The Total Differential and Linear Approximation
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Taylor Polynomials for Functions of Two Variables
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[{"Name":"The Total Differential and Linear Approximation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Total Differential and Linear Approximation","Duration":"7m 23s","ChapterTopicVideoID":29356,"CourseChapterTopicPlaylistID":294462,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.655","Text":"A new topic, possibly,"},{"Start":"00:02.655 ","End":"00:05.505","Text":"you\u0027ve seen it in 1 variable,"},{"Start":"00:05.505 ","End":"00:10.110","Text":"the total differential and linear approximations but we\u0027ll review the,"},{"Start":"00:10.110 ","End":"00:13.515","Text":"1 variable case and then get to the 2 variable case."},{"Start":"00:13.515 ","End":"00:17.235","Text":"Let\u0027s say y is a function of x and f is differentiable,"},{"Start":"00:17.235 ","End":"00:21.420","Text":"and at some point x_0 ft is y_0,"},{"Start":"00:21.420 ","End":"00:24.810","Text":"then the derivative at x_0 on the 1 hand,"},{"Start":"00:24.810 ","End":"00:26.290","Text":"we can write it as d y by dx,"},{"Start":"00:26.290 ","End":"00:31.560","Text":"and the other hand we\u0027d write it as f’ and if we consider this as a fraction,"},{"Start":"00:31.560 ","End":"00:36.480","Text":"then we get that dy=f’ times dx,"},{"Start":"00:36.480 ","End":"00:40.370","Text":"and this is called the total differential in 1 variable,"},{"Start":"00:40.370 ","End":"00:45.910","Text":"we can estimate dy and dx by Delta y and Delta x."},{"Start":"00:45.910 ","End":"00:48.980","Text":"That\u0027s how originally dy by dx came to be,"},{"Start":"00:48.980 ","End":"00:51.605","Text":"it\u0027s Delta y over Delta x in the limit."},{"Start":"00:51.605 ","End":"00:57.485","Text":"This approximation comes out to be Delta y is y minus y_0,"},{"Start":"00:57.485 ","End":"01:01.210","Text":"and Delta x is x minus x_0,"},{"Start":"01:01.210 ","End":"01:05.056","Text":"and y is f of x and y_0 is f(x_0)."},{"Start":"01:05.056 ","End":"01:07.025","Text":"Bringing this to the other side,"},{"Start":"01:07.025 ","End":"01:10.205","Text":"we get the following approximation formula."},{"Start":"01:10.205 ","End":"01:17.680","Text":"That f(x) is approximately equal to f(x_0) plus f\u0027 at x_0 times x minus x_0,"},{"Start":"01:17.680 ","End":"01:19.700","Text":"and the closer x is to x_0,"},{"Start":"01:19.700 ","End":"01:21.620","Text":"the better the approximation."},{"Start":"01:21.620 ","End":"01:23.405","Text":"This is the form we use,"},{"Start":"01:23.405 ","End":"01:24.875","Text":"even though strictly speaking,"},{"Start":"01:24.875 ","End":"01:27.185","Text":"this is the total differential."},{"Start":"01:27.185 ","End":"01:30.515","Text":"This is mostly used in integration,"},{"Start":"01:30.515 ","End":"01:34.380","Text":"in change of variables, for example."},{"Start":"01:34.630 ","End":"01:38.075","Text":"This linear approximation formula,"},{"Start":"01:38.075 ","End":"01:40.254","Text":"it\u0027s often written instead of x_0,"},{"Start":"01:40.254 ","End":"01:43.150","Text":"we write a or c or anything really,"},{"Start":"01:43.150 ","End":"01:45.350","Text":"and let\u0027s do an example now."},{"Start":"01:45.350 ","End":"01:48.500","Text":"Here we want to approximate the square root of 5,"},{"Start":"01:48.500 ","End":"01:53.190","Text":"given the fact that the square root of 4 and 4 is near 5,"},{"Start":"01:53.190 ","End":"01:55.860","Text":"square root of 4 is equal to 2."},{"Start":"01:55.860 ","End":"02:00.125","Text":"What we\u0027ll do is we\u0027ll take the function to be the square root function,"},{"Start":"02:00.125 ","End":"02:02.500","Text":"we\u0027ll take x_0 to be 4."},{"Start":"02:02.500 ","End":"02:05.030","Text":"That\u0027s the point where we know the value of the function,"},{"Start":"02:05.030 ","End":"02:08.749","Text":"and x is where we want to know the value of the function."},{"Start":"02:08.749 ","End":"02:12.505","Text":"Copying this formula, we get the following,"},{"Start":"02:12.505 ","End":"02:16.200","Text":"and in our case, f(4) is the square root of 4,"},{"Start":"02:16.200 ","End":"02:21.270","Text":"which is 2 and f’(x) is 1 over 2 root x,"},{"Start":"02:21.270 ","End":"02:25.380","Text":"so f’(4) comes out to be 1/4,"},{"Start":"02:25.380 ","End":"02:27.030","Text":"and the square root of 5,"},{"Start":"02:27.030 ","End":"02:30.240","Text":"using this formula is 2 from here,"},{"Start":"02:30.240 ","End":"02:35.880","Text":"plus 1/4 times x minus x_0 is 5 minus 4."},{"Start":"02:35.880 ","End":"02:37.830","Text":"That\u0027s the Delta x,"},{"Start":"02:37.830 ","End":"02:42.300","Text":"and that comes out to be 2 plus 1 divided by 4,"},{"Start":"02:42.300 ","End":"02:45.195","Text":"2 and 1/4 2.25,"},{"Start":"02:45.195 ","End":"02:47.570","Text":"and if we use a calculator,"},{"Start":"02:47.570 ","End":"02:49.309","Text":"and nowadays we have calculators,"},{"Start":"02:49.309 ","End":"02:52.370","Text":"so it wouldn\u0027t do this exercise this way."},{"Start":"02:52.370 ","End":"02:54.605","Text":"But in a calculator we get"},{"Start":"02:54.605 ","End":"03:00.805","Text":"2.236 and you can decide whether it\u0027s a good approximation or not."},{"Start":"03:00.805 ","End":"03:03.620","Text":"That\u0027s for 1 variable."},{"Start":"03:03.620 ","End":"03:07.175","Text":"Now let\u0027s go to approximation of functions of 2 variables."},{"Start":"03:07.175 ","End":"03:15.205","Text":"This time we\u0027ll take z as a function of x and y at a particular x_0 y_0 we get z_0,"},{"Start":"03:15.205 ","End":"03:21.680","Text":"and the total differential is defined as dz=df by dx,"},{"Start":"03:21.680 ","End":"03:24.530","Text":"dx plus df by dy, dy."},{"Start":"03:24.530 ","End":"03:28.330","Text":"This is a shorthand way of writing partial derivatives,"},{"Start":"03:28.330 ","End":"03:31.280","Text":"and this gives us an approximation rule,"},{"Start":"03:31.280 ","End":"03:33.680","Text":"if we replace d by Delta,"},{"Start":"03:33.680 ","End":"03:39.430","Text":"Delta z is approximately equal to fx Delta x plus fy Delta y."},{"Start":"03:39.430 ","End":"03:43.200","Text":"Now, Delta x is defined to be x minus x_0,"},{"Start":"03:43.200 ","End":"03:45.770","Text":"and Delta y is y minus y_0."},{"Start":"03:45.770 ","End":"03:48.985","Text":"That\u0027s assuming we have some point x y,"},{"Start":"03:48.985 ","End":"03:50.700","Text":"which is near x_0,"},{"Start":"03:50.700 ","End":"03:55.380","Text":"y_0, and Delta z is z minus z_0,"},{"Start":"03:55.380 ","End":"03:58.515","Text":"which is f(x) y minus f(x_0, y_0.)"},{"Start":"03:58.515 ","End":"04:01.325","Text":"What we get from this is that Delta z,"},{"Start":"04:01.325 ","End":"04:06.970","Text":"which is this is approximately equal to fx Delta x plus fy Delta y."},{"Start":"04:06.970 ","End":"04:08.510","Text":"The fx and fy of course,"},{"Start":"04:08.510 ","End":"04:11.135","Text":"are evaluated at the point x_0, y_0,"},{"Start":"04:11.135 ","End":"04:13.010","Text":"and bring this to the other side,"},{"Start":"04:13.010 ","End":"04:16.895","Text":"and this is the approximation formula that we\u0027ll use."},{"Start":"04:16.895 ","End":"04:19.155","Text":"We won\u0027t really use in this form,"},{"Start":"04:19.155 ","End":"04:23.435","Text":"but we\u0027ll basically be using this approximation formula."},{"Start":"04:23.435 ","End":"04:25.670","Text":"This is a linear approximation formula."},{"Start":"04:25.670 ","End":"04:27.410","Text":"There is also quadratic,"},{"Start":"04:27.410 ","End":"04:29.600","Text":"second degree and higher,"},{"Start":"04:29.600 ","End":"04:31.505","Text":"but not in this clip."},{"Start":"04:31.505 ","End":"04:35.000","Text":"Often we write a and b instead of x_0 and y_0,"},{"Start":"04:35.000 ","End":"04:37.520","Text":"so we would write it like this."},{"Start":"04:37.520 ","End":"04:40.265","Text":"Here\u0027s an illustration."},{"Start":"04:40.265 ","End":"04:44.900","Text":"The approximation is actually the tangent plane."},{"Start":"04:44.900 ","End":"04:46.160","Text":"If P is the point a,"},{"Start":"04:46.160 ","End":"04:48.065","Text":"b or x_0, y_0,"},{"Start":"04:48.065 ","End":"04:49.925","Text":"then if we substitute in this,"},{"Start":"04:49.925 ","End":"04:54.980","Text":"we get the value in the tangent plane rather than on the function itself,"},{"Start":"04:54.980 ","End":"04:56.550","Text":"and when x,"},{"Start":"04:56.550 ","End":"05:01.800","Text":"y is close to the point p or the point below it, in the x,"},{"Start":"05:01.800 ","End":"05:08.750","Text":"y plane then the blue and the light brown are approximately equal to each other."},{"Start":"05:08.750 ","End":"05:12.440","Text":"Now let\u0027s do an exercise, an example."},{"Start":"05:12.440 ","End":"05:16.160","Text":"Let\u0027s approximate this square root using"},{"Start":"05:16.160 ","End":"05:20.270","Text":"the fact that we know that the square root of 3 squared plus 4 squared,"},{"Start":"05:20.270 ","End":"05:21.620","Text":"which is close to this,"},{"Start":"05:21.620 ","End":"05:23.855","Text":"is equal to 5."},{"Start":"05:23.855 ","End":"05:26.150","Text":"We\u0027ll take the function of x,"},{"Start":"05:26.150 ","End":"05:29.120","Text":"y to be square root of x^2 plus y^2,"},{"Start":"05:29.120 ","End":"05:32.000","Text":"and the point x_0, y_0 is 3, 4."},{"Start":"05:32.000 ","End":"05:36.000","Text":"That\u0027s the point where we know the value of the function, is 5,"},{"Start":"05:36.000 ","End":"05:45.600","Text":"3^2 is 9 plus 4^2 is 16 square root of 25 is 5 and we\u0027ll take x y to be the 3.01,3.97,"},{"Start":"05:45.600 ","End":"05:48.110","Text":"and we\u0027ll use this formula from above,"},{"Start":"05:48.110 ","End":"05:52.390","Text":"the derivatives df by dx and df by dy,"},{"Start":"05:52.390 ","End":"05:57.350","Text":"df by dx is 1 over twice the square root times the antiderivative,"},{"Start":"05:57.350 ","End":"06:00.170","Text":"which is 2x and so the 2 cancels with the 2."},{"Start":"06:00.170 ","End":"06:01.385","Text":"The x goes on top."},{"Start":"06:01.385 ","End":"06:03.980","Text":"It\u0027s x over the square root of x^2 plus y^2,"},{"Start":"06:03.980 ","End":"06:07.850","Text":"and similarly, fy is y over the square root."},{"Start":"06:07.850 ","End":"06:11.390","Text":"We need f(x_0, y_0) that\u0027s 5."},{"Start":"06:11.390 ","End":"06:15.040","Text":"We need df by dx at this point,"},{"Start":"06:15.040 ","End":"06:18.270","Text":"which is 3 over square root of 3^2 plus 4^2,"},{"Start":"06:18.270 ","End":"06:19.455","Text":"which is 3/5,"},{"Start":"06:19.455 ","End":"06:22.525","Text":"and df by dy comes out 4/5."},{"Start":"06:22.525 ","End":"06:29.595","Text":"Now we can substitute it all and get 5 plus 3/5 Delta x,"},{"Start":"06:29.595 ","End":"06:31.500","Text":"which is x minus x_0."},{"Start":"06:31.500 ","End":"06:36.440","Text":"This and here, 4/5, y minus y_0."},{"Start":"06:36.440 ","End":"06:44.150","Text":"This comes out to be 5 plus 0.6 times 0.01 plus 0.8 times 0.03,"},{"Start":"06:44.150 ","End":"06:45.589","Text":"but with a minus,"},{"Start":"06:45.589 ","End":"06:53.570","Text":"and that comes out to be 4.982, and that\u0027s the answer."},{"Start":"06:53.570 ","End":"06:54.980","Text":"Now if you\u0027re wondering,"},{"Start":"06:54.980 ","End":"07:00.080","Text":"is it close or not to the real value because we have pocket calculators,"},{"Start":"07:00.080 ","End":"07:05.090","Text":"we wouldn\u0027t do an easy exercise like this with approximation."},{"Start":"07:05.090 ","End":"07:07.160","Text":"If you check on the calculator,"},{"Start":"07:07.160 ","End":"07:12.539","Text":"we get that it\u0027s actually good to 4 decimal places."},{"Start":"07:12.539 ","End":"07:16.505","Text":"That\u0027s pretty good approximation and we\u0027re just lucky here."},{"Start":"07:16.505 ","End":"07:20.525","Text":"Normally you can\u0027t expect to get this good in approximation."},{"Start":"07:20.525 ","End":"07:23.310","Text":"That concludes this clip."}],"ID":30961},{"Watched":false,"Name":"Exercise 1","Duration":"3m 8s","ChapterTopicVideoID":29357,"CourseChapterTopicPlaylistID":294462,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this exercise, we\u0027re going to approximate this expression,"},{"Start":"00:04.290 ","End":"00:08.790","Text":"natural log of 0.01^2 plus 0.99^2"},{"Start":"00:08.790 ","End":"00:14.520","Text":"using the total differential or the linear approximation formula related."},{"Start":"00:14.520 ","End":"00:17.760","Text":"This is the formula that we\u0027ll be using."},{"Start":"00:17.760 ","End":"00:21.900","Text":"We know that if we had here 0 and here 1,"},{"Start":"00:21.900 ","End":"00:23.325","Text":"we would know the answer."},{"Start":"00:23.325 ","End":"00:25.800","Text":"That\u0027s how we get the approximation."},{"Start":"00:25.800 ","End":"00:27.930","Text":"I will take the f(x,"},{"Start":"00:27.930 ","End":"00:32.895","Text":"y) to be natural log of x^2 plus y^2 modeled on this."},{"Start":"00:32.895 ","End":"00:35.205","Text":"Our starting point,"},{"Start":"00:35.205 ","End":"00:38.685","Text":"1 where we know the value is 0,1."},{"Start":"00:38.685 ","End":"00:41.190","Text":"Because when x is 0 and y is 1,"},{"Start":"00:41.190 ","End":"00:44.640","Text":"we get natural log of 1 which is 0."},{"Start":"00:44.640 ","End":"00:50.720","Text":"The x, y that we\u0027re looking for is close to 0,1, it\u0027s this."},{"Start":"00:50.720 ","End":"00:53.030","Text":"I we\u0027ll need the function and its derivatives."},{"Start":"00:53.030 ","End":"00:54.665","Text":"The function we have already,"},{"Start":"00:54.665 ","End":"00:59.480","Text":"the derivative with respect to x is this because the derivative"},{"Start":"00:59.480 ","End":"01:04.565","Text":"of natural log is 1/ x^2 plus y^2."},{"Start":"01:04.565 ","End":"01:08.045","Text":"But the chain rule says we need an anti-derivative."},{"Start":"01:08.045 ","End":"01:11.850","Text":"The anti-derivative with respect to x is 2x,"},{"Start":"01:11.850 ","End":"01:14.175","Text":"so we have a 2x on the numerator."},{"Start":"01:14.175 ","End":"01:18.460","Text":"Similarly, df by dy is 2y/x^2 plus y^2."},{"Start":"01:18.460 ","End":"01:21.650","Text":"Now we want to compute these at the point x_0,"},{"Start":"01:21.650 ","End":"01:24.380","Text":"y_0, which is 0,1."},{"Start":"01:24.380 ","End":"01:28.550","Text":"This 1 comes out to be, like we said, 0."},{"Start":"01:28.550 ","End":"01:34.160","Text":"That\u0027s the point where we know the value of f. The derivative at 0,1,"},{"Start":"01:34.160 ","End":"01:38.030","Text":"x is 0, so there\u0027s no need to compute anymore at 0."},{"Start":"01:38.030 ","End":"01:40.600","Text":"Here, y is 1,"},{"Start":"01:40.600 ","End":"01:45.120","Text":"and the denominator is also 1."},{"Start":"01:45.120 ","End":"01:46.850","Text":"We did that already here."},{"Start":"01:46.850 ","End":"01:49.820","Text":"It\u0027s 1 anyway, 0^2 plus 1^2."},{"Start":"01:49.820 ","End":"01:52.400","Text":"We have twice 1/1,"},{"Start":"01:52.400 ","End":"01:56.725","Text":"which is 2, these 3 numbers."},{"Start":"01:56.725 ","End":"02:01.325","Text":"This 1, this 1 and this 1."},{"Start":"02:01.325 ","End":"02:08.260","Text":"They\u0027re the ones that go here, here and here."},{"Start":"02:08.260 ","End":"02:12.810","Text":"We\u0027re left with the x minus x_0 and y minus y_0."},{"Start":"02:12.810 ","End":"02:16.440","Text":"The x is 0.01 and the x_0 is 0."},{"Start":"02:16.440 ","End":"02:18.915","Text":"Y minus y_0 is this."},{"Start":"02:18.915 ","End":"02:25.535","Text":"Here again are the 0,0 and 2 that we had from here."},{"Start":"02:25.535 ","End":"02:28.090","Text":"We only need to look at the last one,"},{"Start":"02:28.090 ","End":"02:32.790","Text":"0.99 minus 1 is minus 0.01."},{"Start":"02:32.790 ","End":"02:38.310","Text":"This comes out to be minus 0.02."},{"Start":"02:38.310 ","End":"02:41.390","Text":"This is our approximation."},{"Start":"02:41.390 ","End":"02:43.805","Text":"Let\u0027s highlight the answer."},{"Start":"02:43.805 ","End":"02:45.740","Text":"Because nowadays we wouldn\u0027t use"},{"Start":"02:45.740 ","End":"02:49.580","Text":"this approximation because we live in the age of calculators,"},{"Start":"02:49.580 ","End":"02:54.935","Text":"and we could just compute this on the pocket scientific calculator."},{"Start":"02:54.935 ","End":"03:00.100","Text":"What we get is minus 0.0199998."},{"Start":"03:01.130 ","End":"03:04.130","Text":"It happens to be very close to this,"},{"Start":"03:04.130 ","End":"03:09.300","Text":"but normally you can\u0027t expect such good results. That\u0027s it."}],"ID":30962},{"Watched":false,"Name":"Exercise 2","Duration":"4m 32s","ChapterTopicVideoID":29358,"CourseChapterTopicPlaylistID":294462,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.695","Text":"In this exercise, we\u0027re asked to approximate the fourth root of 15.09"},{"Start":"00:07.695 ","End":"00:16.095","Text":"plus 0.99^2 using the total differential or the linear approximation formula."},{"Start":"00:16.095 ","End":"00:22.530","Text":"This is the formula for linear approximation of a function in 2 variables."},{"Start":"00:22.530 ","End":"00:25.365","Text":"In our case, we\u0027re going to take x_0,"},{"Start":"00:25.365 ","End":"00:35.190","Text":"y_0 to be 15,1 because 15 plus 1^2 is 16 and we know its fourth root, which is 2."},{"Start":"00:35.190 ","End":"00:40.795","Text":"The function we\u0027ll take based on this fourth root of x plus y^2."},{"Start":"00:40.795 ","End":"00:43.935","Text":"Again, x_0, y_0 is 15,1."},{"Start":"00:43.935 ","End":"00:50.055","Text":"The point we want is 15.09, 0.99."},{"Start":"00:50.055 ","End":"00:54.770","Text":"We\u0027ll need the derivatives of f with respect to x and y, and for differentiation,"},{"Start":"00:54.770 ","End":"00:59.555","Text":"it\u0027s better to write it as power of a 1/4 rather than fourth root."},{"Start":"00:59.555 ","End":"01:09.185","Text":"Df by dx is 1/4 times this thing to the power of a 1/4 minus 1 is minus 3/4,"},{"Start":"01:09.185 ","End":"01:13.430","Text":"and the inner derivative with respect to x is just 1."},{"Start":"01:13.430 ","End":"01:16.460","Text":"But when we differentiate with respect to y, it\u0027s similar,"},{"Start":"01:16.460 ","End":"01:18.425","Text":"but instead of the 1, we get 2y,"},{"Start":"01:18.425 ","End":"01:21.730","Text":"because we want the inner derivative with respect to y."},{"Start":"01:21.730 ","End":"01:23.670","Text":"At the point x_0,"},{"Start":"01:23.670 ","End":"01:25.905","Text":"y_0, which is 15,1,"},{"Start":"01:25.905 ","End":"01:30.120","Text":"we get fourth root of 15 plus 1 and fourth root of 16 is 2."},{"Start":"01:30.120 ","End":"01:35.135","Text":"That\u0027s why we chose 15,1 because it\u0027s close to what we want"},{"Start":"01:35.135 ","End":"01:41.600","Text":"and we know the answer to this it\u0027s just 2, yeah."},{"Start":"01:41.600 ","End":"01:47.700","Text":"Df by dx at this point is the 1/4 from here,"},{"Start":"01:47.700 ","End":"01:50.925","Text":"x^2 plus y^2 is 16 as we saw."},{"Start":"01:50.925 ","End":"01:55.720","Text":"The power of minus 3/4 means we can take the fourth root,"},{"Start":"01:55.720 ","End":"01:57.440","Text":"which is 2,"},{"Start":"01:57.440 ","End":"01:59.010","Text":"to the minus 1,"},{"Start":"01:59.010 ","End":"02:02.670","Text":"is a 1/2 and to the power of 3 is cubed,"},{"Start":"02:02.670 ","End":"02:06.240","Text":"so it\u0027s a 1/2^3 times a 1/4,"},{"Start":"02:06.240 ","End":"02:09.375","Text":"and that comes out to be 1/32."},{"Start":"02:09.375 ","End":"02:13.840","Text":"Similarly here it\u0027s basically the same thing except at the end instead of 1,"},{"Start":"02:13.840 ","End":"02:15.310","Text":"we have 2y,"},{"Start":"02:15.310 ","End":"02:17.860","Text":"which is 2 times 1."},{"Start":"02:17.860 ","End":"02:20.590","Text":"We\u0027re just going to get double the answer here,"},{"Start":"02:20.590 ","End":"02:23.200","Text":"and that is 1^16."},{"Start":"02:23.200 ","End":"02:26.450","Text":"Now, we have all that\u0027s necessary to substitute in here,"},{"Start":"02:26.450 ","End":"02:30.249","Text":"and what we get is that f at our point,"},{"Start":"02:30.249 ","End":"02:32.395","Text":"which is the expression we want,"},{"Start":"02:32.395 ","End":"02:39.375","Text":"is approximately 2 plus 1/32, that\u0027s this,"},{"Start":"02:39.375 ","End":"02:43.740","Text":"x minus x_0 is 15.09 minus 15,"},{"Start":"02:43.740 ","End":"02:49.145","Text":"and df by dy is 1^16 and y minus y_0 is this."},{"Start":"02:49.145 ","End":"02:51.545","Text":"By the way, if you wanted to make another estimate,"},{"Start":"02:51.545 ","End":"02:52.685","Text":"not at this point,"},{"Start":"02:52.685 ","End":"02:54.500","Text":"but something else. I don\u0027t know."},{"Start":"02:54.500 ","End":"02:58.840","Text":"15.070 and 0.96,"},{"Start":"02:58.840 ","End":"03:01.625","Text":"you wouldn\u0027t have to do the whole work again."},{"Start":"03:01.625 ","End":"03:05.060","Text":"You just have to at this point put in different numbers."},{"Start":"03:05.060 ","End":"03:12.245","Text":"Basically, this template is good for other numbers besides 15.09 and 0.99."},{"Start":"03:12.245 ","End":"03:16.445","Text":"Just mentioning it in case you had several similar estimates to make."},{"Start":"03:16.445 ","End":"03:22.560","Text":"Now, let\u0027s compute this 0.09 here is 9/100,"},{"Start":"03:22.720 ","End":"03:25.900","Text":"which makes it 9/3200,"},{"Start":"03:25.900 ","End":"03:27.629","Text":"if we go for fractions,"},{"Start":"03:27.629 ","End":"03:29.535","Text":"we could have gone a little decimal."},{"Start":"03:29.535 ","End":"03:33.630","Text":"Here, minus 0.01, which is 1/100,"},{"Start":"03:33.630 ","End":"03:38.610","Text":"at the 16th, minus 1/1600."},{"Start":"03:38.610 ","End":"03:40.455","Text":"This is 2/3200,"},{"Start":"03:40.455 ","End":"03:42.750","Text":"9 minus 2 is 7,"},{"Start":"03:42.750 ","End":"03:46.385","Text":"so we get 2 and 7/3200."},{"Start":"03:46.385 ","End":"03:48.190","Text":"Could leave it as a fraction."},{"Start":"03:48.190 ","End":"03:50.320","Text":"Let\u0027s go for decimal."},{"Start":"03:50.320 ","End":"03:53.775","Text":"If you divide 7 by 3200."},{"Start":"03:53.775 ","End":"03:55.450","Text":"Of course I did it on a calculator,"},{"Start":"03:55.450 ","End":"03:59.260","Text":"but it could by hand and we get this."},{"Start":"03:59.260 ","End":"04:02.465","Text":"This is the answer we\u0027re looking for."},{"Start":"04:02.465 ","End":"04:04.580","Text":"We could leave it at that."},{"Start":"04:04.580 ","End":"04:07.295","Text":"But we live in the age of calculators,"},{"Start":"04:07.295 ","End":"04:13.105","Text":"so you can check our result and just punch in this."},{"Start":"04:13.105 ","End":"04:17.460","Text":"What we get is very close to this,"},{"Start":"04:17.460 ","End":"04:22.745","Text":"in fact I took as many decimal places until the digits differed."},{"Start":"04:22.745 ","End":"04:25.190","Text":"Here we have a 5 and here we have a 0."},{"Start":"04:25.190 ","End":"04:27.290","Text":"Happens to be a very good approximation,"},{"Start":"04:27.290 ","End":"04:29.705","Text":"which is not usually to be expected."},{"Start":"04:29.705 ","End":"04:32.520","Text":"That\u0027s the end of this clip."}],"ID":30963},{"Watched":false,"Name":"Exercise 3","Duration":"3m 24s","ChapterTopicVideoID":29359,"CourseChapterTopicPlaylistID":294462,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.240","Text":"In this exercise, we compute the volume of a particular cylinder using the formula,"},{"Start":"00:07.240 ","End":"00:11.990","Text":"volume equals Pi radius squared times height."},{"Start":"00:12.000 ","End":"00:15.400","Text":"We\u0027re not given the actual measurements."},{"Start":"00:15.400 ","End":"00:23.060","Text":"But suppose that the maximum relative error for r is 2 percent,"},{"Start":"00:23.060 ","End":"00:30.399","Text":"which means that the actual value is plus or minus 2 percent of the measured value."},{"Start":"00:30.399 ","End":"00:36.385","Text":"It\u0027s called relative error because it\u0027s a percentage of the measurement itself."},{"Start":"00:36.385 ","End":"00:38.560","Text":"Let\u0027s suppose that the maximum error,"},{"Start":"00:38.560 ","End":"00:43.370","Text":"also relative error in h is plus or minus 4 percent."},{"Start":"00:43.370 ","End":"00:46.520","Text":"We have to use these to figure out what\u0027s"},{"Start":"00:46.520 ","End":"00:50.000","Text":"the maximum relative error in computing the volume."},{"Start":"00:50.000 ","End":"00:54.905","Text":"We know there\u0027s an error in r up to 2 percent and the error in h up to 4 percent,"},{"Start":"00:54.905 ","End":"00:58.330","Text":"up to how many percent can the error in v be?"},{"Start":"00:58.330 ","End":"01:00.705","Text":"We\u0027ll use the total differential."},{"Start":"01:00.705 ","End":"01:04.700","Text":"To use the customary letters x, y, z,"},{"Start":"01:04.700 ","End":"01:07.550","Text":"we\u0027ll write z equals a function of x,"},{"Start":"01:07.550 ","End":"01:09.095","Text":"y, which is Pix squared y,"},{"Start":"01:09.095 ","End":"01:10.790","Text":"so x is the radius,"},{"Start":"01:10.790 ","End":"01:11.945","Text":"y is the height,"},{"Start":"01:11.945 ","End":"01:13.745","Text":"z is the volume."},{"Start":"01:13.745 ","End":"01:16.235","Text":"The total differential,"},{"Start":"01:16.235 ","End":"01:18.170","Text":"and forgive me, I wrote them backwards."},{"Start":"01:18.170 ","End":"01:19.280","Text":"This should be the equal,"},{"Start":"01:19.280 ","End":"01:21.995","Text":"this should be the approximate. Never mind."},{"Start":"01:21.995 ","End":"01:25.565","Text":"This total differential approximates to,"},{"Start":"01:25.565 ","End":"01:27.755","Text":"replace d by Delta,"},{"Start":"01:27.755 ","End":"01:34.879","Text":"we get that Delta z is df by dx Delta x plus df by dy Delta y."},{"Start":"01:34.879 ","End":"01:36.665","Text":"Now, df by dx,"},{"Start":"01:36.665 ","End":"01:40.070","Text":"we can just differentiate is 2Pixy,"},{"Start":"01:40.070 ","End":"01:42.695","Text":"the x squared becomes 2x."},{"Start":"01:42.695 ","End":"01:45.005","Text":"If we differentiate with respect to y,"},{"Start":"01:45.005 ","End":"01:46.100","Text":"the y becomes 1,"},{"Start":"01:46.100 ","End":"01:47.780","Text":"so we get Pix squared."},{"Start":"01:47.780 ","End":"01:50.360","Text":"Notice that both of these are positive."},{"Start":"01:50.360 ","End":"01:52.370","Text":"When we take absolute value,"},{"Start":"01:52.370 ","End":"01:55.760","Text":"we don\u0027t have to take absolute value, they\u0027re already positive."},{"Start":"01:55.760 ","End":"01:59.210","Text":"The absolute value of Delta z is less"},{"Start":"01:59.210 ","End":"02:02.600","Text":"than or equal to the absolute value of this plus the absolute value of this."},{"Start":"02:02.600 ","End":"02:05.330","Text":"Now, fx and fy are positive,"},{"Start":"02:05.330 ","End":"02:08.090","Text":"so there don\u0027t need to be an absolute value."},{"Start":"02:08.090 ","End":"02:12.010","Text":"The 2 percent, 4 percent means that the absolute value of"},{"Start":"02:12.010 ","End":"02:17.495","Text":"Delta x is at most 0.02 times x,"},{"Start":"02:17.495 ","End":"02:19.300","Text":"and the absolute value of Delta y,"},{"Start":"02:19.300 ","End":"02:22.165","Text":"which is the absolute error in y,"},{"Start":"02:22.165 ","End":"02:27.230","Text":"is up to 4 percent of y, 0.04y."},{"Start":"02:27.230 ","End":"02:33.430","Text":"We get that absolute value of Delta z is less than or equal to fx 2Pixy,"},{"Start":"02:33.430 ","End":"02:37.960","Text":"Delta x less than or equal to 0.02x."},{"Start":"02:37.960 ","End":"02:39.610","Text":"Here, this."},{"Start":"02:39.610 ","End":"02:41.105","Text":"Here, this."},{"Start":"02:41.105 ","End":"02:46.451","Text":"Multiplying out 2 times 0.02 is 0.04Pixyx,"},{"Start":"02:46.451 ","End":"02:48.387","Text":"which means Pix squared y."},{"Start":"02:48.387 ","End":"02:50.894","Text":"Here also we get the same thing,"},{"Start":"02:50.894 ","End":"02:52.632","Text":"0.04 Pix squared y."},{"Start":"02:52.632 ","End":"02:58.980","Text":"Collecting together we have 0.08Pix squared y,"},{"Start":"02:58.980 ","End":"03:03.645","Text":"but Pix squared y is the same as z."},{"Start":"03:03.645 ","End":"03:12.500","Text":"We get that the difference in the volume is less than or equal to 0.08 of the volume,"},{"Start":"03:12.500 ","End":"03:14.190","Text":"which is 8 percent of z."},{"Start":"03:14.190 ","End":"03:18.545","Text":"What we can say is the maximum error is 8 percent,"},{"Start":"03:18.545 ","End":"03:21.260","Text":"it\u0027s a maximum relative error."},{"Start":"03:21.260 ","End":"03:24.990","Text":"That\u0027s the answer, 8 percent and we\u0027re done."}],"ID":30964},{"Watched":false,"Name":"Exercise 4","Duration":"3m 24s","ChapterTopicVideoID":29354,"CourseChapterTopicPlaylistID":294462,"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 several things to do."},{"Start":"00:03.930 ","End":"00:09.210","Text":"First of all, we have to compute the length of the diagonal of 10 by"},{"Start":"00:09.210 ","End":"00:11.520","Text":"24 rectangles because we can use"},{"Start":"00:11.520 ","End":"00:17.610","Text":"Pythagoras theorem and then we\u0027re given that the length changes to new dimensions."},{"Start":"00:17.610 ","End":"00:23.250","Text":"We first of all have to estimate the change in the length of the diagonal using"},{"Start":"00:23.250 ","End":"00:30.990","Text":"the total differential and then using a calculator to compute the exact change."},{"Start":"00:30.990 ","End":"00:35.065","Text":"Here\u0027s the rectangle after the change of dimensions."},{"Start":"00:35.065 ","End":"00:37.825","Text":"I\u0027m going to use Pythagoras\u0027 rule,"},{"Start":"00:37.825 ","End":"00:42.020","Text":"and this is what it says if that is the diagonal and x and y are"},{"Start":"00:42.020 ","End":"00:46.585","Text":"the sides then that is the square root of x- squared plus y-squared."},{"Start":"00:46.585 ","End":"00:49.930","Text":"In the case of 10 and 24,"},{"Start":"00:49.930 ","End":"00:52.985","Text":"the diagonal turns out to be a whole number 26."},{"Start":"00:52.985 ","End":"00:55.850","Text":"It\u0027s actually double a famous triangle,"},{"Start":"00:55.850 ","End":"00:57.980","Text":"the 5, 12, 13 triangle,"},{"Start":"00:57.980 ","End":"00:59.070","Text":"and this is just double."},{"Start":"00:59.070 ","End":"01:01.715","Text":"So twice 13 is 26."},{"Start":"01:01.715 ","End":"01:07.370","Text":"This is the formula for the total differential in two variables and it"},{"Start":"01:07.370 ","End":"01:13.588","Text":"approximates to put Delta instead of d and we have the following."},{"Start":"01:13.588 ","End":"01:16.160","Text":"Should really be approximately equal to."},{"Start":"01:16.160 ","End":"01:21.554","Text":"Now we need the partial derivatives df by dx and df by dy."},{"Start":"01:21.554 ","End":"01:25.520","Text":"Df by dx is the derivative of this with respect to x."},{"Start":"01:25.520 ","End":"01:28.370","Text":"The derivative of the square root is"},{"Start":"01:28.370 ","End":"01:32.530","Text":"1 over twice the square root and then the anti-derivative is 2x."},{"Start":"01:32.530 ","End":"01:34.160","Text":"2 with the 2 cancels,"},{"Start":"01:34.160 ","End":"01:35.768","Text":"the x goes on the top."},{"Start":"01:35.768 ","End":"01:37.674","Text":"X over the square root of x squared plus y"},{"Start":"01:37.674 ","End":"01:41.910","Text":"squared and similarly for fy just get y instead of x."},{"Start":"01:41.910 ","End":"01:47.795","Text":"The Deltas are 0.4 because this grew from 10 to 10.4."},{"Start":"01:47.795 ","End":"01:50.915","Text":"This should be Delta y."},{"Start":"01:50.915 ","End":"01:53.220","Text":"Now at the point 10,"},{"Start":"01:53.220 ","End":"02:00.615","Text":"24 the derivative with respect to x= x is 10."},{"Start":"02:00.615 ","End":"02:07.085","Text":"The square root we already know is 26 and df by dy will be 24 over 26."},{"Start":"02:07.085 ","End":"02:10.055","Text":"Then substituting in the formula for Delta z,"},{"Start":"02:10.055 ","End":"02:17.990","Text":"we get that Delta z is 10 over 26 which is fx times Delta x is 0.4 and the other bit;"},{"Start":"02:17.990 ","End":"02:21.065","Text":"minus 24 over 6 times 0.1,"},{"Start":"02:21.065 ","End":"02:23.945","Text":"the minus from the minus 0.1."},{"Start":"02:23.945 ","End":"02:25.985","Text":"If we put all this over 26,"},{"Start":"02:25.985 ","End":"02:29.135","Text":"we have 10 times 0.4 which is 4."},{"Start":"02:29.135 ","End":"02:32.630","Text":"24 times 0.1 is 2.4."},{"Start":"02:32.630 ","End":"02:36.625","Text":"We get 1.6 over 26."},{"Start":"02:36.625 ","End":"02:40.845","Text":"16 over 260 divide everything by 4, 4 over 65."},{"Start":"02:40.845 ","End":"02:42.660","Text":"If you want it in decimal,"},{"Start":"02:42.660 ","End":"02:45.695","Text":"either do long division or go to a calculator."},{"Start":"02:45.695 ","End":"02:48.620","Text":"This is what we have in decimal."},{"Start":"02:48.620 ","End":"02:50.330","Text":"We found two of the three things."},{"Start":"02:50.330 ","End":"02:53.180","Text":"We found the length of the original diagonal,"},{"Start":"02:53.180 ","End":"02:56.735","Text":"we found an estimate of the change,"},{"Start":"02:56.735 ","End":"02:59.900","Text":"and now we want the exact value."},{"Start":"02:59.900 ","End":"03:07.925","Text":"Using a calculator this computes to be 26.0647,"},{"Start":"03:07.925 ","End":"03:12.900","Text":"and then if we subtract the 26 to get just the Delta;"},{"Start":"03:12.900 ","End":"03:19.220","Text":"the difference, we get 0.0647 which is close."},{"Start":"03:19.220 ","End":"03:25.500","Text":"Anyway, this is what it is and that concludes this question."}],"ID":30965},{"Watched":false,"Name":"Exercise 5","Duration":"4m 27s","ChapterTopicVideoID":29355,"CourseChapterTopicPlaylistID":294462,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"This exercise relates to the diagonal of a cuboid,"},{"Start":"00:05.250 ","End":"00:08.985","Text":"sometimes called the rectangular parallelepiped,"},{"Start":"00:08.985 ","End":"00:13.860","Text":"basically a 3D rectangle and there\u0027s a formula for the length of"},{"Start":"00:13.860 ","End":"00:21.720","Text":"the diagonal which generalized Pythagoras in the plane that\u0027s just x^2 plus y^2,"},{"Start":"00:21.720 ","End":"00:27.825","Text":"but in 3D, the formula extends to the square root of x^2 plus y^2 plus z^2,"},{"Start":"00:27.825 ","End":"00:29.850","Text":"where these are the 3 dimensions."},{"Start":"00:29.850 ","End":"00:31.755","Text":"I don\u0027t know the length,"},{"Start":"00:31.755 ","End":"00:33.600","Text":"the width, and the height."},{"Start":"00:33.600 ","End":"00:37.175","Text":"We have to find the maximum relative error in l,"},{"Start":"00:37.175 ","End":"00:41.060","Text":"if the maximum error in each of the sides x, y,"},{"Start":"00:41.060 ","End":"00:45.095","Text":"and z is 5 percent meaning up to"},{"Start":"00:45.095 ","End":"00:49.835","Text":"plus 5 percent and down to minus 5 percent is the range of error."},{"Start":"00:49.835 ","End":"00:52.010","Text":"We\u0027re going to use a total differential,"},{"Start":"00:52.010 ","End":"00:54.665","Text":"but in this question,"},{"Start":"00:54.665 ","End":"00:58.130","Text":"we have a function of 3 variables so we\u0027re going to use"},{"Start":"00:58.130 ","End":"01:01.822","Text":"the 3 variable total differential."},{"Start":"01:01.822 ","End":"01:05.510","Text":"For that I want you remember the differentiation rule."},{"Start":"01:05.510 ","End":"01:08.930","Text":"We\u0027re going to use this a few times and we have to differentiate"},{"Start":"01:08.930 ","End":"01:12.785","Text":"the square root of something and that something is maybe a function of x."},{"Start":"01:12.785 ","End":"01:14.060","Text":"When we differentiate it,"},{"Start":"01:14.060 ","End":"01:17.360","Text":"we get 1 over twice this something square"},{"Start":"01:17.360 ","End":"01:22.400","Text":"root times the anti-derivative is the derivative of what was under the square root."},{"Start":"01:22.400 ","End":"01:24.490","Text":"Our function f(x, y,"},{"Start":"01:24.490 ","End":"01:26.110","Text":"z) which is l,"},{"Start":"01:26.110 ","End":"01:29.779","Text":"use l for the the diagonal."},{"Start":"01:29.779 ","End":"01:36.290","Text":"The formula for the total differential in 3D starts out like 2D,"},{"Start":"01:36.290 ","End":"01:38.195","Text":"but there\u0027s just 1 extra term."},{"Start":"01:38.195 ","End":"01:40.220","Text":"Instead of just having f_x, dx,"},{"Start":"01:40.220 ","End":"01:42.845","Text":"and f_y dy we have plus f_z, d_z."},{"Start":"01:42.845 ","End":"01:49.805","Text":"This approximates to this formula where we replace every d by a delta,"},{"Start":"01:49.805 ","End":"01:54.605","Text":"we get the following and I wrote equals but it\u0027s approximately equal."},{"Start":"01:54.605 ","End":"01:57.970","Text":"When we say that the relative error is 5 percent,"},{"Start":"01:57.970 ","End":"02:00.665","Text":"it means that the error can be"},{"Start":"02:00.665 ","End":"02:05.300","Text":"up to 5 percent over or 5 percent under or anything in between."},{"Start":"02:05.300 ","End":"02:11.470","Text":"We can say that the absolute value of delta x is 5 percent of the original x,"},{"Start":"02:11.470 ","End":"02:13.995","Text":"5 percent is 0.05."},{"Start":"02:13.995 ","End":"02:17.475","Text":"Similarly for delta y and delta z same thing."},{"Start":"02:17.475 ","End":"02:22.580","Text":"We need the partial derivatives with respect to x and using this rule,"},{"Start":"02:22.580 ","End":"02:29.285","Text":"we have differentiating this 1 over 2 the square root of this"},{"Start":"02:29.285 ","End":"02:36.420","Text":"and then the inner derivative of this with respect to x is 2x."},{"Start":"02:36.420 ","End":"02:38.520","Text":"This 2 cancels with this 2,"},{"Start":"02:38.520 ","End":"02:42.380","Text":"we can put the x on top and we get x over the square root"},{"Start":"02:42.380 ","End":"02:46.700","Text":"of x^2 plus y^2 plus z^2 and similarly f_y and f_z,"},{"Start":"02:46.700 ","End":"02:50.620","Text":"the same thing, just replace the x here by y and then by z."},{"Start":"02:50.620 ","End":"02:56.750","Text":"Note that these are all positive quantities because the sides x,"},{"Start":"02:56.750 ","End":"03:00.680","Text":"y, and z are positive and the square root is positive."},{"Start":"03:00.680 ","End":"03:05.730","Text":"Because they\u0027re positive, when we take the absolute value they stay the same."},{"Start":"03:05.730 ","End":"03:08.660","Text":"Now the absolute value of a sum is less than or equal to"},{"Start":"03:08.660 ","End":"03:11.645","Text":"the sum of the absolute values and because f_x,"},{"Start":"03:11.645 ","End":"03:12.830","Text":"f_y, f_z are positive,"},{"Start":"03:12.830 ","End":"03:15.335","Text":"we can take them outside of the absolute value."},{"Start":"03:15.335 ","End":"03:19.865","Text":"Now replace the deltas on the right by their estimates."},{"Start":"03:19.865 ","End":"03:22.325","Text":"Delta x is less than or equal to this,"},{"Start":"03:22.325 ","End":"03:24.140","Text":"delta y less than or equal to this,"},{"Start":"03:24.140 ","End":"03:27.335","Text":"delta z less than or equal to this and then times the f_x,"},{"Start":"03:27.335 ","End":"03:30.900","Text":"f_y, f_z here and here and here."},{"Start":"03:30.900 ","End":"03:32.325","Text":"F_x, f_y,"},{"Start":"03:32.325 ","End":"03:36.779","Text":"and f_z we can replace them by what\u0027s here."},{"Start":"03:36.779 ","End":"03:42.735","Text":"Next we can take 0.05 outside the brackets."},{"Start":"03:42.735 ","End":"03:46.355","Text":"Then what we\u0027re left with is x times x over the square root"},{"Start":"03:46.355 ","End":"03:50.030","Text":"plus y times y over the square root plus z times z over the square root."},{"Start":"03:50.030 ","End":"03:53.845","Text":"In other words, x^2 plus y^2 squared plus z^2 over the square root."},{"Start":"03:53.845 ","End":"03:59.165","Text":"Basic algebra, a over the square root of a is just the square root of a."},{"Start":"03:59.165 ","End":"04:06.715","Text":"This thing comes down to 0.05 square root of x^2 plus y^2 plus z^2,"},{"Start":"04:06.715 ","End":"04:13.335","Text":"but this is exactly l. This is equal to 0.05l."},{"Start":"04:13.335 ","End":"04:20.180","Text":"The error in l is bounded in absolute value by 5 percent of l. We can"},{"Start":"04:20.180 ","End":"04:27.840","Text":"say that the maximum relative error in l is 5 percent and that concludes this exercise."}],"ID":30966}],"Thumbnail":null,"ID":294462},{"Name":"Taylor Polynomials for Functions of Two Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"7m 12s","ChapterTopicVideoID":29365,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/29365.jpeg","UploadDate":"2022-06-28T06:40:12.1400000","DurationForVideoObject":"PT7M12S","Description":null,"MetaTitle":"Introduction: Video + Workbook | Proprep","MetaDescription":"The Total Differential and Taylor Polynomials for Functions of Two Variables - Taylor Polynomials for Functions of Two Variables. 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/the-total-differential-and-taylor-polynomials-for-functions-of-two-variables/taylor-polynomials-for-functions-of-two-variables/vid30975","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"In this clip, we\u0027re going to be talking about"},{"Start":"00:02.790 ","End":"00:06.930","Text":"Taylor polynomials for functions of 2 variables."},{"Start":"00:06.930 ","End":"00:12.300","Text":"But first, we\u0027re going to review Taylor Polynomials for functions of 1 variable."},{"Start":"00:12.300 ","End":"00:14.160","Text":"The first part is a review,"},{"Start":"00:14.160 ","End":"00:16.695","Text":"but let\u0027s say y is a function of x and suppose,"},{"Start":"00:16.695 ","End":"00:23.580","Text":"f has a certain number n of derivatives at a point x=c in the domain,"},{"Start":"00:23.580 ","End":"00:27.600","Text":"then the polynomial as shown here is"},{"Start":"00:27.600 ","End":"00:31.830","Text":"called the nth degree Taylor polynomial for the function f at"},{"Start":"00:31.830 ","End":"00:35.910","Text":"the point c. Taylor polynomials can be used to"},{"Start":"00:35.910 ","End":"00:41.265","Text":"approximate the function f near the point x=c."},{"Start":"00:41.265 ","End":"00:47.090","Text":"I haven\u0027t written it, but the approximations get better and better as n increases,"},{"Start":"00:47.090 ","End":"00:51.380","Text":"and they also improve as x gets closer to"},{"Start":"00:51.380 ","End":"00:57.260","Text":"c. Let\u0027s start with the first-degree Taylor polynomial and"},{"Start":"00:57.260 ","End":"01:03.770","Text":"that\u0027s the same as the linear approximation P_1 (x) but"},{"Start":"01:03.770 ","End":"01:05.180","Text":"that is equal to f(c) plus f\"(c)(x minus c)"},{"Start":"01:05.180 ","End":"01:10.940","Text":"c. It\u0027s just the first couple of terms from here."},{"Start":"01:10.940 ","End":"01:16.860","Text":"That approximation for f(x) see a picture for that."},{"Start":"01:16.860 ","End":"01:20.460","Text":"The line goes through the point a,"},{"Start":"01:20.460 ","End":"01:24.770","Text":"f(a) and has the same slope as f at a,"},{"Start":"01:24.770 ","End":"01:28.160","Text":"meaning that it is the tangent line here."},{"Start":"01:28.160 ","End":"01:33.185","Text":"We see that when x gets close to a,"},{"Start":"01:33.185 ","End":"01:38.180","Text":"the difference between the function and the tangent line gets smaller and smaller."},{"Start":"01:38.180 ","End":"01:41.420","Text":"The tangent is an approximation for the function."},{"Start":"01:41.420 ","End":"01:44.975","Text":"The tangent and the function have 2 things in common."},{"Start":"01:44.975 ","End":"01:50.610","Text":"The value at a is common and the slope at a is common."},{"Start":"01:50.610 ","End":"01:51.950","Text":"That\u0027s first-degree."},{"Start":"01:51.950 ","End":"01:54.080","Text":"Let\u0027s go on to second-degree,"},{"Start":"01:54.080 ","End":"01:56.479","Text":"which is better approximation,"},{"Start":"01:56.479 ","End":"02:02.410","Text":"and it\u0027s called the quadratic approximation or second-degree Taylor polynomial."},{"Start":"02:02.410 ","End":"02:07.020","Text":"For this, we just take n=2,"},{"Start":"02:07.020 ","End":"02:10.665","Text":"which means we stop after this term."},{"Start":"02:10.665 ","End":"02:17.840","Text":"What we get is f(c), f\u0027(c)(x minus c)."},{"Start":"02:17.840 ","End":"02:19.400","Text":"Whoops, I forgot the plus here."},{"Start":"02:19.400 ","End":"02:21.380","Text":"Okay, I\u0027ve added it now."},{"Start":"02:21.380 ","End":"02:28.490","Text":"Now the value of the function and the first derivative and the second derivative"},{"Start":"02:28.490 ","End":"02:31.370","Text":"are the same for f(x) and for"},{"Start":"02:31.370 ","End":"02:35.870","Text":"P_2(x) whereas the first-degree approximation only had 2 things in common."},{"Start":"02:35.870 ","End":"02:38.075","Text":"Let\u0027s show that in a picture."},{"Start":"02:38.075 ","End":"02:41.730","Text":"This is an illustration of y equals e^x."},{"Start":"02:41.730 ","End":"02:45.365","Text":"The first-degree approximation is the line,"},{"Start":"02:45.365 ","End":"02:50.270","Text":"and it has the point in common and the tangent here,"},{"Start":"02:50.270 ","End":"02:53.794","Text":"whereas if we take a second-degree approximation,"},{"Start":"02:53.794 ","End":"02:57.950","Text":"then the point isn\u0027t common and the tangent or"},{"Start":"02:57.950 ","End":"03:02.300","Text":"first derivative and the second derivative, also in common."},{"Start":"03:02.300 ","End":"03:07.750","Text":"It sticks closer to the function than just a linear approximation."},{"Start":"03:07.750 ","End":"03:11.630","Text":"Now, we\u0027ll turn to functions of 2 variables."},{"Start":"03:11.630 ","End":"03:16.045","Text":"Again, we\u0027ll start with first-degree and then go to second-degree."},{"Start":"03:16.045 ","End":"03:20.400","Text":"For a function, I should have said call it f of"},{"Start":"03:20.400 ","End":"03:25.040","Text":"2 variables whose first partial derivatives exist at this point."},{"Start":"03:25.040 ","End":"03:31.820","Text":"The first-degree Taylor polynomial of f for (x,y) near(a,b) is the following,"},{"Start":"03:31.820 ","End":"03:36.380","Text":"L for linear L(x,y) the value of the function at the point plus the"},{"Start":"03:36.380 ","End":"03:42.920","Text":"derivative times (x minus a) plus the derivative respect to y(y minus b)."},{"Start":"03:42.920 ","End":"03:48.280","Text":"This function, L(x,y) approximates f(x,y) as long as x,"},{"Start":"03:48.280 ","End":"03:52.080","Text":"y are near(a,b) the nearer the better,"},{"Start":"03:52.080 ","End":"03:56.200","Text":"although we can\u0027t say how good the approximation is."},{"Start":"03:56.200 ","End":"04:02.165","Text":"This is also called the linear approximation or the tangent plane approximation."},{"Start":"04:02.165 ","End":"04:07.685","Text":"Use a symbol approximately equal to to say that f(x,y) is approximately equal to L(x,y)."},{"Start":"04:07.685 ","End":"04:12.820","Text":"In other words, f(x,y) could be approximated by this expression."},{"Start":"04:12.820 ","End":"04:15.720","Text":"Now some notes. First of all,"},{"Start":"04:15.720 ","End":"04:22.130","Text":"L(x,y) is actually the equation of the tangent plane of f at the point (a,b)."},{"Start":"04:22.130 ","End":"04:23.585","Text":"Illustration, at the end."},{"Start":"04:23.585 ","End":"04:29.045","Text":"Notice that the first partial derivatives of L are just f_x and f_y,"},{"Start":"04:29.045 ","End":"04:35.590","Text":"same as f. The partial derivatives are same and the value at (a,b) is also the same."},{"Start":"04:35.590 ","End":"04:40.525","Text":"L(a,b) is f(a,b) because this becomes 0 and this becomes 0."},{"Start":"04:40.525 ","End":"04:43.940","Text":"Now, the linear approximation is closely"},{"Start":"04:43.940 ","End":"04:47.255","Text":"related to another concept called the total differential,"},{"Start":"04:47.255 ","End":"04:49.610","Text":"which is why in the exercises,"},{"Start":"04:49.610 ","End":"04:53.492","Text":"will only do second-degree Taylor polynomials"},{"Start":"04:53.492 ","End":"04:57.380","Text":"because first-degree will be very similar to total differential,"},{"Start":"04:57.380 ","End":"04:59.285","Text":"which will be a separate topic,"},{"Start":"04:59.285 ","End":"05:02.450","Text":"and to avoid duplication of exercises."},{"Start":"05:02.450 ","End":"05:04.615","Text":"Second-degree."},{"Start":"05:04.615 ","End":"05:10.220","Text":"We assume that f has first and second partial derivatives at"},{"Start":"05:10.220 ","End":"05:15.720","Text":"the point (a,b) and the second-degree Taylor polynomial of f for x,"},{"Start":"05:15.720 ","End":"05:21.100","Text":"y near a, b is Q for quadratic (x,y)."},{"Start":"05:21.100 ","End":"05:28.440","Text":"The same beginning as for the linear and then the extra bit, 1/2 factorial."},{"Start":"05:28.440 ","End":"05:30.770","Text":"There\u0027s a reason for the factorial"},{"Start":"05:30.770 ","End":"05:35.210","Text":"because the third-degree will get 1/3 factorial times something,"},{"Start":"05:35.210 ","End":"05:38.210","Text":"but we won\u0027t get into third-degree here. 1/2 factorial."},{"Start":"05:38.210 ","End":"05:42.455","Text":"This is the partial derivative with respect to x twice,"},{"Start":"05:42.455 ","End":"05:45.050","Text":"here are the mixed partial derivative and here are the partial"},{"Start":"05:45.050 ","End":"05:48.040","Text":"derivative twice with respect to y."},{"Start":"05:48.040 ","End":"05:50.355","Text":"Here we have (x minus a)^2."},{"Start":"05:50.355 ","End":"05:55.380","Text":"Here we have mixture (x minus a) and (y minus b) and here(y minus b)^2."},{"Start":"05:55.380 ","End":"05:57.510","Text":"There\u0027s a 2 here, really,"},{"Start":"05:57.510 ","End":"06:01.800","Text":"because there\u0027s 1 for f_xy and 1 for f_yx, which are the same."},{"Start":"06:01.800 ","End":"06:03.050","Text":"It does in the linear case,"},{"Start":"06:03.050 ","End":"06:06.245","Text":"the quadratic is used to approximate the function"},{"Start":"06:06.245 ","End":"06:09.725","Text":"f. It\u0027s a better approximation than the linear."},{"Start":"06:09.725 ","End":"06:12.565","Text":"In general, the closer (x,y) is to (a,b),"},{"Start":"06:12.565 ","End":"06:14.735","Text":"the better the approximation."},{"Start":"06:14.735 ","End":"06:19.190","Text":"Then we just copy this expression with approximately equal to and put"},{"Start":"06:19.190 ","End":"06:24.370","Text":"f instead of Q and we get this is approximated by this."},{"Start":"06:24.370 ","End":"06:32.195","Text":"As before, the first derivatives and second derivatives are the same for Q as for f,"},{"Start":"06:32.195 ","End":"06:34.760","Text":"as is the value at the point (a,b)."},{"Start":"06:34.760 ","End":"06:38.570","Text":"Now some pictures, I got them from the Internet."},{"Start":"06:38.570 ","End":"06:41.780","Text":"Here\u0027s a picture of the first-degree approximation,"},{"Start":"06:41.780 ","End":"06:43.100","Text":"which is the tangent plane,"},{"Start":"06:43.100 ","End":"06:44.840","Text":"That\u0027s the point (a,b)."},{"Start":"06:44.840 ","End":"06:48.320","Text":"We have some function f(x,y) so the tangent plane"},{"Start":"06:48.320 ","End":"06:54.050","Text":"approximates the function and a quadratic approximation."},{"Start":"06:54.050 ","End":"07:01.415","Text":"The blue one is the function and the orange one is the approximation,"},{"Start":"07:01.415 ","End":"07:02.810","Text":"and it\u0027s not a plane."},{"Start":"07:02.810 ","End":"07:07.370","Text":"It hugs more closely the function than the linear one."},{"Start":"07:07.370 ","End":"07:09.830","Text":"If this helps, good."},{"Start":"07:09.830 ","End":"07:12.904","Text":"Anyway, that concludes this clip."}],"ID":30975},{"Watched":false,"Name":"Exercise 1","Duration":"2m 42s","ChapterTopicVideoID":29366,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we\u0027re asked to find the second degree"},{"Start":"00:04.170 ","End":"00:08.550","Text":"Taylor polynomial for the function f(x,"},{"Start":"00:08.550 ","End":"00:13.320","Text":"y) which is equal to x^3 y plus 3y minus 2 around the point a,"},{"Start":"00:13.320 ","End":"00:15.645","Text":"b, which is 1,2."},{"Start":"00:15.645 ","End":"00:20.775","Text":"The second degree polynomial is also called the quadratic polynomial,"},{"Start":"00:20.775 ","End":"00:22.770","Text":"we call it q(x, y)."},{"Start":"00:22.770 ","End":"00:25.155","Text":"It\u0027s given by the following formula."},{"Start":"00:25.155 ","End":"00:26.460","Text":"That\u0027s in general."},{"Start":"00:26.460 ","End":"00:30.555","Text":"In our case a,b is the point 1,2,"},{"Start":"00:30.555 ","End":"00:32.685","Text":"so we get this."},{"Start":"00:32.685 ","End":"00:35.760","Text":"What we need to find are all these f(1,"},{"Start":"00:35.760 ","End":"00:41.255","Text":"2) f_x(1, 2) and so on."},{"Start":"00:41.255 ","End":"00:48.380","Text":"The function, the first derivatives and the second derivatives of f. Just copying this,"},{"Start":"00:48.380 ","End":"00:52.900","Text":"f is x^2 y plus 3y minus 2."},{"Start":"00:52.900 ","End":"00:57.075","Text":"The first derivative f_x is 2xy."},{"Start":"00:57.075 ","End":"00:59.840","Text":"Y is the constant. This part goes to 0."},{"Start":"00:59.840 ","End":"01:08.660","Text":"We get 2x times y. F_y is x^2 plus the 3 from here."},{"Start":"01:08.660 ","End":"01:11.840","Text":"Now we want second order derivatives."},{"Start":"01:11.840 ","End":"01:16.355","Text":"F_x with respect to x gives us 2y."},{"Start":"01:16.355 ","End":"01:19.055","Text":"If we differentiate this with respect to y,"},{"Start":"01:19.055 ","End":"01:20.480","Text":"we get 2x,"},{"Start":"01:20.480 ","End":"01:24.395","Text":"which is also what you get if you differentiate this with respect to x."},{"Start":"01:24.395 ","End":"01:25.880","Text":"We have one more to go,"},{"Start":"01:25.880 ","End":"01:30.305","Text":"f_yy and that is equal to 0,"},{"Start":"01:30.305 ","End":"01:32.660","Text":"because this is a function of x,"},{"Start":"01:32.660 ","End":"01:34.340","Text":"so its derivative is 0."},{"Start":"01:34.340 ","End":"01:36.185","Text":"Now for each of these,"},{"Start":"01:36.185 ","End":"01:39.890","Text":"we want its value at 1,2."},{"Start":"01:39.890 ","End":"01:42.390","Text":"We put x is 1, y equals 2."},{"Start":"01:42.390 ","End":"01:49.455","Text":"Here we get 1 times 2 plus 3 times 2 minus 2, so it\u0027s 6."},{"Start":"01:49.455 ","End":"01:52.440","Text":"The rest, it\u0027s all just substitutions."},{"Start":"01:52.440 ","End":"01:54.405","Text":"X is 1, y is 2,"},{"Start":"01:54.405 ","End":"01:55.800","Text":"and we get the following."},{"Start":"01:55.800 ","End":"01:59.315","Text":"Now we\u0027re going to substitute these here."},{"Start":"01:59.315 ","End":"02:02.270","Text":"What we get is that q(x,"},{"Start":"02:02.270 ","End":"02:03.950","Text":"y) is 6,"},{"Start":"02:03.950 ","End":"02:07.050","Text":"4, 4, 4, 2, 0."},{"Start":"02:11.220 ","End":"02:13.840","Text":"Now just simplifying."},{"Start":"02:13.840 ","End":"02:15.880","Text":"Here, we have 0.5,"},{"Start":"02:15.880 ","End":"02:18.184","Text":"so this 4 is just a 2,"},{"Start":"02:18.184 ","End":"02:25.190","Text":"and 2 times 2 just becomes 2 and this is still 0, so this disappears."},{"Start":"02:25.190 ","End":"02:27.185","Text":"This is the answer."},{"Start":"02:27.185 ","End":"02:28.735","Text":"Just to remind you,"},{"Start":"02:28.735 ","End":"02:31.210","Text":"this is an approximation for f(x, y)."},{"Start":"02:31.210 ","End":"02:36.270","Text":"F(x, y) is approximately equal to what it says here,"},{"Start":"02:36.270 ","End":"02:42.220","Text":"when x is close to 1 and y is close to 2. We\u0027re done."}],"ID":30976},{"Watched":false,"Name":"Exercise 2","Duration":"3m 36s","ChapterTopicVideoID":29367,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.000","Text":"In this exercise, we\u0027re going to find"},{"Start":"00:03.000 ","End":"00:10.575","Text":"the quadratic Taylor approximation for this function of f(x,y) around the origin."},{"Start":"00:10.575 ","End":"00:18.030","Text":"The general solution for the second degree or quadratic Taylor polynomial is this."},{"Start":"00:18.030 ","End":"00:19.820","Text":"And if we let a,"},{"Start":"00:19.820 ","End":"00:21.720","Text":"b equal 0,"},{"Start":"00:21.720 ","End":"00:25.710","Text":"0 then this becomes just this."},{"Start":"00:25.710 ","End":"00:28.920","Text":"Now we need to find all these f(0,0),"},{"Start":"00:28.920 ","End":"00:34.485","Text":"f_x(0,0) of these partial derivatives."},{"Start":"00:34.485 ","End":"00:39.400","Text":"Here\u0027s the function f itself and at 0,0,"},{"Start":"00:39.400 ","End":"00:44.570","Text":"it\u0027s equal to 0 because y is 0 and x is 0,"},{"Start":"00:44.570 ","End":"00:48.380","Text":"so x minus y is 0 so natural log of 1 is 0."},{"Start":"00:48.380 ","End":"00:51.140","Text":"The derivative with respect to x,"},{"Start":"00:51.140 ","End":"00:59.950","Text":"1 plus y is a constant like and we take 1/1 plus x minus y and the anti-derivative is 1."},{"Start":"00:59.950 ","End":"01:03.455","Text":"So we have this. Then when x and y are 0,"},{"Start":"01:03.455 ","End":"01:07.070","Text":"we get 1/1 which is 1."},{"Start":"01:07.070 ","End":"01:12.630","Text":"F with respect to y use a product rule."},{"Start":"01:13.880 ","End":"01:19.310","Text":"So we have 1 plus y times a derivative of this,"},{"Start":"01:19.310 ","End":"01:23.945","Text":"derivative of this is 1/1 plus x minus y times minus 1,"},{"Start":"01:23.945 ","End":"01:25.900","Text":"that\u0027s this minus here."},{"Start":"01:25.900 ","End":"01:30.980","Text":"Then plus the derivative of this times this with respect to y,"},{"Start":"01:30.980 ","End":"01:34.460","Text":"the derivative of this is 1 and this as is."},{"Start":"01:34.460 ","End":"01:38.720","Text":"So we get this expression for df by dy."},{"Start":"01:38.720 ","End":"01:41.845","Text":"If we put x equals y equals 0,"},{"Start":"01:41.845 ","End":"01:46.500","Text":"we get this is natural log of 1 is 0,"},{"Start":"01:46.500 ","End":"01:48.825","Text":"the denominator is 1,"},{"Start":"01:48.825 ","End":"01:51.240","Text":"the numerator is minus 1,"},{"Start":"01:51.240 ","End":"01:55.260","Text":"get minus 1/1 plus 0 minus 1."},{"Start":"01:55.260 ","End":"01:59.330","Text":"Then second derivative of f with respect to x,"},{"Start":"01:59.330 ","End":"02:02.320","Text":"we get by differentiating this."},{"Start":"02:02.320 ","End":"02:05.840","Text":"And then if we differentiate this with respect to y,"},{"Start":"02:05.840 ","End":"02:08.210","Text":"I\u0027ll leave you to check the details."},{"Start":"02:08.210 ","End":"02:12.695","Text":"It\u0027s using the product rule and then simplifying."},{"Start":"02:12.695 ","End":"02:16.895","Text":"And the last one we get is this with respect to y."},{"Start":"02:16.895 ","End":"02:19.955","Text":"Again, I\u0027ll leave you check the details."},{"Start":"02:19.955 ","End":"02:24.840","Text":"Continue substituting, we get minus 1."},{"Start":"02:24.970 ","End":"02:28.490","Text":"And here, if x is 0 and y is 0,"},{"Start":"02:28.490 ","End":"02:32.140","Text":"we get 2/1 is 2."},{"Start":"02:32.140 ","End":"02:35.480","Text":"And here, when x is 0 and y is 0,"},{"Start":"02:35.480 ","End":"02:38.945","Text":"you know what, let\u0027s just simplify it first before substituting."},{"Start":"02:38.945 ","End":"02:44.025","Text":"And what we get on the numerator here is minus 1,"},{"Start":"02:44.025 ","End":"02:46.375","Text":"minus 1 is minus 2."},{"Start":"02:46.375 ","End":"02:50.720","Text":"We have minus x and then we have plus y minus y,"},{"Start":"02:50.720 ","End":"02:52.550","Text":"so it\u0027s minus 2 minus x."},{"Start":"02:52.550 ","End":"03:01.080","Text":"Now we can substitute x equals y equals 0 and we get here minus 2/1,"},{"Start":"03:01.080 ","End":"03:05.710","Text":"minus 1/1, minus 2 minus 1 is minus 3."},{"Start":"03:05.710 ","End":"03:14.545","Text":"Now we want all of these substituted in the expression we had for Q(x, y)."},{"Start":"03:14.545 ","End":"03:17.210","Text":"And if we substitute over all of them,"},{"Start":"03:17.210 ","End":"03:19.310","Text":"we get the following,"},{"Start":"03:19.310 ","End":"03:24.050","Text":"and we can also say this by writing that f(x,"},{"Start":"03:24.050 ","End":"03:29.720","Text":"y) is approximately equal to up to a quadratic approximation to"},{"Start":"03:29.720 ","End":"03:36.990","Text":"the following polynomial in x and y and that concludes this exercise."}],"ID":30977},{"Watched":false,"Name":"Exercise 3","Duration":"2m 20s","ChapterTopicVideoID":29360,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise you want to find"},{"Start":"00:02.550 ","End":"00:07.457","Text":"the quadratic or second-degree Taylor approximation for f(x,"},{"Start":"00:07.457 ","End":"00:13.920","Text":"y)= e^4y- x^2- y^2 around the origin."},{"Start":"00:13.920 ","End":"00:20.475","Text":"The general solution for such a problem for second degree is the following."},{"Start":"00:20.475 ","End":"00:22.848","Text":"If we let (a,"},{"Start":"00:22.848 ","End":"00:26.088","Text":"b) be (0,"},{"Start":"00:26.088 ","End":"00:30.000","Text":"0) then it reduces to this."},{"Start":"00:30.000 ","End":"00:36.090","Text":"Now we need to find some partial derivative of the function itself at (0, 0)."},{"Start":"00:36.090 ","End":"00:39.286","Text":"We have to find all of these. Let\u0027s see."},{"Start":"00:39.286 ","End":"00:40.515","Text":"Start with f,"},{"Start":"00:40.515 ","End":"00:42.127","Text":"which is this,"},{"Start":"00:42.127 ","End":"00:49.370","Text":"then derivative with respect to x is e to this times the derivative,"},{"Start":"00:49.370 ","End":"00:54.516","Text":"the inner, which is -2x."},{"Start":"00:54.516 ","End":"00:56.475","Text":"With respect to y,"},{"Start":"00:56.475 ","End":"01:02.725","Text":"the inner derivative is 4-2y."},{"Start":"01:02.725 ","End":"01:08.090","Text":"Differentiating this with respect to x and using the product rule we"},{"Start":"01:08.090 ","End":"01:15.430","Text":"get -2x times -2x is 4x^2 and then -2 of this."},{"Start":"01:15.430 ","End":"01:18.240","Text":"I\u0027ll leave you to check the other 2."},{"Start":"01:18.240 ","End":"01:20.280","Text":"Then we have to substitute."},{"Start":"01:20.280 ","End":"01:22.603","Text":"If x and y is (0, 0),"},{"Start":"01:22.603 ","End":"01:25.475","Text":"this gives us e^0, which is 1."},{"Start":"01:25.475 ","End":"01:29.375","Text":"Here when x=0,"},{"Start":"01:29.375 ","End":"01:33.010","Text":"this whole thing becomes 0 because there\u0027s an x here."},{"Start":"01:33.010 ","End":"01:35.295","Text":"When x and y is 0,"},{"Start":"01:35.295 ","End":"01:39.810","Text":"4-2y, is 4 times 1, is 4."},{"Start":"01:39.810 ","End":"01:42.630","Text":"Here 4x^2 is 0-2,"},{"Start":"01:42.630 ","End":"01:45.090","Text":"so we have -2."},{"Start":"01:45.090 ","End":"01:47.310","Text":"Here again, x=0,"},{"Start":"01:47.310 ","End":"01:49.245","Text":"so we get 0."},{"Start":"01:49.245 ","End":"01:54.885","Text":"Here we have 0-0+14, it\u0027s 14."},{"Start":"01:54.885 ","End":"02:02.135","Text":"Now we have to substitute these values here and here."},{"Start":"02:02.135 ","End":"02:03.860","Text":"What we get, well,"},{"Start":"02:03.860 ","End":"02:06.140","Text":"we ignore the ones with the 0 of course."},{"Start":"02:06.140 ","End":"02:08.280","Text":"These are divided by 2,"},{"Start":"02:08.280 ","End":"02:11.310","Text":"so instead of -2 and 14,"},{"Start":"02:11.310 ","End":"02:14.115","Text":"we have -1 and 7."},{"Start":"02:14.115 ","End":"02:17.645","Text":"That\u0027s the approximation."},{"Start":"02:17.645 ","End":"02:20.400","Text":"That concludes this exercise."}],"ID":30978},{"Watched":false,"Name":"Exercise 4","Duration":"9m 18s","ChapterTopicVideoID":29361,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, we\u0027re going to find"},{"Start":"00:02.430 ","End":"00:07.495","Text":"the quadratic Taylor approximation for the function f of xy,"},{"Start":"00:07.495 ","End":"00:10.710","Text":"which is x squared minus y over x plus y squared to"},{"Start":"00:10.710 ","End":"00:15.150","Text":"the power of a 3rd around the 0.2 comma 1."},{"Start":"00:15.150 ","End":"00:21.500","Text":"This we could also write as the cube root instead of the power of a 3rd."},{"Start":"00:21.500 ","End":"00:25.070","Text":"I\u0027ve already written the formula,"},{"Start":"00:25.070 ","End":"00:33.035","Text":"the general 1 for second degree quadratic Taylor approximation as follows."},{"Start":"00:33.035 ","End":"00:35.300","Text":"In our case, we know what a,"},{"Start":"00:35.300 ","End":"00:37.610","Text":"b is, it\u0027s 2 comma 1."},{"Start":"00:37.610 ","End":"00:41.600","Text":"So just replace a and b by 2 and 1,"},{"Start":"00:41.600 ","End":"00:44.050","Text":"and this is what we get."},{"Start":"00:44.050 ","End":"00:48.375","Text":"Now we have to compute 6 things."},{"Start":"00:48.375 ","End":"00:49.845","Text":"F of 2,1,"},{"Start":"00:49.845 ","End":"00:51.150","Text":"and x of 2,1,"},{"Start":"00:51.150 ","End":"00:59.125","Text":"and so on up to fyy of 2,1 will have the coefficients of the Taylor approximation."},{"Start":"00:59.125 ","End":"01:04.880","Text":"Also opened up the brackets here and put a half here and half here,"},{"Start":"01:04.880 ","End":"01:07.980","Text":"and this half with this 2 cancel."},{"Start":"01:07.980 ","End":"01:10.170","Text":"Here again is f,"},{"Start":"01:10.170 ","End":"01:14.064","Text":"and so we can get f of 2, 1,"},{"Start":"01:14.064 ","End":"01:17.370","Text":"2 squared minus 1 is 3,"},{"Start":"01:17.370 ","End":"01:20.790","Text":"and 1, 2 plus 1 squared is 3."},{"Start":"01:20.790 ","End":"01:23.085","Text":"So we get the answer 1."},{"Start":"01:23.085 ","End":"01:25.380","Text":"Let\u0027s go onto the next 1."},{"Start":"01:25.380 ","End":"01:27.420","Text":"Df by dx,"},{"Start":"01:27.420 ","End":"01:30.445","Text":"using the chain rule,"},{"Start":"01:30.445 ","End":"01:39.935","Text":"we get that the derivative is this to the power of minus 2/3, 1/3 times that."},{"Start":"01:39.935 ","End":"01:41.930","Text":"Then the inner derivative,"},{"Start":"01:41.930 ","End":"01:47.675","Text":"which is d by dx of this and here we\u0027ll use the quotient rule."},{"Start":"01:47.675 ","End":"01:53.165","Text":"This stays as is and now the quotient rule is the derivative of the numerator"},{"Start":"01:53.165 ","End":"01:59.240","Text":"2x times the denominator minus the derivative of the denominator,"},{"Start":"01:59.240 ","End":"02:03.910","Text":"which is 1 times the numerator over the denominator squared."},{"Start":"02:03.910 ","End":"02:09.470","Text":"Then just collect like terms on the numerator here and we get this."},{"Start":"02:09.470 ","End":"02:14.905","Text":"If we substitute x equals 2 and y equals 1,"},{"Start":"02:14.905 ","End":"02:16.815","Text":"we get 1 third,"},{"Start":"02:16.815 ","End":"02:18.780","Text":"this again is 3 over 3,"},{"Start":"02:18.780 ","End":"02:22.110","Text":"so it\u0027s 1,1 to the minus 2-thirds is 1,"},{"Start":"02:22.110 ","End":"02:25.845","Text":"1 third, this comes out to be 9."},{"Start":"02:25.845 ","End":"02:27.505","Text":"X plus y squared is 3,"},{"Start":"02:27.505 ","End":"02:29.674","Text":"anyway, comes out to be a third."},{"Start":"02:29.674 ","End":"02:31.850","Text":"Next, df by dy,"},{"Start":"02:31.850 ","End":"02:34.985","Text":"similar process as df by dx."},{"Start":"02:34.985 ","End":"02:37.810","Text":"We apply the chain rule."},{"Start":"02:37.810 ","End":"02:41.030","Text":"Also for the antiderivative,"},{"Start":"02:41.030 ","End":"02:42.230","Text":"we now have a quotient."},{"Start":"02:42.230 ","End":"02:46.770","Text":"So we use the quotient rule and we get,"},{"Start":"02:46.770 ","End":"02:50.690","Text":"again, derivative of the numerator, but with respect to y,"},{"Start":"02:50.690 ","End":"02:54.350","Text":"this time is minus 1 times denominator,"},{"Start":"02:54.350 ","End":"02:58.010","Text":"and then minus the derivative of"},{"Start":"02:58.010 ","End":"03:03.280","Text":"the denominator to y times the numerator over the denominator squared."},{"Start":"03:03.280 ","End":"03:08.255","Text":"Again, we just open the brackets, collect like terms,"},{"Start":"03:08.255 ","End":"03:12.845","Text":"and then substitute 2 and 1 for x and y,"},{"Start":"03:12.845 ","End":"03:15.610","Text":"and we get minus a third."},{"Start":"03:15.610 ","End":"03:22.050","Text":"Back to fx from which we can get fxy and xx."},{"Start":"03:22.050 ","End":"03:28.220","Text":"Fxx, we can get by using the product rule on this,"},{"Start":"03:28.220 ","End":"03:30.650","Text":"on the chain rule on the first bit."},{"Start":"03:30.650 ","End":"03:32.225","Text":"First the product rule,"},{"Start":"03:32.225 ","End":"03:34.455","Text":"which gives us an fx, x,"},{"Start":"03:34.455 ","End":"03:40.670","Text":"is the derivative with respect to x of the first part times the second part as is,"},{"Start":"03:40.670 ","End":"03:42.650","Text":"and then vice versa,"},{"Start":"03:42.650 ","End":"03:47.410","Text":"the derivative of the second part times the first part."},{"Start":"03:47.410 ","End":"03:54.380","Text":"Now, this derivative, we can use again the chain rule."},{"Start":"03:54.380 ","End":"04:01.170","Text":"I get minus 2 thirds times a third is minus 2 ninths and then drop the exponent by 1,"},{"Start":"04:01.170 ","End":"04:06.350","Text":"which makes it minus 5 thirds times the inner derivative,"},{"Start":"04:06.350 ","End":"04:10.700","Text":"which is a quotient but we can reuse what we did before."},{"Start":"04:10.700 ","End":"04:13.160","Text":"This is gotten by the same process,"},{"Start":"04:13.160 ","End":"04:15.545","Text":"so we have again one of these."},{"Start":"04:15.545 ","End":"04:18.185","Text":"Then the second part is a bit messier."},{"Start":"04:18.185 ","End":"04:20.330","Text":"This goes over here."},{"Start":"04:20.330 ","End":"04:25.040","Text":"The derivative of this is by the quotient rule."},{"Start":"04:25.040 ","End":"04:29.570","Text":"The derivative of the numerator with respect 2x,"},{"Start":"04:29.570 ","End":"04:33.785","Text":"2x plus 2y squared times the denominator"},{"Start":"04:33.785 ","End":"04:40.280","Text":"minus the derivative of the denominator with respect to x,"},{"Start":"04:40.280 ","End":"04:43.760","Text":"so it\u0027s twice this and the anti-derivative is 1."},{"Start":"04:43.760 ","End":"04:51.000","Text":"So it\u0027s just twice x plus y squared times the numerator over the denominator squared."},{"Start":"04:51.110 ","End":"04:54.020","Text":"It\u0027s pretty routine from here,"},{"Start":"04:54.020 ","End":"04:57.950","Text":"I\u0027ll leave the computations and let you check them."},{"Start":"04:57.950 ","End":"05:00.125","Text":"Now that we have this,"},{"Start":"05:00.125 ","End":"05:02.510","Text":"we could simplify it further."},{"Start":"05:02.510 ","End":"05:08.820","Text":"We can also substitute right away the 2 comma 1."},{"Start":"05:11.450 ","End":"05:16.150","Text":"This comes out to be like before 3 over 3,"},{"Start":"05:16.150 ","End":"05:19.000","Text":"which is 1 to the minus 5/3 is 1."},{"Start":"05:19.000 ","End":"05:20.755","Text":"So this drops out."},{"Start":"05:20.755 ","End":"05:25.205","Text":"Similarly this 1 to the minus 2/3 drops out."},{"Start":"05:25.205 ","End":"05:28.725","Text":"From here, we get minus 2/9,"},{"Start":"05:28.725 ","End":"05:32.510","Text":"and then this thing here,"},{"Start":"05:32.510 ","End":"05:39.900","Text":"this numerator comes out to be 2 squared plus twice 2 plus 1 is 9."},{"Start":"05:39.900 ","End":"05:46.065","Text":"This is 3 squared is 9,9 over 9 is 1, squared is 1."},{"Start":"05:46.065 ","End":"05:48.630","Text":"Here we have 1/3."},{"Start":"05:48.630 ","End":"05:51.180","Text":"Also this is 1, and this,"},{"Start":"05:51.180 ","End":"05:54.090","Text":"I wrote 4 here, it should be 3."},{"Start":"05:54.090 ","End":"05:57.800","Text":"That is because we can cancel here."},{"Start":"05:57.800 ","End":"06:01.400","Text":"The x plus y squared from here,"},{"Start":"06:01.400 ","End":"06:05.045","Text":"and 1 of the x plus y squared here,"},{"Start":"06:05.045 ","End":"06:09.745","Text":"and 1 of the x plus y squared here."},{"Start":"06:09.745 ","End":"06:11.870","Text":"So this disappears."},{"Start":"06:11.870 ","End":"06:14.870","Text":"This becomes a 1 and this becomes a 3."},{"Start":"06:14.870 ","End":"06:18.365","Text":"Then when we expand, we get this."},{"Start":"06:18.365 ","End":"06:21.440","Text":"Here, x plus y squared cubed,"},{"Start":"06:21.440 ","End":"06:24.530","Text":"the numerator comes out to be 4,"},{"Start":"06:24.530 ","End":"06:27.935","Text":"the denominator, 3 cubed is 27,"},{"Start":"06:27.935 ","End":"06:32.570","Text":"and the answer comes out minus 14 over 81."},{"Start":"06:32.570 ","End":"06:35.920","Text":"Next, we need f x y,"},{"Start":"06:35.920 ","End":"06:41.595","Text":"which we get by differentiating fx with respect to y."},{"Start":"06:41.595 ","End":"06:43.260","Text":"We had a product."},{"Start":"06:43.260 ","End":"06:47.390","Text":"So we apply the product rule same as above,"},{"Start":"06:47.390 ","End":"06:52.535","Text":"except that we have a derivative with respect to y here and here."},{"Start":"06:52.535 ","End":"06:58.580","Text":"The derivative of this is minus 2/9,"},{"Start":"06:58.580 ","End":"07:01.220","Text":"this thing to the power of minus 5/3."},{"Start":"07:01.220 ","End":"07:04.610","Text":"The anti-derivative is this, same as this."},{"Start":"07:04.610 ","End":"07:07.100","Text":"For the second part,"},{"Start":"07:07.100 ","End":"07:13.685","Text":"this part is here and the first part just have to use the quotient rule,"},{"Start":"07:13.685 ","End":"07:20.405","Text":"denominator squared, and then the derivative of this,"},{"Start":"07:20.405 ","End":"07:26.930","Text":"which is 4 xy plus 1 times the denominator and vice versa."},{"Start":"07:26.930 ","End":"07:30.425","Text":"1 of the x plus y squared cancels."},{"Start":"07:30.425 ","End":"07:34.895","Text":"So these 2 disappears and this 4 becomes a 3."},{"Start":"07:34.895 ","End":"07:40.790","Text":"If we collect this numerator together, some stuff cancels."},{"Start":"07:40.790 ","End":"07:46.430","Text":"For example, 4x squared y from here cancels with"},{"Start":"07:46.430 ","End":"07:53.585","Text":"4x squared y. I\u0027ll just write the result of this numerator."},{"Start":"07:53.585 ","End":"07:55.535","Text":"This is here."},{"Start":"07:55.535 ","End":"07:57.560","Text":"The rest is clear."},{"Start":"07:57.560 ","End":"08:01.825","Text":"If we substitute 2 comma 1,"},{"Start":"08:01.825 ","End":"08:06.110","Text":"then I\u0027ll leave you to check the calculations."},{"Start":"08:06.110 ","End":"08:08.045","Text":"This is going on a bit too long,"},{"Start":"08:08.045 ","End":"08:11.210","Text":"and this comes out to be 1/9."},{"Start":"08:11.210 ","End":"08:15.975","Text":"We still have 1 more coefficient to compute. We need the fyy."},{"Start":"08:15.975 ","End":"08:21.450","Text":"So we start with fy and then differentiate with respect to y."},{"Start":"08:21.830 ","End":"08:24.560","Text":"It\u0027s similar to before."},{"Start":"08:24.560 ","End":"08:27.680","Text":"There\u0027s really nothing new here."},{"Start":"08:27.680 ","End":"08:32.450","Text":"Similar computation and cancellation as before."},{"Start":"08:32.450 ","End":"08:41.850","Text":"Substitute 2 comma 1 and we get the result 0."},{"Start":"08:41.850 ","End":"08:44.585","Text":"Okay. To remind you,"},{"Start":"08:44.585 ","End":"08:47.390","Text":"this is the expression we had to compute,"},{"Start":"08:47.390 ","End":"08:53.210","Text":"and we just spent time computing these 6 coefficients."},{"Start":"08:53.210 ","End":"08:56.495","Text":"Let\u0027s just put them in."},{"Start":"08:56.495 ","End":"08:59.825","Text":"We had that, this is 1,"},{"Start":"08:59.825 ","End":"09:01.415","Text":"this was a 1/3,"},{"Start":"09:01.415 ","End":"09:03.395","Text":"this was minus a third,"},{"Start":"09:03.395 ","End":"09:06.485","Text":"this was minus 7 over 81,"},{"Start":"09:06.485 ","End":"09:07.805","Text":"this was a 1/9,"},{"Start":"09:07.805 ","End":"09:10.944","Text":"and this was 0, which gives us this."},{"Start":"09:10.944 ","End":"09:19.040","Text":"This is our approximation for f of xy and that concludes this exercise."}],"ID":30979},{"Watched":false,"Name":"Exercise 5","Duration":"2m 15s","ChapterTopicVideoID":29362,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.470 ","End":"00:07.560","Text":"This exercise is a continuation of a previous exercise where we found"},{"Start":"00:07.560 ","End":"00:16.455","Text":"the quadratic Taylor approximation to the function 1 plus y natural log 1 plus x minus y."},{"Start":"00:16.455 ","End":"00:21.990","Text":"We found that this is approximated by this polynomial."},{"Start":"00:21.990 ","End":"00:29.295","Text":"Our task now is to use this approximation to estimate the natural log of 1.5."},{"Start":"00:29.295 ","End":"00:35.145","Text":"The idea is to find x and y to choose them judiciously so"},{"Start":"00:35.145 ","End":"00:41.300","Text":"that we can get here natural log of 1.5 or something related,"},{"Start":"00:41.300 ","End":"00:42.630","Text":"but also that x,"},{"Start":"00:42.630 ","End":"00:46.135","Text":"y should be close to 0,0."},{"Start":"00:46.135 ","End":"00:49.565","Text":"Well, if we want this to be 1.5,"},{"Start":"00:49.565 ","End":"00:51.950","Text":"then x minus y is 1/2."},{"Start":"00:51.950 ","End":"00:56.000","Text":"It\u0027s best if y is 0 because then this is simplest,"},{"Start":"00:56.000 ","End":"01:01.225","Text":"but I suppose you could take x is 0 and y is minus 1/2,"},{"Start":"01:01.225 ","End":"01:05.300","Text":"could work if you could try that afterwards."},{"Start":"01:05.300 ","End":"01:08.060","Text":"Anyway, we\u0027ll go with x,"},{"Start":"01:08.060 ","End":"01:10.780","Text":"y is 1/2, 0."},{"Start":"01:10.780 ","End":"01:13.275","Text":"Then this bit is 1,"},{"Start":"01:13.275 ","End":"01:15.990","Text":"and this bit is 1.5,"},{"Start":"01:15.990 ","End":"01:22.755","Text":"and so we can get 1 times natural log of 1/2 is according to this,"},{"Start":"01:22.755 ","End":"01:26.160","Text":"x is 1/2 minus y is 0,"},{"Start":"01:26.160 ","End":"01:35.745","Text":"1/2x squared 2xy is 0 because y is 0 and this is 0 again because y is 0."},{"Start":"01:35.745 ","End":"01:39.945","Text":"What we get is 1/2,"},{"Start":"01:39.945 ","End":"01:42.135","Text":"and then this is minus 1/8,"},{"Start":"01:42.135 ","End":"01:44.925","Text":"which gives us 3/8,"},{"Start":"01:44.925 ","End":"01:52.120","Text":"and if we do this on a calculator, that\u0027s 0.375."},{"Start":"01:52.120 ","End":"01:56.155","Text":"We\u0027re estimating natural log of 1.5 is this."},{"Start":"01:56.155 ","End":"02:00.620","Text":"Now again, on the calculator we can look this up exactly,"},{"Start":"02:00.620 ","End":"02:05.734","Text":"and it comes out to be 0.405 something."},{"Start":"02:05.734 ","End":"02:10.070","Text":"I don\u0027t know if it\u0027s good or not good the approximation,"},{"Start":"02:10.070 ","End":"02:11.930","Text":"but this is what it is,"},{"Start":"02:11.930 ","End":"02:15.120","Text":"and that concludes this exercise."}],"ID":30980},{"Watched":false,"Name":"Exercise 6","Duration":"1m 18s","ChapterTopicVideoID":29363,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"You might recall that in the previous exercise,"},{"Start":"00:03.630 ","End":"00:08.520","Text":"we estimated this exponent function as"},{"Start":"00:08.520 ","End":"00:14.385","Text":"follows with a quadratic Taylor approximation around the origin."},{"Start":"00:14.385 ","End":"00:21.255","Text":"In this exercise, we\u0027re going to use this approximation to estimate e^3."},{"Start":"00:21.255 ","End":"00:25.500","Text":"The idea is to pick x and y such that x,"},{"Start":"00:25.500 ","End":"00:27.910","Text":"y is close to 0,"},{"Start":"00:27.910 ","End":"00:30.240","Text":"0, though close is a little relative."},{"Start":"00:30.240 ","End":"00:36.825","Text":"In such a way that this gives us e^3 or something similar."},{"Start":"00:36.825 ","End":"00:38.745","Text":"By trial and error,"},{"Start":"00:38.745 ","End":"00:41.180","Text":"if you let x, y equals 0, 1,"},{"Start":"00:41.180 ","End":"00:50.505","Text":"then this expression here is 4 times 1 minus 0 minus 1, which is 3."},{"Start":"00:50.505 ","End":"01:00.125","Text":"What we get if we substitute in here is e^3 is 1 plus 4 minus 0 plus 7."},{"Start":"01:00.125 ","End":"01:03.020","Text":"This comes out to be 12."},{"Start":"01:03.020 ","End":"01:05.735","Text":"We don\u0027t really know how good this estimate is."},{"Start":"01:05.735 ","End":"01:07.580","Text":"On the calculator,"},{"Start":"01:07.580 ","End":"01:10.600","Text":"e^3 comes out to be around 20."},{"Start":"01:10.600 ","End":"01:13.070","Text":"I would say this is not so good."},{"Start":"01:13.070 ","End":"01:14.570","Text":"Anyway, it is what it is,"},{"Start":"01:14.570 ","End":"01:18.090","Text":"and that concludes this exercise."}],"ID":30981},{"Watched":false,"Name":"Exercise 7","Duration":"2m 16s","ChapterTopicVideoID":29364,"CourseChapterTopicPlaylistID":294461,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.880","Text":"This exercise builds on a previous exercise in which we found"},{"Start":"00:05.880 ","End":"00:09.210","Text":"a quadratic Taylor approximation around"},{"Start":"00:09.210 ","End":"00:15.689","Text":"the point (2,1) for the function cube root (x^2 minus y over x plus y^2)."},{"Start":"00:15.689 ","End":"00:21.140","Text":"We got it to approximate to this expression,"},{"Start":"00:21.140 ","End":"00:24.855","Text":"this polynomial into variables."},{"Start":"00:24.855 ","End":"00:26.540","Text":"In this exercise,"},{"Start":"00:26.540 ","End":"00:31.135","Text":"we\u0027re going to use this approximation to estimate the cube root of 2."},{"Start":"00:31.135 ","End":"00:35.388","Text":"What we\u0027re going to do is look for a point (x,"},{"Start":"00:35.388 ","End":"00:39.000","Text":"y) near (2,1),"},{"Start":"00:39.000 ","End":"00:43.320","Text":"and such that this expression is equal to 2."},{"Start":"00:43.320 ","End":"00:46.550","Text":"A bit of trial and error gives us (x,"},{"Start":"00:46.550 ","End":"00:49.345","Text":"y ) equal (3,1),"},{"Start":"00:49.345 ","End":"00:53.885","Text":"and it\u0027s all relative whether this is close to (2,1) or not."},{"Start":"00:53.885 ","End":"00:56.630","Text":"But anyway, we\u0027ll go with this."},{"Start":"00:56.630 ","End":"01:03.020","Text":"Then, what\u0027s under the cube root is 9 minus 1 over 3 plus 1 is 2,"},{"Start":"01:03.020 ","End":"01:06.100","Text":"so we can use (3,1)."},{"Start":"01:06.100 ","End":"01:10.820","Text":"3 minus 2 is 1 everywhere we have x minus 2,"},{"Start":"01:10.820 ","End":"01:13.895","Text":"and y minus 1 is 0."},{"Start":"01:13.895 ","End":"01:20.055","Text":"So substituting, we get 1/3 times 1,"},{"Start":"01:20.055 ","End":"01:21.825","Text":"this will be 0,"},{"Start":"01:21.825 ","End":"01:25.890","Text":"and here we\u0027ll get minus 7 over 81 times 1,"},{"Start":"01:25.890 ","End":"01:28.289","Text":"and this will also be 0."},{"Start":"01:28.289 ","End":"01:34.470","Text":"What we have for the non-zero times 1 plus 1/3 is 4/3,"},{"Start":"01:34.470 ","End":"01:36.255","Text":"minus 7 over 81."},{"Start":"01:36.255 ","End":"01:42.785","Text":"4/3, if you multiply top and bottom by 27,"},{"Start":"01:42.785 ","End":"01:45.170","Text":"will give us a 108 over 81,"},{"Start":"01:45.170 ","End":"01:48.230","Text":"108 minus 7 is 101."},{"Start":"01:48.230 ","End":"01:49.895","Text":"That\u0027s our estimate,"},{"Start":"01:49.895 ","End":"01:58.130","Text":"in decimal it comes out on the calculator to 3 places to be 1.247."},{"Start":"01:58.130 ","End":"02:06.085","Text":"Whereas if we actually look up cube root of 2 on the calculator, we get 1.260."},{"Start":"02:06.085 ","End":"02:10.340","Text":"Whether this is a good approximation or not, that\u0027s debatable."},{"Start":"02:10.340 ","End":"02:16.890","Text":"Anyway, this is our answer and we are done."}],"ID":30982}],"Thumbnail":null,"ID":294461}]

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