680","Text":"What I\u0027d like to do is to factorize here x minus 1."},{"Start":"03:09.680 ","End":"03:11.810","Text":"I want to factorize the numerator."},{"Start":"03:11.810 ","End":"03:19.230","Text":"I\u0027ll do that in the margin so x squared plus 2x minus 3 is going to be"},{"Start":"03:19.230 ","End":"03:27.255","Text":"equal to x minus x1 times x minus x2."},{"Start":"03:27.255 ","End":"03:31.820","Text":"Where x1 and X2 are the roots of this polynomial,"},{"Start":"03:31.820 ","End":"03:35.615","Text":"which means the solution and what I get when I make it equal to 0."},{"Start":"03:35.615 ","End":"03:40.020","Text":"If I do x squared plus 2x minus 3 equals 0."},{"Start":"03:40.020 ","End":"03:42.410","Text":"I don\u0027t want to bore you with"},{"Start":"03:42.410 ","End":"03:46.145","Text":"all the details of solving a quadratic equation, you can do that."},{"Start":"03:46.145 ","End":"03:48.875","Text":"I\u0027ll just give you the answer right away."},{"Start":"03:48.875 ","End":"03:55.810","Text":"We get that 1 of the x\u0027s anyway is equal to 1 and the other 1,"},{"Start":"03:55.810 ","End":"04:01.090","Text":"let\u0027s call it x2 is equal to minus 3."},{"Start":"04:01.090 ","End":"04:05.060","Text":"You could check by substitution or you could just solve it yourselves,"},{"Start":"04:05.060 ","End":"04:08.255","Text":"which means that I can write this here,"},{"Start":"04:08.255 ","End":"04:12.615","Text":"x minus 1, x plus 3."},{"Start":"04:12.615 ","End":"04:18.090","Text":"Now, here x tends to 1 but it isn\u0027t equal to 1 and this is not x minus 1,"},{"Start":"04:18.090 ","End":"04:21.945","Text":"isn\u0027t 0 so we can cancel."},{"Start":"04:21.945 ","End":"04:30.290","Text":"Now all we have to do is plug in x equals 1 and we get 1 plus 3 is equal to 4."},{"Start":"04:30.290 ","End":"04:35.140","Text":"Now, there was absolutely no difference if I let x go to 1 from the right,"},{"Start":"04:35.140 ","End":"04:38.985","Text":"I\u0027m just going to copy the result here that\u0027s also equal to 4."},{"Start":"04:38.985 ","End":"04:45.260","Text":"Now if I use a bit of color here and highlight that 4,"},{"Start":"04:45.260 ","End":"04:48.110","Text":"it happens to be that way and that\u0027s good,"},{"Start":"04:48.110 ","End":"04:49.970","Text":"that\u0027s 2 out of 3 equal."},{"Start":"04:49.970 ","End":"04:52.295","Text":"The final quantity, the third 1 is k,"},{"Start":"04:52.295 ","End":"04:53.780","Text":"how can I make 4,"},{"Start":"04:53.780 ","End":"04:55.265","Text":"4, and k all equal?"},{"Start":"04:55.265 ","End":"04:56.705","Text":"Well are already are equal."},{"Start":"04:56.705 ","End":"04:59.900","Text":"All I have to do is say,"},{"Start":"04:59.900 ","End":"05:03.015","Text":"if I put k equals 4,"},{"Start":"05:03.015 ","End":"05:07.775","Text":"then all these 3 things are equal and that\u0027s enough for continuity at x equals 1."},{"Start":"05:07.775 ","End":"05:10.085","Text":"Like we said, it\u0027s continuous everywhere else."},{"Start":"05:10.085 ","End":"05:13.860","Text":"Anyway, this is the condition and we\u0027re done."}],"ID":2961},{"Watched":false,"Name":"Exercise 9","Duration":"6m 22s","ChapterTopicVideoID":2950,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/2950.jpeg","UploadDate":"2014-11-25T21:15:47.6970000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"Here again, we have to find k,"},{"Start":"00:01.650 ","End":"00:04.650","Text":"which makes the function f continuous for every x."},{"Start":"00:04.650 ","End":"00:06.270","Text":"Let\u0027s first of all,"},{"Start":"00:06.270 ","End":"00:08.880","Text":"take a look at the domain."},{"Start":"00:08.880 ","End":"00:13.305","Text":"The domain is actually all the real numbers, every x."},{"Start":"00:13.305 ","End":"00:17.490","Text":"For 2, it\u0027s defined explicitly and when it\u0027s not 2,"},{"Start":"00:17.490 ","End":"00:20.190","Text":"let\u0027s see if there are any problems."},{"Start":"00:20.190 ","End":"00:23.190","Text":"Denominator is not 0, fine."},{"Start":"00:23.190 ","End":"00:25.560","Text":"The other thing that could go wrong is we could have"},{"Start":"00:25.560 ","End":"00:28.485","Text":"the square root of a negative number, but here not."},{"Start":"00:28.485 ","End":"00:30.930","Text":"Because x squared is non negative,"},{"Start":"00:30.930 ","End":"00:33.150","Text":"and if you add 5 then it\u0027s at least 5."},{"Start":"00:33.150 ","End":"00:34.950","Text":"In any event, it\u0027s not negative."},{"Start":"00:34.950 ","End":"00:38.150","Text":"It\u0027s actually defined everywhere."},{"Start":"00:38.150 ","End":"00:41.990","Text":"Also, note that when x is not equal to 2"},{"Start":"00:41.990 ","End":"00:46.925","Text":"here that it\u0027s continuous because it\u0027s an elementary function."},{"Start":"00:46.925 ","End":"00:51.470","Text":"It\u0027s just square roots and pluses, minuses, divides."},{"Start":"00:51.470 ","End":"00:57.340","Text":"This is continuous everywhere except for when x is not equal 2."},{"Start":"00:57.340 ","End":"00:59.810","Text":"Just to summarize, so far,"},{"Start":"00:59.810 ","End":"01:04.390","Text":"f is continuous everywhere when x is not equal to 2"},{"Start":"01:04.390 ","End":"01:09.320","Text":"and we only have to check its continuity at x equals 2."},{"Start":"01:09.320 ","End":"01:13.595","Text":"We do that in the usual way by writing the 3 quantities."},{"Start":"01:13.595 ","End":"01:15.365","Text":"Let\u0027s see now,"},{"Start":"01:15.365 ","End":"01:25.040","Text":"we have to write the limit as x goes to 2 from the left limit,"},{"Start":"01:25.040 ","End":"01:27.440","Text":"x goes to 2 from the right."},{"Start":"01:27.440 ","End":"01:29.090","Text":"I\u0027ll finish the line in a minute."},{"Start":"01:29.090 ","End":"01:33.910","Text":"We need to find f of 2."},{"Start":"01:33.910 ","End":"01:37.290","Text":"Here it\u0027s f of x,"},{"Start":"01:37.290 ","End":"01:41.880","Text":"and here it\u0027s also f of x."},{"Start":"01:41.880 ","End":"01:44.280","Text":"Let\u0027s start with the easy 1,"},{"Start":"01:44.280 ","End":"01:47.295","Text":"f of 2 when x is 2,"},{"Start":"01:47.295 ","End":"01:55.275","Text":"f is just k. That\u0027s k. Let\u0027s try 2 from the left."},{"Start":"01:55.275 ","End":"01:59.175","Text":"2 from the left. If we substitute the"},{"Start":"01:59.175 ","End":"02:03.530","Text":"2 and we get a 0 in the denominator and in the numerator,"},{"Start":"02:03.530 ","End":"02:05.435","Text":"2 squared plus 5 is 9."},{"Start":"02:05.435 ","End":"02:07.205","Text":"Square root of 3 minus 3."},{"Start":"02:07.205 ","End":"02:10.595","Text":"In other words, we get 0 over 0 here,"},{"Start":"02:10.595 ","End":"02:13.325","Text":"and we\u0027ll also get 0 over 0 here."},{"Start":"02:13.325 ","End":"02:16.805","Text":"We have to use other techniques."},{"Start":"02:16.805 ","End":"02:21.110","Text":"This looks like a good case for using the conjugate."},{"Start":"02:21.110 ","End":"02:27.265","Text":"We have differences of sum of terms and 1 of them or more of them are the square roots."},{"Start":"02:27.265 ","End":"02:33.170","Text":"The usual technique is to multiply top and bottom by the conjugate."},{"Start":"02:33.770 ","End":"02:38.190","Text":"Here we get is equal to."},{"Start":"02:38.190 ","End":"02:41.300","Text":"Both here and here we\u0027ll have the same formula."},{"Start":"02:41.300 ","End":"02:48.420","Text":"We have the square root of x squared plus 5 minus"},{"Start":"02:48.420 ","End":"02:52.770","Text":"3 over x minus"},{"Start":"02:52.770 ","End":"02:58.085","Text":"2 and to multiply top and bottom by the same thing doesn\u0027t change anything."},{"Start":"02:58.085 ","End":"03:04.275","Text":"We get the square root of x squared plus 5,"},{"Start":"03:04.275 ","End":"03:08.310","Text":"this time with a plus 3 over the same thing."},{"Start":"03:08.310 ","End":"03:13.760","Text":"Square root of x squared plus 5 plus 3."},{"Start":"03:13.760 ","End":"03:17.060","Text":"We multiply top and bottom by the conjugate of this."},{"Start":"03:17.060 ","End":"03:25.670","Text":"I\u0027ll just give you a quick reminder what the conjugate of a minus b is a plus b."},{"Start":"03:25.670 ","End":"03:30.459","Text":"The advantage of a conjugate is that if we multiply them together,"},{"Start":"03:30.459 ","End":"03:32.240","Text":"1 with plus and 1 with minus,"},{"Start":"03:32.240 ","End":"03:34.940","Text":"we get the famous difference of squares formula,"},{"Start":"03:34.940 ","End":"03:36.800","Text":"a squared minus b squared,"},{"Start":"03:36.800 ","End":"03:38.905","Text":"and that gets rid of square roots."},{"Start":"03:38.905 ","End":"03:42.945","Text":"This is equal to x squared plus 5,"},{"Start":"03:42.945 ","End":"03:46.550","Text":"without the square root minus 3 squared."},{"Start":"03:46.550 ","End":"03:49.655","Text":"Just write that as 3 squared for the moment."},{"Start":"03:49.655 ","End":"03:55.805","Text":"In the bottom, x minus 2 times the second thing,"},{"Start":"03:55.805 ","End":"04:04.405","Text":"which is the square root of x squared plus 5 plus 3."},{"Start":"04:04.405 ","End":"04:10.790","Text":"Let\u0027s see, I need a bit more space to continue and then this equals x"},{"Start":"04:10.790 ","End":"04:17.420","Text":"squared plus 5 minus 9 is x squared minus 4."},{"Start":"04:17.420 ","End":"04:22.155","Text":"Here we have the same thing x"},{"Start":"04:22.155 ","End":"04:26.675","Text":"minus 2 times square root"},{"Start":"04:26.675 ","End":"04:33.155","Text":"of x squared plus 5 plus 3."},{"Start":"04:33.155 ","End":"04:36.770","Text":"Because of lack of space here,"},{"Start":"04:36.770 ","End":"04:40.310","Text":"what I\u0027m going to do is continue down here."},{"Start":"04:40.310 ","End":"04:43.850","Text":"This equals, now this is also a difference of squares."},{"Start":"04:43.850 ","End":"04:46.615","Text":"This is x minus 2,"},{"Start":"04:46.615 ","End":"04:51.420","Text":"x plus 2 over x"},{"Start":"04:51.420 ","End":"05:00.175","Text":"minus 2 times square root of x squared plus 5 plus 3."},{"Start":"05:00.175 ","End":"05:03.755","Text":"I forgot to write the limits everywhere."},{"Start":"05:03.755 ","End":"05:07.475","Text":"Here it\u0027s all the same x goes to 2 from the left."},{"Start":"05:07.475 ","End":"05:12.185","Text":"Then the x minus 2 cancels and all we have to do,"},{"Start":"05:12.185 ","End":"05:13.925","Text":"I\u0027ll remind you here,"},{"Start":"05:13.925 ","End":"05:16.310","Text":"x goes to 2 from the left,"},{"Start":"05:16.310 ","End":"05:19.865","Text":"is now all we have to do is substitute x equals 2."},{"Start":"05:19.865 ","End":"05:24.275","Text":"We get 2 plus 2 is 4,"},{"Start":"05:24.275 ","End":"05:30.765","Text":"4 over 6, which equals 2/3."},{"Start":"05:30.765 ","End":"05:32.960","Text":"Sorry about the convoluted way around."},{"Start":"05:32.960 ","End":"05:36.035","Text":"Now, exactly the same thing is going to happen here"},{"Start":"05:36.035 ","End":"05:40.620","Text":"because it\u0027s just nothing is different except where x goes to."},{"Start":"05:40.620 ","End":"05:43.140","Text":"I\u0027m going to reach the same expression and"},{"Start":"05:43.140 ","End":"05:46.245","Text":"a substitute is no difference 2 from the left or right."},{"Start":"05:46.245 ","End":"05:48.780","Text":"This is also going to be 2/3."},{"Start":"05:48.780 ","End":"05:52.040","Text":"Now if I use my coloring,"},{"Start":"05:52.040 ","End":"05:55.280","Text":"then the 3 things I\u0027ve computed,"},{"Start":"05:55.280 ","End":"05:57.140","Text":"this is 2/3,"},{"Start":"05:57.140 ","End":"05:59.900","Text":"this is 2/3,"},{"Start":"05:59.900 ","End":"06:04.055","Text":"and this is k. For it to be continuous everywhere,"},{"Start":"06:04.055 ","End":"06:08.135","Text":"it already is continuous here we want to continue so that x equals 2."},{"Start":"06:08.135 ","End":"06:12.595","Text":"For that condition, all these 3 yellow things have got to be equal."},{"Start":"06:12.595 ","End":"06:17.045","Text":"Ultimately, all I have to require is that k"},{"Start":"06:17.045 ","End":"06:22.680","Text":"is equal to 2/3 and that\u0027s the answer and we\u0027re done."}],"ID":2962},{"Watched":false,"Name":"Exercise 10","Duration":"8m 20s","ChapterTopicVideoID":3296,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3296.jpeg","UploadDate":"2014-12-25T09:55:48.3700000","DurationForVideoObject":"PT8M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this exercise, we have to find the value of k,"},{"Start":"00:03.600 ","End":"00:07.755","Text":"which makes the function f continuous for every x."},{"Start":"00:07.755 ","End":"00:11.505","Text":"Note that it\u0027s defined for all x."},{"Start":"00:11.505 ","End":"00:13.470","Text":"When x is positive,"},{"Start":"00:13.470 ","End":"00:21.045","Text":"any positive number to any power is defined and when x is smaller or equal to 0,"},{"Start":"00:21.045 ","End":"00:22.710","Text":"2x is certainly defined,"},{"Start":"00:22.710 ","End":"00:25.860","Text":"take-away a number k. So it\u0027s defined everywhere."},{"Start":"00:25.860 ","End":"00:29.160","Text":"Now, notice that it\u0027s defined in 2 different"},{"Start":"00:29.160 ","End":"00:34.640","Text":"formulae in different places and there\u0027s like a seam line at x equals 0,"},{"Start":"00:34.640 ","End":"00:36.415","Text":"a seam or a border."},{"Start":"00:36.415 ","End":"00:41.150","Text":"The only problem is going to occur at the seam where we might have a discontinuity."},{"Start":"00:41.150 ","End":"00:46.805","Text":"But in each separate piece in x less than 0, we have continuity."},{"Start":"00:46.805 ","End":"00:49.340","Text":"When x is less than 0,"},{"Start":"00:49.340 ","End":"00:51.000","Text":"we have continuity,"},{"Start":"00:51.000 ","End":"00:53.369","Text":"when x is bigger than 0,"},{"Start":"00:53.369 ","End":"00:55.425","Text":"we have continuity,"},{"Start":"00:55.425 ","End":"00:58.685","Text":"when x equals 0,"},{"Start":"00:58.685 ","End":"01:03.575","Text":"that is the question mark that we have to find out if it\u0027s continuous or not."},{"Start":"01:03.575 ","End":"01:08.825","Text":"Or rather, what value does k have to be to make it continuous?"},{"Start":"01:08.825 ","End":"01:10.715","Text":"The usual method is,"},{"Start":"01:10.715 ","End":"01:14.240","Text":"what we apply here is to compute 3 different things."},{"Start":"01:14.240 ","End":"01:15.920","Text":"Left limit, right limit,"},{"Start":"01:15.920 ","End":"01:18.740","Text":"and the value at the point itself."},{"Start":"01:18.740 ","End":"01:25.400","Text":"What we have to compute is the limit as x goes to"},{"Start":"01:25.400 ","End":"01:32.225","Text":"0 from the left of f of x and see what that equals."},{"Start":"01:32.225 ","End":"01:39.650","Text":"Take the limit as x goes to 0 from the right of f of x and see"},{"Start":"01:39.650 ","End":"01:47.010","Text":"what that equals and then to take f of 0 itself and see what that limit is."},{"Start":"01:47.010 ","End":"01:49.145","Text":"If these 3 numbers are all equal,"},{"Start":"01:49.145 ","End":"01:52.775","Text":"then the function will be continuous at x equals 0."},{"Start":"01:52.775 ","End":"01:55.235","Text":"This first 1 here,"},{"Start":"01:55.235 ","End":"02:00.450","Text":"you take the value of x from where x is negative."},{"Start":"02:00.450 ","End":"02:02.310","Text":"We just substitute."},{"Start":"02:02.310 ","End":"02:09.465","Text":"The answer is equals the limit of 2x minus k as x"},{"Start":"02:09.465 ","End":"02:17.615","Text":"goes to 0 from the left and that just equals minus k is put x equals 0."},{"Start":"02:17.615 ","End":"02:27.840","Text":"Now, f of 0 itself is also taken from this formula and it\u0027s just x equals 0,"},{"Start":"02:27.840 ","End":"02:32.010","Text":"so it\u0027s minus k. That\u0027s already 2 out of 3."},{"Start":"02:32.010 ","End":"02:36.260","Text":"The third 1 is what we have to take the limit as x goes to"},{"Start":"02:36.260 ","End":"02:41.495","Text":"0 from the positive direction of x^2x."},{"Start":"02:41.495 ","End":"02:50.585","Text":"That is the limit x goes to 0 plus of x^2x."},{"Start":"02:50.585 ","End":"02:54.410","Text":"This one\u0027s a little bit more difficult to tackle."},{"Start":"02:54.410 ","End":"02:59.105","Text":"I wrote a little margin here so we can do some calculations and formulae."},{"Start":"02:59.105 ","End":"03:03.800","Text":"What we need to do here is to write an exponent."},{"Start":"03:03.800 ","End":"03:11.745","Text":"In general, a^b is equal to e. I can always put"},{"Start":"03:11.745 ","End":"03:16.010","Text":"an exponent in terms of e and then what I do is I copy the b from"},{"Start":"03:16.010 ","End":"03:20.555","Text":"there and put the natural log in front of what was the base."},{"Start":"03:20.555 ","End":"03:22.700","Text":"That\u0027s how you remember this formula."},{"Start":"03:22.700 ","End":"03:24.260","Text":"In any event, in our case,"},{"Start":"03:24.260 ","End":"03:28.610","Text":"if I take a as x and b as 2x,"},{"Start":"03:28.610 ","End":"03:35.560","Text":"I will get that x^2x is equal to e to the power of,"},{"Start":"03:35.560 ","End":"03:44.205","Text":"the b is 2x and a is also x natural log of x."},{"Start":"03:44.205 ","End":"03:50.150","Text":"This is equal to, it\u0027s the limit of e to the power of something."},{"Start":"03:50.150 ","End":"03:55.760","Text":"But the limit has the property that with most functions other than the exponential,"},{"Start":"03:55.760 ","End":"03:58.070","Text":"we can put the limit inside,"},{"Start":"03:58.070 ","End":"04:03.010","Text":"by which I mean that it\u0027s not just the limit of e to the power of something,"},{"Start":"04:03.010 ","End":"04:06.635","Text":"I can also write it as e to the power of the limit."},{"Start":"04:06.635 ","End":"04:09.830","Text":"In other words, limit and e are interchangeable."},{"Start":"04:09.830 ","End":"04:14.630","Text":"We have e to the power of limit of 2x,"},{"Start":"04:14.630 ","End":"04:17.455","Text":"natural log of x."},{"Start":"04:17.455 ","End":"04:21.735","Text":"The limit is as x goes to 0 plus."},{"Start":"04:21.735 ","End":"04:25.520","Text":"Perhaps I should have written the limit. That\u0027s what it is."},{"Start":"04:25.520 ","End":"04:32.060","Text":"I was going to write the limit of e to the power of whatever."},{"Start":"04:32.060 ","End":"04:39.800","Text":"This is the same thing as e to the power of the limit of this, whatever it was."},{"Start":"04:39.800 ","End":"04:46.605","Text":"Now what I\u0027m going to do is to do this limit as a side exercise."},{"Start":"04:46.605 ","End":"04:48.290","Text":"This portion here,"},{"Start":"04:48.290 ","End":"04:53.600","Text":"let\u0027s say I\u0027ll put it in a little box and I\u0027ll call"},{"Start":"04:53.600 ","End":"05:01.460","Text":"this asterisk and do this calculation over here in the work area."},{"Start":"05:01.460 ","End":"05:07.475","Text":"This asterisk will certainly the 2 makes no difference."},{"Start":"05:07.475 ","End":"05:09.425","Text":"Well, we can take it outside."},{"Start":"05:09.425 ","End":"05:13.260","Text":"This thing here is"},{"Start":"05:13.260 ","End":"05:22.355","Text":"twice the limit as x goes to 0 plus of x natural log of x."},{"Start":"05:22.355 ","End":"05:27.290","Text":"But when x goes to 0 plus we get a 0 times"},{"Start":"05:27.290 ","End":"05:32.495","Text":"minus infinity and then wanted to use L\u0027Hopital and his rule,"},{"Start":"05:32.495 ","End":"05:35.690","Text":"and he works better with quotients than products."},{"Start":"05:35.690 ","End":"05:41.825","Text":"What we can do is quick algebraic manipulation here and say that instead of x,"},{"Start":"05:41.825 ","End":"05:43.355","Text":"natural log of x,"},{"Start":"05:43.355 ","End":"05:48.845","Text":"we can put the natural log of x on the numerator and instead of x on the numerator,"},{"Start":"05:48.845 ","End":"05:52.040","Text":"we could put 1 over x in the denominator."},{"Start":"05:52.040 ","End":"05:53.750","Text":"Why is this good?"},{"Start":"05:53.750 ","End":"05:58.700","Text":"Because now we get a situation of infinity over infinity."},{"Start":"05:58.700 ","End":"06:00.470","Text":"Well, not exactly."},{"Start":"06:00.470 ","End":"06:04.475","Text":"Here we get a minus infinity when x goes to 0 plus,"},{"Start":"06:04.475 ","End":"06:06.485","Text":"and here we get a plus infinity."},{"Start":"06:06.485 ","End":"06:11.630","Text":"But L\u0027Hopital works with minus infinity over infinity also."},{"Start":"06:11.630 ","End":"06:16.250","Text":"What this means is that now I\u0027m going to use his rule and"},{"Start":"06:16.250 ","End":"06:20.810","Text":"we differentiate the numerator and the denominator."},{"Start":"06:20.810 ","End":"06:24.230","Text":"It\u0027s twice the limit of,"},{"Start":"06:24.230 ","End":"06:30.375","Text":"now this derivative is 1 over x over that\u0027s just dividing line."},{"Start":"06:30.375 ","End":"06:36.140","Text":"The 1 over x is minus 1 over x squared,"},{"Start":"06:36.140 ","End":"06:41.360","Text":"which is, this goes to 0 plus and this is equal to,"},{"Start":"06:41.360 ","End":"06:43.490","Text":"1 over x over 1 over x squared,"},{"Start":"06:43.490 ","End":"06:44.690","Text":"if you do the algebra,"},{"Start":"06:44.690 ","End":"06:47.060","Text":"the x squared floats to the top."},{"Start":"06:47.060 ","End":"06:48.695","Text":"X squared over x is x."},{"Start":"06:48.695 ","End":"06:51.085","Text":"It\u0027s just minus x."},{"Start":"06:51.085 ","End":"06:54.990","Text":"The minus I put it in front, minus 2,"},{"Start":"06:54.990 ","End":"07:00.945","Text":"lim x goes to 0 plus of just plain x."},{"Start":"07:00.945 ","End":"07:04.590","Text":"When x goes to 0, x goes to 0."},{"Start":"07:04.590 ","End":"07:14.150","Text":"This thing is going to equal 0 because the x goes to 0 plus or whatever then x goes to 0,"},{"Start":"07:14.150 ","End":"07:18.259","Text":"but this 0 is not our final answer."},{"Start":"07:18.259 ","End":"07:25.065","Text":"This 0, that\u0027s now our asterisk so we have to put it back here."},{"Start":"07:25.065 ","End":"07:31.410","Text":"This answer is equal to e^0 from here,"},{"Start":"07:31.410 ","End":"07:34.055","Text":"and e^0 is 1."},{"Start":"07:34.055 ","End":"07:42.365","Text":"1 is the answer to this second thing and perhaps if I color."},{"Start":"07:42.365 ","End":"07:44.345","Text":"Let\u0027s see, so we first of all,"},{"Start":"07:44.345 ","End":"07:45.950","Text":"got the left limit,"},{"Start":"07:45.950 ","End":"07:47.735","Text":"we got minus k,"},{"Start":"07:47.735 ","End":"07:53.160","Text":"for the value of the function at the point we got minus k,"},{"Start":"07:53.160 ","End":"07:55.995","Text":"and for the right limit we got 1."},{"Start":"07:55.995 ","End":"07:59.465","Text":"How do I make all these 3 things equal?"},{"Start":"07:59.465 ","End":"08:06.530","Text":"Well, all I have to do for those to be equal is that minus k should equal 1."},{"Start":"08:06.530 ","End":"08:11.325","Text":"What I need is that minus k should equal 1,"},{"Start":"08:11.325 ","End":"08:15.720","Text":"or that k is equal to minus 1."},{"Start":"08:15.720 ","End":"08:21.090","Text":"That\u0027s the answer we\u0027re looking for. Now we\u0027re done."}],"ID":3307},{"Watched":false,"Name":"Exercise 11","Duration":"6m 45s","ChapterTopicVideoID":3308,"CourseChapterTopicPlaylistID":84396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3308.jpeg","UploadDate":"2015-01-06T22:44:36.0000000","DurationForVideoObject":"PT6M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.845","Text":"Here we have to find the values of the parameters or constants a and b"},{"Start":"00:04.845 ","End":"00:10.575","Text":"that they used in the definition of f so that it\u0027s continuous everywhere in its domain."},{"Start":"00:10.575 ","End":"00:13.395","Text":"Well, it\u0027s defined everywhere."},{"Start":"00:13.395 ","End":"00:17.175","Text":"Everywhere it\u0027s an elementary function,"},{"Start":"00:17.175 ","End":"00:19.620","Text":"so anywhere inside its domain,"},{"Start":"00:19.620 ","End":"00:23.580","Text":"except possibly for 0 and Pi,"},{"Start":"00:23.580 ","End":"00:27.600","Text":"which are crossover transition points,"},{"Start":"00:27.600 ","End":"00:30.840","Text":"it\u0027s going to be continuous."},{"Start":"00:30.840 ","End":"00:35.690","Text":"Then we\u0027re going to have to figure out what to do with the 0 and with Pi."},{"Start":"00:35.690 ","End":"00:38.000","Text":"I\u0027ll just write that down."},{"Start":"00:38.000 ","End":"00:42.110","Text":"If x is not equal to"},{"Start":"00:42.110 ","End":"00:48.425","Text":"0 and x is not equal to Pi,"},{"Start":"00:48.425 ","End":"00:52.835","Text":"then f of x is continuous."},{"Start":"00:52.835 ","End":"00:56.509","Text":"It\u0027s continuous everywhere except 0 or Pi."},{"Start":"00:56.509 ","End":"01:00.230","Text":"Now what happens when x equals 0?"},{"Start":"01:00.230 ","End":"01:01.280","Text":"I\u0027m going to have to do"},{"Start":"01:01.280 ","End":"01:05.015","Text":"the usual exercise limit from the left and the right and the value at the point,"},{"Start":"01:05.015 ","End":"01:07.640","Text":"then the same again for x equals Pi."},{"Start":"01:07.640 ","End":"01:10.625","Text":"That\u0027s the work we\u0027re going to do now."},{"Start":"01:10.625 ","End":"01:15.935","Text":"Let\u0027s get on to the first case where x is equal to 0."},{"Start":"01:15.935 ","End":"01:19.895","Text":"Let\u0027s take x equals 0 first."},{"Start":"01:19.895 ","End":"01:27.615","Text":"We need the limit as x goes to 0 plus f of x,"},{"Start":"01:27.615 ","End":"01:35.075","Text":"we need the limit as x goes to 0 minus f of X,"},{"Start":"01:35.075 ","End":"01:38.935","Text":"and we need f of 0."},{"Start":"01:38.935 ","End":"01:43.620","Text":"Hope that all these 3 things exist and they\u0027re equal."},{"Start":"01:43.620 ","End":"01:48.585","Text":"On the other side, I\u0027ll just work in 2 columns,"},{"Start":"01:48.585 ","End":"01:53.130","Text":"we\u0027ll also take the x equals Pi and again,"},{"Start":"01:53.130 ","End":"02:00.960","Text":"do the same computations limit as x goes to Pi from the right,"},{"Start":"02:00.960 ","End":"02:03.840","Text":"let\u0027s say, of f of x."},{"Start":"02:03.840 ","End":"02:07.095","Text":"Then from the left limit,"},{"Start":"02:07.095 ","End":"02:12.600","Text":"x goes to Pi minus f of x."},{"Start":"02:12.600 ","End":"02:17.980","Text":"Finally, the value of Pi itself in the function."},{"Start":"02:17.980 ","End":"02:19.730","Text":"Let\u0027s see what we get."},{"Start":"02:19.730 ","End":"02:21.180","Text":"If all these 3 are good,"},{"Start":"02:21.180 ","End":"02:23.420","Text":"then 0 is good."},{"Start":"02:23.420 ","End":"02:26.690","Text":"If all these 3 are equal,"},{"Start":"02:26.690 ","End":"02:28.115","Text":"then this 1 is good,"},{"Start":"02:28.115 ","End":"02:30.200","Text":"meaning it\u0027s continuous there."},{"Start":"02:30.200 ","End":"02:34.530","Text":"Maybe something will come out simple."},{"Start":"02:35.270 ","End":"02:39.000","Text":"The right is sine x over 2x."},{"Start":"02:39.000 ","End":"02:41.620","Text":"That\u0027s the right of 0."},{"Start":"02:43.730 ","End":"02:54.150","Text":"On the left of 0, we\u0027ve got ax plus b and the 0 itself,"},{"Start":"02:54.150 ","End":"02:56.190","Text":"we have from this formula also,"},{"Start":"02:56.190 ","End":"02:58.305","Text":"a times 0 plus b,"},{"Start":"02:58.305 ","End":"03:00.735","Text":"it\u0027s just equal to b."},{"Start":"03:00.735 ","End":"03:02.380","Text":"When x goes to 0,"},{"Start":"03:02.380 ","End":"03:04.780","Text":"it\u0027s the same as substituting x equals 0."},{"Start":"03:04.780 ","End":"03:07.000","Text":"It\u0027ll be the same answer as here,"},{"Start":"03:07.000 ","End":"03:09.330","Text":"and it will be b."},{"Start":"03:09.330 ","End":"03:11.030","Text":"That\u0027s already these 2."},{"Start":"03:11.030 ","End":"03:12.580","Text":"Now what about this?"},{"Start":"03:12.580 ","End":"03:17.210","Text":"I think I\u0027ll do this exercise a bit below."},{"Start":"03:17.210 ","End":"03:20.310","Text":"The limit as, I can write it,"},{"Start":"03:20.310 ","End":"03:22.215","Text":"x goes to 0 here,"},{"Start":"03:22.215 ","End":"03:24.645","Text":"x goes to 0."},{"Start":"03:24.645 ","End":"03:29.960","Text":"This limit, never mind the 1-sided or 2-sided limited exists,"},{"Start":"03:29.960 ","End":"03:32.045","Text":"and we\u0027ve seen it before."},{"Start":"03:32.045 ","End":"03:35.470","Text":"When x goes to 0 of sine x over x."},{"Start":"03:35.470 ","End":"03:37.755","Text":"This thing, the limit,"},{"Start":"03:37.755 ","End":"03:38.960","Text":"the famous limit,"},{"Start":"03:38.960 ","End":"03:46.100","Text":"when x goes to 0 of sine x over x,"},{"Start":"03:46.100 ","End":"03:48.245","Text":"that\u0027s equal to 1."},{"Start":"03:48.245 ","End":"03:49.820","Text":"If that\u0027s equal to 1,"},{"Start":"03:49.820 ","End":"03:53.059","Text":"here we have an extra 2 here at the denominator,"},{"Start":"03:53.059 ","End":"03:55.505","Text":"so that will equal 1/2."},{"Start":"03:55.505 ","End":"03:58.790","Text":"That will hold true from the left or from the right or from both."},{"Start":"03:58.790 ","End":"04:00.320","Text":"Anyway, from all this,"},{"Start":"04:00.320 ","End":"04:04.930","Text":"we\u0027ve got the 3 values we wanted and they all have to be the same."},{"Start":"04:04.930 ","End":"04:08.760","Text":"The 1/2 has to be the same as b."},{"Start":"04:08.760 ","End":"04:10.710","Text":"Here\u0027s our b again."},{"Start":"04:10.710 ","End":"04:16.530","Text":"These 3 things have to be equal and what that will tell"},{"Start":"04:16.530 ","End":"04:24.345","Text":"us is that b is equal to 1/2."},{"Start":"04:24.345 ","End":"04:26.555","Text":"Now that\u0027s just part of the answer."},{"Start":"04:26.555 ","End":"04:29.840","Text":"We haven\u0027t still figured out what a is and I presume"},{"Start":"04:29.840 ","End":"04:33.685","Text":"that the second part with Pi is going to give us that."},{"Start":"04:33.685 ","End":"04:35.935","Text":"Let\u0027s see what does happen."},{"Start":"04:35.935 ","End":"04:39.300","Text":"The limit is"},{"Start":"04:50.480 ","End":"04:54.715","Text":"x goes to Pi plus of a cosine x."},{"Start":"04:54.715 ","End":"04:56.855","Text":"To the left of Pi,"},{"Start":"04:56.855 ","End":"04:59.330","Text":"we had the formula as,"},{"Start":"04:59.330 ","End":"05:04.595","Text":"x goes to Pi from the right of sine x over 2x."},{"Start":"05:04.595 ","End":"05:07.700","Text":"Let\u0027s see what this thing is. I\u0027m sorry."},{"Start":"05:07.700 ","End":"05:11.480","Text":"Cosine of Pi is minus 1,"},{"Start":"05:11.480 ","End":"05:13.710","Text":"doesn\u0027t matter from which side."},{"Start":"05:13.710 ","End":"05:18.260","Text":"That\u0027s minus 1. What we get is that the limit as x goes to Pi of this,"},{"Start":"05:18.260 ","End":"05:22.340","Text":"is just equal to minus a."},{"Start":"05:22.340 ","End":"05:25.369","Text":"On the other side, when x goes to Pi,"},{"Start":"05:25.369 ","End":"05:28.760","Text":"and sine x, which is sine of Pi is 0."},{"Start":"05:28.760 ","End":"05:30.260","Text":"But 2 Pi is not 0,"},{"Start":"05:30.260 ","End":"05:32.750","Text":"so we have 0 over non-zero."},{"Start":"05:32.750 ","End":"05:37.045","Text":"Of course, this is equal to just 0."},{"Start":"05:37.045 ","End":"05:39.525","Text":"As for f of Pi,"},{"Start":"05:39.525 ","End":"05:45.210","Text":"like I said, the Pi part comes from here."},{"Start":"05:45.210 ","End":"05:51.360","Text":"So a cosine Pi is a times cosine Pi is minus 1,"},{"Start":"05:51.360 ","End":"05:52.905","Text":"so that\u0027s minus a."},{"Start":"05:52.905 ","End":"05:55.305","Text":"The minus a comes from the top part."},{"Start":"05:55.305 ","End":"05:59.055","Text":"This is equal to minus a."},{"Start":"05:59.055 ","End":"06:01.275","Text":"Getting back up a bit."},{"Start":"06:01.275 ","End":"06:03.090","Text":"What we have here,"},{"Start":"06:03.090 ","End":"06:05.300","Text":"of these 3 limits, on the 1 hand,"},{"Start":"06:05.300 ","End":"06:07.235","Text":"we have the minus a,"},{"Start":"06:07.235 ","End":"06:09.424","Text":"on the 1 hand, the 0,"},{"Start":"06:09.424 ","End":"06:10.775","Text":"and then the 3rd"},{"Start":"06:10.775 ","End":"06:13.700","Text":"hand we have the minus a."},{"Start":"06:13.700 ","End":"06:17.460","Text":"All these 3 have to be equal to each other."},{"Start":"06:17.470 ","End":"06:22.715","Text":"Since they do, we get the a must also be equal to 0,"},{"Start":"06:22.715 ","End":"06:25.445","Text":"so a is equal to 0."},{"Start":"06:25.445 ","End":"06:29.960","Text":"That means that together with this and this,"},{"Start":"06:29.960 ","End":"06:32.060","Text":"then this function,"},{"Start":"06:32.060 ","End":"06:38.795","Text":"which we defined above is going to be continuous everywhere,"},{"Start":"06:38.795 ","End":"06:40.880","Text":"not just at 1 and Pi."},{"Start":"06:40.880 ","End":"06:46.200","Text":"But for all real numbers it will be continuous. That\u0027s it."}],"ID":3319}],"Thumbnail":null,"ID":84396},{"Name":"Points of Discontinuity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Points of Discontinuity - Part 1","Duration":"8m 26s","ChapterTopicVideoID":13851,"CourseChapterTopicPlaylistID":84397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13851.jpeg","UploadDate":"2019-11-14T06:52:11.4670000","DurationForVideoObject":"PT8M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this clip, we\u0027ll be discussing the different kinds of"},{"Start":"00:02.970 ","End":"00:05.670","Text":"discontinuity of a function at a point."},{"Start":"00:05.670 ","End":"00:08.520","Text":"I\u0027m assuming you\u0027ve already studied continuity,"},{"Start":"00:08.520 ","End":"00:11.115","Text":"but I\u0027ll bring you a reminder anyway."},{"Start":"00:11.115 ","End":"00:17.010","Text":"A function f is called continuous at a point x equals a if"},{"Start":"00:17.010 ","End":"00:22.620","Text":"the following holds: the limit as x goes to a from the right of the function,"},{"Start":"00:22.620 ","End":"00:26.390","Text":"and the limit of x goes to a from the left of the function that these 2 are"},{"Start":"00:26.390 ","End":"00:31.460","Text":"equal and they\u0027re also equal to the value of the function at the point a."},{"Start":"00:31.460 ","End":"00:34.040","Text":"There is something that\u0027s implied but not written,"},{"Start":"00:34.040 ","End":"00:37.230","Text":"we only define continuity for points a that"},{"Start":"00:37.230 ","End":"00:40.490","Text":"are in the domain of f. I\u0027m assuming a is in the domain,"},{"Start":"00:40.490 ","End":"00:45.335","Text":"which means that f is defined at a. f of a is actually equal to some finite number,"},{"Start":"00:45.335 ","End":"00:49.865","Text":"and that implies that these 2 also have to exist and they have to be finite."},{"Start":"00:49.865 ","End":"00:54.965","Text":"Now from the definition of continuity comes the definition of discontinuity."},{"Start":"00:54.965 ","End":"00:58.790","Text":"This brings me to the definition of discontinuity,"},{"Start":"00:58.790 ","End":"01:01.295","Text":"simply, the opposite of continuity."},{"Start":"01:01.295 ","End":"01:04.970","Text":"Thus, a function f is called discontinuous at"},{"Start":"01:04.970 ","End":"01:09.215","Text":"a point x equals a if f is not continuous at x equals a."},{"Start":"01:09.215 ","End":"01:11.480","Text":"But I\u0027d like to emphasize that in both cases of"},{"Start":"01:11.480 ","End":"01:15.185","Text":"continuity and discontinuity at a point x equals a,"},{"Start":"01:15.185 ","End":"01:17.360","Text":"that first f has to be defined there."},{"Start":"01:17.360 ","End":"01:19.340","Text":"In other words, a is in the domain of f,"},{"Start":"01:19.340 ","End":"01:24.335","Text":"otherwise it doesn\u0027t make sense to talk about continuity or discontinuity of f at a."},{"Start":"01:24.335 ","End":"01:25.790","Text":"If f is not continuous,"},{"Start":"01:25.790 ","End":"01:28.070","Text":"it means that somehow this equation is broken"},{"Start":"01:28.070 ","End":"01:31.130","Text":"down and it can break down in any number of ways."},{"Start":"01:31.130 ","End":"01:35.960","Text":"For instance, we might have that this limit exists and is equal to 6,"},{"Start":"01:35.960 ","End":"01:38.000","Text":"and this limit exists and is equal to 6,"},{"Start":"01:38.000 ","End":"01:41.300","Text":"but the value of the function at the point is 4."},{"Start":"01:41.300 ","End":"01:42.905","Text":"That\u0027s 1 way it could break down."},{"Start":"01:42.905 ","End":"01:45.620","Text":"Another way it could break down is that 1 of"},{"Start":"01:45.620 ","End":"01:49.445","Text":"these 2 limits doesn\u0027t exist at all or is infinite,"},{"Start":"01:49.445 ","End":"01:52.625","Text":"because then they can\u0027t equal f of a because that\u0027s finite,"},{"Start":"01:52.625 ","End":"01:56.300","Text":"and another way it could break down is that everything here exists,"},{"Start":"01:56.300 ","End":"02:00.265","Text":"but that this limit maybe is 5 and this limit is 3,"},{"Start":"02:00.265 ","End":"02:02.480","Text":"I don\u0027t even care what f of a is."},{"Start":"02:02.480 ","End":"02:05.375","Text":"There are different ways that this equality could break down,"},{"Start":"02:05.375 ","End":"02:07.310","Text":"either by numbers not being equal,"},{"Start":"02:07.310 ","End":"02:09.335","Text":"or by 1 of these limits being bad,"},{"Start":"02:09.335 ","End":"02:11.480","Text":"meaning infinite or undefined."},{"Start":"02:11.480 ","End":"02:15.680","Text":"This leads us to a classification of different kinds of discontinuity."},{"Start":"02:15.680 ","End":"02:19.150","Text":"We\u0027re actually going to define 3 different kinds of discontinuity,"},{"Start":"02:19.150 ","End":"02:22.715","Text":"and the first 1 will be removable discontinuity."},{"Start":"02:22.715 ","End":"02:24.485","Text":"To explain this concept,"},{"Start":"02:24.485 ","End":"02:26.320","Text":"let me start with an example."},{"Start":"02:26.320 ","End":"02:29.885","Text":"Here, f of x is defined piecewise where"},{"Start":"02:29.885 ","End":"02:33.710","Text":"it\u0027s equal to x squared for x which is smaller than 1,"},{"Start":"02:33.710 ","End":"02:36.320","Text":"it\u0027s equal to 4 when x is exactly 1,"},{"Start":"02:36.320 ","End":"02:40.030","Text":"and it\u0027s equal to 2x minus 1 when x is greater than 1."},{"Start":"02:40.030 ","End":"02:43.740","Text":"I\u0027d like to check continuity where x equals 1."},{"Start":"02:43.740 ","End":"02:48.590","Text":"What we\u0027ll do is to check those 3 quantities that are in the definition."},{"Start":"02:48.590 ","End":"02:51.980","Text":"In other words, we\u0027re going to check what is the limit"},{"Start":"02:51.980 ","End":"02:56.370","Text":"as x goes to 1 from the right of f of x,"},{"Start":"02:56.370 ","End":"03:01.755","Text":"what is the limit as x goes to 1 from the left of f of x,"},{"Start":"03:01.755 ","End":"03:04.730","Text":"and lastly, what is the value of the function at the point,"},{"Start":"03:04.730 ","End":"03:05.915","Text":"what is f of 1?"},{"Start":"03:05.915 ","End":"03:08.690","Text":"Those are the 3 quantities that we always check."},{"Start":"03:08.690 ","End":"03:12.995","Text":"I note that f is defined at x equals 1."},{"Start":"03:12.995 ","End":"03:14.810","Text":"As I mentioned before, that we only check"},{"Start":"03:14.810 ","End":"03:18.260","Text":"continuity or discontinuity at a point in the domain."},{"Start":"03:18.260 ","End":"03:19.670","Text":"The first 1,"},{"Start":"03:19.670 ","End":"03:24.800","Text":"this is equal to the limit as x goes to 1 from the right,"},{"Start":"03:24.800 ","End":"03:28.970","Text":"and here from the right means we take our reading from this last row"},{"Start":"03:28.970 ","End":"03:33.000","Text":"because we\u0027re bigger than 1 of 2x minus 1,"},{"Start":"03:33.000 ","End":"03:35.580","Text":"and I can substitute x equals 1 here to get the answer."},{"Start":"03:35.580 ","End":"03:37.825","Text":"Twice 1 minus 1 is equal to 1."},{"Start":"03:37.825 ","End":"03:38.990","Text":"In the second 1,"},{"Start":"03:38.990 ","End":"03:43.550","Text":"we get the limit as x goes to 1 from the left we\u0027re on the left,"},{"Start":"03:43.550 ","End":"03:44.660","Text":"so we\u0027re less than 1,"},{"Start":"03:44.660 ","End":"03:46.760","Text":"so we take it from this here."},{"Start":"03:46.760 ","End":"03:48.155","Text":"It\u0027s x squared,"},{"Start":"03:48.155 ","End":"03:50.150","Text":"and if I substitute x equals 1,"},{"Start":"03:50.150 ","End":"03:52.210","Text":"I get 1. f of 1 itself,"},{"Start":"03:52.210 ","End":"03:55.550","Text":"I look up at the table, it\u0027s given explicitly for x equals 1,"},{"Start":"03:55.550 ","End":"03:58.010","Text":"the value of the function is 4."},{"Start":"03:58.010 ","End":"04:00.110","Text":"These are the 3 numbers I get, 1,"},{"Start":"04:00.110 ","End":"04:02.540","Text":"1, and 4, and I can\u0027t say they\u0027re all equal."},{"Start":"04:02.540 ","End":"04:04.850","Text":"I started out lucky with 1 equaling 1,"},{"Start":"04:04.850 ","End":"04:06.455","Text":"but then this 4 spoiled it."},{"Start":"04:06.455 ","End":"04:09.200","Text":"Now this is an example of a particular situation,"},{"Start":"04:09.200 ","End":"04:11.750","Text":"I\u0027m leading up to this removable discontinuity."},{"Start":"04:11.750 ","End":"04:17.960","Text":"In general, if I have that the limit as x goes to a from the right,"},{"Start":"04:17.960 ","End":"04:20.190","Text":"where I\u0027m just generalizing in our case it was 1,"},{"Start":"04:20.190 ","End":"04:22.145","Text":"in general will be the point a."},{"Start":"04:22.145 ","End":"04:27.785","Text":"If this happens to equal the limit as x goes to a"},{"Start":"04:27.785 ","End":"04:34.039","Text":"from the left and not equal to the value at the point,"},{"Start":"04:34.039 ","End":"04:35.855","Text":"which is f of a,"},{"Start":"04:35.855 ","End":"04:37.260","Text":"then we\u0027ve got 2 out of 3,"},{"Start":"04:37.260 ","End":"04:39.645","Text":"but that\u0027s not good enough for it to be continuous,"},{"Start":"04:39.645 ","End":"04:44.600","Text":"but the kind of discontinuity is called removable discontinuity."},{"Start":"04:44.600 ","End":"04:50.090","Text":"What I would say would be f has a removable discontinuity at x equals a,"},{"Start":"04:50.090 ","End":"04:54.050","Text":"and it\u0027s removable because it\u0027s so easy to make it continuous."},{"Start":"04:54.050 ","End":"04:56.840","Text":"If I just change the value to 1,"},{"Start":"04:56.840 ","End":"04:58.620","Text":"not to equal 4,"},{"Start":"04:58.620 ","End":"05:00.190","Text":"but to also equal 1,"},{"Start":"05:00.190 ","End":"05:01.580","Text":"then I\u0027d get 1, 1, 1,"},{"Start":"05:01.580 ","End":"05:03.095","Text":"and that all would be well."},{"Start":"05:03.095 ","End":"05:05.825","Text":"Now let\u0027s get onto the next kind."},{"Start":"05:05.825 ","End":"05:08.885","Text":"It\u0027s called the discontinuity of the first kind,"},{"Start":"05:08.885 ","End":"05:14.545","Text":"but it\u0027s also called a jump discontinuity or even a step discontinuity,"},{"Start":"05:14.545 ","End":"05:17.150","Text":"and I\u0027ll feel free to use any 1 of these."},{"Start":"05:17.150 ","End":"05:19.145","Text":"Let\u0027s look at the following function."},{"Start":"05:19.145 ","End":"05:21.245","Text":"f of x equals,"},{"Start":"05:21.245 ","End":"05:23.255","Text":"it\u0027s defined in 3 ways,"},{"Start":"05:23.255 ","End":"05:27.665","Text":"x squared when x is less than 1,"},{"Start":"05:27.665 ","End":"05:30.875","Text":"4 when x is equal to 1,"},{"Start":"05:30.875 ","End":"05:36.150","Text":"and 4x plus 1 when x is greater than 1."},{"Start":"05:36.150 ","End":"05:39.430","Text":"I\u0027d like to know what happens at x equals 1."},{"Start":"05:39.430 ","End":"05:43.070","Text":"If the function is continuous or discontinuous,"},{"Start":"05:43.070 ","End":"05:45.520","Text":"and if it\u0027s discontinuous, what kind?"},{"Start":"05:45.520 ","End":"05:48.005","Text":"We check the usual 3 quantities,"},{"Start":"05:48.005 ","End":"05:56.540","Text":"which is the limit as x goes to 1 from the right of f of x and we see what that is,"},{"Start":"05:56.540 ","End":"06:04.125","Text":"and then we check the limit as x goes to 1 from the left of what f of x is,"},{"Start":"06:04.125 ","End":"06:09.170","Text":"and finally, we check f at the point itself and see what that is."},{"Start":"06:09.170 ","End":"06:10.340","Text":"Because if they\u0027re all the same,"},{"Start":"06:10.340 ","End":"06:11.600","Text":"it\u0027s going to be continuous,"},{"Start":"06:11.600 ","End":"06:15.710","Text":"and if not, it\u0027s discontinuous and we\u0027re going to classify what kind."},{"Start":"06:15.710 ","End":"06:18.320","Text":"f of x as x goes to 1 from the right,"},{"Start":"06:18.320 ","End":"06:21.140","Text":"that means we go on the last row because it\u0027s bigger than 1,"},{"Start":"06:21.140 ","End":"06:26.690","Text":"and so we just have to substitute x equals 1 here and we get 4 times 1 plus 1,"},{"Start":"06:26.690 ","End":"06:28.699","Text":"which is equal to 5."},{"Start":"06:28.699 ","End":"06:32.280","Text":"Next, I go to the limit as x goes to 1 from the left,"},{"Start":"06:32.280 ","End":"06:35.555","Text":"which means that we\u0027re on the top row because it\u0027s smaller than 1,"},{"Start":"06:35.555 ","End":"06:37.820","Text":"and just substitute x equals 1,"},{"Start":"06:37.820 ","End":"06:40.385","Text":"we get 1 squared, which equals 1."},{"Start":"06:40.385 ","End":"06:45.230","Text":"Now, as soon as I see that these 2 1-sided limits are not equal to each other,"},{"Start":"06:45.230 ","End":"06:47.750","Text":"I don\u0027t care what happens at 1 itself."},{"Start":"06:47.750 ","End":"06:49.205","Text":"It happens to equal 4,"},{"Start":"06:49.205 ","End":"06:54.080","Text":"but I don\u0027t care about this because already when I see that these 2 are different,"},{"Start":"06:54.080 ","End":"06:56.720","Text":"then that\u0027s the sign that the function has"},{"Start":"06:56.720 ","End":"07:00.290","Text":"a discontinuity of the first kind or a jump discontinuity."},{"Start":"07:00.290 ","End":"07:02.750","Text":"Let\u0027s write down the formal definition of"},{"Start":"07:02.750 ","End":"07:07.645","Text":"a jump discontinuity and we\u0027ll write it as follows."},{"Start":"07:07.645 ","End":"07:11.060","Text":"Definition. The function f of x has"},{"Start":"07:11.060 ","End":"07:17.375","Text":"a jump discontinuity at the point x equals a if both of the 1-sided limits exist,"},{"Start":"07:17.375 ","End":"07:20.320","Text":"and when I say exist I mean finite, not infinity,"},{"Start":"07:20.320 ","End":"07:21.900","Text":"as x equals a,"},{"Start":"07:21.900 ","End":"07:23.825","Text":"but they\u0027re different from each other."},{"Start":"07:23.825 ","End":"07:24.980","Text":"Like in this example,"},{"Start":"07:24.980 ","End":"07:27.950","Text":"we had the right limit existed and was equal to 5,"},{"Start":"07:27.950 ","End":"07:29.615","Text":"and the left limit existed,"},{"Start":"07:29.615 ","End":"07:31.190","Text":"but equaled 1 and they are different."},{"Start":"07:31.190 ","End":"07:32.800","Text":"5 is not equal to 1."},{"Start":"07:32.800 ","End":"07:34.520","Text":"In more mathematical language,"},{"Start":"07:34.520 ","End":"07:39.200","Text":"f has a jump discontinuity at a if the limit as x goes"},{"Start":"07:39.200 ","End":"07:44.405","Text":"to a from the right of f of x is equal to some finite number,"},{"Start":"07:44.405 ","End":"07:48.230","Text":"say k, and the limit as x goes"},{"Start":"07:48.230 ","End":"07:52.370","Text":"to a from the left of f of x equals another finite number,"},{"Start":"07:52.370 ","End":"07:58.645","Text":"say l, and k is not equal to l. It is important that they be different,"},{"Start":"07:58.645 ","End":"08:00.379","Text":"and that\u0027s the definition."},{"Start":"08:00.379 ","End":"08:02.450","Text":"Of course, instead of jumped discontinuity,"},{"Start":"08:02.450 ","End":"08:06.410","Text":"I could say discontinuity of the first kind or step discontinuity and so on."},{"Start":"08:06.410 ","End":"08:10.160","Text":"That\u0027s basically it except that I\u0027d like to add a note which is"},{"Start":"08:10.160 ","End":"08:15.530","Text":"important that the value of f at a is irrelevant."},{"Start":"08:15.530 ","End":"08:16.700","Text":"I don\u0027t care about it."},{"Start":"08:16.700 ","End":"08:19.280","Text":"As soon as I see that the 2 limits,"},{"Start":"08:19.280 ","End":"08:20.600","Text":"1-sided limits are different,"},{"Start":"08:20.600 ","End":"08:23.645","Text":"I don\u0027t even have to bother checking what the value is."},{"Start":"08:23.645 ","End":"08:26.580","Text":"Let\u0027s move on to the next type."}],"ID":14650},{"Watched":false,"Name":"Points of Discontinuity - Part 2","Duration":"8m 6s","ChapterTopicVideoID":13852,"CourseChapterTopicPlaylistID":84397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/13852.jpeg","UploadDate":"2019-11-14T06:52:17.8600000","DurationForVideoObject":"PT8M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"We now come to the last kind of discontinuity."},{"Start":"00:04.590 ","End":"00:07.065","Text":"Like before it has several names."},{"Start":"00:07.065 ","End":"00:10.140","Text":"It\u0027s called a discontinuity of the second kind."},{"Start":"00:10.140 ","End":"00:14.805","Text":"But I usually see it as an essential discontinuity."},{"Start":"00:14.805 ","End":"00:18.810","Text":"It\u0027s even called infinite discontinuity."},{"Start":"00:18.810 ","End":"00:21.365","Text":"This is actually the easiest to define."},{"Start":"00:21.365 ","End":"00:24.930","Text":"Notice that this is a third kind of discontinuity."},{"Start":"00:24.930 ","End":"00:28.490","Text":"I\u0027ll even just quickly summarize the side here."},{"Start":"00:28.490 ","End":"00:34.085","Text":"What we had, the first 1 we had was removable discontinuity."},{"Start":"00:34.085 ","End":"00:39.815","Text":"Then we had discontinuity of the first kind,"},{"Start":"00:39.815 ","End":"00:45.275","Text":"which we also called a jump discontinuity or a step discontinuity."},{"Start":"00:45.275 ","End":"00:49.190","Text":"Now the last kind is defined as follows."},{"Start":"00:49.190 ","End":"00:53.690","Text":"Any kind of discontinuity that\u0027s not this or this must be of the second kind."},{"Start":"00:53.690 ","End":"00:57.280","Text":"It\u0027s a catch-all for anything that doesn\u0027t fall for the first 2."},{"Start":"00:57.280 ","End":"00:58.670","Text":"It\u0027s the second kind."},{"Start":"00:58.670 ","End":"01:02.345","Text":"So let me just write this more formally. Here we are."},{"Start":"01:02.345 ","End":"01:05.720","Text":"If f has a discontinuity at x equals a,"},{"Start":"01:05.720 ","End":"01:09.425","Text":"which is not removable and not of the first kind,"},{"Start":"01:09.425 ","End":"01:12.605","Text":"then the discontinuity is of the second kind."},{"Start":"01:12.605 ","End":"01:16.145","Text":"There is actually an equivalent definition,"},{"Start":"01:16.145 ","End":"01:18.620","Text":"which is often easier to work with."},{"Start":"01:18.620 ","End":"01:20.795","Text":"Here\u0027s the equivalent definition."},{"Start":"01:20.795 ","End":"01:23.045","Text":"If 1 of the 1-sided limits,"},{"Start":"01:23.045 ","End":"01:25.310","Text":"the ones that we\u0027re constantly evaluating,"},{"Start":"01:25.310 ","End":"01:28.580","Text":"a limit as x goes to a from the right or the left of f of x,"},{"Start":"01:28.580 ","End":"01:30.800","Text":"if 1 of these is not a number,"},{"Start":"01:30.800 ","End":"01:32.240","Text":"what do I mean by not a number?"},{"Start":"01:32.240 ","End":"01:34.910","Text":"I mean it\u0027s either undefined or infinite."},{"Start":"01:34.910 ","End":"01:36.470","Text":"If 1 of these is not a number,"},{"Start":"01:36.470 ","End":"01:40.510","Text":"then already f has a discontinuity of the second kind."},{"Start":"01:40.510 ","End":"01:42.889","Text":"I will see this in the next example."},{"Start":"01:42.889 ","End":"01:49.085","Text":"Our example function f of x is given piecewise as x squared for x less than 1,"},{"Start":"01:49.085 ","End":"01:55.095","Text":"5 when x equals 1 and 1/ x minus 1 for x bigger than 1."},{"Start":"01:55.095 ","End":"02:00.675","Text":"Where I want to find out about f is at x equals 1 is a continuous or not."},{"Start":"02:00.675 ","End":"02:05.285","Text":"Routinely, almost automatically I write down 3 quantities."},{"Start":"02:05.285 ","End":"02:07.235","Text":"The limit on the right,"},{"Start":"02:07.235 ","End":"02:09.170","Text":"the limit on the left,"},{"Start":"02:09.170 ","End":"02:11.345","Text":"of the function, of course."},{"Start":"02:11.345 ","End":"02:15.275","Text":"The third thing is the value of the function at the point."},{"Start":"02:15.275 ","End":"02:18.320","Text":"We check and see continuous if"},{"Start":"02:18.320 ","End":"02:23.275","Text":"all these 3 things are equal and different kinds of discontinuity otherwise."},{"Start":"02:23.275 ","End":"02:25.195","Text":"Let\u0027s begin with the first."},{"Start":"02:25.195 ","End":"02:27.610","Text":"X goes to 1 from the right."},{"Start":"02:27.610 ","End":"02:30.080","Text":"It\u0027s slightly above 1."},{"Start":"02:30.080 ","End":"02:32.325","Text":"We are going for the last line."},{"Start":"02:32.325 ","End":"02:34.710","Text":"If x is going to go to 1 from the right,"},{"Start":"02:34.710 ","End":"02:36.010","Text":"then the limit is 1,"},{"Start":"02:36.010 ","End":"02:41.330","Text":"but we write it as 1 plus to indicate that we\u0027ve come from the right."},{"Start":"02:41.330 ","End":"02:45.080","Text":"The kind of arithmetic here with these pluses,"},{"Start":"02:45.080 ","End":"02:47.600","Text":"it all works out although it looks fishy."},{"Start":"02:47.600 ","End":"02:51.785","Text":"Now, 1 plus minus 1 is just 0 plus,"},{"Start":"02:51.785 ","End":"02:55.640","Text":"no such number, but it means slightly bigger than 0."},{"Start":"02:55.640 ","End":"02:57.815","Text":"It\u0027s a positive 0 if you like,"},{"Start":"02:57.815 ","End":"03:02.225","Text":"which means that this thing is infinity or not minus infinity."},{"Start":"03:02.225 ","End":"03:04.910","Text":"In any event, look already at the first line,"},{"Start":"03:04.910 ","End":"03:08.255","Text":"we\u0027ve discovered that the left limit is not a number."},{"Start":"03:08.255 ","End":"03:11.555","Text":"If it comes at undefined or infinity, it\u0027s not a number."},{"Start":"03:11.555 ","End":"03:13.565","Text":"We don\u0027t have to continue."},{"Start":"03:13.565 ","End":"03:15.875","Text":"Straight away, we can say that"},{"Start":"03:15.875 ","End":"03:21.260","Text":"the discontinuity is of the second kind or an essential discontinuity."},{"Start":"03:21.260 ","End":"03:26.539","Text":"But if you feel you\u0027ve wasted your writing and you want to continue, you can continue."},{"Start":"03:26.539 ","End":"03:30.110","Text":"We can say that f of 1 is 5 from here,"},{"Start":"03:30.110 ","End":"03:33.365","Text":"and the limit as x goes to 1 from the left comes from here,"},{"Start":"03:33.365 ","End":"03:35.930","Text":"it\u0027s 1 squared, which is 1."},{"Start":"03:35.930 ","End":"03:37.010","Text":"But we don\u0027t need this."},{"Start":"03:37.010 ","End":"03:41.075","Text":"This is already tells us all we need to know, essential discontinuity."},{"Start":"03:41.075 ","End":"03:45.170","Text":"At this point, basically we\u0027ve finished the theory part for this clip,"},{"Start":"03:45.170 ","End":"03:48.050","Text":"but don\u0027t go yet because I want to end with"},{"Start":"03:48.050 ","End":"03:53.450","Text":"a nice example which summarizes all the 3 kinds of discontinuities."},{"Start":"03:53.450 ","End":"03:55.895","Text":"That\u0027s coming up right now."},{"Start":"03:55.895 ","End":"03:58.070","Text":"I\u0027ve written it all out for you."},{"Start":"03:58.070 ","End":"04:00.410","Text":"This exercise, which is actually a summary and we\u0027ll"},{"Start":"04:00.410 ","End":"04:02.795","Text":"show the 3 different kinds of discontinuity."},{"Start":"04:02.795 ","End":"04:04.715","Text":"We\u0027re given a function f of x,"},{"Start":"04:04.715 ","End":"04:07.775","Text":"which is defined piecewise as follows."},{"Start":"04:07.775 ","End":"04:09.340","Text":"When x is less than 4,"},{"Start":"04:09.340 ","End":"04:10.875","Text":"it\u0027s a 1/2 x plus 1,"},{"Start":"04:10.875 ","End":"04:12.510","Text":"when x is exactly 4,"},{"Start":"04:12.510 ","End":"04:15.450","Text":"it\u0027s 2, when x is between 4 and 10,"},{"Start":"04:15.450 ","End":"04:17.160","Text":"we have 2x minus 5,"},{"Start":"04:17.160 ","End":"04:19.220","Text":"when x is between 10 and 14,"},{"Start":"04:19.220 ","End":"04:21.440","Text":"x plus 1 and above 14,"},{"Start":"04:21.440 ","End":"04:24.290","Text":"it\u0027s defined as 1/x minus 14."},{"Start":"04:24.290 ","End":"04:28.895","Text":"What we have to do is find all the discontinuities and say what kind they are,"},{"Start":"04:28.895 ","End":"04:32.735","Text":"whether they\u0027re removable, first kind, or second kind."},{"Start":"04:32.735 ","End":"04:36.350","Text":"Now, the only place to look is at the seam lines,"},{"Start":"04:36.350 ","End":"04:41.720","Text":"and these are the points where x is equal to either 4 something happens here,"},{"Start":"04:41.720 ","End":"04:44.850","Text":"10, things change, and 14."},{"Start":"04:44.850 ","End":"04:46.380","Text":"There only 3 points to check,"},{"Start":"04:46.380 ","End":"04:47.595","Text":"and we\u0027ll check each 1."},{"Start":"04:47.595 ","End":"04:49.220","Text":"I can tell you in advance because this is"},{"Start":"04:49.220 ","End":"04:52.180","Text":"a summary exercise that this one will turn out to be removable,"},{"Start":"04:52.180 ","End":"04:53.530","Text":"this one will be of the first kind,"},{"Start":"04:53.530 ","End":"04:54.935","Text":"and this will be of the second kind."},{"Start":"04:54.935 ","End":"04:58.340","Text":"I don\u0027t want to lose the definition, I have an idea."},{"Start":"04:58.340 ","End":"05:02.930","Text":"Let\u0027s just do this in a table form where for x equals 4,"},{"Start":"05:02.930 ","End":"05:05.000","Text":"we need to compute 3 things."},{"Start":"05:05.000 ","End":"05:11.975","Text":"We need to know what is the limit as x goes to 4 from the left of f of x."},{"Start":"05:11.975 ","End":"05:17.930","Text":"We need to know the limit as x goes to 4 from the right of f of x."},{"Start":"05:17.930 ","End":"05:24.020","Text":"We need to know what is f of 4 and see how equal or not equal,"},{"Start":"05:24.020 ","End":"05:25.280","Text":"so on these are."},{"Start":"05:25.280 ","End":"05:27.695","Text":"Same thing for x equals 10."},{"Start":"05:27.695 ","End":"05:32.285","Text":"I need the limit as x goes to 10 from the left,"},{"Start":"05:32.285 ","End":"05:36.225","Text":"limit as x goes to 10 from the right."},{"Start":"05:36.225 ","End":"05:40.070","Text":"As I say I may not need all of them because if it breaks down earlier,"},{"Start":"05:40.070 ","End":"05:41.180","Text":"you don\u0027t need to continue,"},{"Start":"05:41.180 ","End":"05:44.360","Text":"and we need f of 10 itself."},{"Start":"05:44.360 ","End":"05:46.580","Text":"Then for 14, similar thing,"},{"Start":"05:46.580 ","End":"05:48.290","Text":"limit as x goes to,"},{"Start":"05:48.290 ","End":"05:50.075","Text":"I\u0027ll try to make it quickly."},{"Start":"05:50.075 ","End":"05:52.880","Text":"Limit as x goes to,"},{"Start":"05:52.880 ","End":"05:54.230","Text":"this is from the right,"},{"Start":"05:54.230 ","End":"06:02.975","Text":"x goes to 14 from the left of f of x equals of f of x equals,"},{"Start":"06:02.975 ","End":"06:06.935","Text":"and f of 14."},{"Start":"06:06.935 ","End":"06:08.270","Text":"Well, let\u0027s see."},{"Start":"06:08.270 ","End":"06:10.040","Text":"Let\u0027s take the first 1,"},{"Start":"06:10.040 ","End":"06:12.830","Text":"4 from the right and has to be bigger than 4."},{"Start":"06:12.830 ","End":"06:15.140","Text":"Look here it is where x is bigger than 4."},{"Start":"06:15.140 ","End":"06:17.270","Text":"We need 2x minus 5,"},{"Start":"06:17.270 ","End":"06:21.040","Text":"twice 4 minus 5 is 3."},{"Start":"06:21.040 ","End":"06:22.550","Text":"How about from the left?"},{"Start":"06:22.550 ","End":"06:25.205","Text":"From the left, we take it from here, x less than 4."},{"Start":"06:25.205 ","End":"06:28.755","Text":"1/2 of 4 which is 2 plus 1 is 3."},{"Start":"06:28.755 ","End":"06:32.565","Text":"Very good. At 4 itself, it\u0027s only 2."},{"Start":"06:32.565 ","End":"06:35.835","Text":"These 2 are equal on their numbers,"},{"Start":"06:35.835 ","End":"06:38.730","Text":"but they are just not equal to f at 4,"},{"Start":"06:38.730 ","End":"06:41.820","Text":"so this will be a removable discontinuity."},{"Start":"06:41.820 ","End":"06:43.215","Text":"Let\u0027s go on to the next 1."},{"Start":"06:43.215 ","End":"06:45.120","Text":"I\u0027ll write the names down in a moment."},{"Start":"06:45.120 ","End":"06:47.505","Text":"At 10 from the right,"},{"Start":"06:47.505 ","End":"06:51.015","Text":"we\u0027re going to use this bit here, we get,"},{"Start":"06:51.015 ","End":"06:55.064","Text":"if we substitute 10, we get 11. This will be 11."},{"Start":"06:55.064 ","End":"06:57.255","Text":"Now, 10 from the left,"},{"Start":"06:57.255 ","End":"06:59.745","Text":"we\u0027ll need to go less than 10."},{"Start":"06:59.745 ","End":"07:03.350","Text":"Twice 10 minus 5 is 15."},{"Start":"07:03.350 ","End":"07:06.395","Text":"At this point, I can already stop and say,"},{"Start":"07:06.395 ","End":"07:11.720","Text":"I know this is the discontinuity of the first kind or a jump discontinuity."},{"Start":"07:11.720 ","End":"07:13.640","Text":"It jumps from 11 to 15."},{"Start":"07:13.640 ","End":"07:17.360","Text":"This is a jump of first kind and onto the last 1 at 14."},{"Start":"07:17.360 ","End":"07:18.925","Text":"Let\u0027s see what happens there,"},{"Start":"07:18.925 ","End":"07:20.340","Text":"limit on the right,"},{"Start":"07:20.340 ","End":"07:22.020","Text":"we take it from here."},{"Start":"07:22.020 ","End":"07:25.590","Text":"X is 14 plus a little bit."},{"Start":"07:25.590 ","End":"07:28.385","Text":"What we get here is x goes to 14,"},{"Start":"07:28.385 ","End":"07:29.870","Text":"but it\u0027s bigger than 14,"},{"Start":"07:29.870 ","End":"07:33.575","Text":"we get 1/0 plus which is infinity."},{"Start":"07:33.575 ","End":"07:37.909","Text":"That already means that the limit is not a number,"},{"Start":"07:37.909 ","End":"07:40.310","Text":"can be undefined or infinity is not a number."},{"Start":"07:40.310 ","End":"07:42.920","Text":"I don\u0027t need to continue to proceed anymore."},{"Start":"07:42.920 ","End":"07:45.830","Text":"I know that this is of the second kind."},{"Start":"07:45.830 ","End":"07:49.745","Text":"The discontinuity here is removable."},{"Start":"07:49.745 ","End":"07:55.965","Text":"This 1 was of the first kind or jump discontinuity."},{"Start":"07:55.965 ","End":"08:02.450","Text":"The last 1 we have second kind or essential discontinuity."},{"Start":"08:02.450 ","End":"08:07.500","Text":"This exercise summarizes everything and we are done."}],"ID":14651},{"Watched":false,"Name":"Exercise 1","Duration":"3m 44s","ChapterTopicVideoID":3309,"CourseChapterTopicPlaylistID":84397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3309.jpeg","UploadDate":"2014-08-31T14:03:29.7100000","DurationForVideoObject":"PT3M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we\u0027re given a graph of a function f over"},{"Start":"00:04.410 ","End":"00:11.950","Text":"a certain interval and we have to classify any points of discontinuity."},{"Start":"00:12.080 ","End":"00:18.840","Text":"I\u0027m going to just adjust the position here."},{"Start":"00:18.840 ","End":"00:21.070","Text":"Just a second."},{"Start":"00:22.250 ","End":"00:29.040","Text":"Well, if we just look at it,"},{"Start":"00:29.040 ","End":"00:34.865","Text":"we can see that everything goes along smoothly until we get to this point."},{"Start":"00:34.865 ","End":"00:44.045","Text":"This looks like it\u0027s where x equals minus 3."},{"Start":"00:44.045 ","End":"00:47.510","Text":"There\u0027s a jump here, and actually,"},{"Start":"00:47.510 ","End":"00:52.405","Text":"that\u0027s what we call a jump discontinuity."},{"Start":"00:52.405 ","End":"00:55.850","Text":"I won\u0027t write the word discontinuity each time,"},{"Start":"00:55.850 ","End":"01:01.280","Text":"but here we have a jump discontinuity."},{"Start":"01:01.280 ","End":"01:09.360","Text":"It\u0027s actually also sometimes called a step discontinuity."},{"Start":"01:10.300 ","End":"01:13.325","Text":"Choose 1 or the other."},{"Start":"01:13.325 ","End":"01:15.575","Text":"Then we go along here."},{"Start":"01:15.575 ","End":"01:18.905","Text":"There\u0027s a bend but there\u0027s no break and there\u0027s a discontinuity."},{"Start":"01:18.905 ","End":"01:21.350","Text":"Here, looks like it goes up to infinity."},{"Start":"01:21.350 ","End":"01:22.855","Text":"In other words, there is no limit."},{"Start":"01:22.855 ","End":"01:24.220","Text":"Already even at 1 side,"},{"Start":"01:24.220 ","End":"01:29.390","Text":"there\u0027s no limit as x goes to minus 1."},{"Start":"01:30.260 ","End":"01:35.075","Text":"When x equals minus 1,"},{"Start":"01:35.075 ","End":"01:41.910","Text":"then that\u0027s more serious and that\u0027s called an essential discontinuity."},{"Start":"01:44.320 ","End":"01:46.580","Text":"But there\u0027s also another word,"},{"Start":"01:46.580 ","End":"01:48.725","Text":"everything has more than 1 word."},{"Start":"01:48.725 ","End":"01:53.580","Text":"It\u0027s sometimes also called an infinite discontinuity."},{"Start":"01:54.450 ","End":"01:59.110","Text":"But I prefer the word essential because it could"},{"Start":"01:59.110 ","End":"02:03.160","Text":"be essential without being any infinity in there."},{"Start":"02:03.160 ","End":"02:09.935","Text":"Certainly, if it goes to infinity, it\u0027s both essential."},{"Start":"02:09.935 ","End":"02:13.000","Text":"A jump just means that it has a limit"},{"Start":"02:13.000 ","End":"02:16.375","Text":"on the left and a limit on the right but they\u0027re not equal."},{"Start":"02:16.375 ","End":"02:21.700","Text":"Essential means more than that."},{"Start":"02:22.040 ","End":"02:25.560","Text":"Basically, it just means that it\u0027s not removable,"},{"Start":"02:25.560 ","End":"02:27.700","Text":"or a jump singularity, you could say,"},{"Start":"02:27.700 ","End":"02:30.460","Text":"but what it means is that on 1 side the limit doesn\u0027t exist,"},{"Start":"02:30.460 ","End":"02:32.320","Text":"or it goes to infinity,"},{"Start":"02:32.320 ","End":"02:36.615","Text":"or something like that, as opposed to."},{"Start":"02:36.615 ","End":"02:38.980","Text":"Then as we\u0027re going along from minus 1,"},{"Start":"02:38.980 ","End":"02:40.970","Text":"we\u0027re traveling along nice and smoothly,"},{"Start":"02:40.970 ","End":"02:46.060","Text":"we see that there\u0027s a gap here that the function is defined at 2,"},{"Start":"02:46.060 ","End":"02:48.130","Text":"but it\u0027s not where it\u0027s supposed to be."},{"Start":"02:48.130 ","End":"02:51.890","Text":"It come out of place and g1 from here to here."},{"Start":"02:55.730 ","End":"02:59.680","Text":"If it could be fixed by just repairing 1 point,"},{"Start":"02:59.680 ","End":"03:03.145","Text":"then that\u0027s called a removable singularity."},{"Start":"03:03.145 ","End":"03:10.560","Text":"This is known as removable discontinuity."},{"Start":"03:10.560 ","End":"03:13.240","Text":"Did I say singularity?"},{"Start":"03:13.430 ","End":"03:18.370","Text":"I\u0027ll just write the word discontinuity,"},{"Start":"03:23.780 ","End":"03:28.240","Text":"and that\u0027s at x equals 2."},{"Start":"03:28.460 ","End":"03:33.090","Text":"There they are. Removable 1 here,"},{"Start":"03:33.090 ","End":"03:34.875","Text":"a jump 1 here,"},{"Start":"03:34.875 ","End":"03:37.600","Text":"and an essential 1 here."},{"Start":"03:38.470 ","End":"03:43.860","Text":"That\u0027s about all there is to say. That\u0027s it."}],"ID":3320},{"Watched":false,"Name":"Exercise 2","Duration":"6m 44s","ChapterTopicVideoID":3310,"CourseChapterTopicPlaylistID":84397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3310.jpeg","UploadDate":"2017-03-01T09:49:38.3970000","DurationForVideoObject":"PT6M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.095","Text":"This exercise is really 2 separate exercises, a and b."},{"Start":"00:04.095 ","End":"00:07.080","Text":"In each one, we\u0027re given a function and we have to"},{"Start":"00:07.080 ","End":"00:11.475","Text":"find and classify any points of discontinuity."},{"Start":"00:11.475 ","End":"00:16.965","Text":"Let\u0027s begin with a. Scroll up a bit."},{"Start":"00:16.965 ","End":"00:25.865","Text":"The first one is f of x equals x plus 1 over x minus 1."},{"Start":"00:25.865 ","End":"00:33.140","Text":"Now, its domain is everywhere where it\u0027s reasonably defined,"},{"Start":"00:33.140 ","End":"00:36.380","Text":"which means that x can\u0027t be 1 because we can\u0027t have a"},{"Start":"00:36.380 ","End":"00:39.680","Text":"0 on the denominator not equal to 1,"},{"Start":"00:39.680 ","End":"00:43.745","Text":"and everywhere else it is defined and it\u0027s continuous."},{"Start":"00:43.745 ","End":"00:48.035","Text":"The only suspect we have is x equals 1."},{"Start":"00:48.035 ","End":"00:49.985","Text":"Now what about x equals 1?"},{"Start":"00:49.985 ","End":"00:55.940","Text":"Normally, when you discuss the matter of discontinuities,"},{"Start":"00:55.940 ","End":"01:03.330","Text":"you check 3 things: you check the limit as x goes to one side,"},{"Start":"01:03.330 ","End":"01:06.235","Text":"let\u0027s say 1 of the function,"},{"Start":"01:06.235 ","End":"01:09.529","Text":"we check the limit on the other side,"},{"Start":"01:09.529 ","End":"01:12.905","Text":"and f of x equals,"},{"Start":"01:12.905 ","End":"01:16.669","Text":"and we also check the value at the point."},{"Start":"01:16.669 ","End":"01:19.760","Text":"Now, you could do the first 2,"},{"Start":"01:19.760 ","End":"01:21.290","Text":"but if you look at it,"},{"Start":"01:21.290 ","End":"01:25.010","Text":"we should do the last one because this is undefined,"},{"Start":"01:25.010 ","End":"01:27.695","Text":"1 is outside the domain,"},{"Start":"01:27.695 ","End":"01:32.640","Text":"so there\u0027s no point in asking about discontinuity."},{"Start":"01:32.650 ","End":"01:35.425","Text":"This is undefined."},{"Start":"01:35.425 ","End":"01:37.460","Text":"So if you just did it by rote,"},{"Start":"01:37.460 ","End":"01:39.335","Text":"by doing these 3 things,"},{"Start":"01:39.335 ","End":"01:42.620","Text":"you might think you\u0027ll get somewhere, but actually,"},{"Start":"01:42.620 ","End":"01:46.190","Text":"already I could have stopped as soon as it\u0027s undefined."},{"Start":"01:46.190 ","End":"01:50.930","Text":"I\u0027m just doing all this because it has some relevance in part b,"},{"Start":"01:50.930 ","End":"01:52.850","Text":"where it might make more sense to do this,"},{"Start":"01:52.850 ","End":"01:54.335","Text":"but it would still be wrong."},{"Start":"01:54.335 ","End":"01:58.965","Text":"Something outside the domain can\u0027t be continuous or discontinuous,"},{"Start":"01:58.965 ","End":"02:02.490","Text":"this function doesn\u0027t apply there."},{"Start":"02:03.500 ","End":"02:06.870","Text":"These are the 3 things we normally check,"},{"Start":"02:06.870 ","End":"02:09.810","Text":"and this is the one that\u0027s important,"},{"Start":"02:09.810 ","End":"02:11.450","Text":"and because of this one,"},{"Start":"02:11.450 ","End":"02:14.875","Text":"it\u0027s not continuous there because it\u0027s not defined."},{"Start":"02:14.875 ","End":"02:16.725","Text":"But it\u0027s not in the domain,"},{"Start":"02:16.725 ","End":"02:20.000","Text":"so I can\u0027t say it\u0027s discontinuous either."},{"Start":"02:20.000 ","End":"02:22.370","Text":"I can say basically,"},{"Start":"02:22.370 ","End":"02:28.135","Text":"f is continuous where defined,"},{"Start":"02:28.135 ","End":"02:33.420","Text":"ie, x not equal to 1."},{"Start":"02:33.420 ","End":"02:37.735","Text":"That\u0027s the domain and that\u0027s the places where it\u0027s continuous."},{"Start":"02:37.735 ","End":"02:41.890","Text":"But I can\u0027t say it\u0027s discontinuous at x equals 1 either."},{"Start":"02:41.890 ","End":"02:43.840","Text":"It\u0027s neither discontinuous or just continuous,"},{"Start":"02:43.840 ","End":"02:45.595","Text":"it\u0027s just not defined."},{"Start":"02:45.595 ","End":"02:49.375","Text":"That\u0027s the first one, that\u0027s a."},{"Start":"02:49.375 ","End":"02:50.945","Text":"We\u0027re done with a,"},{"Start":"02:50.945 ","End":"02:54.534","Text":"now part b, which I\u0027ve already copied."},{"Start":"02:54.534 ","End":"02:56.620","Text":"Once again, I\u0027m going to ask,"},{"Start":"02:56.620 ","End":"02:58.615","Text":"what the domain is."},{"Start":"02:58.615 ","End":"03:03.745","Text":"The domain is wherever the denominator is not 0,"},{"Start":"03:03.745 ","End":"03:05.520","Text":"x squared can\u0027t be 1."},{"Start":"03:05.520 ","End":"03:12.555","Text":"In other words, wherever x is not equal to plus 1 or minus 1,"},{"Start":"03:12.555 ","End":"03:14.055","Text":"let us write the plus for emphasis,"},{"Start":"03:14.055 ","End":"03:17.580","Text":"doesn\u0027t the 2 points which are outside the domain,"},{"Start":"03:17.580 ","End":"03:21.490","Text":"and everywhere in the domain it\u0027s continuous."},{"Start":"03:21.490 ","End":"03:27.380","Text":"Yet I do see a point just for exercising that we should check."},{"Start":"03:27.380 ","End":"03:29.555","Text":"I\u0027ll just give you an example of one of them."},{"Start":"03:29.555 ","End":"03:31.970","Text":"Let\u0027s say x equals 1."},{"Start":"03:31.970 ","End":"03:37.549","Text":"Why I might want to check the limit from the left,"},{"Start":"03:37.549 ","End":"03:41.945","Text":"the limit from the right,"},{"Start":"03:41.945 ","End":"03:45.185","Text":"and the value of the function at 1."},{"Start":"03:45.185 ","End":"03:48.755","Text":"So f of x and f of x,"},{"Start":"03:48.755 ","End":"03:51.770","Text":"and the one hand, you\u0027d say it\u0027s a waste of time, we know the answer."},{"Start":"03:51.770 ","End":"03:55.030","Text":"It\u0027s continuous everywhere except 1 and minus 1,"},{"Start":"03:55.030 ","End":"03:57.050","Text":"and that\u0027s all there is to it and that\u0027s true,"},{"Start":"03:57.050 ","End":"03:59.930","Text":"but I think there\u0027s something useful to be learned here,"},{"Start":"03:59.930 ","End":"04:03.560","Text":"because it relates somehow to removable discontinuities."},{"Start":"04:03.560 ","End":"04:05.525","Text":"Anyway, if we do do this,"},{"Start":"04:05.525 ","End":"04:09.815","Text":"I\u0027ll just give you the gist of it without all the steps margin here."},{"Start":"04:09.815 ","End":"04:15.030","Text":"If I was to factorize x squared plus x minus 2,"},{"Start":"04:15.030 ","End":"04:18.015","Text":"if I solve the equation, now this equals 0,"},{"Start":"04:18.015 ","End":"04:24.820","Text":"I\u0027d get x is equal to 1 or minus 2,"},{"Start":"04:25.430 ","End":"04:30.785","Text":"this can be factorized as x minus 1,"},{"Start":"04:30.785 ","End":"04:35.555","Text":"x plus 2 would be what this polynomial is,"},{"Start":"04:35.555 ","End":"04:46.235","Text":"and over here, we would get limit x goes to 1 from the left of x minus 1 and x plus 2."},{"Start":"04:46.235 ","End":"04:49.400","Text":"The denominator is clearly x minus 1, x plus 1,"},{"Start":"04:49.400 ","End":"04:53.200","Text":"it\u0027s like the conjugates difference of squares."},{"Start":"04:53.200 ","End":"04:56.250","Text":"Then if x was tending to 1,"},{"Start":"04:56.250 ","End":"04:57.750","Text":"but it wouldn\u0027t equal 1,"},{"Start":"04:57.750 ","End":"04:59.400","Text":"we could cancel these 2,"},{"Start":"04:59.400 ","End":"05:02.090","Text":"and it would actually have a limit because if I put 1 here,"},{"Start":"05:02.090 ","End":"05:04.045","Text":"I\u0027ll get 3 over 2,"},{"Start":"05:04.045 ","End":"05:07.520","Text":"and the same thing here would be exactly the same calculation,"},{"Start":"05:07.520 ","End":"05:10.550","Text":"would also be 3/2."},{"Start":"05:10.550 ","End":"05:13.830","Text":"But if I had defined f of 1 arbitrarily,"},{"Start":"05:13.830 ","End":"05:16.175","Text":"let\u0027s say 17 or something,"},{"Start":"05:16.175 ","End":"05:21.110","Text":"that\u0027s just suppose that I told you in addition over here"},{"Start":"05:21.110 ","End":"05:26.985","Text":"that that\u0027s for x not equal to plus or minus 1,"},{"Start":"05:26.985 ","End":"05:34.460","Text":"and that f of 1 was 17 and f of minus 1 was,"},{"Start":"05:34.460 ","End":"05:37.369","Text":"I don\u0027t know, 92 or anything."},{"Start":"05:37.369 ","End":"05:39.305","Text":"Then it would be defined there,"},{"Start":"05:39.305 ","End":"05:43.265","Text":"then it would be removable discontinuity."},{"Start":"05:43.265 ","End":"05:44.840","Text":"Because in this case,"},{"Start":"05:44.840 ","End":"05:47.359","Text":"if I said f of 1 was 17,"},{"Start":"05:47.359 ","End":"05:51.420","Text":"which it isn\u0027t, it\u0027s actually undefined."},{"Start":"05:51.530 ","End":"05:55.420","Text":"I\u0027m making a point here that because it\u0027s undefined,"},{"Start":"05:55.420 ","End":"05:57.970","Text":"I can\u0027t say it\u0027s a point of discontinuity,"},{"Start":"05:57.970 ","End":"05:59.860","Text":"but if it was defined to be anything,"},{"Start":"05:59.860 ","End":"06:03.775","Text":"any number minus 47.5,"},{"Start":"06:03.775 ","End":"06:08.050","Text":"then because the limit from the left is equal to the limit on the right,"},{"Start":"06:08.050 ","End":"06:11.455","Text":"this continuity would be called removable,"},{"Start":"06:11.455 ","End":"06:14.230","Text":"anything except 3/2 of course,"},{"Start":"06:14.230 ","End":"06:17.030","Text":"because then it would actually be continuous."},{"Start":"06:17.030 ","End":"06:22.030","Text":"This was just some further ramblings that would be helpful in the future."},{"Start":"06:22.030 ","End":"06:26.230","Text":"I took the opportunity to show you what removable discontinuity is."},{"Start":"06:26.230 ","End":"06:28.090","Text":"It\u0027s where they\u0027re defined,"},{"Start":"06:28.090 ","End":"06:29.730","Text":"but something other,"},{"Start":"06:29.730 ","End":"06:34.100","Text":"and that\u0027s different than being totally undefined."},{"Start":"06:34.220 ","End":"06:36.740","Text":"Our answer is just our domain."},{"Start":"06:36.740 ","End":"06:39.950","Text":"The function is defined, is continuous everywhere in the domain,"},{"Start":"06:39.950 ","End":"06:44.640","Text":"which is x not equal to plus or minus 1. Done."}],"ID":3321},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"3m 38s","ChapterTopicVideoID":3311,"CourseChapterTopicPlaylistID":84397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3311.jpeg","UploadDate":"2017-03-01T09:53:07.3230000","DurationForVideoObject":"PT3M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"This exercise is really 3 exercises, a, b,"},{"Start":"00:02.520 ","End":"00:05.850","Text":"and c. I have a function f in each case,"},{"Start":"00:05.850 ","End":"00:13.745","Text":"which is defined piece-wise and we have to find and classify any points of discontinuity."},{"Start":"00:13.745 ","End":"00:16.390","Text":"Let\u0027s start with a."},{"Start":"00:16.550 ","End":"00:22.560","Text":"If we look at this, this consists of 2 ranges."},{"Start":"00:22.560 ","End":"00:24.435","Text":"Everything less than 1,"},{"Start":"00:24.435 ","End":"00:27.990","Text":"everything greater than 1, and a single 0.1."},{"Start":"00:27.990 ","End":"00:31.455","Text":"Something happens around the point x equals 1."},{"Start":"00:31.455 ","End":"00:33.710","Text":"But other than around x equals 1,"},{"Start":"00:33.710 ","End":"00:37.730","Text":"like where x is less than 1 and nothing special happens."},{"Start":"00:37.730 ","End":"00:39.695","Text":"There\u0027s not going to be completely"},{"Start":"00:39.695 ","End":"00:45.200","Text":"continuous because it\u0027s an elementary function polynomial,"},{"Start":"00:45.200 ","End":"00:48.775","Text":"whatever, x squared is continuous for x less than 1."},{"Start":"00:48.775 ","End":"00:54.135","Text":"The same thing, 2 minus x is continuous on x bigger than 1."},{"Start":"00:54.135 ","End":"01:01.520","Text":"The only possible discontinuity might be at the seam line where x equals 1."},{"Start":"01:01.520 ","End":"01:03.575","Text":"Where x equals 1,"},{"Start":"01:03.575 ","End":"01:07.495","Text":"that\u0027s what we\u0027re going to check, those 3 quantities that we usually check."},{"Start":"01:07.495 ","End":"01:10.950","Text":"X equals 1 is the only suspect,"},{"Start":"01:10.950 ","End":"01:14.050","Text":"the function is defined for all x though."},{"Start":"01:14.240 ","End":"01:17.590","Text":"Let\u0027s try this finding the 3 things."},{"Start":"01:17.590 ","End":"01:20.875","Text":"Start off by writing it\u0027s a limit,"},{"Start":"01:20.875 ","End":"01:25.155","Text":"limit and then f of,"},{"Start":"01:25.155 ","End":"01:26.790","Text":"the point is 1."},{"Start":"01:26.790 ","End":"01:29.100","Text":"So it\u0027s f of 1, here,"},{"Start":"01:29.100 ","End":"01:32.969","Text":"it\u0027s x goes to 1 from 1 side,"},{"Start":"01:32.969 ","End":"01:35.930","Text":"x goes to 1 from the other side."},{"Start":"01:35.930 ","End":"01:38.070","Text":"Here it\u0027s always f of x, here,"},{"Start":"01:38.070 ","End":"01:42.485","Text":"it\u0027s always f of x equals, equals, equals."},{"Start":"01:42.485 ","End":"01:45.004","Text":"Those are the 3 quantities that are interesting."},{"Start":"01:45.004 ","End":"01:47.045","Text":"Let\u0027s go with the easiest."},{"Start":"01:47.045 ","End":"01:48.710","Text":"At x equals 1,"},{"Start":"01:48.710 ","End":"01:50.810","Text":"the 1 is just 0."},{"Start":"01:50.810 ","End":"01:54.070","Text":"That\u0027s written up there, plain."},{"Start":"01:54.070 ","End":"02:00.360","Text":"X less than 1 so it\u0027s the limit of x"},{"Start":"02:00.360 ","End":"02:08.010","Text":"squared as x goes to 1 from the left because when we\u0027re going to 1 from the left,"},{"Start":"02:08.010 ","End":"02:09.060","Text":"we\u0027re just below 1,"},{"Start":"02:09.060 ","End":"02:10.800","Text":"and we\u0027re in this formula."},{"Start":"02:10.800 ","End":"02:17.670","Text":"Here we just have to substitute 1 squared, which is 1."},{"Start":"02:17.670 ","End":"02:19.400","Text":"From the other side,"},{"Start":"02:19.400 ","End":"02:26.225","Text":"we get the limit as x goes to 1 from above of f of x,"},{"Start":"02:26.225 ","End":"02:31.730","Text":"but f of x is 2 minus x,"},{"Start":"02:31.730 ","End":"02:34.025","Text":"and x we substitute is 1,"},{"Start":"02:34.025 ","End":"02:41.625","Text":"is 2 minus 1, which is 1."},{"Start":"02:41.625 ","End":"02:45.015","Text":"Now I\u0027m going to highlight these 3."},{"Start":"02:45.015 ","End":"02:47.355","Text":"I\u0027ve got 1 here,"},{"Start":"02:47.355 ","End":"02:48.975","Text":"I\u0027ve got 1 here,"},{"Start":"02:48.975 ","End":"02:51.450","Text":"and I\u0027ve got 0 here."},{"Start":"02:51.450 ","End":"02:55.785","Text":"What I have is I have all those 3 quantities exist,"},{"Start":"02:55.785 ","End":"02:59.405","Text":"the limit from the left is equal to the limit from the right,"},{"Start":"02:59.405 ","End":"03:03.415","Text":"that just not equal to the value of the function at the point."},{"Start":"03:03.415 ","End":"03:05.450","Text":"In the previous exercise,"},{"Start":"03:05.450 ","End":"03:07.460","Text":"I noted that if it wasn\u0027t defined here,"},{"Start":"03:07.460 ","End":"03:08.810","Text":"that would be a different story."},{"Start":"03:08.810 ","End":"03:10.805","Text":"It is defined, it\u0027s just different."},{"Start":"03:10.805 ","End":"03:17.850","Text":"In this case, what we get is what is known as a removable discontinuity."},{"Start":"03:18.530 ","End":"03:21.675","Text":"What is meant by removable?"},{"Start":"03:21.675 ","End":"03:26.885","Text":"If we just changed the f of 1 to equal 1 also,"},{"Start":"03:26.885 ","End":"03:28.580","Text":"just change it at 1 point,"},{"Start":"03:28.580 ","End":"03:31.000","Text":"that this is at a 0 was 1,"},{"Start":"03:31.000 ","End":"03:33.080","Text":"if I just change this number,"},{"Start":"03:33.080 ","End":"03:36.260","Text":"then I could remove the discontinuity."},{"Start":"03:36.260 ","End":"03:38.850","Text":"But that\u0027s the answer."}],"ID":3322},{"Watched":false,"Name":"Exercise 3 - Parts b-c","Duration":"5m 45s","ChapterTopicVideoID":4842,"CourseChapterTopicPlaylistID":84397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/4842.jpeg","UploadDate":"2015-07-21T10:43:39.2500000","DurationForVideoObject":"PT5M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.140","Text":"Now we come to part B and I\u0027ll copy it over here. Here we are."},{"Start":"00:07.140 ","End":"00:13.530","Text":"Notice that it\u0027s the same division of the line as in part A."},{"Start":"00:13.530 ","End":"00:14.700","Text":"In other words, we have less than 1,"},{"Start":"00:14.700 ","End":"00:17.250","Text":"bigger than 1, and equals 1."},{"Start":"00:17.250 ","End":"00:23.000","Text":"Just like before, we expect no trouble when x is less than 1 or when x is bigger than 1."},{"Start":"00:23.000 ","End":"00:25.385","Text":"Everything is polynomial,"},{"Start":"00:25.385 ","End":"00:27.630","Text":"elementary function is continuous."},{"Start":"00:27.630 ","End":"00:32.445","Text":"The only suspect could possibly be x equals 1."},{"Start":"00:32.445 ","End":"00:36.045","Text":"Let\u0027s see what\u0027s happening over there."},{"Start":"00:36.045 ","End":"00:42.825","Text":"Just scroll down a bit to get some more room here. There we go."},{"Start":"00:42.825 ","End":"00:44.970","Text":"Just like in part A,"},{"Start":"00:44.970 ","End":"00:49.695","Text":"there are three things we need to compute and all three of them have to be equal."},{"Start":"00:49.695 ","End":"00:53.835","Text":"That is the limit from the left,"},{"Start":"00:53.835 ","End":"00:58.550","Text":"that is as x goes to 1 from the left of f of x,"},{"Start":"00:58.550 ","End":"01:00.530","Text":"what is that equal to?"},{"Start":"01:00.530 ","End":"01:06.120","Text":"Then we want the limit as x goes to 1 from the right of f of x,"},{"Start":"01:06.120 ","End":"01:07.425","Text":"what is that equal to?"},{"Start":"01:07.425 ","End":"01:10.605","Text":"Finally, what is f equal to at the point 1?"},{"Start":"01:10.605 ","End":"01:12.645","Text":"If all these three are equal,"},{"Start":"01:12.645 ","End":"01:15.350","Text":"fine it\u0027s continuous, and if not,"},{"Start":"01:15.350 ","End":"01:18.995","Text":"then we\u0027ll see which type of discontinuity we have."},{"Start":"01:18.995 ","End":"01:22.475","Text":"Whether it\u0027s removable, whether it\u0027s type 1, type 2."},{"Start":"01:22.475 ","End":"01:24.230","Text":"Let\u0027s just get started."},{"Start":"01:24.230 ","End":"01:26.345","Text":"I\u0027ll go for the easy 1 first,"},{"Start":"01:26.345 ","End":"01:29.475","Text":"f of 1 is 0."},{"Start":"01:29.475 ","End":"01:32.925","Text":"Limit as x goes to 1 from the left,"},{"Start":"01:32.925 ","End":"01:36.165","Text":"this is the range we need x less than 1."},{"Start":"01:36.165 ","End":"01:42.735","Text":"It\u0027s the limit as x goes to 1 from the left of x squared."},{"Start":"01:42.735 ","End":"01:45.060","Text":"X squared is not problematic,"},{"Start":"01:45.060 ","End":"01:50.130","Text":"we just substitute 1 squared which is equal to 1."},{"Start":"01:50.130 ","End":"01:54.110","Text":"Already we see we have a discontinuity of some kind."},{"Start":"01:54.110 ","End":"01:56.405","Text":"Now to the other 1,1 from the right."},{"Start":"01:56.405 ","End":"01:58.415","Text":"From the right we\u0027re bigger than 1,"},{"Start":"01:58.415 ","End":"01:59.765","Text":"so in this case,"},{"Start":"01:59.765 ","End":"02:07.030","Text":"the limit is the limit of minus x squared plus 2x plus 1."},{"Start":"02:07.030 ","End":"02:10.485","Text":"Again we just substitute x equals 1, what do we get?"},{"Start":"02:10.485 ","End":"02:14.145","Text":"Minus 1 plus 2 plus 1."},{"Start":"02:14.145 ","End":"02:16.965","Text":"I make that 2."},{"Start":"02:16.965 ","End":"02:24.095","Text":"Now, both these limits exist,"},{"Start":"02:24.095 ","End":"02:26.045","Text":"but they are different."},{"Start":"02:26.045 ","End":"02:28.355","Text":"1 is not equal to 2,"},{"Start":"02:28.355 ","End":"02:30.430","Text":"I think you\u0027ll agree with me there."},{"Start":"02:30.430 ","End":"02:33.425","Text":"Because both limits exist but they\u0027re not equal,"},{"Start":"02:33.425 ","End":"02:39.205","Text":"this is sometimes called a jump discontinuity or discontinuity of type 1."},{"Start":"02:39.205 ","End":"02:41.470","Text":"I\u0027ll just write it down as"},{"Start":"02:41.470 ","End":"02:52.115","Text":"a jump discontinuity and that answers the question."},{"Start":"02:52.115 ","End":"02:57.225","Text":"I should add at x equals 1."},{"Start":"02:57.225 ","End":"02:59.085","Text":"Now we are done."},{"Start":"02:59.085 ","End":"03:04.265","Text":"Now we come to part C. Once again,"},{"Start":"03:04.265 ","End":"03:09.710","Text":"we have the same division of the line into pieces."},{"Start":"03:09.710 ","End":"03:13.115","Text":"We have less than 1, equal 1, and greater than 1."},{"Start":"03:13.115 ","End":"03:16.130","Text":"At less than 1, we are continuous,"},{"Start":"03:16.130 ","End":"03:19.235","Text":"x squared we\u0027ve had it before, no problem."},{"Start":"03:19.235 ","End":"03:21.470","Text":"1 over x minus 1."},{"Start":"03:21.470 ","End":"03:25.010","Text":"There might\u0027ve been trouble if we\u0027d had x equals 1 here,"},{"Start":"03:25.010 ","End":"03:26.990","Text":"because then the denominator would be 0,"},{"Start":"03:26.990 ","End":"03:29.000","Text":"but no it\u0027s only for x bigger than 1,"},{"Start":"03:29.000 ","End":"03:31.040","Text":"so that\u0027s no problem either."},{"Start":"03:31.040 ","End":"03:33.140","Text":"The only problem, once again,"},{"Start":"03:33.140 ","End":"03:34.790","Text":"the troublemaker, so to speak,"},{"Start":"03:34.790 ","End":"03:36.470","Text":"is x equals 1."},{"Start":"03:36.470 ","End":"03:38.870","Text":"Let\u0027s check if three quantities are equal."},{"Start":"03:38.870 ","End":"03:41.905","Text":"I\u0027ll just copy the exercise down here."},{"Start":"03:41.905 ","End":"03:49.030","Text":"I\u0027d better scroll down a bit. That should do it."},{"Start":"03:49.060 ","End":"03:54.260","Text":"As before we proceed by checking if 3 things are equal."},{"Start":"03:54.260 ","End":"04:00.690","Text":"1 of them is f of the point let\u0027s say f of 1,"},{"Start":"04:00.690 ","End":"04:02.295","Text":"what is that equal?"},{"Start":"04:02.295 ","End":"04:06.735","Text":"The next would be the limit from the left,"},{"Start":"04:06.735 ","End":"04:11.460","Text":"what f of x is equal to as we approach 1 from the left?"},{"Start":"04:11.460 ","End":"04:15.285","Text":"What happens when we approach 1 from the right?"},{"Start":"04:15.285 ","End":"04:18.780","Text":"All these three things have to be equal for continuity,"},{"Start":"04:18.780 ","End":"04:23.160","Text":"but if they\u0027re not, then we classify the type of discontinuity."},{"Start":"04:23.170 ","End":"04:26.990","Text":"Let\u0027s start, f of 1 is 0,"},{"Start":"04:26.990 ","End":"04:28.820","Text":"it says so here."},{"Start":"04:28.820 ","End":"04:31.715","Text":"When x goes to 1 from the left,"},{"Start":"04:31.715 ","End":"04:39.450","Text":"we read from here and that\u0027s the limit as x goes to 1 from the left of x squared."},{"Start":"04:39.450 ","End":"04:45.060","Text":"No problem here, just substitute x equals 1,1 squared is 1."},{"Start":"04:45.060 ","End":"04:47.985","Text":"Now, x goes to 1 from the right,"},{"Start":"04:47.985 ","End":"04:50.250","Text":"so we have to read from here."},{"Start":"04:50.250 ","End":"04:56.930","Text":"It\u0027s the limit as x goes to 1 from the right of 1 over x minus 1."},{"Start":"04:56.930 ","End":"04:59.090","Text":"Now, if x goes to 1 from the right,"},{"Start":"04:59.090 ","End":"05:02.505","Text":"x is 1 and a tiny bit more,"},{"Start":"05:02.505 ","End":"05:05.810","Text":"so x minus 1 is just a little bit above 0."},{"Start":"05:05.810 ","End":"05:09.670","Text":"In fact, we can symbolically say it\u0027s positive 0,"},{"Start":"05:09.670 ","End":"05:11.150","Text":"1 over 0 plus,"},{"Start":"05:11.150 ","End":"05:13.955","Text":"and this is equal to plus infinity."},{"Start":"05:13.955 ","End":"05:18.965","Text":"In any event, this is not defined."},{"Start":"05:18.965 ","End":"05:22.070","Text":"The limit as x goes to 1 from the right is undefined."},{"Start":"05:22.070 ","End":"05:25.020","Text":"If 1 of the 1 sided limit is undefined,"},{"Start":"05:25.020 ","End":"05:30.290","Text":"that\u0027s enough for us to say that this is an essential discontinuity."},{"Start":"05:30.290 ","End":"05:36.475","Text":"An essential discontinuity is also sometimes called a Type 2 discontinuity."},{"Start":"05:36.475 ","End":"05:45.090","Text":"Just write that and we\u0027ll add at x equals 1 and we are done."}],"ID":4842}],"Thumbnail":null,"ID":84397},{"Name":"The Intermediate Value Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Intermediate Value Theorem","Duration":"5m 33s","ChapterTopicVideoID":8255,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8255.jpeg","UploadDate":"2019-11-14T06:52:28.5930000","DurationForVideoObject":"PT5M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.935","Text":"In this clip, I\u0027m going to be talking about the intermediate value theorem,"},{"Start":"00:04.935 ","End":"00:09.060","Text":"which is very tied into the concept of a continuous function."},{"Start":"00:09.060 ","End":"00:11.190","Text":"I\u0027ll introduce the theorem in stages."},{"Start":"00:11.190 ","End":"00:13.335","Text":"First of all, the general outline."},{"Start":"00:13.335 ","End":"00:16.785","Text":"It talks about a function f which satisfies 2 conditions."},{"Start":"00:16.785 ","End":"00:20.490","Text":"I\u0027m going to give you condition 1 and condition 2."},{"Start":"00:20.490 ","End":"00:23.805","Text":"It says that if the function satisfies these 2 conditions,"},{"Start":"00:23.805 ","End":"00:26.085","Text":"then draw some conclusions."},{"Start":"00:26.085 ","End":"00:32.970","Text":"The first condition says that f needs to be continuous on the interval a, b."},{"Start":"00:32.970 ","End":"00:36.590","Text":"An interval with square brackets means that it includes a and b."},{"Start":"00:36.590 ","End":"00:39.745","Text":"f is continuous at all the points in a b."},{"Start":"00:39.745 ","End":"00:42.890","Text":"Condition number 2 says that f of a and f"},{"Start":"00:42.890 ","End":"00:45.920","Text":"of b have to have opposite signs. What does this mean?"},{"Start":"00:45.920 ","End":"00:50.940","Text":"If I apply the function f to a and then to b they\u0027re not 0,"},{"Start":"00:50.940 ","End":"00:52.590","Text":"they have signs but opposite."},{"Start":"00:52.590 ","End":"00:54.290","Text":"One\u0027s positive and one\u0027s negative."},{"Start":"00:54.290 ","End":"00:57.155","Text":"These are the conditions that our f needs to satisfy,"},{"Start":"00:57.155 ","End":"00:59.210","Text":"and if these 2 do occur,"},{"Start":"00:59.210 ","End":"01:05.525","Text":"then the equation f of x equals 0 has a solution in the interval a b."},{"Start":"01:05.525 ","End":"01:07.670","Text":"Now when I say has a solution,"},{"Start":"01:07.670 ","End":"01:13.415","Text":"another way of saying that is that the graph of f of x crosses the x-axis."},{"Start":"01:13.415 ","End":"01:16.175","Text":"Because at the point where it crosses the x-axis,"},{"Start":"01:16.175 ","End":"01:18.950","Text":"it satisfies f of x equals 0."},{"Start":"01:18.950 ","End":"01:23.510","Text":"Also note that this time the interval is with the round brackets,"},{"Start":"01:23.510 ","End":"01:25.550","Text":"which means it doesn\u0027t include a and b."},{"Start":"01:25.550 ","End":"01:30.830","Text":"We can\u0027t expect it to include a and b because f of a and f of b are not 0,"},{"Start":"01:30.830 ","End":"01:32.585","Text":"one\u0027s positive and one\u0027s negative."},{"Start":"01:32.585 ","End":"01:34.405","Text":"That\u0027s that point here."},{"Start":"01:34.405 ","End":"01:37.080","Text":"This concept is best explained with a graph,"},{"Start":"01:37.080 ","End":"01:39.320","Text":"so let me bring out the graph."},{"Start":"01:39.320 ","End":"01:41.570","Text":"f is continuous on the interval a, b."},{"Start":"01:41.570 ","End":"01:42.860","Text":"There\u0027s 2 points."},{"Start":"01:42.860 ","End":"01:45.590","Text":"There\u0027s a and there\u0027s b,"},{"Start":"01:45.590 ","End":"01:49.205","Text":"and the function is defined between a and b."},{"Start":"01:49.205 ","End":"01:53.600","Text":"Well, it could even go beyond and it\u0027s continuous meaning I can sketch"},{"Start":"01:53.600 ","End":"01:57.880","Text":"the path between a and b without taking the pen off the paper."},{"Start":"01:57.880 ","End":"02:00.605","Text":"f of a and f of b have opposite signs."},{"Start":"02:00.605 ","End":"02:03.620","Text":"Here\u0027s f of a, it\u0027s the point that belongs to a,"},{"Start":"02:03.620 ","End":"02:05.585","Text":"but we look at the y value of it,"},{"Start":"02:05.585 ","End":"02:08.180","Text":"and here\u0027s f of b we go up here and here."},{"Start":"02:08.180 ","End":"02:11.540","Text":"In this case, f of b is positive and f of a is negative,"},{"Start":"02:11.540 ","End":"02:13.280","Text":"although it could have been the other way round."},{"Start":"02:13.280 ","End":"02:18.515","Text":"Then the conclusion is that the equation f of x equals 0 has a solution,"},{"Start":"02:18.515 ","End":"02:20.715","Text":"x equals c in this picture."},{"Start":"02:20.715 ","End":"02:26.795","Text":"Like I said, a solution to the equation is exactly where the graph crosses the x-axis."},{"Start":"02:26.795 ","End":"02:29.240","Text":"Now we could have had more than 1 solution,"},{"Start":"02:29.240 ","End":"02:32.450","Text":"doesn\u0027t matter as long as we have at least 1 or example,"},{"Start":"02:32.450 ","End":"02:35.510","Text":"the function might have gone from here upwards, here,"},{"Start":"02:35.510 ","End":"02:38.675","Text":"and across several times and then being like this,"},{"Start":"02:38.675 ","End":"02:40.940","Text":"and then it would have had not only c but"},{"Start":"02:40.940 ","End":"02:43.820","Text":"this point and this point and this point and this point,"},{"Start":"02:43.820 ","End":"02:44.930","Text":"there could be more than 1,"},{"Start":"02:44.930 ","End":"02:47.600","Text":"but I don\u0027t care as long as there\u0027s at least 1."},{"Start":"02:47.600 ","End":"02:51.065","Text":"That\u0027s basically what the intermediate value theorem says."},{"Start":"02:51.065 ","End":"02:53.870","Text":"If I start out below the X-axis"},{"Start":"02:53.870 ","End":"02:56.570","Text":"and I go continuously to a point above"},{"Start":"02:56.570 ","End":"03:00.290","Text":"the x-axis at somewhere in the middle I\u0027ve got to cross the x-axis."},{"Start":"03:00.290 ","End":"03:01.505","Text":"That\u0027s intuitive."},{"Start":"03:01.505 ","End":"03:04.040","Text":"Let\u0027s go on and do an exercise using"},{"Start":"03:04.040 ","End":"03:07.460","Text":"this intermediate value theorem to see if it\u0027s useful."},{"Start":"03:07.460 ","End":"03:10.415","Text":"Here\u0027s the example exercise."},{"Start":"03:10.415 ","End":"03:13.025","Text":"We have to prove that the equation,"},{"Start":"03:13.025 ","End":"03:14.430","Text":"x cubed plus 4,"},{"Start":"03:14.430 ","End":"03:18.275","Text":"x minus 3 equals 0 has at least 1 solution."},{"Start":"03:18.275 ","End":"03:23.270","Text":"This is a cubic equation and we haven\u0027t learned how to solve cubic equations"},{"Start":"03:23.270 ","End":"03:27.610","Text":"so what I\u0027m going to do is use the intermediate value theorem."},{"Start":"03:27.610 ","End":"03:29.930","Text":"I\u0027m going to write it first as a function,"},{"Start":"03:29.930 ","End":"03:34.570","Text":"and then I\u0027m going to show that this function f has properties 1 and 2."},{"Start":"03:34.570 ","End":"03:38.990","Text":"Then immediately I\u0027ll be able to draw the conclusion that it has a solution."},{"Start":"03:38.990 ","End":"03:41.600","Text":"I\u0027ll choose the a and b as convenient for me."},{"Start":"03:41.600 ","End":"03:43.955","Text":"I just wanted to have a solution anywhere."},{"Start":"03:43.955 ","End":"03:45.784","Text":"We have to find the interval."},{"Start":"03:45.784 ","End":"03:47.330","Text":"Let\u0027s try first of all,"},{"Start":"03:47.330 ","End":"03:49.805","Text":"x equals 0 and see what happens."},{"Start":"03:49.805 ","End":"03:51.830","Text":"Well, just try and remember what we have to do."},{"Start":"03:51.830 ","End":"03:54.210","Text":"Remember we have to prove continuity on"},{"Start":"03:54.210 ","End":"03:57.400","Text":"an interval and have to have opposite signs at the end."},{"Start":"03:57.400 ","End":"03:59.645","Text":"But try values that are easy to substitute."},{"Start":"03:59.645 ","End":"04:02.960","Text":"If x equals 0, and I forgot to give this a name,"},{"Start":"04:02.960 ","End":"04:05.030","Text":"but we\u0027ll call this f of x."},{"Start":"04:05.030 ","End":"04:07.010","Text":"When x equals 0,"},{"Start":"04:07.010 ","End":"04:09.110","Text":"we get f of x,"},{"Start":"04:09.110 ","End":"04:11.870","Text":"which is f of 0,"},{"Start":"04:11.870 ","End":"04:17.525","Text":"is equal to 0 cubed plus 4 times 0 minus 3 is minus 3."},{"Start":"04:17.525 ","End":"04:21.650","Text":"That\u0027s negative, and let\u0027s try another value, x equals 1."},{"Start":"04:21.650 ","End":"04:24.470","Text":"We get 1 cubed is 1,"},{"Start":"04:24.470 ","End":"04:28.775","Text":"4 times 1 is 4, 5 minus 3 is 2."},{"Start":"04:28.775 ","End":"04:33.005","Text":"In this case, f of x is f of 1,"},{"Start":"04:33.005 ","End":"04:34.945","Text":"which is plus 2,"},{"Start":"04:34.945 ","End":"04:36.974","Text":"which is bigger than 0."},{"Start":"04:36.974 ","End":"04:40.410","Text":"That\u0027s the part about the a and the b having opposite signs,"},{"Start":"04:40.410 ","End":"04:43.820","Text":"as for continuity, we know that polynomials"},{"Start":"04:43.820 ","End":"04:46.685","Text":"or elementary functions are always continuous."},{"Start":"04:46.685 ","End":"04:48.275","Text":"We have it\u0027s continuous."},{"Start":"04:48.275 ","End":"04:49.890","Text":"This we\u0027re going to take as a and b."},{"Start":"04:49.890 ","End":"04:53.180","Text":"The interval a, b is going to be,"},{"Start":"04:53.180 ","End":"04:54.290","Text":"we going to take for a b,"},{"Start":"04:54.290 ","End":"04:58.220","Text":"the interval 0, 1. f is continuous on a b,"},{"Start":"04:58.220 ","End":"05:02.005","Text":"it\u0027s continuous everywhere so it\u0027s continuous on the interval 0, 1."},{"Start":"05:02.005 ","End":"05:04.340","Text":"At the end points, at 0,"},{"Start":"05:04.340 ","End":"05:08.390","Text":"it\u0027s negative, and here it\u0027s positive."},{"Start":"05:08.390 ","End":"05:10.280","Text":"It has the opposite signs,"},{"Start":"05:10.280 ","End":"05:12.680","Text":"so we can jump to the conclusion,"},{"Start":"05:12.680 ","End":"05:14.000","Text":"it\u0027s still up here,"},{"Start":"05:14.000 ","End":"05:17.660","Text":"that f has a solution between 0 and 1."},{"Start":"05:17.660 ","End":"05:19.175","Text":"The conclusion is,"},{"Start":"05:19.175 ","End":"05:24.350","Text":"that f has a solution specifically in 0,"},{"Start":"05:24.350 ","End":"05:26.630","Text":"1, but we don\u0027t even care where it is."},{"Start":"05:26.630 ","End":"05:30.335","Text":"This is true by the mean value theorem."},{"Start":"05:30.335 ","End":"05:33.960","Text":"That\u0027s it. That\u0027s the example and I\u0027m done."}],"ID":8415},{"Watched":false,"Name":"Exercise 1","Duration":"4m 50s","ChapterTopicVideoID":3312,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3312.jpeg","UploadDate":"2015-01-07T16:54:55.9070000","DurationForVideoObject":"PT4M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"In this exercise, we\u0027re asked to show or prove that the equation cosine x equals x"},{"Start":"00:07.050 ","End":"00:11.610","Text":"has to have at least 1 solution and the tool we\u0027re going to use is"},{"Start":"00:11.610 ","End":"00:14.310","Text":"the intermediate value theorem."},{"Start":"00:14.310 ","End":"00:20.740","Text":"Now there\u0027s a standard way of beginning these problems."},{"Start":"00:20.740 ","End":"00:25.680","Text":"Usually what you do is you define a function f of x,"},{"Start":"00:25.680 ","End":"00:29.085","Text":"which is the difference between these things."},{"Start":"00:29.085 ","End":"00:34.230","Text":"In our case it would be cosine x minus x."},{"Start":"00:34.230 ","End":"00:40.639","Text":"Because then the existence of a solution means a place where f of x is 0."},{"Start":"00:40.639 ","End":"00:42.290","Text":"Otherwise, we were looking for,"},{"Start":"00:42.290 ","End":"00:51.455","Text":"we want a solution to f of x equals 0."},{"Start":"00:51.455 ","End":"00:54.995","Text":"You see what I mean? If f of x is 0,"},{"Start":"00:54.995 ","End":"00:57.140","Text":"then cosine x minus x is 0,"},{"Start":"00:57.140 ","End":"01:00.170","Text":"then cosine x equals x and vice versa."},{"Start":"01:00.170 ","End":"01:05.120","Text":"It\u0027s equivalent to having a 0 of the function f."},{"Start":"01:05.120 ","End":"01:11.410","Text":"The way we do that with the intermediate value is we find 2 points,"},{"Start":"01:11.410 ","End":"01:13.080","Text":"different points to set of x."},{"Start":"01:13.080 ","End":"01:14.850","Text":"Let\u0027s say if we find a point a,"},{"Start":"01:14.850 ","End":"01:17.475","Text":"one which is bigger than 0,"},{"Start":"01:17.475 ","End":"01:23.775","Text":"and we find another value b by guesswork or whatever,"},{"Start":"01:23.775 ","End":"01:26.180","Text":"or divine inspiration, somehow you find"},{"Start":"01:26.180 ","End":"01:31.240","Text":"the values a and b where one\u0027s positive and 1 is negative."},{"Start":"01:31.240 ","End":"01:35.930","Text":"If we find this, then automatically we know"},{"Start":"01:35.930 ","End":"01:39.140","Text":"that by the intermediate value theorem that f of c is"},{"Start":"01:39.140 ","End":"01:46.310","Text":"going to be exactly equal to 0 for some c. If this and this,"},{"Start":"01:46.310 ","End":"01:48.540","Text":"then we have our thing,"},{"Start":"01:48.540 ","End":"01:54.425","Text":"and if f of c is 0, then it means the original thing at c is a solution for this."},{"Start":"01:54.425 ","End":"01:58.760","Text":"All I have to do now is find 2 values, 2 numbers,"},{"Start":"01:58.760 ","End":"02:00.350","Text":"a and b or just 2 numbers,"},{"Start":"02:00.350 ","End":"02:03.680","Text":"one where this positive and 1 where it\u0027s negative."},{"Start":"02:03.680 ","End":"02:06.660","Text":"Now what values of x I\u0027m going to try out?"},{"Start":"02:06.660 ","End":"02:08.185","Text":"I\u0027m going to look for an a and b,"},{"Start":"02:08.185 ","End":"02:11.810","Text":"probably angles whose cosine I know easily."},{"Start":"02:11.810 ","End":"02:13.445","Text":"I know cosine 0,"},{"Start":"02:13.445 ","End":"02:15.260","Text":"I know cosine Pi over 3,"},{"Start":"02:15.260 ","End":"02:18.290","Text":"I know cosine pi over 2 and negative of those."},{"Start":"02:18.290 ","End":"02:21.910","Text":"Let\u0027s just start off with 0 and see what happens."},{"Start":"02:21.910 ","End":"02:26.600","Text":"If I start off with try a equals 0,"},{"Start":"02:26.600 ","End":"02:33.089","Text":"so f of 0 will be cosine 0 minus 0."},{"Start":"02:33.089 ","End":"02:35.335","Text":"We all know what cosine 0 is."},{"Start":"02:35.335 ","End":"02:37.180","Text":"Cosine of 0 is 1,"},{"Start":"02:37.180 ","End":"02:39.895","Text":"1 minus 0 is 1."},{"Start":"02:39.895 ","End":"02:42.165","Text":"F of 0 is 1,"},{"Start":"02:42.165 ","End":"02:45.110","Text":"which is definitely positive."},{"Start":"02:45.110 ","End":"02:47.584","Text":"Now what else am I going to try?"},{"Start":"02:47.584 ","End":"02:50.465","Text":"I know 60 degrees and I know 90 degrees."},{"Start":"02:50.465 ","End":"02:52.570","Text":"Let\u0027s go for the 60 degrees."},{"Start":"02:52.570 ","End":"02:55.430","Text":"Let\u0027s try. But it\u0027s not 60 degrees,"},{"Start":"02:55.430 ","End":"02:56.960","Text":"it\u0027s in radians, of course,"},{"Start":"02:56.960 ","End":"02:58.145","Text":"it\u0027s Pi over 3."},{"Start":"02:58.145 ","End":"03:00.110","Text":"Pi is 180,"},{"Start":"03:00.110 ","End":"03:02.330","Text":"so Pi over 3 is 60."},{"Start":"03:02.330 ","End":"03:04.465","Text":"Let\u0027s see what that gives us."},{"Start":"03:04.465 ","End":"03:12.015","Text":"F of Pi over 3 is cosine Pi over 3 minus Pi over 3."},{"Start":"03:12.015 ","End":"03:13.430","Text":"Cosine Pi over 3,"},{"Start":"03:13.430 ","End":"03:19.190","Text":"which is cosine 60 degrees is 1/2 and minus Pi over 3."},{"Start":"03:19.190 ","End":"03:23.780","Text":"Now, I don\u0027t know exactly what this is,"},{"Start":"03:23.780 ","End":"03:25.040","Text":"but I know it\u0027s negative."},{"Start":"03:25.040 ","End":"03:27.470","Text":"That\u0027s for sure. Well, you could compute it."},{"Start":"03:27.470 ","End":"03:30.290","Text":"But I\u0027ll tell you a good reason why it\u0027s negative."},{"Start":"03:30.290 ","End":"03:33.570","Text":"It\u0027s negative because 1 of"},{"Start":"03:33.570 ","End":"03:38.760","Text":"the easiest ways to see it is that Pi over 3 is bigger than 1."},{"Start":"03:38.760 ","End":"03:40.455","Text":"I hope that\u0027s obvious."},{"Start":"03:40.455 ","End":"03:43.370","Text":"If it\u0027s not obvious, just notice what Pi is."},{"Start":"03:43.370 ","End":"03:46.810","Text":"It\u0027s 3.14 something. Pi is bigger than 3."},{"Start":"03:46.810 ","End":"03:48.990","Text":"That Pi over 3 is bigger than 1."},{"Start":"03:48.990 ","End":"03:54.365","Text":"A half minus something greater than 1 is going to be quite negative,"},{"Start":"03:54.365 ","End":"03:56.615","Text":"certainly lower than minus a half."},{"Start":"03:56.615 ","End":"04:00.115","Text":"In other words, we found f of 0,"},{"Start":"04:00.115 ","End":"04:02.835","Text":"which gives positive answer,"},{"Start":"04:02.835 ","End":"04:04.430","Text":"and f of Pi over 3,"},{"Start":"04:04.430 ","End":"04:06.470","Text":"which gives a negative answer."},{"Start":"04:06.470 ","End":"04:08.780","Text":"We know that there\u0027s some point c."},{"Start":"04:08.780 ","End":"04:21.210","Text":"We know that f of c is going to equal 0 for some c such that 0 is less than c,"},{"Start":"04:21.210 ","End":"04:24.485","Text":"is less than Pi over 3."},{"Start":"04:24.485 ","End":"04:27.020","Text":"Somewhere in the range from 0 to Pi over 3,"},{"Start":"04:27.020 ","End":"04:30.090","Text":"there is a c for which f of c is 0,"},{"Start":"04:30.090 ","End":"04:33.090","Text":"and going back again to remind you what that"},{"Start":"04:33.090 ","End":"04:37.560","Text":"means in terms of f. It means that cosine c minus c,"},{"Start":"04:37.560 ","End":"04:44.090","Text":"this implies that cosine c is equal to c because cosine c minus c is 0,"},{"Start":"04:44.090 ","End":"04:50.390","Text":"and so on and so on. We\u0027re done."}],"ID":3323},{"Watched":false,"Name":"Exercise 2","Duration":"3m 40s","ChapterTopicVideoID":3313,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3313.jpeg","UploadDate":"2015-01-07T17:34:15.1930000","DurationForVideoObject":"PT3M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this exercise, we have to use the intermediate value theorem"},{"Start":"00:04.185 ","End":"00:08.790","Text":"to find the solution to the equation x cubed plus 4x equals 1."},{"Start":"00:08.790 ","End":"00:13.440","Text":"That we have to show that at least such solution exists."},{"Start":"00:13.440 ","End":"00:16.980","Text":"Usual technique is to convert the equation into"},{"Start":"00:16.980 ","End":"00:21.550","Text":"a function and look for a value which makes the function 0."},{"Start":"00:21.550 ","End":"00:23.345","Text":"I\u0027ll show you what I mean."},{"Start":"00:23.345 ","End":"00:24.720","Text":"In this case, let\u0027s bring,"},{"Start":"00:24.720 ","End":"00:26.515","Text":"the 1 to the left-hand side,"},{"Start":"00:26.515 ","End":"00:34.760","Text":"and define a function f being x cubed plus 4x minus 1."},{"Start":"00:34.760 ","End":"00:41.270","Text":"Now, we can see that f of x is equal to 0 is the same thing."},{"Start":"00:41.270 ","End":"00:47.510","Text":"It\u0027s if and only if that x cubed plus 4x is equal to 1."},{"Start":"00:47.510 ","End":"00:52.370","Text":"In other words, a solution to our equation is the same as the value of x,"},{"Start":"00:52.370 ","End":"00:54.335","Text":"which makes f 0."},{"Start":"00:54.335 ","End":"00:55.910","Text":"And now we can use"},{"Start":"00:55.910 ","End":"00:59.975","Text":"the intermediate value theorem because all you have to do is find 2 values."},{"Start":"00:59.975 ","End":"01:01.850","Text":"I don\u0027t know what say a and b,"},{"Start":"01:01.850 ","End":"01:03.280","Text":"if we can find,"},{"Start":"01:03.280 ","End":"01:08.930","Text":"if we can be so lucky or skillful at finding a and b such that f"},{"Start":"01:08.930 ","End":"01:15.455","Text":"of a is less than 0 and f of b is bigger than 0,"},{"Start":"01:15.455 ","End":"01:16.940","Text":"then that\u0027s great,"},{"Start":"01:16.940 ","End":"01:20.540","Text":"because the intermediate value theorem says then that there is"},{"Start":"01:20.540 ","End":"01:30.590","Text":"a c between a and b such that f of c is equal to 0,"},{"Start":"01:30.590 ","End":"01:35.605","Text":"which means that c is a solution to the original equation."},{"Start":"01:35.605 ","End":"01:38.950","Text":"Now, all that remains is to find a and b,"},{"Start":"01:38.950 ","End":"01:43.625","Text":"and my suggestion is just try anything that\u0027s easy to substitute,"},{"Start":"01:43.625 ","End":"01:45.695","Text":"and if we get positive or negative,"},{"Start":"01:45.695 ","End":"01:49.655","Text":"we\u0027ll still be on our way to finding the 2,so let\u0027s try it."},{"Start":"01:49.655 ","End":"01:52.520","Text":"0 is usually the easiest thing to do."},{"Start":"01:52.520 ","End":"01:55.640","Text":"So f of 0 is 0 plus 4,"},{"Start":"01:55.640 ","End":"01:58.580","Text":"0s minus 1 is minus 1,"},{"Start":"01:58.580 ","End":"02:01.055","Text":"which happens to be negative."},{"Start":"02:01.055 ","End":"02:05.090","Text":"All we need now is f of something to be positive, and that\u0027s it."},{"Start":"02:05.090 ","End":"02:07.270","Text":"I could just keep trying values,"},{"Start":"02:07.270 ","End":"02:09.440","Text":"and probably you could do it that way."},{"Start":"02:09.440 ","End":"02:13.220","Text":"I just would like to point out you can do it not so totally random."},{"Start":"02:13.220 ","End":"02:22.295","Text":"Here, for example, I know that when a polynomial is as a positive leading coefficient,"},{"Start":"02:22.295 ","End":"02:24.515","Text":"then when x goes to infinity,"},{"Start":"02:24.515 ","End":"02:26.945","Text":"the polynomial also goes to infinity."},{"Start":"02:26.945 ","End":"02:28.880","Text":"If I can\u0027t find something positive,"},{"Start":"02:28.880 ","End":"02:31.850","Text":"we\u0027ll just try xs which are larger and larger."},{"Start":"02:31.850 ","End":"02:34.569","Text":"Eventually I\u0027ll get 1 that\u0027s positive."},{"Start":"02:34.569 ","End":"02:37.310","Text":"First, we don\u0027t find something which is positive,"},{"Start":"02:37.310 ","End":"02:39.650","Text":"just try larger and larger values."},{"Start":"02:39.650 ","End":"02:45.380","Text":"Anyway, I was just going to try x equals 1 because it\u0027s easy to substitute,"},{"Start":"02:45.380 ","End":"02:47.365","Text":"so let\u0027s see, maybe we get lucky."},{"Start":"02:47.365 ","End":"02:53.510","Text":"f of 1 is 1 cubed plus 4 times 1 minus 1."},{"Start":"02:53.510 ","End":"02:56.674","Text":"1 plus 4 minus 1 is 4,"},{"Start":"02:56.674 ","End":"02:58.260","Text":"which is bigger than 0,"},{"Start":"02:58.260 ","End":"03:00.360","Text":"so yay, that\u0027s good."},{"Start":"03:00.360 ","End":"03:02.040","Text":"We have 1 that\u0027s smaller than 0,"},{"Start":"03:02.040 ","End":"03:04.050","Text":"and we have 1 that\u0027s bigger than 0,"},{"Start":"03:04.050 ","End":"03:09.195","Text":"so we know that between 0 and 1,"},{"Start":"03:09.195 ","End":"03:18.304","Text":"there is an x or a c or whatever letter you want such that f of c is equal to 0."},{"Start":"03:18.304 ","End":"03:20.930","Text":"Hence, an f of c equals 0,"},{"Start":"03:20.930 ","End":"03:24.380","Text":"which implies that, as we said here,"},{"Start":"03:24.380 ","End":"03:27.515","Text":"if something is 0, that x cubed,"},{"Start":"03:27.515 ","End":"03:35.270","Text":"c cubed here, c cubed minus 4 times c is equal to 1,so c,"},{"Start":"03:35.270 ","End":"03:38.435","Text":"the c is the x that we\u0027re looking for."},{"Start":"03:38.435 ","End":"03:41.160","Text":"That\u0027s basically it."}],"ID":3324},{"Watched":false,"Name":"Exercise 3","Duration":"6m 44s","ChapterTopicVideoID":3314,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3314.jpeg","UploadDate":"2015-01-08T00:22:49.1430000","DurationForVideoObject":"PT6M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"Here we have to show that the equation which I\u0027ve underlined here,"},{"Start":"00:04.860 ","End":"00:10.050","Text":"natural log of x equals minus x^2 has to have at least 1 solution."},{"Start":"00:10.050 ","End":"00:14.805","Text":"We\u0027re supposed to use the intermediate value theorem to do this."},{"Start":"00:14.805 ","End":"00:23.535","Text":"I just made a note to myself that this equation is only defined when x is bigger than 0,"},{"Start":"00:23.535 ","End":"00:29.475","Text":"because natural log, it\u0027s argument has to be positive strictly."},{"Start":"00:29.475 ","End":"00:33.495","Text":"The usual technique where a set an equation,"},{"Start":"00:33.495 ","End":"00:36.840","Text":"we turn it into a function that has to be 0."},{"Start":"00:36.840 ","End":"00:39.385","Text":"We subtract 1 side from the other."},{"Start":"00:39.385 ","End":"00:46.640","Text":"We take, define f of x is equal to this side minus this side,"},{"Start":"00:46.640 ","End":"00:53.100","Text":"which is natural log of x minus minus is plus x squared."},{"Start":"00:53.100 ","End":"00:56.960","Text":"We have to show that what we want,"},{"Start":"00:56.960 ","End":"01:00.920","Text":"f of x equals 0."},{"Start":"01:00.920 ","End":"01:03.950","Text":"In other words, show that there is at least 1 such x."},{"Start":"01:03.950 ","End":"01:07.760","Text":"The usual technique is to find,"},{"Start":"01:07.760 ","End":"01:10.340","Text":"say, a pair of values a and b."},{"Start":"01:10.340 ","End":"01:15.350","Text":"If we take, find an a and by luck or"},{"Start":"01:15.350 ","End":"01:21.380","Text":"some other technique which is bigger than 0 and we have another value,"},{"Start":"01:21.380 ","End":"01:25.555","Text":"say b, which is smaller than 0,"},{"Start":"01:25.555 ","End":"01:32.150","Text":"then the intermediate value theorem says that there has to be some x,"},{"Start":"01:32.150 ","End":"01:34.010","Text":"or let\u0027s call it a c in this case."},{"Start":"01:34.010 ","End":"01:39.350","Text":"I\u0027ll write it as f of c is got to equals 0."},{"Start":"01:39.350 ","End":"01:45.815","Text":"For some c, where c is between a and b,"},{"Start":"01:45.815 ","End":"01:51.460","Text":"there exists such as c and that c is the x that we want here."},{"Start":"01:51.460 ","End":"01:58.430","Text":"We have to find such an a and a b. I\u0027m wondering what we should try."},{"Start":"01:58.430 ","End":"02:00.960","Text":"We have to try something positive."},{"Start":"02:00.960 ","End":"02:06.035","Text":"I think we should just go for values that are easy to compute."},{"Start":"02:06.035 ","End":"02:08.630","Text":"We know the natural log, for example,"},{"Start":"02:08.630 ","End":"02:11.480","Text":"of all the powers of e. For example,"},{"Start":"02:11.480 ","End":"02:18.530","Text":"the natural log of e^n is going to be just n,"},{"Start":"02:18.530 ","End":"02:23.180","Text":"because that\u0027s actually what the definition of natural log is."},{"Start":"02:23.180 ","End":"02:26.690","Text":"If a natural log of m,"},{"Start":"02:26.690 ","End":"02:32.840","Text":"let\u0027s say is n then that\u0027s exactly equivalent by definition of saying"},{"Start":"02:32.840 ","End":"02:39.905","Text":"that e^n is equal to m. Because natural log mean,"},{"Start":"02:39.905 ","End":"02:43.850","Text":"you ask yourself e of the power of what gives me this."},{"Start":"02:43.850 ","End":"02:46.700","Text":"In that case, this is the same thing."},{"Start":"02:46.700 ","End":"02:49.050","Text":"Then it follows from here."},{"Start":"02:49.050 ","End":"02:51.450","Text":"If I just put here instead of m,"},{"Start":"02:51.450 ","End":"02:53.295","Text":"I put e^n,"},{"Start":"02:53.295 ","End":"02:58.110","Text":"then I get exactly what is written on this line here, anyway."},{"Start":"02:58.330 ","End":"03:03.020","Text":"I suggest trying different values of n. For example,"},{"Start":"03:03.020 ","End":"03:06.050","Text":"if I put n equals 0,"},{"Start":"03:06.050 ","End":"03:09.765","Text":"that would mean my m is 1."},{"Start":"03:09.765 ","End":"03:16.845","Text":"I\u0027m going to try to substitute x equals 1. f"},{"Start":"03:16.845 ","End":"03:23.835","Text":"of 1 is natural log of 1 plus 1 squared,"},{"Start":"03:23.835 ","End":"03:25.800","Text":"and this is 0 plus 1,"},{"Start":"03:25.800 ","End":"03:27.075","Text":"which equals 1,"},{"Start":"03:27.075 ","End":"03:29.850","Text":"which is definitely positive."},{"Start":"03:29.850 ","End":"03:31.970","Text":"Now what else might I try?"},{"Start":"03:31.970 ","End":"03:35.000","Text":"We got a positive 1. Now we got to try and find something negative."},{"Start":"03:35.000 ","End":"03:36.650","Text":"Well, let\u0027s just try,"},{"Start":"03:36.650 ","End":"03:39.020","Text":"instead of n equals 0,"},{"Start":"03:39.020 ","End":"03:41.225","Text":"we\u0027ll try n equals 1."},{"Start":"03:41.225 ","End":"03:43.400","Text":"That would make m equal to e,"},{"Start":"03:43.400 ","End":"03:45.905","Text":"and that\u0027s what I\u0027ll try substituting next."},{"Start":"03:45.905 ","End":"03:51.300","Text":"Let\u0027s see what is f of e. Well that will equal,"},{"Start":"03:51.300 ","End":"03:54.500","Text":"and I\u0027m just hoping I get lucky and get something negative,"},{"Start":"03:54.500 ","End":"03:56.615","Text":"5 not of, lets keep trying."},{"Start":"03:56.615 ","End":"04:06.085","Text":"f of e is natural log of e plus e squared."},{"Start":"04:06.085 ","End":"04:12.195","Text":"That doesn\u0027t look very good because that\u0027s already, yeah, it\u0027s positive."},{"Start":"04:12.195 ","End":"04:17.860","Text":"It\u0027s 1 plus e^2, still more positive."},{"Start":"04:17.900 ","End":"04:20.380","Text":"If that didn\u0027t work,"},{"Start":"04:20.380 ","End":"04:24.840","Text":"let\u0027s try may be a negative value of n would work."},{"Start":"04:24.840 ","End":"04:27.310","Text":"Because how about n is minus 1?"},{"Start":"04:27.310 ","End":"04:30.220","Text":"In which case m is 1 over e,"},{"Start":"04:30.220 ","End":"04:32.335","Text":"or I mean it\u0027s e to the minus 1."},{"Start":"04:32.335 ","End":"04:35.650","Text":"Let\u0027s see. This is good for us."},{"Start":"04:35.650 ","End":"04:39.595","Text":"This is now not good for us because we want a positive and a negative."},{"Start":"04:39.595 ","End":"04:42.640","Text":"We try f to the e minus 1,"},{"Start":"04:42.640 ","End":"04:49.315","Text":"or 1 over e. That will be the natural log of e to the minus 1,"},{"Start":"04:49.315 ","End":"04:51.635","Text":"is just minus 1."},{"Start":"04:51.635 ","End":"04:56.160","Text":"It\u0027s minus 1 plus 1 over e squared,"},{"Start":"04:56.160 ","End":"04:59.070","Text":"which is 1 over e squared."},{"Start":"04:59.070 ","End":"05:02.595","Text":"That is definitely negative."},{"Start":"05:02.595 ","End":"05:04.490","Text":"The reason that\u0027s negative,"},{"Start":"05:04.490 ","End":"05:07.190","Text":"because e is bigger than 1,"},{"Start":"05:07.190 ","End":"05:11.660","Text":"we also have that e squared is bigger than 1,"},{"Start":"05:11.660 ","End":"05:17.565","Text":"which makes 1 over e squared less than 1."},{"Start":"05:17.565 ","End":"05:19.574","Text":"If that\u0027s the case,"},{"Start":"05:19.574 ","End":"05:22.820","Text":"then if I bring the 1 over to this side,"},{"Start":"05:22.820 ","End":"05:28.455","Text":"I get minus 1 plus 1 over e squared is less than 0."},{"Start":"05:28.455 ","End":"05:32.265","Text":"When we tried f of 1,"},{"Start":"05:32.265 ","End":"05:35.040","Text":"we got something positive."},{"Start":"05:35.040 ","End":"05:38.250","Text":"When we tried 1 over e,"},{"Start":"05:38.250 ","End":"05:43.415","Text":"then we got something negative in the function,"},{"Start":"05:43.415 ","End":"05:45.980","Text":"which means that this is good for us."},{"Start":"05:45.980 ","End":"05:49.100","Text":"We have 1 positive and 1 negative."},{"Start":"05:49.100 ","End":"05:56.510","Text":"By this theorem, it means that there is somewhere between these 2."},{"Start":"05:56.510 ","End":"05:58.655","Text":"This 1 is bigger than this."},{"Start":"05:58.655 ","End":"06:04.880","Text":"There is some value c between 1 over e and 1,"},{"Start":"06:04.880 ","End":"06:10.325","Text":"this exists, such that f of c is equal to 0,"},{"Start":"06:10.325 ","End":"06:12.950","Text":"and that\u0027s the x that we\u0027re looking for."},{"Start":"06:12.950 ","End":"06:16.745","Text":"I\u0027m claiming that c is the solution for this equation."},{"Start":"06:16.745 ","End":"06:19.550","Text":"Because if f of c is 0,"},{"Start":"06:19.550 ","End":"06:25.970","Text":"then natural log of c plus c squared is equal to 0,"},{"Start":"06:25.970 ","End":"06:33.465","Text":"which gives us exactly that natural log of c equals minus c squared."},{"Start":"06:33.465 ","End":"06:36.445","Text":"c is the x,"},{"Start":"06:36.445 ","End":"06:41.255","Text":"so to speak, that we\u0027re looking forward that satisfies this."},{"Start":"06:41.255 ","End":"06:45.480","Text":"That\u0027ll do it. That\u0027s proof enough."}],"ID":3325},{"Watched":false,"Name":"Exercise 4","Duration":"6m 26s","ChapterTopicVideoID":3315,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3315.jpeg","UploadDate":"2015-01-08T00:23:20.8970000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.560","Text":"Here we have to use"},{"Start":"00:01.560 ","End":"00:07.545","Text":"the famous intermediate value theorem to show that a cubic polynomial,"},{"Start":"00:07.545 ","End":"00:12.420","Text":"or specifically x cubed plus some quadratic,"},{"Start":"00:12.420 ","End":"00:19.185","Text":"has to have at least 1 value of x for which it gives 0 when you substitute."},{"Start":"00:19.185 ","End":"00:25.200","Text":"Now, here it\u0027s different than the previous exercises because we don\u0027t know what b, c,"},{"Start":"00:25.200 ","End":"00:29.145","Text":"and d are, so just random substituting,"},{"Start":"00:29.145 ","End":"00:32.925","Text":"we\u0027re going to know we have to get more sophisticated than that."},{"Start":"00:32.925 ","End":"00:36.480","Text":"I\u0027ve seen cubic equations often enough to"},{"Start":"00:36.480 ","End":"00:41.350","Text":"know that what\u0027s important is that 3 is an odd number."},{"Start":"00:41.350 ","End":"00:46.085","Text":"What I\u0027m about to say will work for powers of 5, 7, etc."},{"Start":"00:46.085 ","End":"00:49.650","Text":"If you draw the graph and you don\u0027t have to draw a graph,"},{"Start":"00:49.650 ","End":"00:53.270","Text":"just giving you an idea of rationale."},{"Start":"00:53.270 ","End":"00:59.405","Text":"In general, the polynomials with the x cubed,"},{"Start":"00:59.405 ","End":"01:01.400","Text":"the beginning or x^5,"},{"Start":"01:01.400 ","End":"01:04.445","Text":"they go up and down,"},{"Start":"01:04.445 ","End":"01:07.615","Text":"but they start from minus infinity."},{"Start":"01:07.615 ","End":"01:13.010","Text":"They might have more bumps or less but in the middle,"},{"Start":"01:13.010 ","End":"01:17.540","Text":"but in general they all go something like this."},{"Start":"01:17.540 ","End":"01:23.605","Text":"I\u0027m saying, the important thing is that this goes to infinity in this direction,"},{"Start":"01:23.605 ","End":"01:27.740","Text":"and it goes to minus infinity in that direction."},{"Start":"01:27.740 ","End":"01:33.230","Text":"Basically, if I can just demonstrate to you that when x goes to infinity,"},{"Start":"01:33.230 ","End":"01:37.400","Text":"that this equation, which I called y,"},{"Start":"01:37.400 ","End":"01:41.270","Text":"goes to infinity and in the other direction to minus infinity."},{"Start":"01:41.270 ","End":"01:43.145","Text":"If something goes to infinity,"},{"Start":"01:43.145 ","End":"01:46.295","Text":"it has to go above 0 at some point."},{"Start":"01:46.295 ","End":"01:49.250","Text":"In fact, at some point onwards,"},{"Start":"01:49.250 ","End":"01:52.505","Text":"it\u0027s going to be higher than anything you\u0027d like to say."},{"Start":"01:52.505 ","End":"01:54.200","Text":"In the other direction,"},{"Start":"01:54.200 ","End":"01:58.535","Text":"it will be lower than anything I want, in particular 0."},{"Start":"01:58.535 ","End":"02:03.370","Text":"It has to go up above 0 on the right and it has to go below 0 on the left,"},{"Start":"02:03.370 ","End":"02:08.465","Text":"that gives us our positive and negative doesn\u0027t matter where it can go below the axis."},{"Start":"02:08.465 ","End":"02:11.585","Text":"But from some point on which it\u0027s going to be above"},{"Start":"02:11.585 ","End":"02:15.200","Text":"and also hear from some point and onwards to the left,"},{"Start":"02:15.200 ","End":"02:20.145","Text":"it\u0027s going to be positive here, negative here."},{"Start":"02:20.145 ","End":"02:21.530","Text":"We know in the middle,"},{"Start":"02:21.530 ","End":"02:24.125","Text":"is going to be somewhere where it\u0027s 0."},{"Start":"02:24.125 ","End":"02:28.505","Text":"Actually the 3 places in the middle where it could be."},{"Start":"02:28.505 ","End":"02:30.080","Text":"But we just need 1."},{"Start":"02:30.080 ","End":"02:33.380","Text":"But if here is positive and here it\u0027s negative,"},{"Start":"02:33.380 ","End":"02:35.825","Text":"then it does cross 3 points."},{"Start":"02:35.825 ","End":"02:39.320","Text":"If we start off below the line and ended up above the line,"},{"Start":"02:39.320 ","End":"02:41.390","Text":"it has to cross the line at some point."},{"Start":"02:41.390 ","End":"02:43.760","Text":"The picture is 1 thing,"},{"Start":"02:43.760 ","End":"02:47.870","Text":"but it need to make it more precise, more mathematical."},{"Start":"02:47.870 ","End":"02:51.180","Text":"I claim that the limit,"},{"Start":"02:51.580 ","End":"02:55.095","Text":"I start with the plus infinity side,"},{"Start":"02:55.095 ","End":"03:04.970","Text":"x goes to infinity of x cubed plus bx squared plus cx plus d. Well,"},{"Start":"03:04.970 ","End":"03:07.685","Text":"I\u0027m going to show that this is equal to infinity."},{"Start":"03:07.685 ","End":"03:10.555","Text":"Let\u0027s see how it works."},{"Start":"03:10.555 ","End":"03:12.510","Text":"Equals the limit."},{"Start":"03:12.510 ","End":"03:15.570","Text":"What we\u0027re going to do is our famous trick of taking out"},{"Start":"03:15.570 ","End":"03:19.340","Text":"of the fact to the highest power of x,"},{"Start":"03:19.340 ","End":"03:20.795","Text":"which is the x cubed,"},{"Start":"03:20.795 ","End":"03:28.370","Text":"times 1 plus b over x plus c over x squared,"},{"Start":"03:28.370 ","End":"03:32.345","Text":"plus d over x cubed."},{"Start":"03:32.345 ","End":"03:35.620","Text":"Now this equals basically,"},{"Start":"03:35.620 ","End":"03:42.620","Text":"infinity cubed is infinity and some number over infinity is 0."},{"Start":"03:42.620 ","End":"03:44.810","Text":"Infinity squared is infinity,"},{"Start":"03:44.810 ","End":"03:47.585","Text":"and c over infinity is 0."},{"Start":"03:47.585 ","End":"03:49.430","Text":"Infinity cubed is infinity,"},{"Start":"03:49.430 ","End":"03:52.510","Text":"and d over infinity is 0."},{"Start":"03:52.510 ","End":"03:57.980","Text":"Infinity times 1 is well-defined and it\u0027s equal to infinity,"},{"Start":"03:57.980 ","End":"04:01.985","Text":"which I\u0027m going to write as plus infinity just for emphasis."},{"Start":"04:01.985 ","End":"04:07.925","Text":"Next thing I want to show is that it\u0027s minus infinity on the other side."},{"Start":"04:07.925 ","End":"04:10.580","Text":"The limit for it\u0027s suitable stopping here,"},{"Start":"04:10.580 ","End":"04:11.930","Text":"I\u0027m not continuing from here,"},{"Start":"04:11.930 ","End":"04:19.400","Text":"I\u0027m also going to show that the limit as x goes to minus infinity of"},{"Start":"04:19.400 ","End":"04:27.305","Text":"x cubed plus bx squared plus cx plus d is equal to,"},{"Start":"04:27.305 ","End":"04:29.450","Text":"well, the same expression,"},{"Start":"04:29.450 ","End":"04:36.860","Text":"is equal to the limit as x goes to minus infinity,"},{"Start":"04:36.860 ","End":"04:43.310","Text":"of x cubed, 1 plus b over x plus c over x squared,"},{"Start":"04:43.310 ","End":"04:46.055","Text":"plus d over x cubed."},{"Start":"04:46.055 ","End":"04:49.340","Text":"What\u0027s the difference in the minus infinity case?"},{"Start":"04:49.340 ","End":"04:51.065","Text":"Well, first of all,"},{"Start":"04:51.065 ","End":"04:55.460","Text":"minus infinity cubed is minus infinity."},{"Start":"04:55.460 ","End":"04:59.420","Text":"Infinity times infinity times infinity is still infinity and minus, minus,"},{"Start":"04:59.420 ","End":"05:01.550","Text":"minus because there\u0027s an odd number,"},{"Start":"05:01.550 ","End":"05:02.995","Text":"makes it minus."},{"Start":"05:02.995 ","End":"05:05.030","Text":"It\u0027s minus infinity."},{"Start":"05:05.030 ","End":"05:09.694","Text":"I should also point out that if we have a constant, any number,"},{"Start":"05:09.694 ","End":"05:14.355","Text":"not only is a over infinity equal to 0,"},{"Start":"05:14.355 ","End":"05:20.080","Text":"but a over minus infinity is also equal to 0 like it\u0027s minus 0."},{"Start":"05:20.080 ","End":"05:24.760","Text":"In fact, I could even just write it as plus or minus infinity is 0."},{"Start":"05:24.760 ","End":"05:27.580","Text":"All these 3 things come out to be 0."},{"Start":"05:27.580 ","End":"05:32.045","Text":"Again, we have 1 plus 0 plus 0 plus 0."},{"Start":"05:32.045 ","End":"05:33.954","Text":"It\u0027s the same as before,"},{"Start":"05:33.954 ","End":"05:37.910","Text":"except that there\u0027s a minus there which is minus infinity."},{"Start":"05:38.430 ","End":"05:41.450","Text":"That means that we\u0027re actually done."},{"Start":"05:41.450 ","End":"05:43.075","Text":"Because once we have that,"},{"Start":"05:43.075 ","End":"05:47.125","Text":"this limit is infinity and this limit is minus infinity,"},{"Start":"05:47.125 ","End":"05:49.990","Text":"and what I said before holds goes to infinity,"},{"Start":"05:49.990 ","End":"05:54.520","Text":"so it\u0027s above 0 from some point on and here minus infinity,"},{"Start":"05:54.520 ","End":"05:57.805","Text":"so it\u0027s less than 0 from some point on."},{"Start":"05:57.805 ","End":"06:00.800","Text":"Once we have the positive and the negative in the middle,"},{"Start":"06:00.800 ","End":"06:05.870","Text":"we get the 0 in this case is 3 times, but not necessarily."},{"Start":"06:05.870 ","End":"06:11.475","Text":"I want to emphasize the important thing here is that the 3 is odd."},{"Start":"06:11.475 ","End":"06:14.880","Text":"It would hold also if it was a 5 or a 7,"},{"Start":"06:14.880 ","End":"06:20.180","Text":"but not if it was like x squared because x squared easily"},{"Start":"06:20.180 ","End":"06:26.350","Text":"could be a parabola totally above the x-axis. We\u0027re done."}],"ID":3326},{"Watched":false,"Name":"Exercise 5","Duration":"8m 51s","ChapterTopicVideoID":3316,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3316.jpeg","UploadDate":"2015-01-08T00:24:31.0430000","DurationForVideoObject":"PT8M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have to do what we usually do,"},{"Start":"00:03.480 ","End":"00:05.099","Text":"but it\u0027s slightly different."},{"Start":"00:05.099 ","End":"00:08.235","Text":"Usually, we have to show that some equation"},{"Start":"00:08.235 ","End":"00:12.480","Text":"has a solution using the intermediate value theorem."},{"Start":"00:12.480 ","End":"00:19.305","Text":"In this time, we have to show that it has at least two solutions not just one."},{"Start":"00:19.305 ","End":"00:20.910","Text":"But other than that,"},{"Start":"00:20.910 ","End":"00:23.130","Text":"the technique is similar."},{"Start":"00:23.130 ","End":"00:27.540","Text":"There is also to watch out for where this thing is defined,"},{"Start":"00:27.540 ","End":"00:31.140","Text":"we note that this thing doesn\u0027t make sense for x equals 0,"},{"Start":"00:31.140 ","End":"00:32.925","Text":"so I\u0027ll use that in a minute."},{"Start":"00:32.925 ","End":"00:36.630","Text":"What we\u0027re going to do is use the normal method of converting"},{"Start":"00:36.630 ","End":"00:41.415","Text":"this equation into a function by subtracting one side from the other."},{"Start":"00:41.415 ","End":"00:47.135","Text":"Then it becomes the solution to an equation where function is equal to 0."},{"Start":"00:47.135 ","End":"00:48.530","Text":"What I mean is this,"},{"Start":"00:48.530 ","End":"00:50.555","Text":"when I define a function,"},{"Start":"00:50.555 ","End":"00:54.785","Text":"f of x is equal to what it says here,"},{"Start":"00:54.785 ","End":"01:02.895","Text":"4x cubed plus 5x minus 1 over x equals 0."},{"Start":"01:02.895 ","End":"01:09.935","Text":"Then what this thing says is exactly that f of x equals 0."},{"Start":"01:09.935 ","End":"01:12.620","Text":"Because if this is equal to 0,"},{"Start":"01:12.620 ","End":"01:18.100","Text":"that\u0027s the same as this minus this is 0 which is this equals this, though it\u0027s the same."},{"Start":"01:18.100 ","End":"01:20.960","Text":"Normally, we have to find some value of x."},{"Start":"01:20.960 ","End":"01:25.940","Text":"We don\u0027t actually show what the number is but we show it exists using the theorem."},{"Start":"01:25.940 ","End":"01:30.515","Text":"Usually, we find the value where c which fits or show that it exists."},{"Start":"01:30.515 ","End":"01:32.390","Text":"This time we\u0027re going to have at least two of them."},{"Start":"01:32.390 ","End":"01:36.115","Text":"I\u0027ll call them c and c prime."},{"Start":"01:36.115 ","End":"01:39.590","Text":"Because what we normally do is first of all,"},{"Start":"01:39.590 ","End":"01:44.135","Text":"we find a value a for which f of a is positive and another one,"},{"Start":"01:44.135 ","End":"01:47.000","Text":"b for which f of b is negative."},{"Start":"01:47.000 ","End":"01:50.525","Text":"Then we say that between a and b it\u0027s got to be 0 somewhere."},{"Start":"01:50.525 ","End":"01:53.330","Text":"But here we have to be very careful."},{"Start":"01:53.330 ","End":"02:02.770","Text":"This is tricky because this function is only defined when x is not equal to 0."},{"Start":"02:02.770 ","End":"02:05.340","Text":"X is not equal to 0,"},{"Start":"02:05.340 ","End":"02:10.610","Text":"the function is not defined there and so the intermediate value theorem wouldn\u0027t"},{"Start":"02:10.610 ","End":"02:16.235","Text":"apply if we had a range where the a is on one side of the 0 and b on the other."},{"Start":"02:16.235 ","End":"02:18.485","Text":"We have to have these two cases."},{"Start":"02:18.485 ","End":"02:23.990","Text":"We can either have the first case is that let\u0027s talk about a and b."},{"Start":"02:23.990 ","End":"02:28.470","Text":"You want to have a and b and they\u0027re both positive, let say,"},{"Start":"02:28.470 ","End":"02:33.515","Text":"and then in the next round we\u0027ll have both negative or we want both positive,"},{"Start":"02:33.515 ","End":"02:42.310","Text":"where f of one of them say a comes out negative and f of b comes out positive."},{"Start":"02:42.310 ","End":"02:47.570","Text":"If we have this such that this is true and this is true,"},{"Start":"02:47.570 ","End":"02:56.690","Text":"then what we get from the intermediate value theorem is that f of c is equal to 0 for"},{"Start":"02:56.690 ","End":"03:06.260","Text":"some c between a and b. I\u0027m going to find some values for you."},{"Start":"03:06.260 ","End":"03:11.585","Text":"Then afterwards we\u0027ll look for a different a and b that are both negative,"},{"Start":"03:11.585 ","End":"03:16.200","Text":"and then we\u0027ll find a different c. But let\u0027s just stay with this for a minute."},{"Start":"03:16.840 ","End":"03:21.380","Text":"Find an f of a which is negative."},{"Start":"03:21.380 ","End":"03:23.945","Text":"I think I can find the b first."},{"Start":"03:23.945 ","End":"03:27.050","Text":"Let\u0027s just find the number f of,"},{"Start":"03:27.050 ","End":"03:31.130","Text":"let\u0027s try 1 to substitute anything but 0."},{"Start":"03:31.130 ","End":"03:32.975","Text":"X cannot be 0."},{"Start":"03:32.975 ","End":"03:35.160","Text":"If I put f of 1 to see what happens."},{"Start":"03:35.160 ","End":"03:37.710","Text":"If we get positive we\u0027ll look for the negative,"},{"Start":"03:37.710 ","End":"03:40.140","Text":"if we get negative we\u0027ll look then for the positive."},{"Start":"03:40.140 ","End":"03:41.975","Text":"F of 1 is,"},{"Start":"03:41.975 ","End":"03:43.310","Text":"let\u0027s do it in our heads,"},{"Start":"03:43.310 ","End":"03:48.620","Text":"4 times 1 cubed is 4 plus 5 times 1 is 5."},{"Start":"03:48.620 ","End":"03:50.210","Text":"4 plus 5 minus 1,"},{"Start":"03:50.210 ","End":"03:54.445","Text":"that\u0027s equal to 8 and the 8 is positive."},{"Start":"03:54.445 ","End":"03:57.270","Text":"Actually, this will be my b."},{"Start":"03:57.270 ","End":"03:59.930","Text":"The one that\u0027s positive here I\u0027ll call b."},{"Start":"03:59.930 ","End":"04:05.030","Text":"Then I want to also find something which will come out negative."},{"Start":"04:05.030 ","End":"04:08.614","Text":"What am I going to put here to make it come out negative?"},{"Start":"04:08.614 ","End":"04:11.030","Text":"I want it to equal something,"},{"Start":"04:11.030 ","End":"04:13.535","Text":"which is obviously negative."},{"Start":"04:13.535 ","End":"04:15.920","Text":"Now, if I put large numbers here,"},{"Start":"04:15.920 ","End":"04:17.405","Text":"it\u0027s going to get large."},{"Start":"04:17.405 ","End":"04:19.625","Text":"In order for them to get small,"},{"Start":"04:19.625 ","End":"04:22.280","Text":"I need to have x small because 1 over"},{"Start":"04:22.280 ","End":"04:24.920","Text":"x is enlarged so I\u0027m putting a lot of negative weight."},{"Start":"04:24.920 ","End":"04:27.215","Text":"I\u0027m looking probably something less than 1,"},{"Start":"04:27.215 ","End":"04:29.225","Text":"maybe even less than a 1/2."},{"Start":"04:29.225 ","End":"04:32.770","Text":"I just went for 1 10th I tried it didn\u0027t work."},{"Start":"04:32.770 ","End":"04:36.635","Text":"I did it before and I\u0027ll just show you that it works."},{"Start":"04:36.635 ","End":"04:38.825","Text":"I just keep experimenting."},{"Start":"04:38.825 ","End":"04:42.425","Text":"But the idea is to get x close to 0 or positive,"},{"Start":"04:42.425 ","End":"04:44.030","Text":"then this becomes large,"},{"Start":"04:44.030 ","End":"04:46.520","Text":"these very large as x goes towards 0."},{"Start":"04:46.520 ","End":"04:49.310","Text":"We\u0027re subtracting a large amount and we\u0027ll get negative."},{"Start":"04:49.310 ","End":"04:51.715","Text":"F of 1 10th is,"},{"Start":"04:51.715 ","End":"04:56.205","Text":"of course I can do this side scrap work,"},{"Start":"04:56.205 ","End":"05:00.449","Text":"so f of 1 10th is,"},{"Start":"05:00.449 ","End":"05:04.265","Text":"let\u0027s see, it\u0027s 4 times 1 10th cubed."},{"Start":"05:04.265 ","End":"05:05.830","Text":"It\u0027s 4 over a 1,000,"},{"Start":"05:05.830 ","End":"05:11.965","Text":"0.004 and 5x is 5 over 10,"},{"Start":"05:11.965 ","End":"05:16.020","Text":"minus 1 over a 10th is 10,"},{"Start":"05:16.020 ","End":"05:19.085","Text":"so it\u0027s minus 10 and altogether,"},{"Start":"05:19.085 ","End":"05:24.500","Text":"what I get is minus 9.496."},{"Start":"05:24.500 ","End":"05:26.630","Text":"It\u0027s negative, that\u0027s the point."},{"Start":"05:26.630 ","End":"05:28.550","Text":"That\u0027s going to be my a,"},{"Start":"05:28.550 ","End":"05:32.960","Text":"f of a is negative and f of b is positive."},{"Start":"05:32.960 ","End":"05:38.220","Text":"We know that there is a c such that between 1 10th,"},{"Start":"05:38.220 ","End":"05:40.985","Text":"we have 1 10th less than c,"},{"Start":"05:40.985 ","End":"05:42.420","Text":"less than 1,"},{"Start":"05:42.420 ","End":"05:49.030","Text":"such that f of c is equal to 0."},{"Start":"05:49.400 ","End":"05:52.070","Text":"Now that we\u0027ve done that,"},{"Start":"05:52.070 ","End":"05:58.000","Text":"we need to find pair a and b and I\u0027ll call them,"},{"Start":"05:58.000 ","End":"05:59.330","Text":"a prime and b prime."},{"Start":"05:59.330 ","End":"06:01.535","Text":"In other words, what I\u0027m going to do now,"},{"Start":"06:01.535 ","End":"06:03.560","Text":"is I\u0027m going to do the same thing,"},{"Start":"06:03.560 ","End":"06:05.195","Text":"but different pair of numbers."},{"Start":"06:05.195 ","End":"06:06.845","Text":"I\u0027ll call them different letters,"},{"Start":"06:06.845 ","End":"06:09.650","Text":"a prime and b prime, which are negative."},{"Start":"06:09.650 ","End":"06:12.925","Text":"Like I said before, I\u0027m reminding you,"},{"Start":"06:12.925 ","End":"06:18.330","Text":"they can\u0027t be 0 and the function has to be continuous all the way from a to b,"},{"Start":"06:18.330 ","End":"06:21.980","Text":"so 0 can\u0027t be in the survey that both positive or both negative."},{"Start":"06:21.980 ","End":"06:23.750","Text":"We\u0027ll try for the other case."},{"Start":"06:23.750 ","End":"06:28.065","Text":"We want these negative such that f of"},{"Start":"06:28.065 ","End":"06:36.195","Text":"a\u0027 is negative and f of b prime is positive."},{"Start":"06:36.195 ","End":"06:41.910","Text":"Then we\u0027ll get that f of c prime has to be 0 for"},{"Start":"06:41.910 ","End":"06:48.825","Text":"some c prime is between a prime and b prime."},{"Start":"06:48.825 ","End":"06:52.275","Text":"Same thing here we have to find a and b."},{"Start":"06:52.275 ","End":"06:57.225","Text":"Well, the obvious thing to try would be first of all minus 1 we can\u0027t try 0."},{"Start":"06:57.225 ","End":"07:01.125","Text":"Let\u0027s go with minus 1."},{"Start":"07:01.125 ","End":"07:03.720","Text":"Let\u0027s see what\u0027s f of minus 1?"},{"Start":"07:03.720 ","End":"07:05.385","Text":"That\u0027s equal to,"},{"Start":"07:05.385 ","End":"07:06.800","Text":"we do it in our heads,"},{"Start":"07:06.800 ","End":"07:08.854","Text":"we\u0027re going to get exactly the same,"},{"Start":"07:08.854 ","End":"07:10.340","Text":"just with opposite signs,"},{"Start":"07:10.340 ","End":"07:12.790","Text":"it\u0027s going to come out minus 8."},{"Start":"07:12.790 ","End":"07:14.325","Text":"That got me wondering."},{"Start":"07:14.325 ","End":"07:17.455","Text":"If 1 gives me 8 and minus 1 gives me minus 8,"},{"Start":"07:17.455 ","End":"07:19.790","Text":"I look more closely and I happened to notice this is"},{"Start":"07:19.790 ","End":"07:23.840","Text":"an odd function because it\u0027s got only odd powers x to the 3,"},{"Start":"07:23.840 ","End":"07:25.825","Text":"x to the 1, x to the minus 1."},{"Start":"07:25.825 ","End":"07:27.690","Text":"You don\u0027t have to notice this,"},{"Start":"07:27.690 ","End":"07:30.390","Text":"it\u0027s just the trial and error."},{"Start":"07:30.390 ","End":"07:32.375","Text":"I think what I\u0027m going to do is,"},{"Start":"07:32.375 ","End":"07:35.665","Text":"if that works, why not try minus a 10th?"},{"Start":"07:35.665 ","End":"07:38.955","Text":"If I do f of minus 1 10th,"},{"Start":"07:38.955 ","End":"07:42.710","Text":"I\u0027ll spare you all the arithmetic this time just because it\u0027s"},{"Start":"07:42.710 ","End":"07:47.360","Text":"an odd function and it has to be just the same thing with an opposite sign."},{"Start":"07:47.360 ","End":"07:55.669","Text":"This has to be plus 9.496 or substituted and so this one\u0027s a smaller,"},{"Start":"07:55.669 ","End":"07:57.980","Text":"so that one\u0027s going to be my a prime,"},{"Start":"07:57.980 ","End":"08:00.320","Text":"this one\u0027s going to be my b prime."},{"Start":"08:00.320 ","End":"08:04.825","Text":"I know that my x is this time is going to be c prime,"},{"Start":"08:04.825 ","End":"08:10.490","Text":"which is going to be between minus 1 is going to be on one side of c prime,"},{"Start":"08:10.490 ","End":"08:12.265","Text":"and minus a 10th."},{"Start":"08:12.265 ","End":"08:17.000","Text":"On the other side, c prime is somewhere from between here and here,"},{"Start":"08:17.000 ","End":"08:22.245","Text":"and f of c prime will equal 0."},{"Start":"08:22.245 ","End":"08:24.845","Text":"I don\u0027t know exactly what it is, but it\u0027s in this range."},{"Start":"08:24.845 ","End":"08:26.810","Text":"Now we have two x\u0027s."},{"Start":"08:26.810 ","End":"08:29.750","Text":"This will be my first x."},{"Start":"08:29.750 ","End":"08:33.330","Text":"The c is positive,"},{"Start":"08:33.330 ","End":"08:34.850","Text":"it\u0027s between this and this,"},{"Start":"08:34.850 ","End":"08:36.619","Text":"and the other one is negative,"},{"Start":"08:36.619 ","End":"08:38.625","Text":"which between this and this."},{"Start":"08:38.625 ","End":"08:45.444","Text":"I know that both of these exist and these are my two x\u0027s at which f of x is 0."},{"Start":"08:45.444 ","End":"08:47.630","Text":"That\u0027s all. It sounds a bit convoluted,"},{"Start":"08:47.630 ","End":"08:51.210","Text":"but really quite simple. We\u0027re done."}],"ID":3327},{"Watched":false,"Name":"Exercise 6","Duration":"4m 31s","ChapterTopicVideoID":3339,"CourseChapterTopicPlaylistID":84398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/3339.jpeg","UploadDate":"2015-01-10T23:11:17.5700000","DurationForVideoObject":"PT4M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"This is a familiar format of an exercise."},{"Start":"00:03.630 ","End":"00:05.100","Text":"We\u0027re given an equation,"},{"Start":"00:05.100 ","End":"00:08.940","Text":"and to show using this theorem that it has at least 1 solution,"},{"Start":"00:08.940 ","End":"00:13.995","Text":"what we usually do is start off by defining a function f of x,"},{"Start":"00:13.995 ","End":"00:23.730","Text":"which is the difference between the left minus the right, e^x minus 5x."},{"Start":"00:23.730 ","End":"00:28.440","Text":"To say that this equation has a solution is the same as"},{"Start":"00:28.440 ","End":"00:34.650","Text":"saying that f of x is equal to 0 for some x."},{"Start":"00:34.650 ","End":"00:40.160","Text":"The usual approach is to find points a and b,"},{"Start":"00:40.160 ","End":"00:43.355","Text":"where we look for an a and a b,"},{"Start":"00:43.355 ","End":"00:51.605","Text":"where f of a is less than 0 and a b such that f of b is bigger than 0,"},{"Start":"00:51.605 ","End":"00:56.840","Text":"and then deduce using the intermediate value theorem that f of"},{"Start":"00:56.840 ","End":"01:02.120","Text":"c is equal to 0 for some c. I won\u0027t write all of that."},{"Start":"01:02.120 ","End":"01:06.240","Text":"Say c is between a and b,"},{"Start":"01:06.240 ","End":"01:09.025","Text":"and that c is our x."},{"Start":"01:09.025 ","End":"01:12.060","Text":"But we have to also be careful and make sure that where"},{"Start":"01:12.060 ","End":"01:15.770","Text":"f is defined and whether there are any discontinuities."},{"Start":"01:15.770 ","End":"01:21.320","Text":"Here, there\u0027s absolutely no problem because it\u0027s defined for all x."},{"Start":"01:21.320 ","End":"01:25.130","Text":"It\u0027s also continuous because for example,"},{"Start":"01:25.130 ","End":"01:26.770","Text":"e^x is continuous,"},{"Start":"01:26.770 ","End":"01:29.750","Text":"it\u0027s elementary and so is the polynomial 5x"},{"Start":"01:29.750 ","End":"01:33.305","Text":"and the addition and subtraction of elementaries."},{"Start":"01:33.305 ","End":"01:37.680","Text":"Show if f is defined and it\u0027s even continuous everywhere,"},{"Start":"01:37.680 ","End":"01:41.135","Text":"so we don\u0027t have to watch out for all these tricky things."},{"Start":"01:41.135 ","End":"01:44.405","Text":"We just have to find such values."},{"Start":"01:44.405 ","End":"01:48.260","Text":"Really, it\u0027s trial and error, for example,"},{"Start":"01:48.260 ","End":"01:56.195","Text":"my inclination would be to just to try a 0 and see what happens with that."},{"Start":"01:56.195 ","End":"02:05.850","Text":"Let\u0027s see, f of 0 is e^0 minus 5 zeros is just 1."},{"Start":"02:05.850 ","End":"02:08.070","Text":"That\u0027s positive."},{"Start":"02:08.070 ","End":"02:11.860","Text":"Now, I want to find a value which is negative,"},{"Start":"02:11.860 ","End":"02:15.920","Text":"and really what we can do is trial and error."},{"Start":"02:15.920 ","End":"02:17.585","Text":"We can put some method into it."},{"Start":"02:17.585 ","End":"02:22.970","Text":"But I just messed around a bit and I found that x equals 1 is good."},{"Start":"02:22.970 ","End":"02:31.290","Text":"Let\u0027s see, f of 1 is actually equal to e minus 5."},{"Start":"02:31.750 ","End":"02:35.420","Text":"Well, this is definitely negative."},{"Start":"02:35.420 ","End":"02:43.990","Text":"The reason for this is that e is approximately equal to 2.72."},{"Start":"02:44.720 ","End":"02:48.810","Text":"Even if it was 2.73 or 4,"},{"Start":"02:48.810 ","End":"02:51.370","Text":"it still be less than 5."},{"Start":"02:51.560 ","End":"02:54.330","Text":"This is going to be negative."},{"Start":"02:54.330 ","End":"02:57.330","Text":"The 1 that\u0027s bigger than 0, I called b,"},{"Start":"02:57.330 ","End":"03:02.060","Text":"so let\u0027s call this 1 the 0 is my b"},{"Start":"03:02.060 ","End":"03:06.980","Text":"and where it\u0027s bigger than 0 and where it\u0027s less than 0,"},{"Start":"03:06.980 ","End":"03:12.780","Text":"I call that a. I know that I have somewhere c between a and b,"},{"Start":"03:12.780 ","End":"03:15.465","Text":"and since 0 is less than 1,"},{"Start":"03:15.465 ","End":"03:21.645","Text":"then 0 has to be between c and 1."},{"Start":"03:21.645 ","End":"03:26.340","Text":"F of c is going to be 0, because at 1,"},{"Start":"03:26.340 ","End":"03:29.700","Text":"it\u0027s less than 0,"},{"Start":"03:29.700 ","End":"03:31.860","Text":"its negative, here at 0,"},{"Start":"03:31.860 ","End":"03:36.444","Text":"it\u0027s positive, and at c itself it\u0027s 0."},{"Start":"03:36.444 ","End":"03:41.870","Text":"Once again, it\u0027s the intermediate value which just says that if we have"},{"Start":"03:41.870 ","End":"03:47.305","Text":"a graph and if at some point in this case it\u0027s 0, it\u0027s positive,"},{"Start":"03:47.305 ","End":"03:53.885","Text":"and somewhere else where x is 1, it\u0027s negative,"},{"Start":"03:53.885 ","End":"03:59.360","Text":"say here, and there we connect somehow this with this continuously,"},{"Start":"03:59.360 ","End":"04:02.125","Text":"there has to be a crossing point here."},{"Start":"04:02.125 ","End":"04:07.125","Text":"This is my a, this is b,"},{"Start":"04:07.125 ","End":"04:10.450","Text":"and here it\u0027s going to be my c."},{"Start":"04:10.610 ","End":"04:18.080","Text":"That\u0027s basically all we need to do just to quote the intermediate value theorem,"},{"Start":"04:18.080 ","End":"04:20.524","Text":"and such a c exists."},{"Start":"04:20.524 ","End":"04:24.094","Text":"As I said, if f of c is 0,"},{"Start":"04:24.094 ","End":"04:28.880","Text":"then it just means that this c is a solution to this equation,"},{"Start":"04:28.880 ","End":"04:31.410","Text":"1 more to add, we\u0027re done."}],"ID":3350}],"Thumbnail":null,"ID":84398}]

[{"ID":84396,"Videos":[8414,90,91,92,2957,2958,2959,2960,2961,2962,3307,3319]},{"ID":84397,"Videos":[14650,14651,3320,3321,3322,4842]},{"ID":84398,"Videos":[8415,3323,3324,3325,3326,3327,3350]}];

[8414,90,91,92,2957,2958,2959,2960,2961,2962,3307,3319];

1.1

1

Get unlimited access to **1500 subjects** including **personalised modules**

Start your free trial
We couldn't find any results for

Upload your syllabus now and our team will create a customised module especially for you!

Alert

and we will create a personalised module (just for you) in less than **48 hours...**