Differentiability of Piecewise Functions
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- Differentiability - Definition and Piecewise Functions
- Exercise 1 - Part 1
- Exercise 1 - Part 2
- Exercise 1 - Part 3
- Exercise 2 Part 1 - Via theorem
- Exercise 2 Part 1 - Via definition
- Exercise 2 Part 2 - via Theorem
- Exercise 2 Part 2 - via Definition
- Exercise 2 Part 3 - Via theorem
- Exercise 2 Part 3 - Via definition
- Exercise 3 - Part 1
- Exercise 3 - Part 2
- Exercise 4 - Part 1
- Exercise 4 - Part 2
- Exercise 4 - Part 3

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[{"Name":"Differentiability of Piecewise Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Differentiability - Definition and Piecewise Functions","Duration":"22m 55s","ChapterTopicVideoID":1489,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/1489.jpeg","UploadDate":"2019-11-14T07:04:19.8770000","DurationForVideoObject":"PT22M55S","Description":null,"MetaTitle":"Differentiability - Definition and Piecewise Functions: Video + Workbook | Proprep","MetaDescription":"Differentiability - Differentiability of Piecewise Functions. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/differentiability/differentiability-of-piecewise-functions/vid1473","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this clip, I\u0027ll be talking about the differentiability of"},{"Start":"00:04.860 ","End":"00:10.950","Text":"piecewise-defined functions at the same-line using 2 different methods."},{"Start":"00:10.950 ","End":"00:15.555","Text":"Sometimes piecewise-defined functions are called split functions."},{"Start":"00:15.555 ","End":"00:17.835","Text":"Same-line is an informal word,"},{"Start":"00:17.835 ","End":"00:19.725","Text":"sometimes also same point,"},{"Start":"00:19.725 ","End":"00:23.865","Text":"and it\u0027s the place where the function changes formula."},{"Start":"00:23.865 ","End":"00:26.220","Text":"You\u0027ll see in the example."},{"Start":"00:26.220 ","End":"00:28.740","Text":"I\u0027ll be teaching this concept through the use of"},{"Start":"00:28.740 ","End":"00:32.730","Text":"a single example which will follow us all the way through."},{"Start":"00:32.730 ","End":"00:41.585","Text":"This example will be as follows: f of x is equal to, you see the split,"},{"Start":"00:41.585 ","End":"00:46.105","Text":"x squared plus 8x,"},{"Start":"00:46.105 ","End":"00:50.210","Text":"whenever x is bigger or equal to 2,"},{"Start":"00:50.210 ","End":"00:57.830","Text":"and x cubed plus 12 whenever x is less than 2."},{"Start":"00:57.830 ","End":"01:06.090","Text":"The first method, we\u0027ll be using the definition of a derivative,"},{"Start":"01:06.350 ","End":"01:08.820","Text":"and the second method,"},{"Start":"01:08.820 ","End":"01:12.465","Text":"we\u0027ll be using a theorem."},{"Start":"01:12.465 ","End":"01:16.400","Text":"At the end, I\u0027ll say a few words about when to"},{"Start":"01:16.400 ","End":"01:24.345","Text":"use each of these methods for finding the differentiability."},{"Start":"01:24.345 ","End":"01:29.005","Text":"I\u0027m going to start by showing method 1."},{"Start":"01:29.005 ","End":"01:32.975","Text":"It\u0027s probably best to start on a new page."},{"Start":"01:32.975 ","End":"01:40.625","Text":"Recall that method 1 is using the definition of the derivative."},{"Start":"01:40.625 ","End":"01:43.785","Text":"What this says is as follows,"},{"Start":"01:43.785 ","End":"01:47.365","Text":"if you want to check differentiability,"},{"Start":"01:47.365 ","End":"01:51.415","Text":"the point x equals a,"},{"Start":"01:51.415 ","End":"01:54.110","Text":"which is a same-line,"},{"Start":"01:54.110 ","End":"01:57.670","Text":"then we compute 2 quantities."},{"Start":"01:57.670 ","End":"02:01.570","Text":"The first is called the right derivative,"},{"Start":"02:01.570 ","End":"02:05.120","Text":"and it\u0027s written with a little plus here,"},{"Start":"02:05.120 ","End":"02:15.470","Text":"and it\u0027s defined as the limit as h goes to 0 from the right of f of a plus"},{"Start":"02:15.470 ","End":"02:19.945","Text":"h minus f of a all over"},{"Start":"02:19.945 ","End":"02:28.375","Text":"h. The second quantity we compute is the left derivative."},{"Start":"02:28.375 ","End":"02:35.150","Text":"This is the limit as h goes to 0 from the left of f of a"},{"Start":"02:35.150 ","End":"02:42.110","Text":"plus h minus f of a over h. Now,"},{"Start":"02:42.110 ","End":"02:45.340","Text":"if these 2 are equal,"},{"Start":"02:46.610 ","End":"02:52.100","Text":"the right derivative and the left derivative,"},{"Start":"02:52.100 ","End":"02:57.975","Text":"then f is differentiable at x equals a,"},{"Start":"02:57.975 ","End":"03:03.080","Text":"otherwise not, by which I mean it\u0027s not differentiable otherwise."},{"Start":"03:03.080 ","End":"03:05.435","Text":"There\u0027s something I have to stress here,"},{"Start":"03:05.435 ","End":"03:06.725","Text":"may not be obvious,"},{"Start":"03:06.725 ","End":"03:08.945","Text":"when I say that these limits are equal,"},{"Start":"03:08.945 ","End":"03:11.945","Text":"I mean that there are also finite, not infinite."},{"Start":"03:11.945 ","End":"03:15.360","Text":"What I need are finite limits."},{"Start":"03:15.360 ","End":"03:21.125","Text":"Please remember that infinity does not equal infinity necessarily, infinity is out."},{"Start":"03:21.125 ","End":"03:27.325","Text":"Now, it\u0027s time for our example using our function and I\u0027ll do it on a fresh page."},{"Start":"03:27.325 ","End":"03:31.035","Text":"Here\u0027s our function as from before."},{"Start":"03:31.035 ","End":"03:35.250","Text":"The same-line is 2 and that\u0027s what we want to check differentiability."},{"Start":"03:35.250 ","End":"03:37.625","Text":"According to what I wrote,"},{"Start":"03:37.625 ","End":"03:40.475","Text":"we first of all have to check the right derivative,"},{"Start":"03:40.475 ","End":"03:47.435","Text":"which is called the derivative with a little plus here at the point 2."},{"Start":"03:47.435 ","End":"03:50.575","Text":"This equals, according to the definition,"},{"Start":"03:50.575 ","End":"03:56.115","Text":"to the limit as h goes to 0 of f of 2 plus h"},{"Start":"03:56.115 ","End":"04:02.190","Text":"minus f of 2 over h. The h goes to 0 from the right,"},{"Start":"04:02.190 ","End":"04:05.420","Text":"of course and then we continue."},{"Start":"04:05.420 ","End":"04:09.270","Text":"This equals the limit as h goes to 0,"},{"Start":"04:09.270 ","End":"04:10.985","Text":"again, from the right."},{"Start":"04:10.985 ","End":"04:14.045","Text":"Now, the question is what to use for f?"},{"Start":"04:14.045 ","End":"04:20.370","Text":"Well, 2 plus h is slightly bigger than 2 because h is slightly positive."},{"Start":"04:20.370 ","End":"04:22.250","Text":"We\u0027re using this definition,"},{"Start":"04:22.250 ","End":"04:24.210","Text":"which is x squared plus 8x."},{"Start":"04:24.210 ","End":"04:34.295","Text":"In other words, we get 2 plus h squared plus 8 times 2 plus h minus."},{"Start":"04:34.295 ","End":"04:37.850","Text":"Now as for 2, it certainly belongs here,"},{"Start":"04:37.850 ","End":"04:48.325","Text":"so we also use 2 squared plus 8 times 2 over h. Let\u0027s use the same coloring."},{"Start":"04:48.325 ","End":"04:52.265","Text":"We\u0027ll continue with the algebra. What do we get?"},{"Start":"04:52.265 ","End":"04:56.270","Text":"Limit as h goes to 0 plus,"},{"Start":"04:56.270 ","End":"05:01.870","Text":"the square of this is 4 plus 4h plus h squared,"},{"Start":"05:01.870 ","End":"05:08.630","Text":"and from here, plus 16 plus 8h minus,"},{"Start":"05:08.630 ","End":"05:15.780","Text":"this whole thing is 20 all over h, which equals limit."},{"Start":"05:15.780 ","End":"05:17.860","Text":"Same thing."},{"Start":"05:17.860 ","End":"05:20.150","Text":"If we combine terms together,"},{"Start":"05:20.150 ","End":"05:24.660","Text":"4 and 16 is 20 minus 20, that cancels,"},{"Start":"05:24.660 ","End":"05:28.110","Text":"and all we\u0027re left with is 4h plus 8h,"},{"Start":"05:28.110 ","End":"05:37.345","Text":"which is 12h and plus h squared all over h,"},{"Start":"05:37.345 ","End":"05:46.275","Text":"which we can cancel from top and bottom and get limit as h goes to 0 plus,"},{"Start":"05:46.275 ","End":"05:49.440","Text":"that 12h over h is just 12."},{"Start":"05:49.440 ","End":"05:54.534","Text":"Then all we\u0027re left with is plus h. When h goes to 0,"},{"Start":"05:54.534 ","End":"05:58.280","Text":"this expression just goes to 12."},{"Start":"05:58.280 ","End":"06:01.770","Text":"We got the limit from the right is 12."},{"Start":"06:01.770 ","End":"06:06.565","Text":"Let\u0027s see what happens with the limit from the left and see if they\u0027re equal or not."},{"Start":"06:06.565 ","End":"06:08.855","Text":"Here I\u0027ve written our function again,"},{"Start":"06:08.855 ","End":"06:11.750","Text":"and this time we\u0027re going to find the left derivative,"},{"Start":"06:11.750 ","End":"06:16.025","Text":"which we write as f prime with a little minus here,"},{"Start":"06:16.025 ","End":"06:17.955","Text":"again at the point 2."},{"Start":"06:17.955 ","End":"06:21.245","Text":"What this is equal to is just like before,"},{"Start":"06:21.245 ","End":"06:25.940","Text":"except that h goes to 0 from the left, the same thing,"},{"Start":"06:25.940 ","End":"06:33.835","Text":"f of 2 plus h minus f of 2 over h,"},{"Start":"06:33.835 ","End":"06:44.745","Text":"which equals the limit as h goes to 0 from the left."},{"Start":"06:44.745 ","End":"06:49.555","Text":"This time, f might go from a different definition."},{"Start":"06:49.555 ","End":"06:52.090","Text":"We can\u0027t be sure it\u0027s the top 1 again. In fact, it\u0027s not."},{"Start":"06:52.090 ","End":"06:56.370","Text":"It\u0027s the bottom 1 because h is slightly negative,"},{"Start":"06:56.370 ","End":"06:58.920","Text":"so 2 plus h is slightly below 2,"},{"Start":"06:58.920 ","End":"07:00.895","Text":"so we\u0027re going to use this definition,"},{"Start":"07:00.895 ","End":"07:04.160","Text":"which means that what we have is"},{"Start":"07:04.160 ","End":"07:15.120","Text":"2 plus h to the power of 3 plus 12."},{"Start":"07:15.120 ","End":"07:18.905","Text":"This is the f of 2 plus h. F of 2,"},{"Start":"07:18.905 ","End":"07:21.790","Text":"just like before, we computed it as 20,"},{"Start":"07:21.790 ","End":"07:23.830","Text":"so it remains 20,"},{"Start":"07:23.830 ","End":"07:33.755","Text":"and all this is over h. You might have forgotten the formula for a binomial cubed,"},{"Start":"07:33.755 ","End":"07:39.770","Text":"a plus b to the power of 3 equals a cubed plus 3a"},{"Start":"07:39.770 ","End":"07:47.435","Text":"squared b plus 3ab squared plus b cubed."},{"Start":"07:47.435 ","End":"07:50.525","Text":"If we use that here,"},{"Start":"07:50.525 ","End":"08:00.850","Text":"then we\u0027ll get, I\u0027ll spare you all the minor details, 8 plus 6."},{"Start":"08:01.260 ","End":"08:10.405","Text":"Plus 12 h plus"},{"Start":"08:10.405 ","End":"08:19.390","Text":"6 h squared plus h cubed plus 12 minus 20,"},{"Start":"08:19.390 ","End":"08:24.445","Text":"all over h, which equals,"},{"Start":"08:24.445 ","End":"08:28.030","Text":"the 8 goes with the 12 and the 20,"},{"Start":"08:28.030 ","End":"08:30.280","Text":"and if we take h out,"},{"Start":"08:30.280 ","End":"08:33.640","Text":"we get the limit as h goes to 0 from the left."},{"Start":"08:33.640 ","End":"08:35.965","Text":"H will cancel and to all of these things,"},{"Start":"08:35.965 ","End":"08:44.620","Text":"and we\u0027ll just get 12 plus 6 h plus h squared,"},{"Start":"08:44.620 ","End":"08:47.740","Text":"and as h goes to 0,"},{"Start":"08:47.740 ","End":"08:52.270","Text":"these 2 terms go to 0 and all we\u0027re left with is 12,"},{"Start":"08:52.270 ","End":"08:55.900","Text":"and this is very good,"},{"Start":"08:55.900 ","End":"09:01.480","Text":"because what we got previously for the right derivative was also 12."},{"Start":"09:01.480 ","End":"09:03.924","Text":"Since 12 equals 12,"},{"Start":"09:03.924 ","End":"09:07.240","Text":"then f is indeed differentiable,"},{"Start":"09:07.240 ","End":"09:09.280","Text":"and that\u0027s not the only thing we can say."},{"Start":"09:09.280 ","End":"09:13.360","Text":"We can even say that we know what the derivative is because the derivative of"},{"Start":"09:13.360 ","End":"09:18.640","Text":"f at 2 is now the same value of the 12 from the left and 12 from the right,"},{"Start":"09:18.640 ","End":"09:21.650","Text":"so this is also equal 12."},{"Start":"09:22.080 ","End":"09:25.270","Text":"As a matter of fact, there\u0027s something else interesting."},{"Start":"09:25.270 ","End":"09:29.125","Text":"I could actually show you a formula for the complete derivative"},{"Start":"09:29.125 ","End":"09:33.805","Text":"of f. Just let me make some room by erasing some of this stuff,"},{"Start":"09:33.805 ","End":"09:40.150","Text":"and then I can show you that we could make"},{"Start":"09:40.150 ","End":"09:46.270","Text":"a split function for"},{"Start":"09:46.270 ","End":"09:52.719","Text":"f prime and get that f prime of x equals."},{"Start":"09:52.719 ","End":"09:55.780","Text":"There was never any problem outside of the seam line,"},{"Start":"09:55.780 ","End":"09:56.905","Text":"x equals 2,"},{"Start":"09:56.905 ","End":"09:58.660","Text":"so for x bigger than 2,"},{"Start":"09:58.660 ","End":"10:04.640","Text":"we just differentiate regular and get from this 2 x plus 8."},{"Start":"10:06.120 ","End":"10:09.370","Text":"This will be for x bigger than 2,"},{"Start":"10:09.370 ","End":"10:12.625","Text":"and there was also no doubt that for x less than 2,"},{"Start":"10:12.625 ","End":"10:14.710","Text":"this thing is differentiable as usual,"},{"Start":"10:14.710 ","End":"10:19.430","Text":"but the formula 3 x squared is all we get here."},{"Start":"10:19.650 ","End":"10:25.075","Text":"The only thing we had a problem with was 2 itself and that we found was 12,"},{"Start":"10:25.075 ","End":"10:26.980","Text":"and finally or not finally,"},{"Start":"10:26.980 ","End":"10:31.405","Text":"it turns out to be equal to both of these when x is 2 so we can pick."},{"Start":"10:31.405 ","End":"10:34.255","Text":"But since this was in this form,"},{"Start":"10:34.255 ","End":"10:36.430","Text":"I\u0027ll put the equal sign over here,"},{"Start":"10:36.430 ","End":"10:40.495","Text":"so here we have a nice formula for the derivative."},{"Start":"10:40.495 ","End":"10:44.530","Text":"Last thing I want to do is just summarize what we said here."},{"Start":"10:44.530 ","End":"10:51.460","Text":"To summarize, since the right derivative at 2 equals the left derivative at 2,"},{"Start":"10:51.460 ","End":"10:54.025","Text":"and if I remember this was equal to 12,"},{"Start":"10:54.025 ","End":"10:56.725","Text":"then f is differentiable at x equals 2."},{"Start":"10:56.725 ","End":"11:01.795","Text":"Furthermore, the derivative of f of 2 is this common value of 12."},{"Start":"11:01.795 ","End":"11:07.345","Text":"Now we come to Method 2 for differentiability using a theorem."},{"Start":"11:07.345 ","End":"11:08.845","Text":"What is this theorem?"},{"Start":"11:08.845 ","End":"11:11.420","Text":"Let me present it to you."},{"Start":"11:11.640 ","End":"11:20.905","Text":"It says, if f is continuous at x equals a, and the limits,"},{"Start":"11:20.905 ","End":"11:25.690","Text":"limit as x goes to a from the right of f prime of x,"},{"Start":"11:25.690 ","End":"11:30.970","Text":"and limit as x goes to a from the left of f prime of x exists both of them,"},{"Start":"11:30.970 ","End":"11:33.175","Text":"and they\u0027re equal to each other,"},{"Start":"11:33.175 ","End":"11:36.025","Text":"and this also implies that they\u0027re not infinity,"},{"Start":"11:36.025 ","End":"11:38.604","Text":"they exist and they\u0027re finite."},{"Start":"11:38.604 ","End":"11:41.230","Text":"If these 2 conditions hold,"},{"Start":"11:41.230 ","End":"11:48.115","Text":"then what we get is that f is differentiable at the point x equals a,"},{"Start":"11:48.115 ","End":"11:49.660","Text":"and not only that,"},{"Start":"11:49.660 ","End":"11:54.175","Text":"but we actually know the value of the derivative at a,"},{"Start":"11:54.175 ","End":"11:58.570","Text":"which is equal to these 2 limits above that were already equal,"},{"Start":"11:58.570 ","End":"12:01.315","Text":"so that\u0027s basically what the theorem says."},{"Start":"12:01.315 ","End":"12:07.135","Text":"Now let\u0027s go on to demonstrate it using an example,"},{"Start":"12:07.135 ","End":"12:11.950","Text":"and the example will be our favorite example that we\u0027ve been dragging along with us,"},{"Start":"12:11.950 ","End":"12:14.390","Text":"but I\u0027ll do it on a new page."},{"Start":"12:14.820 ","End":"12:20.920","Text":"The example we\u0027ve been using all along is f of x is equal"},{"Start":"12:20.920 ","End":"12:30.070","Text":"to x squared plus 8 x for x that are bigger or equal 2,"},{"Start":"12:31.340 ","End":"12:39.435","Text":"and x cubed plus 12 for x less than 2."},{"Start":"12:39.435 ","End":"12:41.410","Text":"Our task, of course,"},{"Start":"12:41.410 ","End":"12:45.655","Text":"is to check differentiability at x equals 2."},{"Start":"12:45.655 ","End":"12:49.900","Text":"The first step is to check that f is continuous."},{"Start":"12:49.900 ","End":"12:52.510","Text":"If we find that f is not continuous,"},{"Start":"12:52.510 ","End":"12:55.000","Text":"then we can immediately stop the process and just"},{"Start":"12:55.000 ","End":"12:57.880","Text":"declare f is not differentiable, so again,"},{"Start":"12:57.880 ","End":"13:01.540","Text":"in writing, Step 1,"},{"Start":"13:01.540 ","End":"13:04.090","Text":"verify f is continuous,"},{"Start":"13:04.090 ","End":"13:06.445","Text":"and I hope you remember how to do that."},{"Start":"13:06.445 ","End":"13:08.020","Text":"If not, I\u0027ll remind you."},{"Start":"13:08.020 ","End":"13:10.495","Text":"If it isn\u0027t continuous,"},{"Start":"13:10.495 ","End":"13:15.235","Text":"f is not differentiable, and we\u0027re done."},{"Start":"13:15.235 ","End":"13:19.030","Text":"To remind you how we show a function is continuous,"},{"Start":"13:19.030 ","End":"13:23.245","Text":"we show that it satisfies a triple equality,"},{"Start":"13:23.245 ","End":"13:25.855","Text":"that the value of the function at the point."},{"Start":"13:25.855 ","End":"13:30.925","Text":"Of course, we\u0027re still talking about the point x equals 2 has to equal the limit"},{"Start":"13:30.925 ","End":"13:37.225","Text":"from the right of the function,"},{"Start":"13:37.225 ","End":"13:45.580","Text":"and it\u0027s also going to equal the limit from the left of the function as x goes to 2,"},{"Start":"13:45.580 ","End":"13:47.605","Text":"of course, in either direction."},{"Start":"13:47.605 ","End":"13:49.720","Text":"Let\u0027s just do this in our heads."},{"Start":"13:49.720 ","End":"13:51.685","Text":"This is not difficult."},{"Start":"13:51.685 ","End":"13:58.705","Text":"What we get here is that f of 2 is we take the definition from here,"},{"Start":"13:58.705 ","End":"14:02.305","Text":"2 squared plus 8 times 2 is 20,"},{"Start":"14:02.305 ","End":"14:08.210","Text":"so we have 20, and let\u0027s see."},{"Start":"14:10.560 ","End":"14:14.320","Text":"X goes to 2 from the right,"},{"Start":"14:14.320 ","End":"14:18.160","Text":"2 from the right is on this side of the 2, so again,"},{"Start":"14:18.160 ","End":"14:23.335","Text":"it\u0027s this and we can just substitute, so it\u0027s 20."},{"Start":"14:23.335 ","End":"14:26.390","Text":"On the left,"},{"Start":"14:27.120 ","End":"14:31.030","Text":"we use this definition, and again,"},{"Start":"14:31.030 ","End":"14:35.005","Text":"we just substitute to find the limit, so we substitute 2,"},{"Start":"14:35.005 ","End":"14:37.390","Text":"2 cubed is 8 plus 12 is 20,"},{"Start":"14:37.390 ","End":"14:39.400","Text":"so 20 equals 20 equals 20."},{"Start":"14:39.400 ","End":"14:42.295","Text":"So yes, on continuous."},{"Start":"14:42.295 ","End":"14:45.050","Text":"That means we need to continue."},{"Start":"14:46.200 ","End":"14:48.880","Text":"Now Step 2."},{"Start":"14:48.880 ","End":"14:56.920","Text":"Step 2 is to check the limits as in the theorem."},{"Start":"14:56.920 ","End":"14:59.125","Text":"What we have to do is to check,"},{"Start":"14:59.125 ","End":"15:03.700","Text":"and these were the 2 one sided limits of the derivative,"},{"Start":"15:03.700 ","End":"15:09.190","Text":"so the limit as x goes to"},{"Start":"15:09.190 ","End":"15:18.220","Text":"2 from the right of f prime of x,"},{"Start":"15:18.220 ","End":"15:23.080","Text":"and the limit as x goes to 2 from the left of f prime of x."},{"Start":"15:23.080 ","End":"15:25.675","Text":"But what is f prime of x?"},{"Start":"15:25.675 ","End":"15:33.115","Text":"Let me make a little table of f prime of x at the side here,"},{"Start":"15:33.115 ","End":"15:34.885","Text":"f prime of x is equal to,"},{"Start":"15:34.885 ","End":"15:38.800","Text":"and it\u0027s going to be piece-wise defined itself."},{"Start":"15:38.800 ","End":"15:44.695","Text":"What we know, let me get this straight off the definition of f,"},{"Start":"15:44.695 ","End":"15:50.840","Text":"the left prime is equal to 2 x plus 8."},{"Start":"15:51.840 ","End":"15:54.820","Text":"Whenever x is bigger than 2,"},{"Start":"15:54.820 ","End":"15:58.960","Text":"there we have no problems of differentiability, and also,"},{"Start":"15:58.960 ","End":"16:02.710","Text":"we have no problems when x is less than 2 and then we just take"},{"Start":"16:02.710 ","End":"16:07.195","Text":"it from the formula and it\u0027s 3x squared."},{"Start":"16:07.195 ","End":"16:10.780","Text":"We don\u0027t know what\u0027s going on when x equals 2,"},{"Start":"16:10.780 ","End":"16:12.760","Text":"and that\u0027s what we\u0027re here to find out."},{"Start":"16:12.760 ","End":"16:16.975","Text":"On the other hand, we don\u0027t need this for checking"},{"Start":"16:16.975 ","End":"16:23.350","Text":"these 2 limits because when x goes to 2 from the right,"},{"Start":"16:23.350 ","End":"16:24.580","Text":"it doesn\u0027t equal 2."},{"Start":"16:24.580 ","End":"16:29.770","Text":"This is actually not important to us at this stage. Let\u0027s check."},{"Start":"16:29.770 ","End":"16:37.375","Text":"To check these, we just have to substitute x equals 2 in the right place."},{"Start":"16:37.375 ","End":"16:39.445","Text":"We can do this in our heads."},{"Start":"16:39.445 ","End":"16:41.575","Text":"If we take the limit from the right,"},{"Start":"16:41.575 ","End":"16:47.110","Text":"then we use this definition and then we just substitute x equals 2,"},{"Start":"16:47.110 ","End":"16:49.735","Text":"so we get twice 2 plus 8,"},{"Start":"16:49.735 ","End":"16:51.880","Text":"and that is equal to 12."},{"Start":"16:51.880 ","End":"16:54.310","Text":"This limit here is 12."},{"Start":"16:54.310 ","End":"16:55.930","Text":"The other 1 from the left,"},{"Start":"16:55.930 ","End":"16:58.690","Text":"so we use the lower definition and again,"},{"Start":"16:58.690 ","End":"17:00.520","Text":"just substitute x equals 2,"},{"Start":"17:00.520 ","End":"17:03.580","Text":"2 squared is 4 times 3 is 12."},{"Start":"17:03.580 ","End":"17:07.810","Text":"Indeed, these 2 limits are equal."},{"Start":"17:07.810 ","End":"17:11.740","Text":"Basically, we satisfied all the conditions of the theorem."},{"Start":"17:11.740 ","End":"17:15.190","Text":"Above, we showed the continuity and here,"},{"Start":"17:15.190 ","End":"17:19.105","Text":"we showed existence inequality of these 2 limits."},{"Start":"17:19.105 ","End":"17:22.465","Text":"We conclude that yes,"},{"Start":"17:22.465 ","End":"17:27.145","Text":"that f is differentiable at x equals 2 and we even know"},{"Start":"17:27.145 ","End":"17:32.020","Text":"that f prime of 2 is equal to 12 also."},{"Start":"17:32.020 ","End":"17:35.950","Text":"Let\u0027s summarize this. In Part 1,"},{"Start":"17:35.950 ","End":"17:39.250","Text":"we showed that f is continuous and here we showed"},{"Start":"17:39.250 ","End":"17:43.030","Text":"that these 2 one-sided limits exist and are equal,"},{"Start":"17:43.030 ","End":"17:45.580","Text":"and they also happen to equal 12,"},{"Start":"17:45.580 ","End":"17:51.970","Text":"and we conclude from the theorem that f is differentiable at x equals 2,"},{"Start":"17:51.970 ","End":"17:56.050","Text":"and what\u0027s more, we even know the value of the derivative at 2,"},{"Start":"17:56.050 ","End":"17:58.915","Text":"which is 12, the same as these 2."},{"Start":"17:58.915 ","End":"18:02.080","Text":"I also note, and this is informal,"},{"Start":"18:02.080 ","End":"18:08.845","Text":"but I must note that this method seems a lot easier than the previous method."},{"Start":"18:08.845 ","End":"18:11.095","Text":"I made a note of that here,"},{"Start":"18:11.095 ","End":"18:14.500","Text":"but we\u0027ll be discussing it when we discuss which method,"},{"Start":"18:14.500 ","End":"18:17.350","Text":"1 or 2, we should be using."},{"Start":"18:17.350 ","End":"18:22.080","Text":"What I wanted to do before that is just cleaning up,"},{"Start":"18:22.080 ","End":"18:25.230","Text":"just want to simplify what we wrote as f prime of x."},{"Start":"18:25.230 ","End":"18:28.335","Text":"Because we know that f prime of 2 is 12,"},{"Start":"18:28.335 ","End":"18:30.300","Text":"I could write 12 here,"},{"Start":"18:30.300 ","End":"18:32.580","Text":"but I can simplify it still further,"},{"Start":"18:32.580 ","End":"18:34.590","Text":"because when x equals 2,"},{"Start":"18:34.590 ","End":"18:36.630","Text":"you plug 2 in here, you also get 12,"},{"Start":"18:36.630 ","End":"18:38.315","Text":"so we can write it like this,"},{"Start":"18:38.315 ","End":"18:43.690","Text":"and now let\u0027s go to the next page and discuss which method to use in practice."},{"Start":"18:43.690 ","End":"18:50.335","Text":"Which method should we use in a homework question or an exam?"},{"Start":"18:50.335 ","End":"18:54.055","Text":"Method 1 or method 2?"},{"Start":"18:54.055 ","End":"18:55.360","Text":"But I won\u0027t call them that here,"},{"Start":"18:55.360 ","End":"18:58.509","Text":"I\u0027ll call them definition and theorem,"},{"Start":"18:58.509 ","End":"19:00.805","Text":"and I\u0027ll keep them in this color,"},{"Start":"19:00.805 ","End":"19:08.090","Text":"so you\u0027ll know that I\u0027m talking about these 2 methods of differentiability."},{"Start":"19:08.490 ","End":"19:14.185","Text":"In general, we prefer to work with theorem."},{"Start":"19:14.185 ","End":"19:19.990","Text":"You saw the theorem and the definition in our example, theorem\u0027s much easier."},{"Start":"19:19.990 ","End":"19:22.645","Text":"But sometimes, we have to work with"},{"Start":"19:22.645 ","End":"19:27.325","Text":"definition and there are at least 3 cases I can think of."},{"Start":"19:27.325 ","End":"19:30.760","Text":"Number 1, in the exercise,"},{"Start":"19:30.760 ","End":"19:35.270","Text":"we were specifically asked to use the definition method."},{"Start":"19:35.430 ","End":"19:38.980","Text":"It\u0027s reasonable to ask it."},{"Start":"19:38.980 ","End":"19:43.090","Text":"Reason 2, the teacher or professor has only"},{"Start":"19:43.090 ","End":"19:46.960","Text":"taught the definition method and in his course,"},{"Start":"19:46.960 ","End":"19:50.035","Text":"you\u0027re not allowed to use any method that he hasn\u0027t taught."},{"Start":"19:50.035 ","End":"19:52.390","Text":"There are professors like that."},{"Start":"19:52.390 ","End":"19:57.579","Text":"Number 3, possible that the theorem isn\u0027t applicable,"},{"Start":"19:57.579 ","End":"20:02.485","Text":"that we can\u0027t meet the conditions of the theorem in order to get the conclusions."},{"Start":"20:02.485 ","End":"20:07.449","Text":"For example, at least 1 of the limits."},{"Start":"20:07.449 ","End":"20:09.340","Text":"These 2 that we know about,"},{"Start":"20:09.340 ","End":"20:11.335","Text":"the 2 one-sided limits doesn\u0027t exist."},{"Start":"20:11.335 ","End":"20:12.370","Text":"When I say doesn\u0027t exist,"},{"Start":"20:12.370 ","End":"20:14.600","Text":"there\u0027s a finite number."},{"Start":"20:16.440 ","End":"20:18.714","Text":"In all these cases,"},{"Start":"20:18.714 ","End":"20:23.245","Text":"we can\u0027t use the theorem"},{"Start":"20:23.245 ","End":"20:27.700","Text":"and there are actual examples of this and if you want to see some at the moment,"},{"Start":"20:27.700 ","End":"20:33.655","Text":"they\u0027re in Exercises 7 and 8 examples to follow."},{"Start":"20:33.655 ","End":"20:38.410","Text":"I\u0027d like to say a few more words about Case 3 where the theorem isn\u0027t"},{"Start":"20:38.410 ","End":"20:44.545","Text":"applicable and I\u0027ll scroll up a bit to get to the place I want."},{"Start":"20:44.545 ","End":"20:46.660","Text":"Okay, here it is."},{"Start":"20:46.660 ","End":"20:50.425","Text":"When we checked the right limit and the left limit,"},{"Start":"20:50.425 ","End":"20:54.490","Text":"we got both of them to be finite and they were equal numbers."},{"Start":"20:54.490 ","End":"20:58.645","Text":"We had 12, which is equal to 12."},{"Start":"20:58.645 ","End":"21:04.270","Text":"Now, the theorem talks about this case and normally,"},{"Start":"21:04.270 ","End":"21:05.950","Text":"if we didn\u0027t have this case,"},{"Start":"21:05.950 ","End":"21:07.990","Text":"we couldn\u0027t use the theorem,"},{"Start":"21:07.990 ","End":"21:10.400","Text":"it would be inapplicable."},{"Start":"21:10.590 ","End":"21:18.775","Text":"For example, if I had here 12 and here infinite or undefined,"},{"Start":"21:18.775 ","End":"21:20.230","Text":"or the other way around,"},{"Start":"21:20.230 ","End":"21:24.160","Text":"or both of them undefined or infinite,"},{"Start":"21:24.160 ","End":"21:25.584","Text":"in all these cases,"},{"Start":"21:25.584 ","End":"21:27.340","Text":"we can\u0027t use the theorem."},{"Start":"21:27.340 ","End":"21:29.995","Text":"The theorem doesn\u0027t provide for it."},{"Start":"21:29.995 ","End":"21:34.930","Text":"However, there is 1 thing that\u0027s not been covered or mentioned and it"},{"Start":"21:34.930 ","End":"21:39.925","Text":"could be very useful to you and that is the case where both of these exist,"},{"Start":"21:39.925 ","End":"21:41.980","Text":"but they\u0027re just not equal to each other,"},{"Start":"21:41.980 ","End":"21:44.560","Text":"like here\u0027s 12 and here 20."},{"Start":"21:44.560 ","End":"21:53.810","Text":"In that case, we can conclude definitely that f is not differentiable at that point."},{"Start":"21:53.880 ","End":"21:56.365","Text":"I\u0027d like to repeat."},{"Start":"21:56.365 ","End":"21:59.680","Text":"If we get 2 numbers that are equal,"},{"Start":"21:59.680 ","End":"22:04.375","Text":"fine, we\u0027re differentiable and we can use the theorem."},{"Start":"22:04.375 ","End":"22:09.445","Text":"If these 2 are finite but they\u0027re not equal,"},{"Start":"22:09.445 ","End":"22:15.580","Text":"you\u0027re guaranteed that the function is not differentiable at that point."},{"Start":"22:15.580 ","End":"22:17.980","Text":"Anything else that you get,"},{"Start":"22:17.980 ","End":"22:21.355","Text":"one of these being undefined or infinite,"},{"Start":"22:21.355 ","End":"22:29.170","Text":"you just can\u0027t use the theorem and we don\u0027t know and you\u0027ll use another method."},{"Start":"22:29.170 ","End":"22:31.990","Text":"For example, the definition."},{"Start":"22:31.990 ","End":"22:36.580","Text":"Before I end this clip,"},{"Start":"22:36.580 ","End":"22:38.965","Text":"I just like to remind you that this is"},{"Start":"22:38.965 ","End":"22:42.684","Text":"very important material and it\u0027s also very delicate."},{"Start":"22:42.684 ","End":"22:47.830","Text":"I strongly advise you to go over this lecture if you like."},{"Start":"22:47.830 ","End":"22:51.385","Text":"Again, I recommend,"},{"Start":"22:51.385 ","End":"22:56.240","Text":"you can do what you like. We\u0027re done."}],"ID":1473},{"Watched":false,"Name":"Exercise 1 - Part 1","Duration":"2m 44s","ChapterTopicVideoID":2926,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This exercise is really 3 and 1."},{"Start":"00:02.820 ","End":"00:08.505","Text":"In each one, we have to find a point of non-differentiability of the function"},{"Start":"00:08.505 ","End":"00:14.040","Text":"and to find the derivative of f at the point x equals 2 if possible."},{"Start":"00:14.040 ","End":"00:16.020","Text":"We\u0027ll begin with number 1,"},{"Start":"00:16.020 ","End":"00:18.385","Text":"and I\u0027ll start by copying it."},{"Start":"00:18.385 ","End":"00:21.530","Text":"We know that the function f as written is"},{"Start":"00:21.530 ","End":"00:25.490","Text":"differentiable everywhere except possibly at x equals 2,"},{"Start":"00:25.490 ","End":"00:28.250","Text":"which is the seam line where the function changes formulas."},{"Start":"00:28.250 ","End":"00:32.210","Text":"That\u0027s the only point we need to investigate, x equals 2."},{"Start":"00:32.210 ","End":"00:35.450","Text":"I\u0027d like to use the theorem which states that if f is"},{"Start":"00:35.450 ","End":"00:39.455","Text":"continuous and some other condition at a point,"},{"Start":"00:39.455 ","End":"00:41.885","Text":"then it\u0027s going to be differentiable."},{"Start":"00:41.885 ","End":"00:44.150","Text":"I want to just try the continuity first."},{"Start":"00:44.150 ","End":"00:47.630","Text":"Let\u0027s see if f is continuous at x equals 2,"},{"Start":"00:47.630 ","End":"00:49.670","Text":"because I have a feeling that it might not be."},{"Start":"00:49.670 ","End":"00:53.780","Text":"In order to check that f is continuous at x equals 2,"},{"Start":"00:53.780 ","End":"00:56.795","Text":"we have to show that 3 things are equal."},{"Start":"00:56.795 ","End":"00:58.745","Text":"That f at the point 2,"},{"Start":"00:58.745 ","End":"01:02.000","Text":"and the right limit, that is,"},{"Start":"01:02.000 ","End":"01:08.055","Text":"limit as x goes to 2 from the right of f of x,"},{"Start":"01:08.055 ","End":"01:09.815","Text":"and the left limit,"},{"Start":"01:09.815 ","End":"01:14.675","Text":"limit as x goes to 2 from the left of f of x."},{"Start":"01:14.675 ","End":"01:19.435","Text":"All these 3 things have to be equal in order for it to be continuous."},{"Start":"01:19.435 ","End":"01:22.080","Text":"Let\u0027s see what each one is equal to."},{"Start":"01:22.080 ","End":"01:24.960","Text":"f of 2, I just substitute 2 in here."},{"Start":"01:24.960 ","End":"01:27.075","Text":"Now, 2 belongs in this part."},{"Start":"01:27.075 ","End":"01:30.010","Text":"2 squared minus 4 times 2,"},{"Start":"01:30.010 ","End":"01:32.040","Text":"which is 4 minus 8,"},{"Start":"01:32.040 ","End":"01:34.545","Text":"this is minus 4."},{"Start":"01:34.545 ","End":"01:37.850","Text":"Limit as x goes to 2 from the right of f of x."},{"Start":"01:37.850 ","End":"01:41.335","Text":"I\u0027m looking at the same place where I put the x equals 2,"},{"Start":"01:41.335 ","End":"01:42.960","Text":"and when x goes to 2,"},{"Start":"01:42.960 ","End":"01:44.325","Text":"we just substitute it."},{"Start":"01:44.325 ","End":"01:46.985","Text":"We still get 2 squared minus 4 times 2"},{"Start":"01:46.985 ","End":"01:50.750","Text":"and the same answer, minus 4."},{"Start":"01:50.750 ","End":"01:54.080","Text":"Last one, x goes to 2 from the left,"},{"Start":"01:54.080 ","End":"01:56.120","Text":"then we\u0027re lower than 2."},{"Start":"01:56.120 ","End":"01:58.355","Text":"We\u0027re reading off this definition,"},{"Start":"01:58.355 ","End":"02:00.184","Text":"x cubed minus 14."},{"Start":"02:00.184 ","End":"02:02.855","Text":"If you put 2 in, you get 2 cubed minus 14,"},{"Start":"02:02.855 ","End":"02:06.875","Text":"which is 8 minus 14, which is minus 6."},{"Start":"02:06.875 ","End":"02:08.990","Text":"We\u0027ve got 2 out of 3, but that\u0027s not good enough."},{"Start":"02:08.990 ","End":"02:12.650","Text":"They all 3 have to be equal in order for it to be continuous."},{"Start":"02:12.650 ","End":"02:19.675","Text":"We have that, f is not continuous at x equals 2,"},{"Start":"02:19.675 ","End":"02:23.780","Text":"and hence, it\u0027s not differentiable at x equals 2."},{"Start":"02:23.780 ","End":"02:26.155","Text":"I can write the answer,"},{"Start":"02:26.155 ","End":"02:30.015","Text":"which asked about where f is non-differentiable,"},{"Start":"02:30.015 ","End":"02:34.595","Text":"that f is not differentiable,"},{"Start":"02:34.595 ","End":"02:36.470","Text":"but only at, because everywhere else it is,"},{"Start":"02:36.470 ","End":"02:45.120","Text":"so I\u0027ll write only at x equals 2, and we\u0027re done."}],"ID":2938},{"Watched":false,"Name":"Exercise 1 - Part 2","Duration":"16m 40s","ChapterTopicVideoID":1491,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:02.520","Text":"Okay. We\u0027re done with number 1,"},{"Start":"00:02.520 ","End":"00:07.500","Text":"so I just erased everything and we\u0027re starting a new with number 2."},{"Start":"00:07.500 ","End":"00:10.050","Text":"This is the function for number 2,"},{"Start":"00:10.050 ","End":"00:14.100","Text":"also piece-wise with a seamline at x equals 2."},{"Start":"00:14.100 ","End":"00:16.770","Text":"The theorem I kept."},{"Start":"00:16.770 ","End":"00:25.500","Text":"What we have to do is to check that f is continuous at x equals a."},{"Start":"00:25.500 ","End":"00:26.790","Text":"If we want to use the theorem,"},{"Start":"00:26.790 ","End":"00:28.545","Text":"we need to check these 2 conditions."},{"Start":"00:28.545 ","End":"00:31.420","Text":"First of all, the continuity."},{"Start":"00:31.970 ","End":"00:35.250","Text":"The continuity basically means that"},{"Start":"00:35.250 ","End":"00:39.360","Text":"the left limit and the right limit and the value at the point,"},{"Start":"00:39.360 ","End":"00:41.310","Text":"all those 3 are equal."},{"Start":"00:41.310 ","End":"00:46.685","Text":"What we need for continuity is"},{"Start":"00:46.685 ","End":"00:54.325","Text":"that limit as x goes to 2,"},{"Start":"00:54.325 ","End":"00:56.580","Text":"doesn\u0027t matter which one first,"},{"Start":"00:56.580 ","End":"01:05.660","Text":"from the left of f of x is equal to the limit as x goes to 2 from the right of f of x."},{"Start":"01:05.660 ","End":"01:09.170","Text":"This is going to also equal f of 2."},{"Start":"01:09.170 ","End":"01:12.845","Text":"All these 3 things have to be equal. Let\u0027s see."},{"Start":"01:12.845 ","End":"01:14.540","Text":"The first one,"},{"Start":"01:14.540 ","End":"01:17.105","Text":"I think we can even do it in our heads."},{"Start":"01:17.105 ","End":"01:20.555","Text":"The limit as x goes to 2 from the left of f of x."},{"Start":"01:20.555 ","End":"01:23.195","Text":"From the left means we\u0027re smaller than 2,"},{"Start":"01:23.195 ","End":"01:25.660","Text":"so it means we\u0027re taking it from here."},{"Start":"01:25.660 ","End":"01:28.700","Text":"Here, we just substitute x equals 2."},{"Start":"01:28.700 ","End":"01:31.940","Text":"We get 2 cubed minus 14,"},{"Start":"01:31.940 ","End":"01:36.105","Text":"8 minus 14, which is minus 6."},{"Start":"01:36.105 ","End":"01:37.710","Text":"This is minus 6."},{"Start":"01:37.710 ","End":"01:39.285","Text":"Now, how about this?"},{"Start":"01:39.285 ","End":"01:41.265","Text":"x goes to 2 from the right,"},{"Start":"01:41.265 ","End":"01:43.220","Text":"so we\u0027re using this definition."},{"Start":"01:43.220 ","End":"01:45.290","Text":"2 squared is 4,"},{"Start":"01:45.290 ","End":"01:47.150","Text":"minus 5 times 2 is 10."},{"Start":"01:47.150 ","End":"01:50.415","Text":"4 minus 10 is minus 6."},{"Start":"01:50.415 ","End":"01:52.260","Text":"The value at 2, well,"},{"Start":"01:52.260 ","End":"01:56.125","Text":"2 because of the greater than or equal to 2."},{"Start":"01:56.125 ","End":"01:59.300","Text":"Again, we just put substitute 2 here,"},{"Start":"01:59.300 ","End":"02:01.310","Text":"but we already done that here."},{"Start":"02:01.310 ","End":"02:03.185","Text":"It\u0027s also minus 6."},{"Start":"02:03.185 ","End":"02:05.465","Text":"All these 3 things are equal."},{"Start":"02:05.465 ","End":"02:15.125","Text":"So yes, it is continuous at x equals 2 to be precise."},{"Start":"02:15.125 ","End":"02:17.465","Text":"That\u0027s the first condition."},{"Start":"02:17.465 ","End":"02:20.570","Text":"The second condition that we have to check is"},{"Start":"02:20.570 ","End":"02:24.755","Text":"that the left limit and the right limit of the derivative."},{"Start":"02:24.755 ","End":"02:29.100","Text":"Remember, the derivatives exist because these are just polynomials."},{"Start":"02:30.790 ","End":"02:38.210","Text":"In fact, I might even write it over here that the derivative exists f prime of x."},{"Start":"02:38.210 ","End":"02:41.370","Text":"We don\u0027t know about x equals 2."},{"Start":"02:44.090 ","End":"02:53.730","Text":"This is equal to the derivative of x squared minus 5x is 2x minus 5."},{"Start":"02:54.920 ","End":"03:00.645","Text":"If x is strictly bigger than 2, it\u0027s equal to,"},{"Start":"03:00.645 ","End":"03:06.960","Text":"we don\u0027t know what if x equals 2,"},{"Start":"03:06.960 ","End":"03:09.180","Text":"that\u0027s what we\u0027re trying to find out."},{"Start":"03:09.180 ","End":"03:11.750","Text":"It\u0027s equal to the derivative of this,"},{"Start":"03:11.750 ","End":"03:20.070","Text":"which is just 3x squared when x is less than 2."},{"Start":"03:21.100 ","End":"03:27.020","Text":"We need to check what it says here."},{"Start":"03:27.020 ","End":"03:30.290","Text":"The limit as x goes to 2 from"},{"Start":"03:30.290 ","End":"03:35.945","Text":"1 side of f prime of x and the limit as x goes to 2 from the other side,"},{"Start":"03:35.945 ","End":"03:38.285","Text":"doesn\u0027t really matter in what order."},{"Start":"03:38.285 ","End":"03:42.709","Text":"See if I can keep something on the screen at the same time."},{"Start":"03:42.709 ","End":"03:44.695","Text":"We\u0027re checking next."},{"Start":"03:44.695 ","End":"03:53.140","Text":"Does the limit as x goes."},{"Start":"03:54.620 ","End":"03:57.230","Text":"Sorry, the handwriting, got a bit messy."},{"Start":"03:57.230 ","End":"03:58.715","Text":"I just wrote it out."},{"Start":"03:58.715 ","End":"04:01.340","Text":"This is what we have to check."},{"Start":"04:01.340 ","End":"04:06.140","Text":"Does the limit from the right equals the limit from the left?"},{"Start":"04:06.140 ","End":"04:09.485","Text":"This 1 was 3x squared."},{"Start":"04:09.485 ","End":"04:19.394","Text":"We have to basically say does the limit as x goes to 2 plus"},{"Start":"04:19.394 ","End":"04:26.445","Text":"of 3x squared equal the limit as x goes to 2"},{"Start":"04:26.445 ","End":"04:34.390","Text":"minus of 2x minus 5?"},{"Start":"04:34.390 ","End":"04:36.685","Text":"Well, let\u0027s check."},{"Start":"04:36.685 ","End":"04:38.645","Text":"All we have to do here is substitute."},{"Start":"04:38.645 ","End":"04:40.775","Text":"When x equals 2,"},{"Start":"04:40.775 ","End":"04:42.740","Text":"we get 3 times 2 squared,"},{"Start":"04:42.740 ","End":"04:46.380","Text":"which is 3 times 4, which is 12."},{"Start":"04:47.420 ","End":"04:50.220","Text":"If x goes to 2,"},{"Start":"04:50.220 ","End":"04:53.010","Text":"we just substitute x equals 2 here."},{"Start":"04:53.010 ","End":"04:55.170","Text":"We have 2 times 2 is 4,"},{"Start":"04:55.170 ","End":"04:58.915","Text":"minus 5 is minus 1."},{"Start":"04:58.915 ","End":"05:03.260","Text":"If I ask, does 12 equals to minus 1?"},{"Start":"05:03.260 ","End":"05:08.010","Text":"Well, I have to say the answer is no."},{"Start":"05:09.440 ","End":"05:12.510","Text":"In the exercise 1,"},{"Start":"05:12.510 ","End":"05:16.425","Text":"we failed on the continuity."},{"Start":"05:16.425 ","End":"05:18.650","Text":"In exercise part 2,"},{"Start":"05:18.650 ","End":"05:20.180","Text":"we failed on this."},{"Start":"05:20.180 ","End":"05:22.160","Text":"As soon as we fail, we stop."},{"Start":"05:22.160 ","End":"05:30.640","Text":"That means that f of x is"},{"Start":"05:30.640 ","End":"05:36.600","Text":"not differentiable"},{"Start":"05:41.350 ","End":"05:44.965","Text":"at x equals 2."},{"Start":"05:44.965 ","End":"05:49.685","Text":"That answers the original question to find where f is not differentiable."},{"Start":"05:49.685 ","End":"05:52.930","Text":"Now, as I mentioned before in part 1,"},{"Start":"05:52.930 ","End":"05:57.860","Text":"normally, we would end here by using the theorem."},{"Start":"05:57.860 ","End":"06:00.380","Text":"But we\u0027re going to, for the practice,"},{"Start":"06:00.380 ","End":"06:07.730","Text":"also use the definition of the derivative to do the calculation, just extra work."},{"Start":"06:07.730 ","End":"06:13.140","Text":"If we now go and use the definition,"},{"Start":"06:17.630 ","End":"06:20.625","Text":"this part is over."},{"Start":"06:20.625 ","End":"06:22.820","Text":"We did it by the theorem,"},{"Start":"06:22.820 ","End":"06:26.480","Text":"that was the theorem and now we do it by the definition."},{"Start":"06:26.480 ","End":"06:35.405","Text":"What we get is that the derivative of f at the point 2,"},{"Start":"06:35.405 ","End":"06:38.630","Text":"if it exists and we know it doesn\u0027t exist,"},{"Start":"06:38.630 ","End":"06:41.645","Text":"but we\u0027re starting a fresh as if we really don\u0027t know,"},{"Start":"06:41.645 ","End":"06:48.845","Text":"is equal to the limit as h goes to 0 of"},{"Start":"06:48.845 ","End":"06:56.655","Text":"f of x plus h minus f of, sorry."},{"Start":"06:56.655 ","End":"06:58.575","Text":"x we know is 2,"},{"Start":"06:58.575 ","End":"07:01.710","Text":"just hang on a second. Fix that."},{"Start":"07:01.710 ","End":"07:05.980","Text":"f of 2 plus h minus f of 2,"},{"Start":"07:08.660 ","End":"07:12.315","Text":"I\u0027ll just leave that little mess there,"},{"Start":"07:12.315 ","End":"07:20.160","Text":"over h. If we try doing this,"},{"Start":"07:20.160 ","End":"07:24.100","Text":"we have a problem because when h is positive,"},{"Start":"07:24.100 ","End":"07:29.060","Text":"f of 2 plus h is going to be 1 formula."},{"Start":"07:29.060 ","End":"07:31.024","Text":"If h is negative,"},{"Start":"07:31.024 ","End":"07:34.070","Text":"f of 2 plus h is going to be a different formula,"},{"Start":"07:34.070 ","End":"07:39.410","Text":"because 2 is 1 of those seam points around which the definition changes."},{"Start":"07:39.410 ","End":"07:47.085","Text":"We have to figure out the left limit and the right limit separately."},{"Start":"07:47.085 ","End":"07:50.800","Text":"Let\u0027s try the left limit first."},{"Start":"07:52.100 ","End":"07:54.260","Text":"If they both equal,"},{"Start":"07:54.260 ","End":"07:56.780","Text":"then this will be this limit."},{"Start":"07:56.780 ","End":"08:00.800","Text":"It equals the left limit and the right limit as I said,"},{"Start":"08:00.800 ","End":"08:01.970","Text":"but only if they\u0027re equal."},{"Start":"08:01.970 ","End":"08:03.710","Text":"So h goes to,"},{"Start":"08:03.710 ","End":"08:05.255","Text":"what should we take first?"},{"Start":"08:05.255 ","End":"08:08.674","Text":"Let\u0027s take 0 plus,"},{"Start":"08:08.674 ","End":"08:18.570","Text":"and this equals f of, now,"},{"Start":"08:18.570 ","End":"08:22.800","Text":"what is 2 plus h if h is slightly bigger than 0?"},{"Start":"08:22.800 ","End":"08:25.775","Text":"If h is slightly bigger than 0,"},{"Start":"08:25.775 ","End":"08:31.980","Text":"then we have 2 plus h is slightly bigger than 2,"},{"Start":"08:31.980 ","End":"08:35.110","Text":"so we get x squared minus 5x."},{"Start":"08:43.170 ","End":"08:46.045","Text":"I don\u0027t actually need the f,"},{"Start":"08:46.045 ","End":"08:50.515","Text":"if you give me a minute, I\u0027ll just erase that."},{"Start":"08:50.515 ","End":"08:57.715","Text":"What we need is the definition of f of 2 plus h,"},{"Start":"08:57.715 ","End":"09:06.820","Text":"which is x squared minus 5x."},{"Start":"09:06.820 ","End":"09:15.340","Text":"We have 2 plus h"},{"Start":"09:15.340 ","End":"09:21.310","Text":"squared minus 5(2 plus h) for this bit."},{"Start":"09:21.310 ","End":"09:25.280","Text":"Less, again, x squared."},{"Start":"09:29.550 ","End":"09:32.290","Text":"2 belongs to this also,"},{"Start":"09:32.290 ","End":"09:36.175","Text":"to the part with the x squared minus 5x,"},{"Start":"09:36.175 ","End":"09:39.040","Text":"if you remember, it was for bigger or equal to 2."},{"Start":"09:39.040 ","End":"09:44.710","Text":"So it\u0027s 2 squared minus 5 times"},{"Start":"09:44.710 ","End":"09:53.125","Text":"2 all over h. Now, what do we get here?"},{"Start":"09:53.125 ","End":"10:00.805","Text":"This equals, and those still on the right limit,"},{"Start":"10:00.805 ","End":"10:06.924","Text":"h goes to 0 plus 2 plus h squared is 2 squared"},{"Start":"10:06.924 ","End":"10:15.130","Text":"is 4 and we know that,"},{"Start":"10:15.130 ","End":"10:19.075","Text":"plus 4h plus h squared,"},{"Start":"10:19.075 ","End":"10:20.950","Text":"this a plus b squared formula,"},{"Start":"10:20.950 ","End":"10:22.765","Text":"we just should know it by heart."},{"Start":"10:22.765 ","End":"10:28.825","Text":"Minus 5 times 2 is 10 minus 5h."},{"Start":"10:28.825 ","End":"10:35.530","Text":"Here, we have 2 squared is 4 minus 10 is minus 6, but it\u0027s a minus,"},{"Start":"10:35.530 ","End":"10:44.140","Text":"so it\u0027s a plus 6 and all this is over h. Now, look,"},{"Start":"10:44.140 ","End":"10:47.275","Text":"the minus 10 goes with the 6 and the 4."},{"Start":"10:47.275 ","End":"10:49.690","Text":"After we divide by h,"},{"Start":"10:49.690 ","End":"10:59.695","Text":"basically, what we get is the limit as h goes to 0 plus,"},{"Start":"10:59.695 ","End":"11:08.605","Text":"of h squared over h is just h plus 4h over h is plus 4, and this limit,"},{"Start":"11:08.605 ","End":"11:12.295","Text":"when h goes to 0 is just 4."},{"Start":"11:12.295 ","End":"11:15.595","Text":"So that\u0027s the right limit of"},{"Start":"11:15.595 ","End":"11:25.120","Text":"the expression from the definition of derivative."},{"Start":"11:25.120 ","End":"11:26.665","Text":"This was the right."},{"Start":"11:26.665 ","End":"11:30.445","Text":"Now we move on to the left."},{"Start":"11:30.445 ","End":"11:32.620","Text":"That was the plus."},{"Start":"11:32.620 ","End":"11:40.990","Text":"That was the right limit of the definition of the derivative at x equals 2."},{"Start":"11:40.990 ","End":"11:43.000","Text":"Now, let\u0027s try the left limit,"},{"Start":"11:43.000 ","End":"11:50.275","Text":"which was the limit as h goes to,"},{"Start":"11:50.275 ","End":"11:52.210","Text":"now, we\u0027re trying the left,"},{"Start":"11:52.210 ","End":"11:54.950","Text":"0 from the left."},{"Start":"11:56.580 ","End":"11:59.170","Text":"Once again, I\u0027ll just repeat it."},{"Start":"11:59.170 ","End":"12:05.215","Text":"It\u0027s f of 2 plus h minus f"},{"Start":"12:05.215 ","End":"12:11.785","Text":"of 2 all over h. In this case,"},{"Start":"12:11.785 ","End":"12:16.750","Text":"when h is slightly less than 0,"},{"Start":"12:16.750 ","End":"12:20.320","Text":"that\u0027s just, remember, scroll back up."},{"Start":"12:20.320 ","End":"12:22.780","Text":"As from the left,"},{"Start":"12:22.780 ","End":"12:28.345","Text":"then the answer was x cubed minus 14."},{"Start":"12:28.345 ","End":"12:31.840","Text":"We have to take x cubed minus 14,"},{"Start":"12:31.840 ","End":"12:35.870","Text":"but f of 2,"},{"Start":"12:36.090 ","End":"12:40.330","Text":"we take from, where was it?"},{"Start":"12:40.330 ","End":"12:41.965","Text":"We had f of 2."},{"Start":"12:41.965 ","End":"12:49.670","Text":"f of 2 turned out to be minus 6."},{"Start":"12:49.740 ","End":"12:54.110","Text":"f of 2 is minus 6."},{"Start":"12:57.780 ","End":"13:00.580","Text":"Lets go back up here,"},{"Start":"13:00.580 ","End":"13:09.715","Text":"so this is equal to the limit as h goes to 0 from the left."},{"Start":"13:09.715 ","End":"13:12.310","Text":"Now, x cubed minus 14,"},{"Start":"13:12.310 ","End":"13:21.225","Text":"which is 2 plus h cubed minus 14 minus,"},{"Start":"13:21.225 ","End":"13:25.680","Text":"now here, as I say, we already figured this one out as minus 6."},{"Start":"13:25.680 ","End":"13:28.200","Text":"We actually minus minus 6,"},{"Start":"13:28.200 ","End":"13:30.940","Text":"we get plus 6"},{"Start":"13:33.020 ","End":"13:40.955","Text":"over h. As h goes to 0 plus."},{"Start":"13:40.955 ","End":"13:43.255","Text":"Now, if we open this,"},{"Start":"13:43.255 ","End":"13:45.145","Text":"I\u0027ll just write the limit again."},{"Start":"13:45.145 ","End":"13:48.325","Text":"Limit h goes to 0 from the left."},{"Start":"13:48.325 ","End":"13:56.775","Text":"Here we have 4 plus 4h."},{"Start":"13:56.775 ","End":"13:59.505","Text":"Wait a minute, it was x cubed."},{"Start":"13:59.505 ","End":"14:01.410","Text":"I\u0027m terribly sorry."},{"Start":"14:01.410 ","End":"14:06.700","Text":"I\u0027ll just go and correct that. Corrected it."},{"Start":"14:06.700 ","End":"14:12.580","Text":"Let\u0027s remember the formula for a plus b cubed,"},{"Start":"14:12.580 ","End":"14:19.120","Text":"what we have is that a plus b cubed is"},{"Start":"14:19.120 ","End":"14:29.514","Text":"a cubed plus 3a squared b plus 3ab squared plus b cubed."},{"Start":"14:29.514 ","End":"14:32.365","Text":"If I apply that to this,"},{"Start":"14:32.365 ","End":"14:33.610","Text":"I get 2 cubed,"},{"Start":"14:33.610 ","End":"14:35.050","Text":"which is 8,"},{"Start":"14:35.050 ","End":"14:39.040","Text":"plus 3 times 2 squared,"},{"Start":"14:39.040 ","End":"14:46.075","Text":"which is 12. h plus 3 times 2,"},{"Start":"14:46.075 ","End":"14:49.510","Text":"which is 6h squared,"},{"Start":"14:49.510 ","End":"15:00.820","Text":"plus h cubed minus 14 plus 6,"},{"Start":"15:00.820 ","End":"15:08.095","Text":"all over h. Let\u0027s see what happens with the numbers."},{"Start":"15:08.095 ","End":"15:17.785","Text":"The 8 plus 6 is 14 minus 14 so this disappears."},{"Start":"15:17.785 ","End":"15:22.240","Text":"Then we get, if we take h divided by h,"},{"Start":"15:22.240 ","End":"15:27.880","Text":"we get the limit as h goes to 0"},{"Start":"15:27.880 ","End":"15:34.195","Text":"from the left of h cubed over h is h squared."},{"Start":"15:34.195 ","End":"15:39.880","Text":"6h squared over h is 6h and 12h over"},{"Start":"15:39.880 ","End":"15:45.865","Text":"h is plus 12 and the limit as h goes to 0,"},{"Start":"15:45.865 ","End":"15:52.540","Text":"just put 0 and 0 here is equal to 12 and 12 was,"},{"Start":"15:52.540 ","End":"15:55.255","Text":"just like we wrote 4 the 4 was the right limit,"},{"Start":"15:55.255 ","End":"15:58.220","Text":"this is the left limit."},{"Start":"16:00.540 ","End":"16:07.460","Text":"I ask you, does 4 equals 12?"},{"Start":"16:08.430 ","End":"16:18.130","Text":"Well, already know the answer is no and that means that there is no limit."},{"Start":"16:18.130 ","End":"16:21.850","Text":"There is no derivative at x equals 2."},{"Start":"16:21.850 ","End":"16:24.175","Text":"Now, we already had that before."},{"Start":"16:24.175 ","End":"16:27.520","Text":"I\u0027m just repeating again that f of x is not"},{"Start":"16:27.520 ","End":"16:31.540","Text":"differentiable at x equals 2 and instead of using the theorem,"},{"Start":"16:31.540 ","End":"16:36.565","Text":"we use the definition here only to get the same conclusion which was to be expected."},{"Start":"16:36.565 ","End":"16:40.220","Text":"That\u0027s the end of number 2."}],"ID":1475},{"Watched":false,"Name":"Exercise 1 - Part 3","Duration":"14m 13s","ChapterTopicVideoID":1492,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:04.800","Text":"Now, we come to Part 3 and I\u0027ve modified accordingly."},{"Start":"00:04.800 ","End":"00:07.440","Text":"The f of x is given this way."},{"Start":"00:07.440 ","End":"00:11.310","Text":"f prime of x, we don\u0027t know what happens at x equals 2,"},{"Start":"00:11.310 ","End":"00:13.020","Text":"but when x is bigger than 2,"},{"Start":"00:13.020 ","End":"00:14.685","Text":"it\u0027s the derivative of this."},{"Start":"00:14.685 ","End":"00:16.920","Text":"Here it\u0027s the derivative of that."},{"Start":"00:16.920 ","End":"00:19.620","Text":"This is what we\u0027re going to try and find out."},{"Start":"00:19.620 ","End":"00:22.180","Text":"Now, I kept the theorem."},{"Start":"00:27.350 ","End":"00:30.390","Text":"Well, A, I should\u0027ve changed it to everywhere"},{"Start":"00:30.390 ","End":"00:35.250","Text":"because in fact in all the exercises it was 2,"},{"Start":"00:35.250 ","End":"00:37.830","Text":"change it in every little place."},{"Start":"00:37.830 ","End":"00:43.340","Text":"But if f is continuous at x equal 2 and something else,"},{"Start":"00:43.340 ","End":"00:47.465","Text":"then we\u0027re differentiable and we\u0027ve even got a formula for the derivative."},{"Start":"00:47.465 ","End":"00:49.280","Text":"Let\u0027s check one at a time."},{"Start":"00:49.280 ","End":"00:50.915","Text":"Let\u0027s see if we\u0027re continuous."},{"Start":"00:50.915 ","End":"00:53.465","Text":"Now, continuous at x equals 2,"},{"Start":"00:53.465 ","End":"00:55.835","Text":"means that 3 things are equal."},{"Start":"00:55.835 ","End":"01:03.130","Text":"That the limit as x goes to 2 from the right"},{"Start":"01:03.130 ","End":"01:11.765","Text":"of the function is equal to the limit as x goes to 2 from the left of the function,"},{"Start":"01:11.765 ","End":"01:16.235","Text":"and it\u0027s also equal to f at the point itself too."},{"Start":"01:16.235 ","End":"01:18.575","Text":"Let\u0027s see if these 3 things are equal."},{"Start":"01:18.575 ","End":"01:21.025","Text":"I think we can do it in our heads."},{"Start":"01:21.025 ","End":"01:23.280","Text":"2 plus means we\u0027re going from the right,"},{"Start":"01:23.280 ","End":"01:24.855","Text":"we\u0027re a bit bigger than 2,"},{"Start":"01:24.855 ","End":"01:27.390","Text":"we\u0027re bigger than 2 it means we\u0027re here,"},{"Start":"01:27.390 ","End":"01:29.340","Text":"and if we put in 2,"},{"Start":"01:29.340 ","End":"01:32.360","Text":"2 squared plus 8 times 2,"},{"Start":"01:32.360 ","End":"01:35.065","Text":"that gives us 20."},{"Start":"01:35.065 ","End":"01:38.790","Text":"In fact, we can do the last 1 also."},{"Start":"01:38.790 ","End":"01:43.670","Text":"The last 1 also will be 20 because it goes from the same definition."},{"Start":"01:43.670 ","End":"01:46.985","Text":"Let\u0027s see, if x goes to 2 from the right,"},{"Start":"01:46.985 ","End":"01:49.000","Text":"x has got to be in here,"},{"Start":"01:49.000 ","End":"01:52.730","Text":"and if we plug in x equals 2,"},{"Start":"01:52.730 ","End":"01:54.395","Text":"we get 2 cubed plus 12,"},{"Start":"01:54.395 ","End":"01:57.140","Text":"8 plus 12, which is also 20."},{"Start":"01:57.140 ","End":"02:00.530","Text":"Yes, 20 equals 20 equals 20."},{"Start":"02:00.530 ","End":"02:05.160","Text":"Very good, so we are continuous. There\u0027s hope."},{"Start":"02:05.160 ","End":"02:11.000","Text":"Now, we need 1 more condition that when we take this formula for the derivative,"},{"Start":"02:11.000 ","End":"02:13.050","Text":"which does not include 2,"},{"Start":"02:13.050 ","End":"02:18.185","Text":"if we have the limit as x going to 2 from the left and from the right being equal,"},{"Start":"02:18.185 ","End":"02:21.195","Text":"then we\u0027re in good shape and we get all these goodies."},{"Start":"02:21.195 ","End":"02:23.840","Text":"Let\u0027s see what happens;"},{"Start":"02:23.840 ","End":"02:26.910","Text":"if the limit,"},{"Start":"02:27.210 ","End":"02:30.290","Text":"I\u0027ll scroll a bit,"},{"Start":"02:31.300 ","End":"02:38.290","Text":"as x goes to 2 from the right of f prime of x"},{"Start":"02:38.290 ","End":"02:45.370","Text":"is equal to the limit as x goes to 2 from the left of f prime of x,"},{"Start":"02:45.370 ","End":"02:46.950","Text":"then we\u0027re in good shape."},{"Start":"02:46.950 ","End":"02:50.305","Text":"Meanwhile, I\u0027ll put a question mark here and just say,"},{"Start":"02:50.305 ","End":"02:53.030","Text":"does this equal this?"},{"Start":"02:53.340 ","End":"02:58.210","Text":"We take the 1 from the right and we\u0027ll see if we"},{"Start":"02:58.210 ","End":"03:02.635","Text":"can maybe do that in our head, maybe not."},{"Start":"03:02.635 ","End":"03:04.540","Text":"From the right,"},{"Start":"03:04.540 ","End":"03:08.360","Text":"we need 2x plus 8."},{"Start":"03:08.690 ","End":"03:14.505","Text":"Let me just see, here we have the limit"},{"Start":"03:14.505 ","End":"03:24.015","Text":"as x goes to 2 from the right of 2x plus 8."},{"Start":"03:24.015 ","End":"03:32.280","Text":"Is this equal to the limit when x goes to 2 from"},{"Start":"03:32.280 ","End":"03:35.115","Text":"the left of"},{"Start":"03:35.115 ","End":"03:42.720","Text":"3x squared?"},{"Start":"03:42.720 ","End":"03:47.640","Text":"Well, let\u0027s see, there should be a simple to just substitute;"},{"Start":"03:47.640 ","End":"03:52.815","Text":"2 times 2 is 4 plus 8 is 12,"},{"Start":"03:52.815 ","End":"03:57.315","Text":"and 3 times 2 squared is 3 times 4 is 12."},{"Start":"03:57.315 ","End":"03:59.565","Text":"Does 12 equal 12?"},{"Start":"03:59.565 ","End":"04:02.200","Text":"Well, I say, yes."},{"Start":"04:02.510 ","End":"04:06.225","Text":"We\u0027ve satisfied these 2 conditions,"},{"Start":"04:06.225 ","End":"04:08.990","Text":"now we get the benefit of the then."},{"Start":"04:08.990 ","End":"04:12.460","Text":"First of all, f is differentiable at x equals 2,"},{"Start":"04:12.460 ","End":"04:14.760","Text":"and we already know it\u0027s differentiable everywhere else,"},{"Start":"04:14.760 ","End":"04:17.180","Text":"so it\u0027s differentiable everywhere."},{"Start":"04:17.180 ","End":"04:20.795","Text":"But we also know what the derivative at 2 is equal to."},{"Start":"04:20.795 ","End":"04:23.945","Text":"It\u0027s equal to this common value where these 2 things were equal."},{"Start":"04:23.945 ","End":"04:26.300","Text":"It\u0027s equal to either this 12 or this 12,"},{"Start":"04:26.300 ","End":"04:27.560","Text":"whichever you want,"},{"Start":"04:27.560 ","End":"04:29.225","Text":"but it\u0027s equal to 12,"},{"Start":"04:29.225 ","End":"04:32.750","Text":"which means that this mystery here,"},{"Start":"04:32.750 ","End":"04:36.985","Text":"I can now replace it the 3 question marks with a 12."},{"Start":"04:36.985 ","End":"04:38.890","Text":"Now, here it goes."},{"Start":"04:38.890 ","End":"04:43.369","Text":"There we are, there\u0027s no more question marks. The answer is 12."},{"Start":"04:43.369 ","End":"04:45.125","Text":"I would say we\u0027re finished,"},{"Start":"04:45.125 ","End":"04:47.960","Text":"but we only did it using the theorem and I mentioned that"},{"Start":"04:47.960 ","End":"04:51.290","Text":"since we\u0027re practicing and learning,"},{"Start":"04:51.290 ","End":"04:53.140","Text":"we\u0027re going to do it by both methods."},{"Start":"04:53.140 ","End":"04:55.340","Text":"This was with the theorem method."},{"Start":"04:55.340 ","End":"04:58.140","Text":"Then next thing we\u0027ll do is"},{"Start":"04:58.140 ","End":"05:05.020","Text":"we\u0027ll do this method via the definition."},{"Start":"05:11.140 ","End":"05:16.650","Text":"Which means that what we have to do is to check"},{"Start":"05:16.960 ","End":"05:25.930","Text":"what is the limit as x,"},{"Start":"05:25.930 ","End":"05:29.160","Text":"sorry not x, pardon me."},{"Start":"05:29.160 ","End":"05:31.485","Text":"In fact, let me start the line again."},{"Start":"05:31.485 ","End":"05:35.645","Text":"If there is a derivative at x equals 2,"},{"Start":"05:35.645 ","End":"05:38.405","Text":"then its value is going to be the limit,"},{"Start":"05:38.405 ","End":"05:41.010","Text":"if such a limit exists,"},{"Start":"05:41.420 ","End":"05:51.045","Text":"as h goes to 0 of f of 2 plus h minus f of"},{"Start":"05:51.045 ","End":"05:56.300","Text":"2 all over h. Here, we"},{"Start":"05:56.300 ","End":"06:01.700","Text":"come to a slight problem because h goes to 0,"},{"Start":"06:01.700 ","End":"06:04.160","Text":"we don\u0027t know if h is positive or negative."},{"Start":"06:04.160 ","End":"06:05.660","Text":"Now, why is that important?"},{"Start":"06:05.660 ","End":"06:09.765","Text":"Because if h is positive, then this x,"},{"Start":"06:09.765 ","End":"06:12.405","Text":"which is 2 plus h is bigger than 2,"},{"Start":"06:12.405 ","End":"06:15.640","Text":"and we take the definition from 1 place,"},{"Start":"06:15.640 ","End":"06:18.285","Text":"which is from here."},{"Start":"06:18.285 ","End":"06:20.405","Text":"But if h is negative,"},{"Start":"06:20.405 ","End":"06:25.400","Text":"then 2 plus h is less than 2 and we have to use this definition."},{"Start":"06:25.400 ","End":"06:30.970","Text":"We\u0027re not sure whether to use x squared plus 8x or x cubed plus 12."},{"Start":"06:30.970 ","End":"06:32.975","Text":"I\u0027ll just write those here."},{"Start":"06:32.975 ","End":"06:38.240","Text":"There was x. I just made a small note to"},{"Start":"06:38.240 ","End":"06:40.850","Text":"myself at the side just because I don\u0027t"},{"Start":"06:40.850 ","End":"06:44.530","Text":"want to have to keep scrolling up and down constantly."},{"Start":"06:44.730 ","End":"06:51.610","Text":"We have to figure out separately the limit on the left and the limit on the right."},{"Start":"06:51.610 ","End":"07:00.130","Text":"The limit as h goes to 0 plus of f of"},{"Start":"07:00.130 ","End":"07:08.245","Text":"2 plus h minus f of 2 over h is going to equal."},{"Start":"07:08.245 ","End":"07:09.520","Text":"Now, on the left,"},{"Start":"07:09.520 ","End":"07:12.730","Text":"we are x squared plus 8x."},{"Start":"07:12.730 ","End":"07:16.220","Text":"We\u0027ll take x squared plus 8x for this,"},{"Start":"07:16.220 ","End":"07:20.325","Text":"2 plus h squared plus 8,"},{"Start":"07:20.325 ","End":"07:22.275","Text":"2 plus h,"},{"Start":"07:22.275 ","End":"07:25.875","Text":"minus what happens when it\u0027s 2,"},{"Start":"07:25.875 ","End":"07:32.830","Text":"which is just 2 squared plus 8 times 2."},{"Start":"07:33.620 ","End":"07:37.960","Text":"Oh, I forgot to write the limit here,"},{"Start":"07:37.960 ","End":"07:39.700","Text":"I\u0027ll write in tiny."},{"Start":"07:39.700 ","End":"07:42.925","Text":"Limit and the same as there,"},{"Start":"07:42.925 ","End":"07:51.170","Text":"all over h. Now, if we expand this,"},{"Start":"07:51.930 ","End":"07:55.570","Text":"we will get the limit."},{"Start":"07:55.570 ","End":"08:01.075","Text":"This time, I have room to write h goes to 0 plus of, let\u0027s see,"},{"Start":"08:01.075 ","End":"08:07.330","Text":"2 plus h squared is 4 plus 4,"},{"Start":"08:07.330 ","End":"08:10.960","Text":"h plus h squared."},{"Start":"08:10.960 ","End":"08:19.705","Text":"Here we have plus 16 plus 8h minus,"},{"Start":"08:19.705 ","End":"08:22.240","Text":"this whole thing, 2 squared is 4,"},{"Start":"08:22.240 ","End":"08:23.335","Text":"16 is 20,"},{"Start":"08:23.335 ","End":"08:30.820","Text":"minus 20, all over h. Let\u0027s see now."},{"Start":"08:30.820 ","End":"08:33.245","Text":"4 and 16 is 20,"},{"Start":"08:33.245 ","End":"08:35.515","Text":"cancels with the 20."},{"Start":"08:35.515 ","End":"08:40.675","Text":"If we take h outside the brackets here,"},{"Start":"08:40.675 ","End":"08:44.215","Text":"we divide the top by h,"},{"Start":"08:44.215 ","End":"08:51.580","Text":"we get the limit of"},{"Start":"08:51.580 ","End":"09:04.840","Text":"4 plus h"},{"Start":"09:04.840 ","End":"09:11.320","Text":"plus 8h,"},{"Start":"09:11.320 ","End":"09:13.295","Text":"which is over h is 8."},{"Start":"09:13.295 ","End":"09:20.750","Text":"It looks h goes to 0 plus of h plus 12."},{"Start":"09:21.330 ","End":"09:24.880","Text":"If h goes to 0, then it\u0027s just 4 plus 8,"},{"Start":"09:24.880 ","End":"09:26.195","Text":"it\u0027s just the 12."},{"Start":"09:26.195 ","End":"09:30.400","Text":"The answer is 12 for the limit on the right."},{"Start":"09:31.290 ","End":"09:34.355","Text":"Now, let\u0027s see what happens with the limit on the left."},{"Start":"09:34.355 ","End":"09:37.735","Text":"I see we\u0027re just crossing a page here."},{"Start":"09:37.735 ","End":"09:46.500","Text":"The limit as h goes to 0 from the left of f of"},{"Start":"09:46.500 ","End":"09:55.910","Text":"2 plus h minus f of 2 all over h is going to equal."},{"Start":"09:58.640 ","End":"10:02.255","Text":"Previously, we had x squared plus 8x,"},{"Start":"10:02.255 ","End":"10:09.565","Text":"now, we\u0027re going to have x cubed plus 12."},{"Start":"10:09.565 ","End":"10:12.850","Text":"If we have x cubed plus 12,"},{"Start":"10:12.850 ","End":"10:22.975","Text":"then it means that we have 2 plus h cubed, just 3,"},{"Start":"10:22.975 ","End":"10:28.415","Text":"cubed plus 12 minus,"},{"Start":"10:28.415 ","End":"10:30.485","Text":"and just from the 2,"},{"Start":"10:30.485 ","End":"10:36.800","Text":"2 cubed plus 12,"},{"Start":"10:37.760 ","End":"10:43.500","Text":"all over h. Oh, and silly me,"},{"Start":"10:43.500 ","End":"10:45.275","Text":"again, forgot to write the limit,"},{"Start":"10:45.275 ","End":"10:46.630","Text":"write it in small,"},{"Start":"10:46.630 ","End":"10:49.255","Text":"h goes to 0 plus."},{"Start":"10:49.255 ","End":"10:58.925","Text":"Very well this equals the limit h goes to 0 minus."},{"Start":"10:58.925 ","End":"11:05.960","Text":"Now, we come to some formula from algebra which you may have forgotten,"},{"Start":"11:05.960 ","End":"11:08.660","Text":"probably have, may actually remember it."},{"Start":"11:08.660 ","End":"11:16.255","Text":"That is, that a plus b to the power of 3 is equal,"},{"Start":"11:16.255 ","End":"11:18.150","Text":"and this comes from Newton\u0027s binomial,"},{"Start":"11:18.150 ","End":"11:19.270","Text":"if you\u0027ve heard of that, if not,"},{"Start":"11:19.270 ","End":"11:20.959","Text":"just take this as a formula,"},{"Start":"11:20.959 ","End":"11:31.745","Text":"a cubed plus 3a squared b plus 3ab squared plus b cubed."},{"Start":"11:31.745 ","End":"11:39.525","Text":"Going back here, we get 2 plus h cubed is,"},{"Start":"11:39.525 ","End":"11:47.700","Text":"2 cubed is 8,"},{"Start":"11:47.700 ","End":"11:55.030","Text":"plus 3a squared is 3 times 2 squared, which is 12."},{"Start":"11:55.740 ","End":"12:03.935","Text":"With a b, that\u0027s an h. Then 3ab squared is 3 times 2 is 6."},{"Start":"12:03.935 ","End":"12:05.990","Text":"But h squared."},{"Start":"12:05.990 ","End":"12:09.445","Text":"Finally, h cubed."},{"Start":"12:09.445 ","End":"12:11.200","Text":"That\u0027s this part."},{"Start":"12:11.200 ","End":"12:14.320","Text":"Then plus 12 from here."},{"Start":"12:14.320 ","End":"12:21.100","Text":"Then minus, this bit here we can just do in our heads,"},{"Start":"12:21.100 ","End":"12:23.320","Text":"2 cubed is 8 plus, well,"},{"Start":"12:23.320 ","End":"12:27.470","Text":"I\u0027ll just write it as 8 plus 12 anyway."},{"Start":"12:28.230 ","End":"12:33.490","Text":"All this over h, which equals."},{"Start":"12:33.490 ","End":"12:41.245","Text":"Now, let\u0027s see, 8 and 12 go with the minus 8 and the 12."},{"Start":"12:41.245 ","End":"12:45.220","Text":"Everything else divides by h. What we\u0027re left with is"},{"Start":"12:45.220 ","End":"12:54.485","Text":"12 plus 6h plus h squared."},{"Start":"12:54.485 ","End":"12:58.600","Text":"Oh and again I forgot to write the limit."},{"Start":"12:58.600 ","End":"13:05.005","Text":"Limit as h goes to 0."},{"Start":"13:05.005 ","End":"13:08.710","Text":"If h goes to 0 these 2 terms are 0,"},{"Start":"13:08.710 ","End":"13:12.535","Text":"so we\u0027re left with just the 12."},{"Start":"13:12.535 ","End":"13:17.110","Text":"Now, here we have 12 and here we have 12."},{"Start":"13:17.730 ","End":"13:21.665","Text":"What I asked myself is,"},{"Start":"13:21.665 ","End":"13:27.100","Text":"well, does 12 equals 12? Now, yes."},{"Start":"13:27.100 ","End":"13:30.430","Text":"The left limit and the right limit are equal,"},{"Start":"13:30.430 ","End":"13:34.535","Text":"which means that if I take the 2-sided limit,"},{"Start":"13:34.535 ","End":"13:37.715","Text":"this will now also equal 12."},{"Start":"13:37.715 ","End":"13:41.905","Text":"If the left side and the right side are equal and that\u0027s the limit and it\u0027s 12."},{"Start":"13:41.905 ","End":"13:45.640","Text":"In fact, 12 is exactly the number we got in"},{"Start":"13:45.640 ","End":"13:49.575","Text":"the previous exercise when we said what is f prime of 2,"},{"Start":"13:49.575 ","End":"13:55.105","Text":"where f prime of 2 means x equals 2 here."},{"Start":"13:55.105 ","End":"13:57.115","Text":"We get the same answer both ways."},{"Start":"13:57.115 ","End":"13:58.790","Text":"That\u0027s very encouraging."},{"Start":"13:58.790 ","End":"14:01.580","Text":"Anyway, we\u0027re done with this exercise."},{"Start":"14:01.580 ","End":"14:03.510","Text":"In summary to the question,"},{"Start":"14:03.510 ","End":"14:09.100","Text":"there are no places at which f is not differentiable, it\u0027s differentiable everywhere."},{"Start":"14:09.100 ","End":"14:13.460","Text":"The derivative at 2 is 12. Done."}],"ID":1483},{"Watched":false,"Name":"Exercise 2 Part 1 - Via theorem","Duration":"7m 22s","ChapterTopicVideoID":8270,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we have to find out where f is not"},{"Start":"00:03.930 ","End":"00:10.725","Text":"differentiable and we have to write a formula for the derivative of f prime of x."},{"Start":"00:10.725 ","End":"00:13.305","Text":"Let\u0027s begin with number 1 and here,"},{"Start":"00:13.305 ","End":"00:19.110","Text":"f is piecewise defined and it\u0027s defined for x bigger or equal to 0 in"},{"Start":"00:19.110 ","End":"00:26.310","Text":"one way and it\u0027s defined for x less than 0 and bigger than minus 0.5 in another way."},{"Start":"00:26.310 ","End":"00:27.585","Text":"Could have 3 cases."},{"Start":"00:27.585 ","End":"00:34.485","Text":"Either x is bigger than 0 or we could have x equals 0 and that\u0027s the big question mark."},{"Start":"00:34.485 ","End":"00:38.190","Text":"Or we could have x less than 0,"},{"Start":"00:38.190 ","End":"00:42.210","Text":"less important, but I\u0027ll write it anyway minus a 0.5."},{"Start":"00:42.210 ","End":"00:46.095","Text":"For x bigger than 0, f of x is just a polynomial."},{"Start":"00:46.095 ","End":"00:48.555","Text":"This is definitely differentiable."},{"Start":"00:48.555 ","End":"00:50.025","Text":"For x equals 0,"},{"Start":"00:50.025 ","End":"00:53.130","Text":"well, that\u0027s what we\u0027re going to try and find out."},{"Start":"00:53.130 ","End":"00:55.335","Text":"I\u0027m putting a question mark meanwhile."},{"Start":"00:55.335 ","End":"00:57.454","Text":"For x in this area,"},{"Start":"00:57.454 ","End":"01:01.100","Text":"I claim that it is differentiable and it follows from"},{"Start":"01:01.100 ","End":"01:04.730","Text":"the fact that we could almost generalize it as a function."},{"Start":"01:04.730 ","End":"01:09.185","Text":"Let\u0027s take h of x equals 1 plus 2x."},{"Start":"01:09.185 ","End":"01:13.625","Text":"Now in general, I claim that if we have a function,"},{"Start":"01:13.625 ","End":"01:15.215","Text":"not necessarily this one,"},{"Start":"01:15.215 ","End":"01:21.290","Text":"if h of x is positive in its domain and differentiable,"},{"Start":"01:21.290 ","End":"01:28.500","Text":"then the natural logarithm of h of x is differentiable too."},{"Start":"01:28.500 ","End":"01:30.130","Text":"I\u0027ll just write is too."},{"Start":"01:30.130 ","End":"01:37.190","Text":"The reason is because the composition of 2 differentiable functions is differentiable."},{"Start":"01:37.190 ","End":"01:42.470","Text":"H is differentiable and natural log is differentiable."},{"Start":"01:42.470 ","End":"01:46.670","Text":"The only thing is that we have to worry about is that it has to even be defined because"},{"Start":"01:46.670 ","End":"01:51.920","Text":"the natural logarithm is only defined for the argument to be bigger than 0."},{"Start":"01:51.920 ","End":"01:55.100","Text":"I need to make sure h of x is bigger than 0."},{"Start":"01:55.100 ","End":"01:59.675","Text":"Now, y is h of x bigger than 0, in our case,"},{"Start":"01:59.675 ","End":"02:05.820","Text":"minus 0.5 is less than x so,"},{"Start":"02:05.820 ","End":"02:08.895","Text":"if you multiply by 2,"},{"Start":"02:08.895 ","End":"02:14.010","Text":"we get minus 1 is less than 2x."},{"Start":"02:14.010 ","End":"02:16.200","Text":"Then if you add 1,"},{"Start":"02:16.200 ","End":"02:21.375","Text":"then you get 0 is less than 2x plus 1,"},{"Start":"02:21.375 ","End":"02:28.820","Text":"but that equals our h of x. H is bigger than 0 and differentiable so therefore,"},{"Start":"02:28.820 ","End":"02:32.635","Text":"natural logarithm of 1 plus 2x is also differentiable."},{"Start":"02:32.635 ","End":"02:35.885","Text":"We can put a check mark here."},{"Start":"02:35.885 ","End":"02:39.335","Text":"The only problem is what happens when x equals 0?"},{"Start":"02:39.335 ","End":"02:41.690","Text":"Here\u0027s where we pull out a theorem."},{"Start":"02:41.690 ","End":"02:45.030","Text":"Let\u0027s take number 1 here on the if clause,"},{"Start":"02:45.030 ","End":"02:51.125","Text":"we have to show that f is continuous at the point where in this case x equals 0."},{"Start":"02:51.125 ","End":"02:54.425","Text":"For f to be continuous at x equals 0,"},{"Start":"02:54.425 ","End":"03:00.530","Text":"the definition of the continuity says that we have to have that the limit on"},{"Start":"03:00.530 ","End":"03:08.285","Text":"the right of the function has got to be equal to the limit on the left of the function."},{"Start":"03:08.285 ","End":"03:12.890","Text":"It\u0027s also got to equal the value of the function at the point."},{"Start":"03:12.890 ","End":"03:16.400","Text":"Then we know that f is continuous at 0."},{"Start":"03:16.400 ","End":"03:21.240","Text":"The limit on the right means that we take from here and"},{"Start":"03:21.240 ","End":"03:26.990","Text":"so we substitute 0 squared plus twice 0 is 0."},{"Start":"03:26.990 ","End":"03:29.540","Text":"For the left, we take from here,"},{"Start":"03:29.540 ","End":"03:31.310","Text":"put x equals 0 here,"},{"Start":"03:31.310 ","End":"03:33.050","Text":"1 plus twice 0 is 1,"},{"Start":"03:33.050 ","End":"03:35.850","Text":"natural log of 1 is 0."},{"Start":"03:35.850 ","End":"03:38.205","Text":"The same thing for 0 itself,"},{"Start":"03:38.205 ","End":"03:39.230","Text":"we take from here,"},{"Start":"03:39.230 ","End":"03:40.760","Text":"which is this 0."},{"Start":"03:40.760 ","End":"03:45.290","Text":"In any event, 0 equals 0, equals 0."},{"Start":"03:45.290 ","End":"03:54.005","Text":"We\u0027re okay. We are continuous at x equals 0."},{"Start":"03:54.005 ","End":"03:55.580","Text":"That\u0027s the first step."},{"Start":"03:55.580 ","End":"04:01.490","Text":"Now, the second step is to show that this limit holds."},{"Start":"04:01.490 ","End":"04:07.510","Text":"I\u0027m going to put 3 cases."},{"Start":"04:07.510 ","End":"04:12.140","Text":"I\u0027m going to put the x equals 0 case here because that\u0027s what we\u0027re trying to"},{"Start":"04:12.140 ","End":"04:17.435","Text":"find out and we absolutely don\u0027t know when x equals 0."},{"Start":"04:17.435 ","End":"04:21.880","Text":"However, when x is between 0.5 and 0,"},{"Start":"04:21.880 ","End":"04:24.334","Text":"we have the derivative of this."},{"Start":"04:24.334 ","End":"04:29.690","Text":"If we differentiate this natural logarithm means 1 over this,"},{"Start":"04:29.690 ","End":"04:32.030","Text":"internal derivative is 2."},{"Start":"04:32.030 ","End":"04:37.475","Text":"What we get is 2 over 1 plus 2x."},{"Start":"04:37.475 ","End":"04:43.660","Text":"This is for the case where minus 0.5 is less than x,"},{"Start":"04:43.660 ","End":"04:46.605","Text":"is less than 0."},{"Start":"04:46.605 ","End":"04:52.385","Text":"The final case where x is bigger than 0, just differentiate this."},{"Start":"04:52.385 ","End":"04:54.680","Text":"That\u0027s 2x plus 2."},{"Start":"04:54.680 ","End":"05:00.470","Text":"It\u0027s 2x plus 2 when x is bigger than 0."},{"Start":"05:00.470 ","End":"05:04.390","Text":"Now, I need f prime of x because I want to show the number 2,"},{"Start":"05:04.390 ","End":"05:06.830","Text":"so we have to show that the limit on the left,"},{"Start":"05:06.830 ","End":"05:11.330","Text":"the limit on the right of the derivative function are equal."},{"Start":"05:11.330 ","End":"05:13.340","Text":"What we needed to do, I should have written it with"},{"Start":"05:13.340 ","End":"05:14.970","Text":"question marks because that\u0027s what we\u0027re"},{"Start":"05:14.970 ","End":"05:19.535","Text":"checking and likewise here we will be asking the question,"},{"Start":"05:19.535 ","End":"05:26.900","Text":"is the limit as x goes to 0 on the right of f prime of x."},{"Start":"05:26.900 ","End":"05:28.460","Text":"Is it equal?"},{"Start":"05:28.460 ","End":"05:32.400","Text":"I\u0027ll put a question mark here to the limit as"},{"Start":"05:32.400 ","End":"05:37.430","Text":"x goes to 0 from the left also of f prime of x."},{"Start":"05:37.430 ","End":"05:39.380","Text":"We\u0027ve done showing number 1."},{"Start":"05:39.380 ","End":"05:42.500","Text":"Now, I\u0027m going to show number 2 and then we\u0027ll get all these benefits."},{"Start":"05:42.500 ","End":"05:45.620","Text":"Once again, I think we can do this in our heads."},{"Start":"05:45.620 ","End":"05:47.840","Text":"If x goes to 0 from above,"},{"Start":"05:47.840 ","End":"05:50.645","Text":"then we\u0027re in the case x bigger than 0,"},{"Start":"05:50.645 ","End":"05:57.115","Text":"and we substitute 0 and we get twice 0 plus 2 is 2."},{"Start":"05:57.115 ","End":"05:59.059","Text":"If we go from below,"},{"Start":"05:59.059 ","End":"06:06.800","Text":"then we\u0027re going to be using this definition and we get 2 over 1 plus twice 0 is also 2."},{"Start":"06:06.800 ","End":"06:08.990","Text":"Does 2 equal 2?"},{"Start":"06:08.990 ","End":"06:10.720","Text":"Well, yes."},{"Start":"06:10.720 ","End":"06:14.380","Text":"We\u0027ve met both those conditions and now,"},{"Start":"06:14.380 ","End":"06:20.410","Text":"we can conclude the part that says then that f is differentiable at x equals 0."},{"Start":"06:20.410 ","End":"06:24.250","Text":"More than that, that we know that the derivative of"},{"Start":"06:24.250 ","End":"06:28.705","Text":"f at 0 is equal to these 2 common values, which is 2."},{"Start":"06:28.705 ","End":"06:36.030","Text":"In other words, f is differentiable at x equals"},{"Start":"06:36.030 ","End":"06:44.235","Text":"0 and we know that f prime of 0 is equal to 2."},{"Start":"06:44.235 ","End":"06:46.180","Text":"Now that we know this,"},{"Start":"06:46.180 ","End":"06:52.145","Text":"we can actually give the formula for f prime because the mystery is no longer a mystery."},{"Start":"06:52.145 ","End":"06:56.060","Text":"What I can do is replace this by the number 2 there."},{"Start":"06:56.060 ","End":"07:01.760","Text":"Now, we can shorten this a little bit because notice that if you put x equals 0 here,"},{"Start":"07:01.760 ","End":"07:03.260","Text":"you also get 2,"},{"Start":"07:03.260 ","End":"07:06.215","Text":"so we could combine these cases and make"},{"Start":"07:06.215 ","End":"07:09.620","Text":"this greater or equal to and get rid of this line."},{"Start":"07:09.620 ","End":"07:13.730","Text":"What I mean is that from here we could get something like this."},{"Start":"07:13.730 ","End":"07:15.470","Text":"Just put the greater or equal to,"},{"Start":"07:15.470 ","End":"07:18.530","Text":"got rid of the middle line and that answers the questions to"},{"Start":"07:18.530 ","End":"07:23.160","Text":"write down the formula for the derivative function."}],"ID":8432},{"Watched":false,"Name":"Exercise 2 Part 1 - Via definition","Duration":"4m 14s","ChapterTopicVideoID":8269,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"What we\u0027re going to do is go to the definition."},{"Start":"00:02.565 ","End":"00:06.480","Text":"We\u0027re going to figure out what f prime of 0 is,"},{"Start":"00:06.480 ","End":"00:09.555","Text":"that\u0027s what we were worried about, using the definition."},{"Start":"00:09.555 ","End":"00:15.150","Text":"One way to do it is just do it directly and see what is f prime of 0,"},{"Start":"00:15.150 ","End":"00:17.550","Text":"and that is equal to."},{"Start":"00:17.550 ","End":"00:23.100","Text":"Now, the problem in computing this is that we have this expression f of h here."},{"Start":"00:23.100 ","End":"00:26.960","Text":"How do we know what f of h is when h is near 0?"},{"Start":"00:26.960 ","End":"00:28.400","Text":"Because if it\u0027s slightly over 0,"},{"Start":"00:28.400 ","End":"00:29.590","Text":"we\u0027re using this formula."},{"Start":"00:29.590 ","End":"00:31.065","Text":"If it\u0027s slightly under 0,"},{"Start":"00:31.065 ","End":"00:32.475","Text":"then we\u0027re using this formula."},{"Start":"00:32.475 ","End":"00:35.705","Text":"What we\u0027ll do is compute separately the left and right limits,"},{"Start":"00:35.705 ","End":"00:37.640","Text":"and if they\u0027re equal, then that\u0027s the limit."},{"Start":"00:37.640 ","End":"00:40.520","Text":"Let\u0027s take the right limit first."},{"Start":"00:40.520 ","End":"00:46.835","Text":"The right limit would be the limit as h goes to 0 from the right,"},{"Start":"00:46.835 ","End":"00:50.270","Text":"f of h minus f of 0 over h."},{"Start":"00:50.270 ","End":"00:53.750","Text":"Then put f from using this formula."},{"Start":"00:53.750 ","End":"00:56.660","Text":"We have h squared plus 2h."},{"Start":"00:56.660 ","End":"00:58.750","Text":"If you put 0 into here,"},{"Start":"00:58.750 ","End":"01:07.370","Text":"it\u0027s less 0 squared plus twice 0, all this over h."},{"Start":"01:07.370 ","End":"01:11.350","Text":"Now what you get here is just h squared plus 2h."},{"Start":"01:11.350 ","End":"01:12.580","Text":"This is nothing."},{"Start":"01:12.580 ","End":"01:14.565","Text":"If we divide that by h,"},{"Start":"01:14.565 ","End":"01:17.715","Text":"h squared plus 2h becomes h plus 2."},{"Start":"01:17.715 ","End":"01:20.060","Text":"The limit of h plus 2."},{"Start":"01:20.060 ","End":"01:24.310","Text":"When h goes to 0, this is just equal to 2."},{"Start":"01:24.310 ","End":"01:26.585","Text":"Now let\u0027s do it from the left."},{"Start":"01:26.585 ","End":"01:30.440","Text":"So we need the limit as h goes to 0 from the left."},{"Start":"01:30.440 ","End":"01:35.929","Text":"Again, f of h minus f of 0 over h,"},{"Start":"01:35.929 ","End":"01:41.855","Text":"which equals the limit as h goes to 0 from the left."},{"Start":"01:41.855 ","End":"01:46.330","Text":"This time the function is going to be natural log of 1 plus 2x."},{"Start":"01:46.330 ","End":"01:55.530","Text":"It\u0027s natural logarithm of 1 plus twice h minus the same thing,"},{"Start":"01:55.530 ","End":"01:56.860","Text":"but for x equals 0,"},{"Start":"01:56.860 ","End":"02:05.420","Text":"minus the natural log of 1 plus twice 0 and all this over h."},{"Start":"02:05.420 ","End":"02:12.095","Text":"What we get if we simplify all this is the limit h goes to 0,"},{"Start":"02:12.095 ","End":"02:16.940","Text":"the natural logarithm of 1 plus 2h."},{"Start":"02:16.940 ","End":"02:20.430","Text":"This twice 0 is 0 plus 1 is 1,"},{"Start":"02:20.430 ","End":"02:23.880","Text":"natural logarithm of 1 is 0."},{"Start":"02:23.880 ","End":"02:25.785","Text":"There is nothing left here."},{"Start":"02:25.785 ","End":"02:33.030","Text":"It\u0027s just this thing over h. If we put h equals 0 or 0 minus in this,"},{"Start":"02:33.030 ","End":"02:35.735","Text":"we get the expression of 0 over 0."},{"Start":"02:35.735 ","End":"02:39.050","Text":"We have here at L\u0027Hopital of 0 over 0."},{"Start":"02:39.050 ","End":"02:41.120","Text":"Let\u0027s apply L\u0027Hopital\u0027s Rule,"},{"Start":"02:41.120 ","End":"02:43.650","Text":"which is equal to limit."},{"Start":"02:43.650 ","End":"02:46.730","Text":"Well, equals have to put an L at the bottom"},{"Start":"02:46.730 ","End":"02:50.550","Text":"and indicate that it\u0027s the 0 over 0 case,"},{"Start":"02:50.550 ","End":"02:54.200","Text":"and then we\u0027re applying L\u0027Hopital at the same thing here."},{"Start":"02:54.200 ","End":"02:56.690","Text":"But we now differentiate top and bottom."},{"Start":"02:56.690 ","End":"02:59.690","Text":"The denominator is easier to differentiate. That\u0027s just 1."},{"Start":"02:59.690 ","End":"03:03.890","Text":"The top, natural logarithm of something is 1 over that something,"},{"Start":"03:03.890 ","End":"03:08.640","Text":"it\u0027s 1 over 1 plus 2h times"},{"Start":"03:08.640 ","End":"03:13.635","Text":"the anti-derivative from the chain rule of 1 plus 2h is just 2."},{"Start":"03:13.635 ","End":"03:15.690","Text":"The 1 disappears and 2h gives 2,"},{"Start":"03:15.690 ","End":"03:17.945","Text":"so multiplied by 2."},{"Start":"03:17.945 ","End":"03:22.880","Text":"Basically what that gives us is the limit 2 over 1 plus 2h."},{"Start":"03:22.880 ","End":"03:26.615","Text":"At this point, we can just directly substitute h."},{"Start":"03:26.615 ","End":"03:31.750","Text":"2h is 0, we get 2 over 1, and this equals 2."},{"Start":"03:31.750 ","End":"03:36.455","Text":"We see that this limit on the right equals this limit on the left."},{"Start":"03:36.455 ","End":"03:43.660","Text":"That means that this f prime of 0 also exists and must equal also 2."},{"Start":"03:43.660 ","End":"03:45.510","Text":"That\u0027s the answer we got before."},{"Start":"03:45.510 ","End":"03:48.170","Text":"That means that f is differentiable at 0."},{"Start":"03:48.170 ","End":"03:51.095","Text":"Then the rest of it is the same as it was before,"},{"Start":"03:51.095 ","End":"03:55.025","Text":"that once we had the thing was equal to 0 by the theorem,"},{"Start":"03:55.025 ","End":"03:56.735","Text":"and then we got the formula."},{"Start":"03:56.735 ","End":"04:00.665","Text":"The only difference was in computing the differentiability."},{"Start":"04:00.665 ","End":"04:04.460","Text":"In one case, we got the theorem to get f prime of 0 equals 2,"},{"Start":"04:04.460 ","End":"04:05.840","Text":"and the second time, we"},{"Start":"04:05.840 ","End":"04:11.210","Text":"got that f prime of 0 equals 2 from the definition directly."},{"Start":"04:11.210 ","End":"04:13.310","Text":"We\u0027ve got the same answer in both cases,"},{"Start":"04:13.310 ","End":"04:15.390","Text":"and now we\u0027re done."}],"ID":8431},{"Watched":false,"Name":"Exercise 2 Part 2 - via Theorem","Duration":"11m 50s","ChapterTopicVideoID":1498,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"Next is number 2,"},{"Start":"00:03.075 ","End":"00:06.840","Text":"and I\u0027ve copied number 2 from here."},{"Start":"00:06.840 ","End":"00:14.700","Text":"This is where f of x is 2 plus 4 times the absolute value of x minus 1."},{"Start":"00:15.020 ","End":"00:19.480","Text":"Let\u0027s scroll down. I\u0027ve copied the exercise."},{"Start":"00:20.150 ","End":"00:23.010","Text":"It\u0027s better if we get rid of"},{"Start":"00:23.010 ","End":"00:29.294","Text":"the absolute value and write it as a split definition function."},{"Start":"00:29.294 ","End":"00:34.560","Text":"We can\u0027t differentiate absolute value,"},{"Start":"00:34.560 ","End":"00:35.925","Text":"it\u0027s harder to deal with,"},{"Start":"00:35.925 ","End":"00:39.010","Text":"so it\u0027s better to write it as follows."},{"Start":"00:39.010 ","End":"00:47.400","Text":"That f of x is equal to 2 plus."},{"Start":"00:47.400 ","End":"00:54.100","Text":"Now, what we do here is we use a definition of absolute value."},{"Start":"00:54.100 ","End":"00:55.720","Text":"Let\u0027s use a different color."},{"Start":"00:55.720 ","End":"01:00.160","Text":"Let\u0027s say absolute value of some number, not x."},{"Start":"01:00.160 ","End":"01:02.140","Text":"Let\u0027s choose a,"},{"Start":"01:02.140 ","End":"01:05.065","Text":"in general is equal to,"},{"Start":"01:05.065 ","End":"01:06.969","Text":"it\u0027s just the a itself."},{"Start":"01:06.969 ","End":"01:10.505","Text":"If a is bigger than 0,"},{"Start":"01:10.505 ","End":"01:14.490","Text":"in fact, it even could equal 0."},{"Start":"01:14.490 ","End":"01:16.850","Text":"That\u0027s like if a is 7,"},{"Start":"01:16.850 ","End":"01:18.535","Text":"absolute value of a is 7,"},{"Start":"01:18.535 ","End":"01:20.950","Text":"but if a was minus 7,"},{"Start":"01:20.950 ","End":"01:23.290","Text":"the absolute value of a would be plus 7,"},{"Start":"01:23.290 ","End":"01:25.195","Text":"which is actually minus a,"},{"Start":"01:25.195 ","End":"01:27.425","Text":"so it\u0027s equal to minus a,"},{"Start":"01:27.425 ","End":"01:30.615","Text":"when a is less than 0."},{"Start":"01:30.615 ","End":"01:34.980","Text":"It could have been less than or equal to but we put the equal with this 1."},{"Start":"01:34.980 ","End":"01:40.245","Text":"In this case, we have to write,"},{"Start":"01:40.245 ","End":"01:45.195","Text":"better, even start off writing it separately."},{"Start":"01:45.195 ","End":"01:47.650","Text":"What I\u0027ll do is this,"},{"Start":"01:48.200 ","End":"01:53.910","Text":"erase that 2 plus and"},{"Start":"01:53.910 ","End":"02:00.940","Text":"just write it as 1 curly bracket with 2 definitions."}],"ID":1479},{"Watched":false,"Name":"Exercise 2 Part 2 - via Definition","Duration":"2m 59s","ChapterTopicVideoID":1497,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"Now, we\u0027re going to use the definition to see if we"},{"Start":"00:02.760 ","End":"00:06.135","Text":"can find out what happens at x equals 1."},{"Start":"00:06.135 ","End":"00:09.270","Text":"If it\u0027s differentiable at x equals 1,"},{"Start":"00:09.270 ","End":"00:13.110","Text":"we can find the derivative there by using the definition that"},{"Start":"00:13.110 ","End":"00:18.000","Text":"f prime of 1 is equal to the limit as h goes to"},{"Start":"00:18.000 ","End":"00:27.370","Text":"0 of f of 1 plus h minus f of 1 all over h. Now, where\u0027s the problem?"},{"Start":"00:27.370 ","End":"00:32.495","Text":"I\u0027ll tell you where. This h can be either positive or negative."},{"Start":"00:32.495 ","End":"00:34.670","Text":"If we look at f of x here,"},{"Start":"00:34.670 ","End":"00:38.180","Text":"or rather here, then if h is slightly positive,"},{"Start":"00:38.180 ","End":"00:40.760","Text":"our x is going to be slightly bigger than 1,"},{"Start":"00:40.760 ","End":"00:44.100","Text":"so we\u0027re going to be taking the definition from here."},{"Start":"00:44.100 ","End":"00:46.430","Text":"If h is a bit less than 0,"},{"Start":"00:46.430 ","End":"00:48.650","Text":"then x is a bit less than 1,"},{"Start":"00:48.650 ","End":"00:51.095","Text":"we\u0027ll be taking our definition from here."},{"Start":"00:51.095 ","End":"00:53.780","Text":"What we\u0027d better do to find the limit is to find"},{"Start":"00:53.780 ","End":"00:56.360","Text":"the left limit and the right limit and if they\u0027re equal,"},{"Start":"00:56.360 ","End":"00:58.550","Text":"and that\u0027s the limit but if not, they\u0027re not."},{"Start":"00:58.550 ","End":"01:00.050","Text":"Let\u0027s work on 1 of them,"},{"Start":"01:00.050 ","End":"01:03.125","Text":"let\u0027s say the right limit,"},{"Start":"01:03.125 ","End":"01:09.290","Text":"the limit as h goes to 0 from the right of the same thing."},{"Start":"01:09.290 ","End":"01:11.495","Text":"Since it is the same thing,"},{"Start":"01:11.495 ","End":"01:13.340","Text":"then I\u0027ll just throw it away,"},{"Start":"01:13.340 ","End":"01:15.905","Text":"use from the right with the working here."},{"Start":"01:15.905 ","End":"01:18.860","Text":"We need 4x minus 2 and this is our x."},{"Start":"01:18.860 ","End":"01:25.835","Text":"We need 4,1 plus h minus 2 minus f of 1,"},{"Start":"01:25.835 ","End":"01:30.830","Text":"which is 4 of 1 minus 2 all"},{"Start":"01:30.830 ","End":"01:36.770","Text":"over h pulls the limit as h goes to 0,"},{"Start":"01:36.770 ","End":"01:38.270","Text":"let\u0027s see if we can work it all out."},{"Start":"01:38.270 ","End":"01:42.690","Text":"It\u0027s 4 h\u0027s from here and let\u0027s see how many free numbers we have,"},{"Start":"01:42.690 ","End":"01:48.180","Text":"4 minus 2 minus 4 plus 2."},{"Start":"01:48.180 ","End":"01:54.375","Text":"This comes out to be 0 and this is over h."},{"Start":"01:54.375 ","End":"01:57.590","Text":"The h with the h cancels so we just have"},{"Start":"01:57.590 ","End":"02:01.780","Text":"the limit of 4 and the limit of a constant is just that constant."},{"Start":"02:01.780 ","End":"02:05.540","Text":"Now, I\u0027m going to just save you some time because if you do"},{"Start":"02:05.540 ","End":"02:09.830","Text":"the same thing for 6 minus 4x and do this substitution,"},{"Start":"02:09.830 ","End":"02:11.495","Text":"it also comes out,"},{"Start":"02:11.495 ","End":"02:14.440","Text":"everything cancels and because of the minus 4,"},{"Start":"02:14.440 ","End":"02:22.340","Text":"we\u0027ll get that the limit as h goes to 0 from the left of f of 1 of whatever it is,"},{"Start":"02:22.340 ","End":"02:30.710","Text":"f of 1 plus h minus f of 1 over h is going to equal that."},{"Start":"02:30.710 ","End":"02:32.060","Text":"By the same techniques,"},{"Start":"02:32.060 ","End":"02:40.130","Text":"we\u0027ll get minus 4 and since 4 is obviously not equal to minus 4,"},{"Start":"02:40.130 ","End":"02:43.360","Text":"then there is no limit."},{"Start":"02:43.360 ","End":"02:46.790","Text":"This limit does not exist and hence f is not"},{"Start":"02:46.790 ","End":"02:50.600","Text":"differentiable at x equals 1 and that\u0027s what we had before,"},{"Start":"02:50.600 ","End":"02:54.560","Text":"but we just did it differently with the definition instead of with the theorem."},{"Start":"02:54.560 ","End":"03:00.510","Text":"We\u0027re done with this part 2 of 3 of this question."}],"ID":1478},{"Watched":false,"Name":"Exercise 2 Part 3 - Via theorem","Duration":"6m 12s","ChapterTopicVideoID":8265,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"We\u0027ve arrived at number 3 and I\u0027ve rewritten it."},{"Start":"00:04.185 ","End":"00:06.200","Text":"What we have to do is the same as before,"},{"Start":"00:06.200 ","End":"00:08.640","Text":"find where f is non-differentiable and where it"},{"Start":"00:08.640 ","End":"00:11.820","Text":"is write the formula for the derivative function."},{"Start":"00:11.820 ","End":"00:15.420","Text":"Here we noticed that there is an absolute value of x."},{"Start":"00:15.420 ","End":"00:17.110","Text":"With this question,"},{"Start":"00:17.110 ","End":"00:19.110","Text":"the absolute values it\u0027s better to deal with"},{"Start":"00:19.110 ","End":"00:21.810","Text":"the function as a piecewise defined function,"},{"Start":"00:21.810 ","End":"00:24.000","Text":"just like absolute value of x is."},{"Start":"00:24.000 ","End":"00:29.460","Text":"If you remember, absolute value of x is equal to 1 of 2 different things."},{"Start":"00:29.460 ","End":"00:31.440","Text":"If a positive, everything is okay."},{"Start":"00:31.440 ","End":"00:37.185","Text":"Absolute value is just x itself when x is positive or even when x is 0,"},{"Start":"00:37.185 ","End":"00:39.555","Text":"and what remains is less than 0."},{"Start":"00:39.555 ","End":"00:42.440","Text":"Here, absolute value of x is equal to minus x."},{"Start":"00:42.440 ","End":"00:46.010","Text":"For example, absolute value of minus 7 is plus 7,"},{"Start":"00:46.010 ","End":"00:47.945","Text":"which is minus of minus 7."},{"Start":"00:47.945 ","End":"00:49.670","Text":"If we apply that here,"},{"Start":"00:49.670 ","End":"00:54.920","Text":"we can also write f of x as a piecewise-defined function without absolute value."},{"Start":"00:54.920 ","End":"00:58.585","Text":"So we have f of x is equal to,"},{"Start":"00:58.585 ","End":"01:00.285","Text":"putting the 2 cases,"},{"Start":"01:00.285 ","End":"01:04.880","Text":"1 of them is x bigger or equal to 0 and 1 x less than 0."},{"Start":"01:04.880 ","End":"01:09.230","Text":"I\u0027ll have to do is substitute the right form of the absolute value of x in each."},{"Start":"01:09.230 ","End":"01:12.935","Text":"For this case, we put x and for the other case, we put minus x."},{"Start":"01:12.935 ","End":"01:20.730","Text":"In other words, we just copy this as 3x squared plus x times just x itself plus 1."},{"Start":"01:20.730 ","End":"01:27.660","Text":"Here, 3x squared plus x times minus x plus 1."},{"Start":"01:27.660 ","End":"01:29.555","Text":"If we simplify all this,"},{"Start":"01:29.555 ","End":"01:35.990","Text":"what we get is that f of x is equal to 4x squared plus 1 for"},{"Start":"01:35.990 ","End":"01:44.015","Text":"x bigger or equal to 0 or 2x squared plus 1 for x less than 0."},{"Start":"01:44.015 ","End":"01:46.910","Text":"It\u0027s easy to see that if f is not 0,"},{"Start":"01:46.910 ","End":"01:48.170","Text":"0 is the border line,"},{"Start":"01:48.170 ","End":"01:50.375","Text":"the scene point, or whatever."},{"Start":"01:50.375 ","End":"01:52.990","Text":"At 0, something funny happens, the formula changes."},{"Start":"01:52.990 ","End":"01:57.065","Text":"We don\u0027t know about x equals 0 because those situations are indeterminate."},{"Start":"01:57.065 ","End":"02:00.110","Text":"But we do know that for x bigger than 0,"},{"Start":"02:00.110 ","End":"02:02.830","Text":"4x squared is differentiable."},{"Start":"02:02.830 ","End":"02:08.240","Text":"I can even write that as the f prime of x is equal to,"},{"Start":"02:08.240 ","End":"02:14.365","Text":"when x is bigger than 0, the derivative of this is 8x so it\u0027s just 8x."},{"Start":"02:14.365 ","End":"02:19.100","Text":"4x for x less than 0,"},{"Start":"02:19.100 ","End":"02:21.095","Text":"the only question is,"},{"Start":"02:21.095 ","End":"02:23.420","Text":"what happens when x equals 0?"},{"Start":"02:23.420 ","End":"02:25.820","Text":"Everything changes that the seam point."},{"Start":"02:25.820 ","End":"02:28.430","Text":"So I\u0027ll leave 3 question marks here for the moment,"},{"Start":"02:28.430 ","End":"02:30.050","Text":"and that\u0027s what we\u0027re going to find out."},{"Start":"02:30.050 ","End":"02:32.885","Text":"Now we have 2 ways of continuing from here."},{"Start":"02:32.885 ","End":"02:34.955","Text":"There is a theorem that helps us,"},{"Start":"02:34.955 ","End":"02:36.880","Text":"and generally the theorem is easier,"},{"Start":"02:36.880 ","End":"02:40.010","Text":"and there\u0027s also differentiation from the definition."},{"Start":"02:40.010 ","End":"02:41.330","Text":"The former is easy,"},{"Start":"02:41.330 ","End":"02:43.640","Text":"the latter is used less often,"},{"Start":"02:43.640 ","End":"02:46.405","Text":"but for good practice will do both."},{"Start":"02:46.405 ","End":"02:49.175","Text":"Hopefully, they\u0027ll give us the same conclusions."},{"Start":"02:49.175 ","End":"02:54.905","Text":"Let\u0027s first of all do it using the theorem which I\u0027ve already pre-written."},{"Start":"02:54.905 ","End":"02:59.000","Text":"If f is continuous at x equals a, in our case,"},{"Start":"02:59.000 ","End":"03:03.110","Text":"a is 0 and the limit from the right and the limit of"},{"Start":"03:03.110 ","End":"03:08.555","Text":"the left of f prime are equal as x tends to 0 in our case,"},{"Start":"03:08.555 ","End":"03:09.900","Text":"then we get 2 things."},{"Start":"03:09.900 ","End":"03:12.800","Text":"That f is differentiable at that point, x equals a,"},{"Start":"03:12.800 ","End":"03:15.410","Text":"and the value of the derivative at"},{"Start":"03:15.410 ","End":"03:19.080","Text":"that point is equal to that common value which we saw here."},{"Start":"03:19.080 ","End":"03:20.840","Text":"Is equal to this or this,"},{"Start":"03:20.840 ","End":"03:23.690","Text":"because they\u0027re both be the same. Let\u0027s get on with it."},{"Start":"03:23.690 ","End":"03:25.805","Text":"I\u0027ve underlined the word continuous,"},{"Start":"03:25.805 ","End":"03:27.260","Text":"that\u0027s the first thing we\u0027ll check,"},{"Start":"03:27.260 ","End":"03:29.435","Text":"and then we\u0027ll check this equality here."},{"Start":"03:29.435 ","End":"03:33.200","Text":"For the continuity, what we do is we"},{"Start":"03:33.200 ","End":"03:36.710","Text":"show that the limit from the left and the right of the function,"},{"Start":"03:36.710 ","End":"03:39.020","Text":"not the derivative of the function are equal and they\u0027re"},{"Start":"03:39.020 ","End":"03:42.350","Text":"also equal to the value of the function at the point."},{"Start":"03:42.350 ","End":"03:45.920","Text":"By the way, the domain of definition of f is all of x."},{"Start":"03:45.920 ","End":"03:47.299","Text":"On with the continuity,"},{"Start":"03:47.299 ","End":"03:50.810","Text":"we need to see that the limit as x goes to 0,"},{"Start":"03:50.810 ","End":"03:54.350","Text":"let\u0027s say from the right first is equal to"},{"Start":"03:54.350 ","End":"03:59.780","Text":"the limit as x goes to 0 from the left of the function,"},{"Start":"03:59.780 ","End":"04:04.025","Text":"and is also equal to the value of f at the point itself, which is 0."},{"Start":"04:04.025 ","End":"04:07.250","Text":"F of x, when x goes to 0 from the right,"},{"Start":"04:07.250 ","End":"04:08.810","Text":"will be gotten from here."},{"Start":"04:08.810 ","End":"04:11.610","Text":"So just put in 0 and you\u0027ll get the answer,"},{"Start":"04:11.610 ","End":"04:13.115","Text":"this is equal to 4."},{"Start":"04:13.115 ","End":"04:15.800","Text":"Forehand that should have really written it with a question mark."},{"Start":"04:15.800 ","End":"04:17.500","Text":"This is what I want to check."},{"Start":"04:17.500 ","End":"04:20.735","Text":"Let\u0027s see if it really is true. What have I done?"},{"Start":"04:20.735 ","End":"04:24.335","Text":"I\u0027ve substituted the wrong value, terribly sorry."},{"Start":"04:24.335 ","End":"04:26.345","Text":"When we substitute 0 here,"},{"Start":"04:26.345 ","End":"04:29.735","Text":"we also get twice 0 plus 1 which is 1,"},{"Start":"04:29.735 ","End":"04:33.605","Text":"and f of 0 itself from here we can get it, x equals 0."},{"Start":"04:33.605 ","End":"04:37.220","Text":"Again, 4 times 0 plus 1 is again 1."},{"Start":"04:37.220 ","End":"04:38.930","Text":"This is equal to this,"},{"Start":"04:38.930 ","End":"04:39.950","Text":"which is equal to this."},{"Start":"04:39.950 ","End":"04:41.030","Text":"The answer is yes,"},{"Start":"04:41.030 ","End":"04:42.680","Text":"and we are continuous."},{"Start":"04:42.680 ","End":"04:44.870","Text":"That\u0027s the first part. But that\u0027s not enough."},{"Start":"04:44.870 ","End":"04:46.700","Text":"You have to also check that this thing is true."},{"Start":"04:46.700 ","End":"04:52.715","Text":"The second thing is the limit as x goes to 0 from the right is"},{"Start":"04:52.715 ","End":"04:59.200","Text":"equal to the limit as x goes to 0 from the left of f prime of x,"},{"Start":"04:59.200 ","End":"05:01.400","Text":"and I\u0027ll put a question mark above here."},{"Start":"05:01.400 ","End":"05:02.825","Text":"That\u0027s what we want to know."},{"Start":"05:02.825 ","End":"05:07.700","Text":"Well, let\u0027s see. We have the formula for f prime of x."},{"Start":"05:07.700 ","End":"05:11.915","Text":"If we take the limit from the right means x goes to 0,"},{"Start":"05:11.915 ","End":"05:14.670","Text":"so we get 0."},{"Start":"05:14.670 ","End":"05:16.965","Text":"If we put x going from the left,"},{"Start":"05:16.965 ","End":"05:18.705","Text":"we put 0 in here,"},{"Start":"05:18.705 ","End":"05:20.655","Text":"4x is also 0."},{"Start":"05:20.655 ","End":"05:24.175","Text":"In other words, this thing computes to 0."},{"Start":"05:24.175 ","End":"05:25.860","Text":"This thing computes to 0."},{"Start":"05:25.860 ","End":"05:27.780","Text":"They are indeed equal."},{"Start":"05:27.780 ","End":"05:30.650","Text":"We\u0027ve got another V. Now we can get"},{"Start":"05:30.650 ","End":"05:35.585","Text":"the conclusions that f is differentiable at x equals a."},{"Start":"05:35.585 ","End":"05:43.700","Text":"Furthermore, we know that f prime of 0 is equal to this common value is equal to 0."},{"Start":"05:43.700 ","End":"05:45.875","Text":"Basically, we\u0027ve answered the question,"},{"Start":"05:45.875 ","End":"05:49.640","Text":"almost f is differentiable everywhere even at 0,"},{"Start":"05:49.640 ","End":"05:52.600","Text":"so we don\u0027t have any point of non-differentiability."},{"Start":"05:52.600 ","End":"05:58.240","Text":"Furthermore, the formula for the derivative of f prime is equal to this, this,"},{"Start":"05:58.240 ","End":"06:03.050","Text":"and at 0, we now know that it\u0027s equal to 0,"},{"Start":"06:03.050 ","End":"06:06.500","Text":"and there it is, and that\u0027s the formula for f prime."},{"Start":"06:06.500 ","End":"06:08.570","Text":"We\u0027re almost done. We just have to use"},{"Start":"06:08.570 ","End":"06:12.990","Text":"the definition of the derivative to solve the same problem."}],"ID":8427},{"Watched":false,"Name":"Exercise 2 Part 3 - Via definition","Duration":"2m 51s","ChapterTopicVideoID":8264,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.695","Text":"Well, we just found out what happens in exercise number 3."},{"Start":"00:04.695 ","End":"00:07.600","Text":"When x is 0, what happens to the derivative?"},{"Start":"00:07.600 ","End":"00:09.180","Text":"The answer was that it\u0027s 0."},{"Start":"00:09.180 ","End":"00:10.725","Text":"We did this using the theorem."},{"Start":"00:10.725 ","End":"00:13.710","Text":"Now we\u0027re going to do it again with the definition."},{"Start":"00:13.710 ","End":"00:18.915","Text":"We\u0027re back to the point where we don\u0027t know what happens at x equals 0."},{"Start":"00:18.915 ","End":"00:22.355","Text":"We\u0027re going to use the definition of the derivative,"},{"Start":"00:22.355 ","End":"00:25.160","Text":"in this case that x equals 0,and what we get"},{"Start":"00:25.160 ","End":"00:28.340","Text":"using the definition is that the derivative at"},{"Start":"00:28.340 ","End":"00:34.970","Text":"0 is equal to the limit as h goes to 0 of f,"},{"Start":"00:34.970 ","End":"00:42.290","Text":"in general a plus h. But a in our case is 0 minus f of a,"},{"Start":"00:42.290 ","End":"00:47.915","Text":"which is f of 0 all over h. The problem with this is,"},{"Start":"00:47.915 ","End":"00:50.090","Text":"is that f of h,"},{"Start":"00:50.090 ","End":"00:52.655","Text":"it\u0027s equal to either this or this,"},{"Start":"00:52.655 ","End":"00:57.485","Text":"depending on when h is bigger or equal to 0 or h is less than 0,"},{"Start":"00:57.485 ","End":"00:59.030","Text":"and it\u0027s just bigger than 0."},{"Start":"00:59.030 ","End":"01:00.800","Text":"We don\u0027t take h equals 0."},{"Start":"01:00.800 ","End":"01:02.570","Text":"We\u0027ll get to different things."},{"Start":"01:02.570 ","End":"01:07.595","Text":"What we\u0027d better do is take the left limit and then the right limit separately."},{"Start":"01:07.595 ","End":"01:12.455","Text":"If they\u0027re equal, then we\u0027re all good and we have a limit as h goes to 0 here."},{"Start":"01:12.455 ","End":"01:20.600","Text":"Limit h goes to 0 on the right of f of h. Now h is bigger than 0,"},{"Start":"01:20.600 ","End":"01:24.305","Text":"so we\u0027re going to be using the top formula,"},{"Start":"01:24.305 ","End":"01:26.755","Text":"and it\u0027s 4h squared plus 1,"},{"Start":"01:26.755 ","End":"01:29.145","Text":"his just x in our case,"},{"Start":"01:29.145 ","End":"01:31.360","Text":"so it\u0027s 4h squared,"},{"Start":"01:31.360 ","End":"01:34.970","Text":"0 plus h of course I can write as just h everywhere."},{"Start":"01:34.970 ","End":"01:41.910","Text":"It\u0027s 4h squared plus 1 minus 0 squared plus 1."},{"Start":"01:41.910 ","End":"01:44.900","Text":"One time I\u0027m putting in 0 plus h or h,"},{"Start":"01:44.900 ","End":"01:50.675","Text":"and onetime I\u0027m putting in just 0 itself over h. Now,"},{"Start":"01:50.675 ","End":"01:52.205","Text":"if we figure this out,"},{"Start":"01:52.205 ","End":"01:54.740","Text":"0 squared plus 1 is just 1,"},{"Start":"01:54.740 ","End":"01:56.645","Text":"and 1 and the 1 cancel,"},{"Start":"01:56.645 ","End":"02:00.470","Text":"so 4h over h is just equal to 4."},{"Start":"02:00.470 ","End":"02:04.565","Text":"The limit is just 4h only 1 of the h\u0027s cancels."},{"Start":"02:04.565 ","End":"02:08.150","Text":"Now that certainly goes to 0,"},{"Start":"02:08.150 ","End":"02:11.870","Text":"write the limit as h goes to 0 plus."},{"Start":"02:11.870 ","End":"02:19.415","Text":"Similarly limit as h goes to 0 minus using now this formula 2 x squared plus 1."},{"Start":"02:19.415 ","End":"02:25.140","Text":"We get twice h squared plus 1 minus the same 0"},{"Start":"02:25.140 ","End":"02:31.605","Text":"squared plus 1 all over h. Just like we got 4h before,"},{"Start":"02:31.605 ","End":"02:34.710","Text":"we get 2h and the limit of 2h,"},{"Start":"02:34.710 ","End":"02:36.765","Text":"and this goes also to 0."},{"Start":"02:36.765 ","End":"02:39.680","Text":"When h goes to 0 even from the right."},{"Start":"02:39.680 ","End":"02:45.619","Text":"All together because the left limit and the right limit gave us the same answer,"},{"Start":"02:45.619 ","End":"02:48.455","Text":"then we can write that this limit exists,"},{"Start":"02:48.455 ","End":"02:52.350","Text":"and is also equal to 0."}],"ID":8426},{"Watched":false,"Name":"Exercise 3 - Part 1","Duration":"15m 1s","ChapterTopicVideoID":1484,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"This pair of exercises,"},{"Start":"00:02.070 ","End":"00:06.660","Text":"we have to find points where f is not differentiable and if possible,"},{"Start":"00:06.660 ","End":"00:08.835","Text":"to find the derivative of f at 0."},{"Start":"00:08.835 ","End":"00:11.130","Text":"Well, let\u0027s start with the first 1."},{"Start":"00:11.130 ","End":"00:13.080","Text":"Well it\u0027s too wet methods to go about it."},{"Start":"00:13.080 ","End":"00:17.625","Text":"We have the method using the theorem and the method straight from the definition."},{"Start":"00:17.625 ","End":"00:20.805","Text":"Usually we go for the theorem because it\u0027s easier."},{"Start":"00:20.805 ","End":"00:23.040","Text":"But I can tell you in advance in this example,"},{"Start":"00:23.040 ","End":"00:27.045","Text":"we\u0027re going to have a failure with the theorem until be forced to use the definition."},{"Start":"00:27.045 ","End":"00:30.060","Text":"But let\u0027s start naively as if we don\u0027t know anything."},{"Start":"00:30.060 ","End":"00:31.649","Text":"Here\u0027s the theorem."},{"Start":"00:31.649 ","End":"00:32.790","Text":"We\u0027re familiar with it."},{"Start":"00:32.790 ","End":"00:34.980","Text":"If f satisfies 2 conditions, A,"},{"Start":"00:34.980 ","End":"00:37.820","Text":"that it\u0027s continuous at the point and that"},{"Start":"00:37.820 ","End":"00:41.060","Text":"the left limit and right limit of the derivatives are equal,"},{"Start":"00:41.060 ","End":"00:43.430","Text":"then we know that f is differentiable at"},{"Start":"00:43.430 ","End":"00:46.595","Text":"the point and we even have a formula for the derivative."},{"Start":"00:46.595 ","End":"00:50.765","Text":"I\u0027m going over this quickly because we\u0027re going to fail. I know this."},{"Start":"00:50.765 ","End":"00:57.230","Text":"The first thing we would do would be to show that f is continuous at x equals a,"},{"Start":"00:57.230 ","End":"00:59.879","Text":"which in our case is 0."},{"Start":"01:01.300 ","End":"01:03.890","Text":"Well, let\u0027s go back a little bit."},{"Start":"01:03.890 ","End":"01:13.215","Text":"What we do know is that f is differentiable when x is less than 0."},{"Start":"01:13.215 ","End":"01:20.300","Text":"In fact it\u0027s differentiable when x is bigger than 0 because 1 over x is simple function,"},{"Start":"01:20.300 ","End":"01:22.010","Text":"a primitive it\u0027s differentiable."},{"Start":"01:22.010 ","End":"01:24.950","Text":"The sine function is differentiable."},{"Start":"01:24.950 ","End":"01:32.285","Text":"The composition of 2 differentiable functions is differentiable,"},{"Start":"01:32.285 ","End":"01:36.170","Text":"what I mean, the reciprocal of the sine and the product of 2 differentiable."},{"Start":"01:36.170 ","End":"01:37.925","Text":"Basically, there\u0027s no problem."},{"Start":"01:37.925 ","End":"01:40.090","Text":"We can even compute f prime,"},{"Start":"01:40.090 ","End":"01:44.700","Text":"f prime of x is equal to,"},{"Start":"01:44.700 ","End":"01:49.630","Text":"well, it certainly 0 when x is less than 0."},{"Start":"01:50.500 ","End":"01:53.540","Text":"The emphasis is on x equals 0."},{"Start":"01:53.540 ","End":"01:58.030","Text":"We\u0027re going to investigate when x equals 0."},{"Start":"01:58.030 ","End":"02:02.920","Text":"For x bigger than 0, we have a formula."},{"Start":"02:12.370 ","End":"02:15.110","Text":"By the product rule,"},{"Start":"02:15.110 ","End":"02:17.330","Text":"we get the derivative of the first,"},{"Start":"02:17.330 ","End":"02:26.930","Text":"which is 1 times the 2nd as it is sine of 1/x plus x as it is,"},{"Start":"02:26.930 ","End":"02:28.790","Text":"times the derivative of the 2nd,"},{"Start":"02:28.790 ","End":"02:33.410","Text":"which is cosine of 1/x,"},{"Start":"02:33.410 ","End":"02:39.000","Text":"but times the internal derivative times minus 1 over x squared."},{"Start":"02:41.080 ","End":"02:45.845","Text":"Now, what we have to do is,"},{"Start":"02:45.845 ","End":"02:48.700","Text":"leave that alone for the moment."},{"Start":"02:48.830 ","End":"02:54.175","Text":"We\u0027ll just go and say why f is continuous?"},{"Start":"02:54.175 ","End":"03:04.840","Text":"F is continuous because we have to check that the f of 0"},{"Start":"03:04.840 ","End":"03:11.260","Text":"is equal to the limit as x goes to 0 plus of f of x is equal"},{"Start":"03:11.260 ","End":"03:17.980","Text":"to the limit x goes to 0 minus f of x."},{"Start":"03:17.980 ","End":"03:20.380","Text":"If these 2 things hold,"},{"Start":"03:20.380 ","End":"03:21.910","Text":"then we have continuity."},{"Start":"03:21.910 ","End":"03:26.330","Text":"F of 0 is certainly 0."},{"Start":"03:28.700 ","End":"03:34.770","Text":"The limit as f goes to 0 from the left is also 0."},{"Start":"03:34.770 ","End":"03:37.625","Text":"If f goes to 0 from the right,"},{"Start":"03:37.625 ","End":"03:42.500","Text":"then what we have is sine of 1/x is bounded."},{"Start":"03:42.500 ","End":"03:49.070","Text":"Cosine is always between minus 1 and 1 and x goes to 0 when x goes to 0."},{"Start":"03:49.070 ","End":"03:52.480","Text":"0 times something bounded is also 0."},{"Start":"03:52.480 ","End":"03:55.690","Text":"We do indeed have 0 equals 0 equals 0."},{"Start":"03:55.690 ","End":"03:57.680","Text":"We do have continuous."},{"Start":"03:57.680 ","End":"03:59.540","Text":"But now let\u0027s go and check."},{"Start":"03:59.540 ","End":"04:02.580","Text":"The 2cd thing is where we\u0027re going to fall."},{"Start":"04:03.410 ","End":"04:05.900","Text":"What we have to do,"},{"Start":"04:05.900 ","End":"04:10.670","Text":"and just going to copy f prime of x, maybe not."},{"Start":"04:10.670 ","End":"04:13.265","Text":"I\u0027ll just go up and get it each time."},{"Start":"04:13.265 ","End":"04:21.440","Text":"Let\u0027s see now. What we need to check here is that"},{"Start":"04:21.440 ","End":"04:29.870","Text":"the limit on the left or to the right 1 first,"},{"Start":"04:29.870 ","End":"04:37.880","Text":"the limit as x goes to 0 on the right of sine 1/x."},{"Start":"04:37.880 ","End":"04:44.090","Text":"Put it in brackets. Sine 1/x minus 1/x,"},{"Start":"04:44.090 ","End":"04:52.600","Text":"cosine 1/x. That\u0027s the limit from"},{"Start":"04:52.600 ","End":"04:55.210","Text":"the right has got to equal the limit from"},{"Start":"04:55.210 ","End":"04:59.845","Text":"the left. Yeah, there we are."},{"Start":"04:59.845 ","End":"05:05.275","Text":"Limit from the right equals the limit from the left has got to equal question mark."},{"Start":"05:05.275 ","End":"05:08.875","Text":"Limit on the left is just 0."},{"Start":"05:08.875 ","End":"05:18.840","Text":"Well, just write it as limit as x goes to 0 from the left of 0."},{"Start":"05:18.840 ","End":"05:22.270","Text":"That\u0027s the question. What is this limit equal to?"},{"Start":"05:22.270 ","End":"05:24.505","Text":"In actual fact, there is no limit."},{"Start":"05:24.505 ","End":"05:26.980","Text":"But to show you this more easily,"},{"Start":"05:26.980 ","End":"05:32.485","Text":"perhaps a substitution, t equals 1/x."},{"Start":"05:32.485 ","End":"05:34.795","Text":"If t is 1/x,"},{"Start":"05:34.795 ","End":"05:39.260","Text":"then x goes to 0 plus that\u0027s a substitution."},{"Start":"05:41.930 ","End":"05:47.510","Text":"We substitute and the limit to which is infinity."},{"Start":"05:47.510 ","End":"05:51.160","Text":"Limit t goes to infinity."},{"Start":"05:51.160 ","End":"05:54.850","Text":"You see if x goes to 0, positive 0,"},{"Start":"05:54.850 ","End":"05:58.945","Text":"t keeps getting larger and larger 1 over something very small and very large."},{"Start":"05:58.945 ","End":"06:02.150","Text":"It\u0027s plus infinity of"},{"Start":"06:02.310 ","End":"06:11.890","Text":"sine t minus t cosine t as t goes to infinity."},{"Start":"06:11.890 ","End":"06:13.615","Text":"Now what is this limit?"},{"Start":"06:13.615 ","End":"06:18.400","Text":"Well, actually it doesn\u0027t exist."},{"Start":"06:18.930 ","End":"06:23.525","Text":"I\u0027ll just write it doesn\u0027t exist."},{"Start":"06:23.525 ","End":"06:33.475","Text":"I\u0027ll explain why it does not exist."},{"Start":"06:33.475 ","End":"06:37.445","Text":"Made a mess here. Never mind does not exist."},{"Start":"06:37.445 ","End":"06:43.535","Text":"I\u0027ll show you an example of values of t you could plug in that would show it."},{"Start":"06:43.535 ","End":"06:49.675","Text":"Let\u0027s suppose we let t is equal to multiples of Pi say n Pi,"},{"Start":"06:49.675 ","End":"06:53.340","Text":"where n goes from 1,"},{"Start":"06:53.340 ","End":"06:56.935","Text":"2, 3, 4, etc."},{"Start":"06:56.935 ","End":"07:00.590","Text":"T would eventually get to infinity this way if I have 4,"},{"Start":"07:00.590 ","End":"07:01.610","Text":"Pi 5, Pi 6,"},{"Start":"07:01.610 ","End":"07:03.875","Text":"Pi 7, Pi 8, Pi and so on."},{"Start":"07:03.875 ","End":"07:07.100","Text":"Let\u0027s see what is this thing?"},{"Start":"07:07.100 ","End":"07:08.660","Text":"This function of t,"},{"Start":"07:08.660 ","End":"07:10.675","Text":"what\u0027s in the square brackets."},{"Start":"07:10.675 ","End":"07:12.575","Text":"If t is this,"},{"Start":"07:12.575 ","End":"07:19.175","Text":"then sine t minus t cosine t for these t will equal."},{"Start":"07:19.175 ","End":"07:22.249","Text":"Now if t is a multiple of Pi,"},{"Start":"07:22.249 ","End":"07:24.620","Text":"then sine t is 0."},{"Start":"07:24.620 ","End":"07:28.610","Text":"On 0, 180 degrees to 360 degrees and so on."},{"Start":"07:28.610 ","End":"07:32.015","Text":"The sine is 0. If you look at its graph, that\u0027s from 0."},{"Start":"07:32.015 ","End":"07:34.230","Text":"This part is 0."},{"Start":"07:36.180 ","End":"07:40.930","Text":"T is equal to n Pi"},{"Start":"07:40.930 ","End":"07:48.850","Text":"and cosine t is"},{"Start":"07:48.850 ","End":"07:54.880","Text":"equal to either 1 or minus 1 depending on where n is even or odd."},{"Start":"07:54.880 ","End":"07:58.090","Text":"For example, the cosine of 0 is 1,"},{"Start":"07:58.090 ","End":"08:01.570","Text":"but the cosine of Pi is minus 1."},{"Start":"08:01.570 ","End":"08:03.940","Text":"Then the cosine of 2 Pi is 1,"},{"Start":"08:03.940 ","End":"08:06.370","Text":"cosine of 3 Pi is minus 1."},{"Start":"08:06.370 ","End":"08:14.350","Text":"Altogether, we get that it\u0027s plus or minus this."},{"Start":"08:19.100 ","End":"08:22.620","Text":"It\u0027s actually equal more precisely."},{"Start":"08:22.620 ","End":"08:28.600","Text":"We don\u0027t need it more precisely just that it oscillates between plus Pi,"},{"Start":"08:28.600 ","End":"08:31.600","Text":"minus 2 Pi, plus 3 Pi, or the other way round."},{"Start":"08:31.600 ","End":"08:33.340","Text":"I can write it exactly that,"},{"Start":"08:33.340 ","End":"08:40.180","Text":"if n is odd, then for example,"},{"Start":"08:40.180 ","End":"08:47.020","Text":"n equals Pi, then we have cosine of Pi is minus 1,"},{"Start":"08:47.020 ","End":"08:49.885","Text":"we get plus n Pi."},{"Start":"08:49.885 ","End":"08:56.440","Text":"It\u0027s n Pi when n is odd and it\u0027s minus n Pi when n is even."},{"Start":"08:56.440 ","End":"08:59.140","Text":"Basically what we get here is a sequence."},{"Start":"08:59.140 ","End":"09:01.750","Text":"If we plug in t of these values,"},{"Start":"09:01.750 ","End":"09:11.545","Text":"we get 1 Pi minus 2 Pi plus 3 Pi minus 4 Pi,"},{"Start":"09:11.545 ","End":"09:13.735","Text":"5 Pi, etc,"},{"Start":"09:13.735 ","End":"09:15.280","Text":"and obviously this has no limit."},{"Start":"09:15.280 ","End":"09:21.460","Text":"It oscillates wildly getting larger positive and even larger negative and so on."},{"Start":"09:21.460 ","End":"09:24.610","Text":"This shows that this doesn\u0027t have a limit because there\u0027s"},{"Start":"09:24.610 ","End":"09:26.950","Text":"a theorem that if it had a limit and you"},{"Start":"09:26.950 ","End":"09:36.370","Text":"could go through a sequence of values that go into infinity."},{"Start":"09:36.370 ","End":"09:38.635","Text":"Anyway, I\u0027m not going to get it bogged down by this."},{"Start":"09:38.635 ","End":"09:40.240","Text":"It doesn\u0027t exist this limit."},{"Start":"09:40.240 ","End":"09:41.845","Text":"Now we\u0027re really stuck,"},{"Start":"09:41.845 ","End":"09:43.975","Text":"at least if we\u0027re using the theorem."},{"Start":"09:43.975 ","End":"09:46.315","Text":"We really have to use the definition."},{"Start":"09:46.315 ","End":"09:55.430","Text":"Let me just go back and erase this stuff and we\u0027ll work from the definition."},{"Start":"09:55.440 ","End":"10:00.230","Text":"Before I do that, perhaps I should clarify what\u0027s going on here."},{"Start":"10:00.510 ","End":"10:05.635","Text":"What happens is that when we checking this equality,"},{"Start":"10:05.635 ","End":"10:08.815","Text":"if both these limits exist,"},{"Start":"10:08.815 ","End":"10:10.795","Text":"whether they\u0027re finite or infinite,"},{"Start":"10:10.795 ","End":"10:12.670","Text":"and they are not equal,"},{"Start":"10:12.670 ","End":"10:15.130","Text":"then f is not differentiable."},{"Start":"10:15.130 ","End":"10:17.440","Text":"But if 1 of them doesn\u0027t even exist,"},{"Start":"10:17.440 ","End":"10:19.210","Text":"then we just don\u0027t know what\u0027s happening."},{"Start":"10:19.210 ","End":"10:22.990","Text":"We can\u0027t conclude and that\u0027s why we\u0027re going to have to use an eraser theorem."},{"Start":"10:22.990 ","End":"10:24.940","Text":"We\u0027re going to work straight from the definition."},{"Start":"10:24.940 ","End":"10:31.090","Text":"Wait. We\u0027re going to work straight from"},{"Start":"10:31.090 ","End":"10:34.030","Text":"the definition of the derivative to try and see what\u0027s"},{"Start":"10:34.030 ","End":"10:37.855","Text":"going on here with this question mark."},{"Start":"10:37.855 ","End":"10:46.840","Text":"Now what it is that we need is f prime of 0 and this equals,"},{"Start":"10:46.840 ","End":"10:49.880","Text":"if we work straight off the definition."},{"Start":"10:50.310 ","End":"10:58.090","Text":"Well, in any event it\u0027s equal to the limit as h goes to 0 of f"},{"Start":"10:58.090 ","End":"11:06.085","Text":"of a plus h and a in our case is 0 plus h minus f of a,"},{"Start":"11:06.085 ","End":"11:13.285","Text":"f of 0 from the formula over h. The thing is that our x,"},{"Start":"11:13.285 ","End":"11:16.195","Text":"which is 0 plus 8, which is just h,"},{"Start":"11:16.195 ","End":"11:21.565","Text":"is given by 1 of 2 formula either this or this."},{"Start":"11:21.565 ","End":"11:24.880","Text":"We have to separately do the limit from the left and the limit from"},{"Start":"11:24.880 ","End":"11:29.080","Text":"the right and if both of those are equal,"},{"Start":"11:29.080 ","End":"11:33.230","Text":"then f prime exists."},{"Start":"11:33.450 ","End":"11:38.185","Text":"Then we have the limit, the 2-sided limit."},{"Start":"11:38.185 ","End":"11:42.230","Text":"If we take the limit from the right,"},{"Start":"11:44.610 ","End":"11:56.620","Text":"the limit on the left is obviously 0."},{"Start":"11:56.620 ","End":"12:00.895","Text":"I get the limit as h goes to 0"},{"Start":"12:00.895 ","End":"12:05.380","Text":"plus maybe I\u0027ll just say straight away that on the left we have"},{"Start":"12:05.380 ","End":"12:09.410","Text":"the limit of h."},{"Start":"12:11.490 ","End":"12:18.040","Text":"H goes to 0 from the left of f of h. Well,"},{"Start":"12:18.040 ","End":"12:19.450","Text":"if h is less than 0,"},{"Start":"12:19.450 ","End":"12:24.100","Text":"then f of h is 0."},{"Start":"12:24.100 ","End":"12:28.870","Text":"Well, first of all, copy the formula first,"},{"Start":"12:28.870 ","End":"12:33.400","Text":"f of 0 plus h minus f of 0 over h. Same thing,"},{"Start":"12:33.400 ","End":"12:39.710","Text":"but in the case of the h being slightly negative, we get this as 0"},{"Start":"12:48.840 ","End":"12:59.140","Text":"and this is 0 always over h. Limit of that rather, the limit,"},{"Start":"12:59.140 ","End":"13:07.105","Text":"yeah, which equals the limit as h goes to 0 from the left of 0,"},{"Start":"13:07.105 ","End":"13:08.950","Text":"which is just 0."},{"Start":"13:08.950 ","End":"13:10.510","Text":"That was the easy 1."},{"Start":"13:10.510 ","End":"13:16.195","Text":"Now the limit on the right is copying the same expression,"},{"Start":"13:16.195 ","End":"13:22.870","Text":"0 plus h minus f of 0 over h. Yeah,"},{"Start":"13:22.870 ","End":"13:24.880","Text":"I could have just replaced all these by h,"},{"Start":"13:24.880 ","End":"13:27.680","Text":"instead of 0 plus h never mind."},{"Start":"13:29.940 ","End":"13:35.920","Text":"Well, here it\u0027s important to me instead of looking at it as 0 plus h,"},{"Start":"13:35.920 ","End":"13:41.620","Text":"I\u0027ll just call it f of h. If that\u0027s f of h,"},{"Start":"13:41.620 ","End":"13:46.930","Text":"what I get is equal to the limit."},{"Start":"13:46.930 ","End":"13:50.770","Text":"Same thing h goes to 0 from the right."},{"Start":"13:50.770 ","End":"13:57.250","Text":"F of h is h sine 1 over h and here minus f of 0,"},{"Start":"13:57.250 ","End":"14:01.015","Text":"f of 0 is 0, all over h,"},{"Start":"14:01.015 ","End":"14:03.115","Text":"which equals 0 is nothing."},{"Start":"14:03.115 ","End":"14:04.780","Text":"The h with the h cancels."},{"Start":"14:04.780 ","End":"14:09.670","Text":"It\u0027s just the limit as h goes to 0 from the right of"},{"Start":"14:09.670 ","End":"14:16.135","Text":"sine 1 over h. Now this actually does not exist."},{"Start":"14:16.135 ","End":"14:18.910","Text":"Let me try to explain as h goes to 0 from"},{"Start":"14:18.910 ","End":"14:24.040","Text":"the right 1 over h is going to infinity and if we have sine of something,"},{"Start":"14:24.040 ","End":"14:25.525","Text":"it\u0027s going to infinity."},{"Start":"14:25.525 ","End":"14:28.780","Text":"Let\u0027s call this x. If x goes to infinity,"},{"Start":"14:28.780 ","End":"14:31.150","Text":"then the sine goes is oscillating."},{"Start":"14:31.150 ","End":"14:32.950","Text":"It goes to 1 minus 1,"},{"Start":"14:32.950 ","End":"14:34.900","Text":"1 minus 1, and everything in between."},{"Start":"14:34.900 ","End":"14:37.240","Text":"Sine of infinity does not exist."},{"Start":"14:37.240 ","End":"14:39.490","Text":"We have no limit on the right,"},{"Start":"14:39.490 ","End":"14:42.175","Text":"and therefore we have no limit at all."},{"Start":"14:42.175 ","End":"14:44.830","Text":"If this thing does not have a limit,"},{"Start":"14:44.830 ","End":"14:52.195","Text":"this is non-existent or so and so f is not differentiable at x equals 0."},{"Start":"14:52.195 ","End":"14:55.810","Text":"What I can do as far as the question goes,"},{"Start":"14:55.810 ","End":"14:59.590","Text":"I can\u0027t give the answer of f prime of 0 as they asked,"},{"Start":"14:59.590 ","End":"15:02.300","Text":"or I can say it does not exist."}],"ID":1482},{"Watched":false,"Name":"Exercise 3 - Part 2","Duration":"13m 13s","ChapterTopicVideoID":1485,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.055","Text":"Here we are with number 2,"},{"Start":"00:02.055 ","End":"00:03.840","Text":"very similar to Number 1."},{"Start":"00:03.840 ","End":"00:07.290","Text":"The only difference is that here x and here it\u0027s x squared,"},{"Start":"00:07.290 ","End":"00:09.375","Text":"but it makes all the difference in the answer."},{"Start":"00:09.375 ","End":"00:14.865","Text":"The previous case, it wasn\u0027t differentiable at x equals 0."},{"Start":"00:14.865 ","End":"00:17.280","Text":"Here, we\u0027ll see what happens."},{"Start":"00:17.280 ","End":"00:23.550","Text":"As usual, we\u0027ll start with the theorem rather than by definition,"},{"Start":"00:23.550 ","End":"00:29.940","Text":"because it\u0027s usually easier and we have no reason to believe that it won\u0027t work."},{"Start":"00:29.940 ","End":"00:33.350","Text":"The first thing we have to do if the theorem\u0027s going to"},{"Start":"00:33.350 ","End":"00:37.370","Text":"work and I won\u0027t go over it because we already have seen this before."},{"Start":"00:37.370 ","End":"00:39.980","Text":"We have to show that it\u0027s continuous and this limit equals this limit."},{"Start":"00:39.980 ","End":"00:44.780","Text":"Then we get the differentiable and the value at a and so on."},{"Start":"00:44.780 ","End":"00:46.700","Text":"We\u0027ve seen this before,"},{"Start":"00:46.700 ","End":"00:48.635","Text":"so I won\u0027t go into detail."},{"Start":"00:48.635 ","End":"00:50.880","Text":"What we have is to,"},{"Start":"00:50.880 ","End":"00:52.670","Text":"first of all, show continuity."},{"Start":"00:52.670 ","End":"00:54.965","Text":"To show the continuity,"},{"Start":"00:54.965 ","End":"00:58.520","Text":"we have to show that f at the point itself is"},{"Start":"00:58.520 ","End":"01:02.135","Text":"equal to the limit on the left is equal to the limit on the right."},{"Start":"01:02.135 ","End":"01:08.210","Text":"Other words, I have to show the continuity in whatever order that f of 0 equals this."},{"Start":"01:08.210 ","End":"01:14.720","Text":"What I\u0027m going to check the limit as x goes to 0 from the right of f of x,"},{"Start":"01:14.720 ","End":"01:19.760","Text":"which is equal to the limit as x goes to 0 from the left of f of x."},{"Start":"01:19.760 ","End":"01:21.425","Text":"That\u0027s the first thing we show."},{"Start":"01:21.425 ","End":"01:24.005","Text":"It\u0027s clear that on the left,"},{"Start":"01:24.005 ","End":"01:25.880","Text":"it\u0027s certainly equal to 0."},{"Start":"01:25.880 ","End":"01:28.055","Text":"In other words on the left,"},{"Start":"01:28.055 ","End":"01:31.450","Text":"I have 0 because I\u0027m going to 0 through 0."},{"Start":"01:31.450 ","End":"01:34.430","Text":"Therefore, 0 itself is also included as 0."},{"Start":"01:34.430 ","End":"01:39.290","Text":"What remains to be seen is whether this is 0"},{"Start":"01:39.290 ","End":"01:45.230","Text":"also the limit as x goes to 0 from the right of this thing."},{"Start":"01:45.230 ","End":"01:49.610","Text":"What we basically get is what we have to"},{"Start":"01:49.610 ","End":"01:57.370","Text":"show is what is the limit as x goes to 0 from the right."},{"Start":"01:57.370 ","End":"02:01.830","Text":"On the right it\u0027s equal to x squared sine 1 over x."},{"Start":"02:02.140 ","End":"02:10.290","Text":"We have to see what is x squared sine 1 over x. Hopefully,"},{"Start":"02:10.290 ","End":"02:12.275","Text":"this will be 0 also."},{"Start":"02:12.275 ","End":"02:14.960","Text":"Well, just a second."},{"Start":"02:14.960 ","End":"02:16.300","Text":"There\u0027s an end missing there,"},{"Start":"02:16.300 ","End":"02:18.840","Text":"hang on, sorry."},{"Start":"02:18.840 ","End":"02:21.485","Text":"Well, this is equal"},{"Start":"02:21.485 ","End":"02:29.810","Text":"to the product of something which goes to 0."},{"Start":"02:29.810 ","End":"02:36.170","Text":"In other words, the x squared is 1 part and this is bounded."},{"Start":"02:36.170 ","End":"02:41.970","Text":"This part is bounded because it never gets above 1 or below 0,"},{"Start":"02:41.970 ","End":"02:43.845","Text":"even though x goes to 0,"},{"Start":"02:43.845 ","End":"02:46.035","Text":"so 1 over x goes to infinity,"},{"Start":"02:46.035 ","End":"02:49.040","Text":"but in any event, sine is between 1 and minus 1."},{"Start":"02:49.040 ","End":"02:54.125","Text":"Something that goes to 0 times something that\u0027s bounded is also 0."},{"Start":"02:54.125 ","End":"02:57.540","Text":"That\u0027s 0, I can write as 0 also here."},{"Start":"02:57.540 ","End":"02:59.870","Text":"This is equal to this is equal to this."},{"Start":"02:59.870 ","End":"03:01.670","Text":"Yes, it is continuous."},{"Start":"03:01.670 ","End":"03:04.790","Text":"We have the continuity."},{"Start":"03:04.790 ","End":"03:08.570","Text":"Now, we need the second part before we can get"},{"Start":"03:08.570 ","End":"03:14.660","Text":"the conclusions that the limit from the left and the limit from the right are equal."},{"Start":"03:14.660 ","End":"03:22.010","Text":"Let\u0027s see what is these limits on the left and the right."},{"Start":"03:22.010 ","End":"03:27.465","Text":"In fact, I should have mentioned that f is differentiable when we\u0027re bigger than 0,"},{"Start":"03:27.465 ","End":"03:30.705","Text":"and when we\u0027re strictly less than 0, certainly differentiable."},{"Start":"03:30.705 ","End":"03:34.579","Text":"Just like before, this function is differentiable, the same reasoning,"},{"Start":"03:34.579 ","End":"03:39.930","Text":"1 over x is differentiable because it\u0027s an elementary function."},{"Start":"03:40.090 ","End":"03:43.100","Text":"Sine function is differentiable."},{"Start":"03:43.100 ","End":"03:46.460","Text":"Composition of functions, the product of differentiable,"},{"Start":"03:46.460 ","End":"03:47.900","Text":"basically all the same reasons."},{"Start":"03:47.900 ","End":"03:53.630","Text":"We know that f is differentiable and we can even give the formula."},{"Start":"03:53.630 ","End":"04:01.100","Text":"The formula is 0,"},{"Start":"04:01.100 ","End":"04:06.500","Text":"certainly for x less than 0."},{"Start":"04:06.500 ","End":"04:13.250","Text":"We have a big question mark or 3 question marks when x actually equals 0."},{"Start":"04:13.250 ","End":"04:14.900","Text":"When x is bigger than 0,"},{"Start":"04:14.900 ","End":"04:22.955","Text":"we just have to derive this to see what happens when x is bigger than 0. We use this."},{"Start":"04:22.955 ","End":"04:26.180","Text":"Here\u0027s the product rule,"},{"Start":"04:26.180 ","End":"04:32.865","Text":"x squared derivative is derived is 2x sine f_1 over x,"},{"Start":"04:32.865 ","End":"04:39.140","Text":"and then the other bit, plus x squared derivative of this is cosine of 1 over"},{"Start":"04:39.140 ","End":"04:46.400","Text":"x. Cosine 1 over x times the internal derivative,"},{"Start":"04:46.400 ","End":"04:50.110","Text":"which is, let\u0027s put a bracket here,"},{"Start":"04:50.110 ","End":"04:54.934","Text":"times minus 1 over x squared."},{"Start":"04:54.934 ","End":"04:57.515","Text":"Okay, let us remember where we are."},{"Start":"04:57.515 ","End":"05:00.230","Text":"We are about to use the theorem."},{"Start":"05:00.230 ","End":"05:03.080","Text":"We\u0027ve done the first part of the continuous, and now,"},{"Start":"05:03.080 ","End":"05:07.819","Text":"we\u0027re about to establish that this equality holds."},{"Start":"05:07.819 ","End":"05:14.705","Text":"Perhaps I\u0027ll just make myself some space here and continue over here,"},{"Start":"05:14.705 ","End":"05:16.970","Text":"because I won\u0027t be able to copy it and everything."},{"Start":"05:16.970 ","End":"05:18.980","Text":"Let us see,"},{"Start":"05:18.980 ","End":"05:22.920","Text":"we have to show that the limit on the right equals the limit on the left."},{"Start":"05:22.920 ","End":"05:25.950","Text":"Let\u0027s start with the limit on the left, it\u0027s rather easier."},{"Start":"05:25.950 ","End":"05:37.175","Text":"The limit as x goes to 0 on the left of f prime of x, which is just 0."},{"Start":"05:37.175 ","End":"05:47.200","Text":"Let\u0027s just write it. F prime of x equals the limit as x goes to 0 of,"},{"Start":"05:47.200 ","End":"05:48.500","Text":"when we\u0027re on the left,"},{"Start":"05:48.500 ","End":"05:54.509","Text":"which has 0 limit of 0 and the limit of a constant is just 0."},{"Start":"05:55.820 ","End":"05:59.710","Text":"Well, I said this before, but did it formally now that\u0027s 0."},{"Start":"05:59.710 ","End":"06:08.590","Text":"Now, let\u0027s hope the other side also gives us 0 limit as x goes to 0 from the right of"},{"Start":"06:08.590 ","End":"06:17.980","Text":"f prime of x is equal to the limit as x goes to 0 from the right of this thing,"},{"Start":"06:17.980 ","End":"06:24.615","Text":"which is 2x times sine"},{"Start":"06:24.615 ","End":"06:31.260","Text":"of 1 over x plus x squared."},{"Start":"06:31.260 ","End":"06:34.590","Text":"The minus 1 over x squared cancel."},{"Start":"06:34.590 ","End":"06:39.400","Text":"Actually, we have minus x squared cosine 1 over x."},{"Start":"06:39.400 ","End":"06:46.430","Text":"Quickly just replace this with a minus."},{"Start":"06:47.220 ","End":"06:56.825","Text":"We have yes, cosine 1 over x. Oops,"},{"Start":"06:56.825 ","End":"06:58.795","Text":"terribly sorry about that pop-up."},{"Start":"06:58.795 ","End":"07:08.030","Text":"Anyway, we were here cosine of 1 over x limit as x goes to 0 from the right."},{"Start":"07:08.160 ","End":"07:12.385","Text":"Anyway, this thing doesn\u0027t have a limit."},{"Start":"07:12.385 ","End":"07:15.085","Text":"This does not exist,"},{"Start":"07:15.085 ","End":"07:18.950","Text":"non-existent, as we write it."},{"Start":"07:19.400 ","End":"07:21.585","Text":"I\u0027ll explain why."},{"Start":"07:21.585 ","End":"07:24.710","Text":"The first part doesn\u0027t have a limit because the sine is bounded"},{"Start":"07:24.710 ","End":"07:28.520","Text":"between 1 and minus 1 and x goes to 0,"},{"Start":"07:28.520 ","End":"07:31.670","Text":"and 0 times something bounded is 0, so this part is 0."},{"Start":"07:31.670 ","End":"07:36.485","Text":"The thing is the cosine of 1 over x as x goes to 0 from the right,"},{"Start":"07:36.485 ","End":"07:38.765","Text":"1 over x goes to infinity."},{"Start":"07:38.765 ","End":"07:42.050","Text":"We\u0027ve got the cosine of something going to infinity,"},{"Start":"07:42.050 ","End":"07:44.375","Text":"and we get the whole range of the cosine."},{"Start":"07:44.375 ","End":"07:48.425","Text":"It oscillates from 1 to minus 1 and everything in between 1."},{"Start":"07:48.425 ","End":"07:51.200","Text":"Wave-like, it doesn\u0027t actually have a limit and"},{"Start":"07:51.200 ","End":"07:54.110","Text":"x goes to infinity, cosine keeps oscillating."},{"Start":"07:54.110 ","End":"07:55.790","Text":"This is non-existent,"},{"Start":"07:55.790 ","End":"07:58.240","Text":"and if it\u0027s non-existent,"},{"Start":"07:58.240 ","End":"08:00.690","Text":"doesn\u0027t say so in the theorem,"},{"Start":"08:00.690 ","End":"08:02.310","Text":"but if 1 of them is nonexistent,"},{"Start":"08:02.310 ","End":"08:04.030","Text":"then we don\u0027t know."},{"Start":"08:04.030 ","End":"08:09.380","Text":"In fact, if both exist finite or infinite and are equal,"},{"Start":"08:09.380 ","End":"08:11.270","Text":"then that is the limit,"},{"Start":"08:11.270 ","End":"08:14.660","Text":"but if 1 doesn\u0027t exist, you just don\u0027t know."},{"Start":"08:14.660 ","End":"08:19.620","Text":"We\u0027re going to have to revert to using the definition and not this clip theorem which,"},{"Start":"08:19.620 ","End":"08:21.300","Text":"unfortunately, didn\u0027t help us here."},{"Start":"08:21.300 ","End":"08:23.840","Text":"Well, usually makes life easier."},{"Start":"08:23.840 ","End":"08:26.990","Text":"Let\u0027s go ahead and I\u0027ll erase"},{"Start":"08:26.990 ","End":"08:31.080","Text":"what\u0027s not necessary and we\u0027ll do it from definition. Hang on."},{"Start":"08:32.120 ","End":"08:39.380","Text":"What remains to be done is to compute the f prime of 0."},{"Start":"08:39.380 ","End":"08:42.960","Text":"In other words, to find out what these question marks are."},{"Start":"08:43.540 ","End":"08:49.640","Text":"F prime of 0 is"},{"Start":"08:49.640 ","End":"08:55.865","Text":"equal to the limit as h goes to 0."},{"Start":"08:55.865 ","End":"09:02.875","Text":"The formula for a point a is f of a plus h minus f of a."},{"Start":"09:02.875 ","End":"09:07.380","Text":"In our case, if a is 0,"},{"Start":"09:07.380 ","End":"09:15.180","Text":"then a plus h is just h. We have f of a plus h,"},{"Start":"09:15.180 ","End":"09:18.855","Text":"which is just f of h minus f of a,"},{"Start":"09:18.855 ","End":"09:25.710","Text":"which is f of 0 and f of 0 is 0,"},{"Start":"09:25.710 ","End":"09:32.970","Text":"but I\u0027ll just write it as f of 0 first over h. Let\u0027s leave it like that for a moment."},{"Start":"09:33.730 ","End":"09:39.860","Text":"Unfortunately, it\u0027s hard to say what f of h is because when h is close to 0,"},{"Start":"09:39.860 ","End":"09:42.620","Text":"it could be on 1 side or on the other side."},{"Start":"09:42.620 ","End":"09:45.040","Text":"What we have to do is, in this case,"},{"Start":"09:45.040 ","End":"09:47.990","Text":"is work a little bit harder and see if we can compute"},{"Start":"09:47.990 ","End":"09:51.530","Text":"the left derivative and the right derivative of 0,"},{"Start":"09:51.530 ","End":"09:55.520","Text":"1 of them is written as f prime of 0 with"},{"Start":"09:55.520 ","End":"09:59.555","Text":"a plus and that\u0027s equal to the same thing as this,"},{"Start":"09:59.555 ","End":"10:05.210","Text":"just h going to 0 plus the same thing,"},{"Start":"10:05.210 ","End":"10:14.600","Text":"f of h minus f of 0 over h, which equals,"},{"Start":"10:14.600 ","End":"10:17.420","Text":"now if h is to the right of 0,"},{"Start":"10:17.420 ","End":"10:20.970","Text":"then f of h we take from here,"},{"Start":"10:20.970 ","End":"10:25.680","Text":"but f of 0 we take from here and this part is 0,"},{"Start":"10:25.680 ","End":"10:30.330","Text":"so we just need the f of h which is h squared sine 1 over"},{"Start":"10:30.330 ","End":"10:38.480","Text":"h. I forgot to"},{"Start":"10:38.480 ","End":"10:40.565","Text":"write limit of space here."},{"Start":"10:40.565 ","End":"10:44.590","Text":"Limit h goes to 0 from the right,"},{"Start":"10:44.590 ","End":"10:46.915","Text":"f of h cosine 1 over h,"},{"Start":"10:46.915 ","End":"10:49.630","Text":"all over h from here,"},{"Start":"10:49.630 ","End":"10:52.580","Text":"which is just the limit."},{"Start":"10:52.580 ","End":"10:55.240","Text":"One of the h\u0027s will cancel."},{"Start":"10:55.240 ","End":"11:01.375","Text":"I\u0027ll have limit of h sine of 1 over h. Now,"},{"Start":"11:01.375 ","End":"11:04.505","Text":"this limit is a product of 2 things,"},{"Start":"11:04.505 ","End":"11:08.980","Text":"h and sine 1 over h. This goes to 0,"},{"Start":"11:08.980 ","End":"11:10.390","Text":"and this thing is bounded."},{"Start":"11:10.390 ","End":"11:12.220","Text":"In other words, it\u0027s not infinity,"},{"Start":"11:12.220 ","End":"11:14.170","Text":"would only be a problem for the infinity or something,"},{"Start":"11:14.170 ","End":"11:16.745","Text":"but it\u0027s bounded between minus 1 and 1."},{"Start":"11:16.745 ","End":"11:21.919","Text":"Something that goes to 0 times something it\u0027s bounded is just 0."},{"Start":"11:22.500 ","End":"11:29.655","Text":"That\u0027s the right limit and the left"},{"Start":"11:29.655 ","End":"11:36.800","Text":"derivative rather at 0 is equal to the limit as h goes to 0 from the left."},{"Start":"11:36.800 ","End":"11:42.470","Text":"Everything is 0. It\u0027s just f of h is"},{"Start":"11:42.470 ","End":"11:50.644","Text":"0 minus f of 0 is also 0 over h,"},{"Start":"11:50.644 ","End":"11:53.065","Text":"which is the limit."},{"Start":"11:53.065 ","End":"11:57.930","Text":"As h goes to 0 of 0,"},{"Start":"11:57.930 ","End":"12:00.960","Text":"which is a constant that\u0027s just 0."},{"Start":"12:00.960 ","End":"12:07.640","Text":"If these 2, if this 0 and this is 0,"},{"Start":"12:07.640 ","End":"12:12.720","Text":"then we can say that this thing exists and is also equal to 0."},{"Start":"12:13.940 ","End":"12:20.400","Text":"In brief, we have basically solved our mystery that these 3 question marks are 0."},{"Start":"12:20.960 ","End":"12:24.470","Text":"While we\u0027re at it, we might as well simplify what we did before."},{"Start":"12:24.470 ","End":"12:25.980","Text":"This x squared and this x squared,"},{"Start":"12:25.980 ","End":"12:28.445","Text":"they cancel each other out and put a minus in front."},{"Start":"12:28.445 ","End":"12:32.040","Text":"This comes out to be there, that\u0027s simplified."},{"Start":"12:32.040 ","End":"12:34.025","Text":"We can do 1 more simplification,"},{"Start":"12:34.025 ","End":"12:35.730","Text":"since it\u0027s 0 here and here,"},{"Start":"12:35.730 ","End":"12:41.160","Text":"why not combine these 2 into 1 and say x less than or equal to 0?"},{"Start":"12:41.960 ","End":"12:47.775","Text":"There we go and as far as answering the question,"},{"Start":"12:47.775 ","End":"12:51.935","Text":"the original question was the determined point of non-differentiability."},{"Start":"12:51.935 ","End":"12:56.810","Text":"Well, there aren\u0027t any because we had no problems bigger than 0 and less than 0."},{"Start":"12:56.810 ","End":"13:02.515","Text":"At 0 we found it was also differentiable and f prime of 0,"},{"Start":"13:02.515 ","End":"13:04.910","Text":"well, we got that at 0."},{"Start":"13:04.910 ","End":"13:08.360","Text":"That\u0027s all the stuff is solved and"},{"Start":"13:08.360 ","End":"13:13.470","Text":"we\u0027re done with number 2 and basically with everything."}],"ID":1484},{"Watched":false,"Name":"Exercise 4 - Part 1","Duration":"4m 42s","ChapterTopicVideoID":8266,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.775","Text":"This exercise is really 3 in1."},{"Start":"00:02.775 ","End":"00:05.430","Text":"In each case we are given a function f of x,"},{"Start":"00:05.430 ","End":"00:07.275","Text":"which is defined piecewise,"},{"Start":"00:07.275 ","End":"00:08.955","Text":"in other words, in a split fashion."},{"Start":"00:08.955 ","End":"00:12.150","Text":"If you notice there are parameters a and b, in each of them,"},{"Start":"00:12.150 ","End":"00:15.810","Text":"we have to find out what values of the constants a"},{"Start":"00:15.810 ","End":"00:20.070","Text":"and b will make f differentiable for all values of x."},{"Start":"00:20.070 ","End":"00:22.220","Text":"We\u0027ll start with number 1."},{"Start":"00:22.220 ","End":"00:25.595","Text":"We\u0027ll need a theorem that will help us here."},{"Start":"00:25.595 ","End":"00:27.255","Text":"The theorem says,"},{"Start":"00:27.255 ","End":"00:32.210","Text":"that if we have a function f and it\u0027s continuous at some point x equals a,"},{"Start":"00:32.210 ","End":"00:38.950","Text":"and if we also have that the limit as x goes to a of f prime of x exists,"},{"Start":"00:38.950 ","End":"00:43.700","Text":"then we can conclude that f is differentiable at the point x equals a,"},{"Start":"00:43.700 ","End":"00:45.530","Text":"after some function f of x."},{"Start":"00:45.530 ","End":"00:49.430","Text":"Actually, if we know that f is differentiable at x equals a,"},{"Start":"00:49.430 ","End":"00:52.985","Text":"then it also follows that the first of these 2 condition holds,"},{"Start":"00:52.985 ","End":"00:56.510","Text":"and if f is continuous, differentiable implies continuous."},{"Start":"00:56.510 ","End":"01:01.790","Text":"In that case, if we know that f is differentiable at x equals 2,"},{"Start":"01:01.790 ","End":"01:04.375","Text":"it\u0027s also continuous at x equals 2."},{"Start":"01:04.375 ","End":"01:07.340","Text":"I should\u0027ve mentioned, the reason I\u0027m concerned with x equals 2,"},{"Start":"01:07.340 ","End":"01:10.700","Text":"is that obviously everywhere else it\u0027s a simple polynomial,"},{"Start":"01:10.700 ","End":"01:13.280","Text":"that\u0027s a continuous differentiable,"},{"Start":"01:13.280 ","End":"01:14.420","Text":"all the good things."},{"Start":"01:14.420 ","End":"01:17.510","Text":"The only problem is that the seam line x equals 2,"},{"Start":"01:17.510 ","End":"01:19.750","Text":"so that\u0027s all I\u0027m concerned with."},{"Start":"01:19.750 ","End":"01:23.660","Text":"Continuity at x equals 2 for that,"},{"Start":"01:23.660 ","End":"01:28.025","Text":"what we need is that the left and the right limit are equal to the value at the point."},{"Start":"01:28.025 ","End":"01:33.980","Text":"In other words, we need that the limit as x goes to 2 from the right of f of x,"},{"Start":"01:33.980 ","End":"01:39.545","Text":"is equal to the limit as x goes to 2 from the left of f of x,"},{"Start":"01:39.545 ","End":"01:42.580","Text":"which also has to equal f of 2,"},{"Start":"01:42.580 ","End":"01:44.510","Text":"then that\u0027s what\u0027s continuity is."},{"Start":"01:44.510 ","End":"01:47.165","Text":"I mean, I don\u0027t know that these whole, I\u0027m going to check."},{"Start":"01:47.165 ","End":"01:49.475","Text":"Let\u0027s see the limit from the right."},{"Start":"01:49.475 ","End":"01:51.920","Text":"We take our definition from here,"},{"Start":"01:51.920 ","End":"01:54.335","Text":"and we can just substitute x equals 2."},{"Start":"01:54.335 ","End":"01:57.255","Text":"2 squared is 4 plus 2a."},{"Start":"01:57.255 ","End":"02:00.075","Text":"So it\u0027s 2a plus 4 here."},{"Start":"02:00.075 ","End":"02:05.555","Text":"At 2 itself, it\u0027s also 2a plus 4 because this definition includes the 2."},{"Start":"02:05.555 ","End":"02:07.270","Text":"If we go from the right,"},{"Start":"02:07.270 ","End":"02:10.040","Text":"then we are going to be using the values of x from here,"},{"Start":"02:10.040 ","End":"02:12.250","Text":"and so we get 2 cubed plus b,"},{"Start":"02:12.250 ","End":"02:13.885","Text":"which is 8 plus b."},{"Start":"02:13.885 ","End":"02:15.520","Text":"First and last are already equal,"},{"Start":"02:15.520 ","End":"02:20.075","Text":"so the only extra condition that we need is for this to equal this."},{"Start":"02:20.075 ","End":"02:25.695","Text":"2a plus 4 must equal 8 plus b,"},{"Start":"02:25.695 ","End":"02:27.530","Text":"or if we want to put it more simply,"},{"Start":"02:27.530 ","End":"02:30.420","Text":"that\u0027s 2a minus b,"},{"Start":"02:30.420 ","End":"02:32.075","Text":"is equal to 4."},{"Start":"02:32.075 ","End":"02:34.490","Text":"That\u0027s the first of 2 equations."},{"Start":"02:34.490 ","End":"02:37.070","Text":"What we are looking for is 2 equations in 2 unknowns,"},{"Start":"02:37.070 ","End":"02:39.525","Text":"in a and b, and then we can find them."},{"Start":"02:39.525 ","End":"02:42.410","Text":"This continuity implies this."},{"Start":"02:42.410 ","End":"02:45.275","Text":"Now we\u0027d like to use the theorem."},{"Start":"02:45.275 ","End":"02:50.590","Text":"Let\u0027s see what happens with the limit as x goes to a of f prime of x."},{"Start":"02:50.590 ","End":"02:54.500","Text":"We can compute it when x is not equal to 2, x equals 2,"},{"Start":"02:54.500 ","End":"02:55.670","Text":"there might be some problem,"},{"Start":"02:55.670 ","End":"02:59.225","Text":"but if x is bigger or less than and there is no problem."},{"Start":"02:59.225 ","End":"03:02.870","Text":"So here we get 2x plus a,"},{"Start":"03:02.870 ","End":"03:05.839","Text":"and this is true for x bigger than 2,"},{"Start":"03:05.839 ","End":"03:07.960","Text":"for x actually equal to 2,"},{"Start":"03:07.960 ","End":"03:10.580","Text":"that\u0027s the thing we don\u0027t know what\u0027s happening there."},{"Start":"03:10.580 ","End":"03:13.415","Text":"But certainly for x less than 2,"},{"Start":"03:13.415 ","End":"03:15.290","Text":"we again know what it is,"},{"Start":"03:15.290 ","End":"03:17.360","Text":"and it\u0027s just 3x squared."},{"Start":"03:17.360 ","End":"03:19.460","Text":"Now we want this theorem to help us."},{"Start":"03:19.460 ","End":"03:24.275","Text":"We want the limit of the derivative f prime to exist."},{"Start":"03:24.275 ","End":"03:26.630","Text":"For that, we just have to make sure that"},{"Start":"03:26.630 ","End":"03:30.095","Text":"the left limit and right limit exist and are equal."},{"Start":"03:30.095 ","End":"03:32.285","Text":"We have to take 2x plus a,"},{"Start":"03:32.285 ","End":"03:35.780","Text":"from the right and 3x squared from the left."},{"Start":"03:35.780 ","End":"03:44.750","Text":"We would like for limit as x goes to 2 from the right of 2x plus a,"},{"Start":"03:44.750 ","End":"03:51.809","Text":"to equal the limit as x goes to 2 from the left of 3x squared."},{"Start":"03:51.809 ","End":"03:55.785","Text":"Now, this limit you just have to put x equals 2 in here."},{"Start":"03:55.785 ","End":"03:58.500","Text":"We get for this when x is 2,"},{"Start":"03:58.500 ","End":"04:00.255","Text":"this is a plus 4,"},{"Start":"04:00.255 ","End":"04:02.280","Text":"and this when x equals 2,"},{"Start":"04:02.280 ","End":"04:03.650","Text":"is 3 times 2 squared,"},{"Start":"04:03.650 ","End":"04:06.275","Text":"which is 3 times 4, which is 12."},{"Start":"04:06.275 ","End":"04:11.540","Text":"In other words, this gives us the equation, a equals 8."},{"Start":"04:11.540 ","End":"04:15.545","Text":"Now if I take this together with the 1 I had previously,"},{"Start":"04:15.545 ","End":"04:17.990","Text":"let\u0027s say this is equation 1,"},{"Start":"04:17.990 ","End":"04:21.350","Text":"and this is equation 2 and 2 unknowns,"},{"Start":"04:21.350 ","End":"04:25.750","Text":"that\u0027s easy to solve because this gives us that a equals 8."},{"Start":"04:25.750 ","End":"04:29.460","Text":"If a is 8, 2 times 8 is 16,"},{"Start":"04:29.460 ","End":"04:31.065","Text":"minus b is 4."},{"Start":"04:31.065 ","End":"04:33.210","Text":"So 16 minus b is 4,"},{"Start":"04:33.210 ","End":"04:35.985","Text":"then b has to be 12."},{"Start":"04:35.985 ","End":"04:39.940","Text":"This would be the answer for a and b,"},{"Start":"04:39.940 ","End":"04:42.900","Text":"and that\u0027s it, I guess."}],"ID":8428},{"Watched":false,"Name":"Exercise 4 - Part 2","Duration":"2m 46s","ChapterTopicVideoID":8267,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.685","Text":"Fine, we\u0027re done with this 1."},{"Start":"00:02.685 ","End":"00:04.050","Text":"Next is Number 2,"},{"Start":"00:04.050 ","End":"00:06.330","Text":"which I\u0027ve written over here."},{"Start":"00:06.330 ","End":"00:09.510","Text":"I\u0027ve kept the theorem that we had from the previous time,"},{"Start":"00:09.510 ","End":"00:12.510","Text":"which will come in useful in 2 and in 3."},{"Start":"00:12.510 ","End":"00:15.960","Text":"Just to briefly say that if we show continuity at"},{"Start":"00:15.960 ","End":"00:20.370","Text":"a point and that the limit of the derivative exists,"},{"Start":"00:20.370 ","End":"00:22.920","Text":"where a here is just 0,"},{"Start":"00:22.920 ","End":"00:26.010","Text":"then we know that f is differentiable at that point."},{"Start":"00:26.010 ","End":"00:30.690","Text":"I must say that this a here is not the same as the a here."},{"Start":"00:30.690 ","End":"00:33.060","Text":"This is just from the theorem. This is a different a."},{"Start":"00:33.060 ","End":"00:34.185","Text":"These are parameters,"},{"Start":"00:34.185 ","End":"00:35.970","Text":"so don\u0027t get confused."},{"Start":"00:35.970 ","End":"00:39.460","Text":"The continuity at x equals 0,"},{"Start":"00:39.460 ","End":"00:41.510","Text":"we have to show that the left limit,"},{"Start":"00:41.510 ","End":"00:44.180","Text":"the right limit, and the value at the point are all equal."},{"Start":"00:44.180 ","End":"00:50.795","Text":"In other words, that the limit as x goes to 0 from the right of the function is equal to"},{"Start":"00:50.795 ","End":"00:54.330","Text":"limit as x goes to 0 from the left of"},{"Start":"00:54.330 ","End":"00:58.130","Text":"the function and it\u0027s equal to actually the function at the point."},{"Start":"00:58.130 ","End":"00:59.645","Text":"F of x,"},{"Start":"00:59.645 ","End":"01:01.685","Text":"when x goes to 0 from the right,"},{"Start":"01:01.685 ","End":"01:06.305","Text":"we\u0027re going to use this definition and then we\u0027ll get b."},{"Start":"01:06.305 ","End":"01:09.080","Text":"If we go to 0 from the left,"},{"Start":"01:09.080 ","End":"01:11.225","Text":"we\u0027re going to be using e^x,"},{"Start":"01:11.225 ","End":"01:13.720","Text":"put in 0, we get 1."},{"Start":"01:13.720 ","End":"01:16.580","Text":"Likewise, f of 0 itself comes from this formula,"},{"Start":"01:16.580 ","End":"01:25.520","Text":"which is also equal to 1. B equals 1 will be our first equation."},{"Start":"01:25.520 ","End":"01:27.935","Text":"This condition, the second 1,"},{"Start":"01:27.935 ","End":"01:31.220","Text":"has got to give us constraint on a. Hopefully,"},{"Start":"01:31.220 ","End":"01:33.185","Text":"we can find a and b."},{"Start":"01:33.185 ","End":"01:37.940","Text":"What we should do now is write down f prime of x."},{"Start":"01:37.940 ","End":"01:39.440","Text":"For x less than 0,"},{"Start":"01:39.440 ","End":"01:43.345","Text":"it\u0027s still e^x for x less than 0."},{"Start":"01:43.345 ","End":"01:47.060","Text":"We don\u0027t know what it is when x equals 0."},{"Start":"01:47.060 ","End":"01:49.310","Text":"For x bigger than 0,"},{"Start":"01:49.310 ","End":"01:52.190","Text":"it\u0027s just the derivative of ax plus b,"},{"Start":"01:52.190 ","End":"01:54.340","Text":"which is just a."},{"Start":"01:54.340 ","End":"02:01.235","Text":"Now we want to see about the condition that the limit as x goes to 0 of f prime exists."},{"Start":"02:01.235 ","End":"02:04.610","Text":"For it to exist, I could instead take the left and right limit."},{"Start":"02:04.610 ","End":"02:07.130","Text":"That\u0027s going to be more convenient or what I have to"},{"Start":"02:07.130 ","End":"02:09.950","Text":"do anyway because on the left of 0 is defined 1 way,"},{"Start":"02:09.950 ","End":"02:11.315","Text":"on the right, the other way."},{"Start":"02:11.315 ","End":"02:14.770","Text":"Let\u0027s take this limit as the left side and the right side."},{"Start":"02:14.770 ","End":"02:19.050","Text":"We would like for the left limit to equal the right limit and for both to exist."},{"Start":"02:19.050 ","End":"02:25.190","Text":"We want the limit as x goes to 0 from the right of f prime of"},{"Start":"02:25.190 ","End":"02:32.975","Text":"x to equal the limit as x goes to 0 from the left of f prime of x."},{"Start":"02:32.975 ","End":"02:40.650","Text":"The limit as x goes to 0 from the right is just e^0, which is 1."},{"Start":"02:40.720 ","End":"02:46.650","Text":"What if I got it backwards from the right? A."}],"ID":8429},{"Watched":false,"Name":"Exercise 4 - Part 3","Duration":"3m 1s","ChapterTopicVideoID":8268,"CourseChapterTopicPlaylistID":1573,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"After number 2 comes number 3,"},{"Start":"00:02.400 ","End":"00:04.260","Text":"I just copied it from above."},{"Start":"00:04.260 ","End":"00:06.390","Text":"Normally, the first thing I would do is continuous,"},{"Start":"00:06.390 ","End":"00:09.120","Text":"but it\u0027s very tempting since this is so close. You know what?"},{"Start":"00:09.120 ","End":"00:11.900","Text":"Let\u0027s just fill in the value of f prime of x."},{"Start":"00:11.900 ","End":"00:13.530","Text":"At x equals z, we don\u0027t know,"},{"Start":"00:13.530 ","End":"00:14.940","Text":"but for x bigger than e,"},{"Start":"00:14.940 ","End":"00:16.980","Text":"then this is an elementary function."},{"Start":"00:16.980 ","End":"00:20.535","Text":"Natural log of x is defined and its cube is defined."},{"Start":"00:20.535 ","End":"00:29.025","Text":"What we basically get is that the derivative is 3 natural log of x squared."},{"Start":"00:29.025 ","End":"00:32.595","Text":"The 2 is put here other than outside the natural log of x."},{"Start":"00:32.595 ","End":"00:38.880","Text":"But that\u0027s not all because this was chain rule in a function is natural log of x,"},{"Start":"00:38.880 ","End":"00:42.145","Text":"so we have to put an extra 1 over x here."},{"Start":"00:42.145 ","End":"00:43.970","Text":"For x less than e,"},{"Start":"00:43.970 ","End":"00:46.570","Text":"we have the derivative of this which is a."},{"Start":"00:46.570 ","End":"00:48.485","Text":"Let\u0027s just leave that there for a moment."},{"Start":"00:48.485 ","End":"00:52.280","Text":"Now, continuous is what we need at x equals a,"},{"Start":"00:52.280 ","End":"00:53.360","Text":"which is e,"},{"Start":"00:53.360 ","End":"00:56.960","Text":"the right limit equals the limit as x goes to e from"},{"Start":"00:56.960 ","End":"01:01.780","Text":"the left of the function and it\u0027s also equal to the value of the function at the point."},{"Start":"01:01.780 ","End":"01:04.290","Text":"Limit from the right of f of x,"},{"Start":"01:04.290 ","End":"01:06.075","Text":"we going from here,"},{"Start":"01:06.075 ","End":"01:09.575","Text":"it\u0027s natural log of e cubed."},{"Start":"01:09.575 ","End":"01:12.060","Text":"Natural log of e is 1 cubed,"},{"Start":"01:12.060 ","End":"01:13.320","Text":"so this is just 1."},{"Start":"01:13.320 ","End":"01:14.445","Text":"On the left,"},{"Start":"01:14.445 ","End":"01:16.640","Text":"we\u0027re going to get using this formula,"},{"Start":"01:16.640 ","End":"01:18.320","Text":"just substitute x equals e,"},{"Start":"01:18.320 ","End":"01:20.150","Text":"it\u0027s ae plus b,"},{"Start":"01:20.150 ","End":"01:21.970","Text":"and f of e,"},{"Start":"01:21.970 ","End":"01:24.540","Text":"I used the top thing, it\u0027s just 1."},{"Start":"01:24.540 ","End":"01:27.300","Text":"What this gives us is our first equation,"},{"Start":"01:27.300 ","End":"01:30.365","Text":"ae plus b is equal to 1,"},{"Start":"01:30.365 ","End":"01:32.750","Text":"it\u0027s first equation in a and b. Hopefully,"},{"Start":"01:32.750 ","End":"01:34.190","Text":"we get another equation."},{"Start":"01:34.190 ","End":"01:36.980","Text":"We will, we get it from this limit existing."},{"Start":"01:36.980 ","End":"01:39.605","Text":"Let\u0027s check what happens here."},{"Start":"01:39.605 ","End":"01:44.930","Text":"The limit as x goes to e from the right of f prime of x is"},{"Start":"01:44.930 ","End":"01:51.020","Text":"equal to the limit as x goes to e from the left of f prime of x."},{"Start":"01:51.020 ","End":"01:53.225","Text":"Let\u0027s see. On the right,"},{"Start":"01:53.225 ","End":"01:55.985","Text":"we need it from here."},{"Start":"01:55.985 ","End":"01:59.495","Text":"From here, if we put x equals e,"},{"Start":"01:59.495 ","End":"02:01.880","Text":"we get natural log of e is 1,"},{"Start":"02:01.880 ","End":"02:05.275","Text":"we\u0027ve done this thing, 1 squared is 1, 3 times 1 is 3,"},{"Start":"02:05.275 ","End":"02:10.909","Text":"3 over e. This part is 3 over e,"},{"Start":"02:10.909 ","End":"02:15.095","Text":"and the other part is just a constant, that\u0027s a."},{"Start":"02:15.095 ","End":"02:17.450","Text":"We get our second equation,"},{"Start":"02:17.450 ","End":"02:21.710","Text":"a equals 3 over e,"},{"Start":"02:21.710 ","End":"02:25.805","Text":"this is where we have 2 equations and 2 unknowns,"},{"Start":"02:25.805 ","End":"02:27.920","Text":"that shouldn\u0027t be too hard to solve."},{"Start":"02:27.920 ","End":"02:31.655","Text":"Suppose, I substitute the value of a here,"},{"Start":"02:31.655 ","End":"02:38.375","Text":"I will get 3 over e times e is just 3 plus b equals 1,"},{"Start":"02:38.375 ","End":"02:42.790","Text":"that gives us that b equals minus 2."},{"Start":"02:42.790 ","End":"02:51.440","Text":"The answer will be a equals 3 over e and b equals minus 2."},{"Start":"02:51.440 ","End":"02:57.010","Text":"These are the values of a and b that will make f differentiable at x equals 0."},{"Start":"02:57.010 ","End":"03:02.440","Text":"That\u0027s it. We\u0027re done for this part 3 and for the whole exercise."}],"ID":8430}],"Thumbnail":null,"ID":1573}]

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