[{"Name":"Extrema in 2 Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"6m 12s","ChapterTopicVideoID":8722,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8722.jpeg","UploadDate":"2017-02-13T11:49:14.3200000","DurationForVideoObject":"PT6M12S","Description":null,"MetaTitle":"Exercise 1 - Extrema in 2 Variables: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on Extrema in 2 Variables practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/advanced-topics-with-extrema-in-2-variables/extrema-in-2-variables/vid9028","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise, we have to find the critical points of this function of 2 variables."},{"Start":"00:06.000 ","End":"00:12.610","Text":"Then classify means to say whether it\u0027s maximum, minimum, or saddle."},{"Start":"00:13.010 ","End":"00:19.095","Text":"Let\u0027s start with the usual method of finding the partial derivatives first."},{"Start":"00:19.095 ","End":"00:23.850","Text":"We take f with respect to x,"},{"Start":"00:23.850 ","End":"00:27.210","Text":"and that will equal, see from here,"},{"Start":"00:27.210 ","End":"00:33.900","Text":"to y minus 2x and f with respect to"},{"Start":"00:33.900 ","End":"00:41.595","Text":"y will be 2x minus 2y."},{"Start":"00:41.595 ","End":"00:47.860","Text":"Now, before I even continue,"},{"Start":"00:47.860 ","End":"00:54.760","Text":"let me find possible critical points and then we\u0027ll continue later with fxx,"},{"Start":"00:54.760 ","End":"00:57.880","Text":"and fxy, and fyy."},{"Start":"00:57.880 ","End":"01:00.980","Text":"I just want to see."},{"Start":"01:00.980 ","End":"01:03.250","Text":"If I take 2 equations,"},{"Start":"01:03.250 ","End":"01:06.115","Text":"this equals 0 and this equals 0,"},{"Start":"01:06.115 ","End":"01:10.740","Text":"and try to solve it with 2 equations and 2 unknowns,"},{"Start":"01:10.740 ","End":"01:13.410","Text":"I run into a bit of difficulty,"},{"Start":"01:13.410 ","End":"01:16.770","Text":"because the first 1 gives me 2y equals 2x."},{"Start":"01:16.770 ","End":"01:18.359","Text":"The second 1 also,"},{"Start":"01:18.359 ","End":"01:22.000","Text":"in both cases, I get that x equals y."},{"Start":"01:22.000 ","End":"01:25.275","Text":"So I don\u0027t have enough information."},{"Start":"01:25.275 ","End":"01:29.240","Text":"In other words, I could let y be anything."},{"Start":"01:29.240 ","End":"01:36.080","Text":"Say, I could let y equal t and then x being equal to y,"},{"Start":"01:36.080 ","End":"01:41.835","Text":"x could also equal t. The general point,"},{"Start":"01:41.835 ","End":"01:43.740","Text":"if I write it point terms,"},{"Start":"01:43.740 ","End":"01:45.150","Text":"it\u0027s like t, t,"},{"Start":"01:45.150 ","End":"01:52.080","Text":"there\u0027s an infinite number of solutions for every t, that\u0027s a real number."},{"Start":"01:52.080 ","End":"01:53.780","Text":"Will give me a solution."},{"Start":"01:53.780 ","End":"01:57.800","Text":"This is actually the 45-degree line going through the origin."},{"Start":"01:57.800 ","End":"02:00.630","Text":"Any point on that gives me a critical point."},{"Start":"02:00.630 ","End":"02:02.510","Text":"I\u0027ve got an infinite number of critical points."},{"Start":"02:02.510 ","End":"02:05.360","Text":"Now, let\u0027s see if we can classify them."},{"Start":"02:05.360 ","End":"02:12.480","Text":"Fxx is this with respect to x is minus 2."},{"Start":"02:12.480 ","End":"02:16.770","Text":"Fxy is this with respect to y, which is 2."},{"Start":"02:16.770 ","End":"02:20.465","Text":"It happens to also be same as Fyx also 2."},{"Start":"02:20.465 ","End":"02:26.330","Text":"I\u0027d like to check that it\u0027s an extra check that I\u0027m on the right path."},{"Start":"02:26.330 ","End":"02:29.330","Text":"Fyy is minus 2."},{"Start":"02:29.330 ","End":"02:31.645","Text":"These are all constants."},{"Start":"02:31.645 ","End":"02:35.010","Text":"For whatever critical point I substitute,"},{"Start":"02:35.010 ","End":"02:37.795","Text":"when I compute this quantity D,"},{"Start":"02:37.795 ","End":"02:41.555","Text":"which is this times this minus this squared,"},{"Start":"02:41.555 ","End":"02:46.110","Text":"It\u0027s minus 2 times minus 2 minus 2 squared."},{"Start":"02:46.110 ","End":"02:50.685","Text":"This comes out to be 0, 4 minus 4."},{"Start":"02:50.685 ","End":"02:53.515","Text":"When we get a 0 for D,"},{"Start":"02:53.515 ","End":"02:56.180","Text":"according to this test, we don\u0027t know."},{"Start":"02:56.180 ","End":"02:59.975","Text":"We\u0027re going to have to use some tricks"},{"Start":"02:59.975 ","End":"03:05.370","Text":"to try and figure out what kind of critical points these are."},{"Start":"03:06.560 ","End":"03:08.690","Text":"The more tricks you know,"},{"Start":"03:08.690 ","End":"03:11.105","Text":"the better you are."},{"Start":"03:11.105 ","End":"03:14.480","Text":"When you see an that kind of exercise that you get stuck on,"},{"Start":"03:14.480 ","End":"03:18.245","Text":"you basically have to use ingenuity plus experience."},{"Start":"03:18.245 ","End":"03:20.030","Text":"I\u0027m giving you some experience."},{"Start":"03:20.030 ","End":"03:24.115","Text":"I look at this and I see that,"},{"Start":"03:24.115 ","End":"03:26.600","Text":"if I just reordered this,"},{"Start":"03:26.600 ","End":"03:31.880","Text":"I can see an x squared plus 2xy plus y squared or minus 2xy plus y squared in here,"},{"Start":"03:31.880 ","End":"03:35.720","Text":"that reminds me of the formula for the special binomial expansion."},{"Start":"03:35.720 ","End":"03:37.870","Text":"Let me show you what I mean."},{"Start":"03:37.870 ","End":"03:43.020","Text":"I can rewrite fxy as 1."},{"Start":"03:43.020 ","End":"03:47.045","Text":"Now, I want it to be x squared minus 2xy plus y squared."},{"Start":"03:47.045 ","End":"03:50.905","Text":"If I put a minus here and I reorder,"},{"Start":"03:50.905 ","End":"03:53.415","Text":"the signs change and so do the order."},{"Start":"03:53.415 ","End":"03:58.905","Text":"Plus x squared minus 2xy and plus y squared."},{"Start":"03:58.905 ","End":"04:00.650","Text":"Now, what good is this?"},{"Start":"04:00.650 ","End":"04:04.340","Text":"Because this is a complete square."},{"Start":"04:04.340 ","End":"04:10.150","Text":"This is the formula for x minus y squared."},{"Start":"04:10.150 ","End":"04:16.320","Text":"Now, I have 1 minus this bit squared now."},{"Start":"04:16.320 ","End":"04:17.670","Text":"At a critical point,"},{"Start":"04:17.670 ","End":"04:19.800","Text":"if I let x,"},{"Start":"04:19.800 ","End":"04:22.380","Text":"y be equal to t,"},{"Start":"04:22.380 ","End":"04:24.275","Text":"t for whatever t,"},{"Start":"04:24.275 ","End":"04:26.915","Text":"then t minus t is 0."},{"Start":"04:26.915 ","End":"04:28.625","Text":"In other words, f of t,"},{"Start":"04:28.625 ","End":"04:31.470","Text":"t is equal to 1."},{"Start":"04:31.600 ","End":"04:35.375","Text":"Now, this thing here,"},{"Start":"04:35.375 ","End":"04:40.139","Text":"it\u0027s a square, so it\u0027s always bigger or equal to 0."},{"Start":"04:40.880 ","End":"04:48.515","Text":"1 minus something non-negative is going to be less than or equal to 1."},{"Start":"04:48.515 ","End":"04:53.045","Text":"In other words, what I\u0027m saying is that 1 is a maximum,"},{"Start":"04:53.045 ","End":"04:58.220","Text":"because here it\u0027s 1 and everywhere else it\u0027s less than or equal to 1,"},{"Start":"04:58.220 ","End":"05:00.710","Text":"because 1 minus this is less than or equal to 1."},{"Start":"05:00.710 ","End":"05:04.890","Text":"So this is a maximum."},{"Start":"05:05.440 ","End":"05:10.580","Text":"I should mention that maximum doesn\u0027t mean absolute biggest."},{"Start":"05:10.580 ","End":"05:12.365","Text":"It could be bigger or equal to."},{"Start":"05:12.365 ","End":"05:15.030","Text":"There\u0027s nothing bigger than it."},{"Start":"05:15.040 ","End":"05:18.170","Text":"I mean, if I take a point, then around it,"},{"Start":"05:18.170 ","End":"05:21.595","Text":"there might be other points that are also equal to 1,"},{"Start":"05:21.595 ","End":"05:24.095","Text":"but nothing greater than 1."},{"Start":"05:24.095 ","End":"05:27.690","Text":"This thing starts from 1 and decreases."},{"Start":"05:28.060 ","End":"05:31.885","Text":"I might also add that the maximum value"},{"Start":"05:31.885 ","End":"05:41.789","Text":"of f is 1."},{"Start":"05:41.789 ","End":"05:46.475","Text":"It\u0027s 1 whenever at all of these critical points,"},{"Start":"05:46.475 ","End":"05:48.560","Text":"because f of t,"},{"Start":"05:48.560 ","End":"05:51.109","Text":"t is 1 and it\u0027s the maximum."},{"Start":"05:51.109 ","End":"05:57.455","Text":"So it occurs in an infinite number of points."},{"Start":"05:57.455 ","End":"06:03.050","Text":"These are the critical points and they are all of type maximum."},{"Start":"06:03.050 ","End":"06:05.420","Text":"That maximum value is 1."},{"Start":"06:05.420 ","End":"06:08.545","Text":"I\u0027ve answered more than they asked for."},{"Start":"06:08.545 ","End":"06:11.680","Text":"That was a useful trick."}],"ID":9028},{"Watched":false,"Name":"Exercise 2","Duration":"5m 3s","ChapterTopicVideoID":8723,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.385","Text":"In this exercise, we have to classify the critical points of this function."},{"Start":"00:05.385 ","End":"00:08.250","Text":"That means to find the critical points and say,"},{"Start":"00:08.250 ","End":"00:13.680","Text":"whether each 1 is maximum, minimum, or subtle."},{"Start":"00:13.680 ","End":"00:20.010","Text":"Notice that the domain of f is all of x and y."},{"Start":"00:20.010 ","End":"00:23.280","Text":"There\u0027s no problem in substituting where we have x and"},{"Start":"00:23.280 ","End":"00:26.320","Text":"y. X squared plus y squared will be bigger or equal to 0,"},{"Start":"00:26.320 ","End":"00:29.190","Text":"so we can take the square roots so no problem."},{"Start":"00:29.190 ","End":"00:35.510","Text":"The usual technique is to look for the derivatives of 1st and 2nd order,"},{"Start":"00:35.510 ","End":"00:42.250","Text":"first the 1st order fx and fy with respect to x, the constant disappears."},{"Start":"00:42.250 ","End":"00:48.545","Text":"The derivative of the square root is 1 over twice the square root of that thing,"},{"Start":"00:48.545 ","End":"00:53.120","Text":"which in this case is x squared plus y squared."},{"Start":"00:53.120 ","End":"00:56.870","Text":"Then the inner derivative of what\u0027s under the square root,"},{"Start":"00:56.870 ","End":"00:58.730","Text":"which in this case is 2x,"},{"Start":"00:58.730 ","End":"01:00.710","Text":"and we had a minus to start with,"},{"Start":"01:00.710 ","End":"01:02.620","Text":"so that stays there."},{"Start":"01:02.620 ","End":"01:04.980","Text":"The 2 cancels with the 2,"},{"Start":"01:04.980 ","End":"01:08.415","Text":"so it\u0027s minus x over this square root."},{"Start":"01:08.415 ","End":"01:11.450","Text":"Similarly for y exactly the same."},{"Start":"01:11.450 ","End":"01:19.835","Text":"We get minus y over the square root of x squared plus y squared."},{"Start":"01:19.835 ","End":"01:26.820","Text":"Now, before we go on to do the 2nd derivatives,"},{"Start":"01:26.820 ","End":"01:31.865","Text":"let\u0027s pause this time at this stage and find critical points."},{"Start":"01:31.865 ","End":"01:34.790","Text":"If I let each of them equals 0,"},{"Start":"01:34.790 ","End":"01:40.250","Text":"I get, and I can ignore the minus."},{"Start":"01:40.250 ","End":"01:49.450","Text":"I get x over square root of x squared plus y squared is 0,"},{"Start":"01:49.450 ","End":"01:59.490","Text":"and I also get y over the square root of x squared plus y squared is 0."},{"Start":"02:00.060 ","End":"02:05.530","Text":"Now, if I try to solve this, I say, okay,"},{"Start":"02:05.530 ","End":"02:08.980","Text":"fraction is 0, so the numerators got to be 0,"},{"Start":"02:08.980 ","End":"02:11.890","Text":"so this gives me x equals 0."},{"Start":"02:11.890 ","End":"02:16.494","Text":"Similarly here, I would conclude that y has to be 0."},{"Start":"02:16.494 ","End":"02:22.915","Text":"However, notice that if x is 0 and y is 0 together,"},{"Start":"02:22.915 ","End":"02:27.040","Text":"these 2 will give me that the square root of x squared plus"},{"Start":"02:27.040 ","End":"02:31.705","Text":"y squared is 0 but there was a denominator is 0."},{"Start":"02:31.705 ","End":"02:35.500","Text":"That\u0027s a problem because it comes out to be 0 over 0,"},{"Start":"02:35.500 ","End":"02:36.790","Text":"which may not be 0,"},{"Start":"02:36.790 ","End":"02:38.280","Text":"at least it\u0027s undefined,"},{"Start":"02:38.280 ","End":"02:40.660","Text":"so we\u0027ve run into a problem."},{"Start":"02:40.660 ","End":"02:48.335","Text":"The original function is defined for 0,0."},{"Start":"02:48.335 ","End":"02:54.480","Text":"Let me just write that, f of 0,0 is defined."},{"Start":"02:54.480 ","End":"03:00.150","Text":"No problem, it\u0027s defined and it\u0027s equal to 4 because it\u0027s 4 minus 0."},{"Start":"03:00.150 ","End":"03:06.465","Text":"But fx and fy"},{"Start":"03:06.465 ","End":"03:15.960","Text":"are undefined at 0,0."},{"Start":"03:15.960 ","End":"03:19.245","Text":"Now, this is a new critical point."},{"Start":"03:19.245 ","End":"03:24.005","Text":"If the function is defined but the partial derivatives are not,"},{"Start":"03:24.005 ","End":"03:30.800","Text":"then we also say that 0,0 is a critical point,"},{"Start":"03:30.800 ","End":"03:34.205","Text":"not the kind we used to have when these 2 are equal to 0."},{"Start":"03:34.205 ","End":"03:36.170","Text":"But when they\u0027re both undefined,"},{"Start":"03:36.170 ","End":"03:37.879","Text":"or even 1 of them is undefined,"},{"Start":"03:37.879 ","End":"03:39.950","Text":"but the original function is defined."},{"Start":"03:39.950 ","End":"03:49.070","Text":"We\u0027re going to have to use some other strategy here than the usual 2nd order and so on,"},{"Start":"03:49.070 ","End":"03:55.804","Text":"because we can\u0027t differentiate 2nd order at 0,0 because the 1st order is not defined."},{"Start":"03:55.804 ","End":"04:04.955","Text":"What we can do is notice that this square root is always bigger or equal to 0,"},{"Start":"04:04.955 ","End":"04:09.739","Text":"and it\u0027s only 0 when x and y are both 0."},{"Start":"04:09.739 ","End":"04:13.385","Text":"In other words, f of x and y"},{"Start":"04:13.385 ","End":"04:21.065","Text":"is 4 minus something which is non-negative."},{"Start":"04:21.065 ","End":"04:22.970","Text":"Let me just write non-negative,"},{"Start":"04:22.970 ","End":"04:25.980","Text":"in other words, it\u0027s positive or 0."},{"Start":"04:27.050 ","End":"04:32.345","Text":"Therefore it has to be less than or equal to 4."},{"Start":"04:32.345 ","End":"04:39.515","Text":"On the other hand, f of 0,0 is 4 minus 0, which equals 4."},{"Start":"04:39.515 ","End":"04:42.530","Text":"It looks like 0,0 is"},{"Start":"04:42.530 ","End":"04:46.660","Text":"a maximum point because here it\u0027s 4 everywhere else it\u0027s less than or equal to."},{"Start":"04:46.660 ","End":"04:52.055","Text":"We can say that 0,0 is a maximum"},{"Start":"04:52.055 ","End":"04:59.075","Text":"just using simple algebraic techniques rather than calculus techniques."},{"Start":"04:59.075 ","End":"05:04.890","Text":"That is it, 1 critical point and maximum."}],"ID":9029},{"Watched":false,"Name":"Exercise 3","Duration":"11m 37s","ChapterTopicVideoID":8724,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.389","Text":"This exercise is a little bit different than your usual critical point problem."},{"Start":"00:06.389 ","End":"00:12.120","Text":"Because z is not given as a function of x, y explicitly,"},{"Start":"00:12.120 ","End":"00:20.145","Text":"it\u0027s given implicitly by a relationship between z and x and y, something equals 0."},{"Start":"00:20.145 ","End":"00:25.830","Text":"What we do, since it\u0027s already written in the form something equals 0,"},{"Start":"00:25.830 ","End":"00:29.805","Text":"is we let that something be a function of 3 variables."},{"Start":"00:29.805 ","End":"00:31.650","Text":"F of x, y,"},{"Start":"00:31.650 ","End":"00:35.810","Text":"and z is equal to z cubed plus"},{"Start":"00:35.810 ","End":"00:43.395","Text":"z plus xy minus 2x minus y plus 2."},{"Start":"00:43.395 ","End":"00:50.570","Text":"Essentially the implicit equation just says that F equals 0."},{"Start":"00:50.570 ","End":"00:52.954","Text":"When we have it in that form,"},{"Start":"00:52.954 ","End":"00:58.610","Text":"then we can compute the partial derivatives using a formula."},{"Start":"00:58.610 ","End":"01:05.165","Text":"I want to compute z with respect to x and z with respect to y."},{"Start":"01:05.165 ","End":"01:06.560","Text":"Now here\u0027s the way it works,"},{"Start":"01:06.560 ","End":"01:09.095","Text":"if you can remember there\u0027s a minus,"},{"Start":"01:09.095 ","End":"01:12.299","Text":"then there\u0027s a dividing line."},{"Start":"01:12.350 ","End":"01:19.155","Text":"There\u0027s an F in both cases and there\u0027s an F here."},{"Start":"01:19.155 ","End":"01:21.760","Text":"Now here\u0027s the thing,"},{"Start":"01:21.760 ","End":"01:27.050","Text":"we see there\u0027s a Z here and Z here, the big letter."},{"Start":"01:27.050 ","End":"01:29.900","Text":"We put the Z at the bottom."},{"Start":"01:29.900 ","End":"01:38.195","Text":"The big one goes at the bottom and maybe I\u0027ll just highlight that so we see Z and Z."},{"Start":"01:38.195 ","End":"01:41.510","Text":"Then here at the bottom we have an x,"},{"Start":"01:41.510 ","End":"01:45.565","Text":"and here we have a y and the ones at the bottom go on the numerator."},{"Start":"01:45.565 ","End":"01:49.810","Text":"Here it\u0027s x and here it\u0027s y."},{"Start":"01:49.820 ","End":"01:53.375","Text":"Now all we have to do is compute this."},{"Start":"01:53.375 ","End":"01:59.810","Text":"Let\u0027s see, minus I\u0027m working on them both simultaneously,"},{"Start":"01:59.810 ","End":"02:02.505","Text":"F, z appears in both."},{"Start":"02:02.505 ","End":"02:07.570","Text":"With respect to z, the z part is just the first 2 terms,"},{"Start":"02:07.570 ","End":"02:10.330","Text":"so it\u0027s 3z squared plus 1."},{"Start":"02:10.330 ","End":"02:14.680","Text":"Here we have 3z squared plus 1 and same here,"},{"Start":"02:14.680 ","End":"02:18.215","Text":"3z squared plus 1."},{"Start":"02:18.215 ","End":"02:21.670","Text":"I also note that the denominator will never be 0."},{"Start":"02:21.670 ","End":"02:25.160","Text":"This is bigger or equal to 0, and this is plus 1,"},{"Start":"02:25.160 ","End":"02:26.850","Text":"so the whole thing is bigger or equal to 1,"},{"Start":"02:26.850 ","End":"02:29.175","Text":"no problem with dividing by 0."},{"Start":"02:29.175 ","End":"02:31.690","Text":"Now, F with respect to x,"},{"Start":"02:31.690 ","End":"02:36.250","Text":"let\u0027s see we have an x here and here and so the derivative will"},{"Start":"02:36.250 ","End":"02:42.500","Text":"be y minus 2 and with respect to y,"},{"Start":"02:42.500 ","End":"02:44.960","Text":"I\u0027m looking at this term and this term,"},{"Start":"02:44.960 ","End":"02:49.530","Text":"I will get x minus 1."},{"Start":"02:49.820 ","End":"02:54.000","Text":"Sometimes I keep going with the second order derivative,"},{"Start":"02:54.000 ","End":"02:58.175","Text":"sometimes I like to stop and see what the critical points might be."},{"Start":"02:58.175 ","End":"02:59.780","Text":"Let\u0027s do that here,"},{"Start":"02:59.780 ","End":"03:01.910","Text":"because it\u0027s so easy."},{"Start":"03:01.910 ","End":"03:10.145","Text":"The critical points when I get an equation where this equals 0 and this equals 0,"},{"Start":"03:10.145 ","End":"03:16.080","Text":"I can throw the minus out to say I have in this color."},{"Start":"03:16.080 ","End":"03:17.760","Text":"Rather than copying it let\u0027s say this is 0,"},{"Start":"03:17.760 ","End":"03:20.595","Text":"this is 0, the numerators have to be 0."},{"Start":"03:20.595 ","End":"03:25.640","Text":"That gives us the y equals 2 and x equals"},{"Start":"03:25.640 ","End":"03:32.430","Text":"1 or if you prefer the critical point is the point 1, 2."},{"Start":"03:32.960 ","End":"03:35.820","Text":"We don\u0027t know what critical point it is,"},{"Start":"03:35.820 ","End":"03:40.100","Text":"let\u0027s continue with the second order derivatives."},{"Start":"03:40.100 ","End":"03:45.750","Text":"Let\u0027s go to Zxx,"},{"Start":"03:45.750 ","End":"03:49.500","Text":"we want to differentiate this here."},{"Start":"03:49.500 ","End":"03:57.430","Text":"Let me take the minus y minus 2 because that\u0027s a constant as far as x goes."},{"Start":"03:57.430 ","End":"04:06.625","Text":"I want the derivative of 1 over 3z squared plus 1 derivative with respect to x."},{"Start":"04:06.625 ","End":"04:10.310","Text":"Let\u0027s suppose you could also use the quotient rule here."},{"Start":"04:10.310 ","End":"04:16.600","Text":"Anyway, this is equal to minus y minus 2."},{"Start":"04:16.600 ","End":"04:25.550","Text":"Now the derivative of 1 over something is minus 1 over that something squared,"},{"Start":"04:25.550 ","End":"04:31.250","Text":"in this case, 3z squared plus 1 times the inner derivative."},{"Start":"04:31.250 ","End":"04:35.855","Text":"I\u0027m going to just erase the 1 and put the inner derivative,"},{"Start":"04:35.855 ","End":"04:38.960","Text":"which at first appears to be"},{"Start":"04:38.960 ","End":"04:43.700","Text":"6z only the derivative is not with respect to z, it\u0027s with respect to x."},{"Start":"04:43.700 ","End":"04:51.900","Text":"We need an inner z with respect to x."},{"Start":"04:52.040 ","End":"04:56.980","Text":"Actually I could cancel the minus with the minus."},{"Start":"04:57.020 ","End":"05:00.975","Text":"Next I\u0027ll to Zyy at the end I\u0027ll do Zxy,"},{"Start":"05:00.975 ","End":"05:08.045","Text":"Zyy is this differentiated with respect to y,"},{"Start":"05:08.045 ","End":"05:09.830","Text":"I\u0027ll use the same idea."},{"Start":"05:09.830 ","End":"05:13.010","Text":"The minus x minus 1 is a constant."},{"Start":"05:13.010 ","End":"05:17.250","Text":"I have 1 over 3z squared plus 1."},{"Start":"05:17.750 ","End":"05:24.305","Text":"I\u0027m going to skip a stage and say that the same thing with respect to y,"},{"Start":"05:24.305 ","End":"05:28.205","Text":"I would just get minus"},{"Start":"05:28.205 ","End":"05:35.100","Text":"1 over 3z squared plus 1 squared."},{"Start":"05:35.100 ","End":"05:38.075","Text":"I didn\u0027t write the 1 because a multiplied by the inner derivative,"},{"Start":"05:38.075 ","End":"05:42.200","Text":"which in this case is 6z like here,"},{"Start":"05:42.200 ","End":"05:46.255","Text":"but times Zy this time."},{"Start":"05:46.255 ","End":"05:50.220","Text":"Again, the minuses can be thrown out,"},{"Start":"05:50.220 ","End":"05:52.380","Text":"they cancel each other."},{"Start":"05:52.380 ","End":"05:56.020","Text":"We\u0027re left with Zxy,"},{"Start":"05:57.060 ","End":"06:02.620","Text":"which we differentiate this with respect to y."},{"Start":"06:02.620 ","End":"06:04.840","Text":"This time I will use the quotient rule."},{"Start":"06:04.840 ","End":"06:09.385","Text":"I\u0027ll leave the minus outside and here\u0027s the dividing line from the quotient rule."},{"Start":"06:09.385 ","End":"06:12.220","Text":"I\u0027ll start with the denominator squared,"},{"Start":"06:12.220 ","End":"06:20.145","Text":"3z squared plus 1 squared and then it\u0027s the derivative of the numerator."},{"Start":"06:20.145 ","End":"06:22.860","Text":"Remember it\u0027s with respect to y,"},{"Start":"06:22.860 ","End":"06:29.370","Text":"derivative of the numerator is 1 times denominator as is 3z squared plus"},{"Start":"06:29.370 ","End":"06:33.530","Text":"1 minus numerator as is and that\u0027s the y minus"},{"Start":"06:33.530 ","End":"06:38.435","Text":"2 times derivative of the denominator with respect to y."},{"Start":"06:38.435 ","End":"06:43.290","Text":"It\u0027s going to be times 6z,"},{"Start":"06:43.290 ","End":"06:47.740","Text":"but the inner derivative z with respect to y, the chain rule."},{"Start":"06:47.780 ","End":"06:50.910","Text":"That\u0027s these 3 quantities."},{"Start":"06:50.910 ","End":"06:57.645","Text":"What we need are the values of these at the point 1, 2."},{"Start":"06:57.645 ","End":"07:01.740","Text":"Let\u0027s see Zxx of 1,"},{"Start":"07:01.740 ","End":"07:06.990","Text":"2 and then I\u0027ll need Zyy at 1,"},{"Start":"07:06.990 ","End":"07:13.560","Text":"2 and then I\u0027ll need Zxy at 1, 2."},{"Start":"07:13.560 ","End":"07:16.410","Text":"The thing is, I know what x and y are,"},{"Start":"07:16.410 ","End":"07:19.730","Text":"I know that x is 1 and y is 2."},{"Start":"07:19.730 ","End":"07:27.810","Text":"But these could also contain z and they also contain Zx and Zy,"},{"Start":"07:27.810 ","End":"07:30.060","Text":"but let\u0027s do them 1 at a time."},{"Start":"07:30.060 ","End":"07:32.060","Text":"X\u0027s and y\u0027s, we have z,"},{"Start":"07:32.060 ","End":"07:35.100","Text":"we don\u0027t have yet, let\u0027s go for z."},{"Start":"07:35.100 ","End":"07:40.075","Text":"To find z, all I do is in this equation,"},{"Start":"07:40.075 ","End":"07:42.010","Text":"the original equation where F is 0,"},{"Start":"07:42.010 ","End":"07:46.870","Text":"I just put x equals 1, y equals 2."},{"Start":"07:46.870 ","End":"07:55.450","Text":"From here, I would get z cubed plus z plus,"},{"Start":"07:55.450 ","End":"07:58.480","Text":"now x is 1 and y is 2,"},{"Start":"07:58.480 ","End":"08:08.415","Text":"xy is 1 times 2 is 2 minus 2x is minus 2 times 1 is minus 2."},{"Start":"08:08.415 ","End":"08:17.235","Text":"Next I get minus 2 and then plus 2 equals 0."},{"Start":"08:17.235 ","End":"08:23.510","Text":"All these 2s cancel each other out in pairs and if I take z out,"},{"Start":"08:23.510 ","End":"08:28.685","Text":"I get z times z squared plus 1 equals 0."},{"Start":"08:28.685 ","End":"08:31.910","Text":"This is never 0, it\u0027s always bigger or equal to 1,"},{"Start":"08:31.910 ","End":"08:33.980","Text":"it\u0027s 1 plus the non-negative."},{"Start":"08:33.980 ","End":"08:41.470","Text":"The only way for getting 0 is when z is equal to 0."},{"Start":"08:41.470 ","End":"08:45.005","Text":"We have x, we have y,"},{"Start":"08:45.005 ","End":"08:49.640","Text":"we have z, but we still need other quantities."},{"Start":"08:49.640 ","End":"08:51.100","Text":"We have here a Zx,"},{"Start":"08:51.100 ","End":"08:55.520","Text":"and here Zy, we can substitute those here."},{"Start":"08:55.950 ","End":"09:01.320","Text":"Zx at the point 1,"},{"Start":"09:01.320 ","End":"09:06.790","Text":"2 will equal, but y is 2 already makes the numerator 0,"},{"Start":"09:06.790 ","End":"09:11.035","Text":"so that is equal to 0."},{"Start":"09:11.035 ","End":"09:19.905","Text":"We also have that Zy at this point 1, 2."},{"Start":"09:19.905 ","End":"09:21.810","Text":"Well here x is 1,"},{"Start":"09:21.810 ","End":"09:27.990","Text":"so this is 0, I have x, y,"},{"Start":"09:27.990 ","End":"09:32.410","Text":"z, and Zx and Zy,"},{"Start":"09:33.590 ","End":"09:36.790","Text":"all these 3 are 0, these 2 are not."},{"Start":"09:36.790 ","End":"09:40.460","Text":"Let\u0027s see what we get if we substitute in here."},{"Start":"09:41.670 ","End":"09:45.905","Text":"Y minus 2 is 0,"},{"Start":"09:45.905 ","End":"09:49.605","Text":"it doesn\u0027t matter what the rest of it is."},{"Start":"09:49.605 ","End":"09:53.750","Text":"This comes out 0, let\u0027s see Zyy,"},{"Start":"09:53.750 ","End":"09:58.070","Text":"well x minus 1 is 0 because x is 1,"},{"Start":"09:58.070 ","End":"10:00.620","Text":"multiplying that is also 0."},{"Start":"10:00.620 ","End":"10:03.030","Text":"What happens with Zxy,"},{"Start":"10:06.040 ","End":"10:09.860","Text":"that\u0027s not going to be 0 now, Zy is 0,"},{"Start":"10:09.860 ","End":"10:14.700","Text":"which means that that last term after the minus is 0."},{"Start":"10:16.360 ","End":"10:22.305","Text":"Z squared is also 0 because z is 0,"},{"Start":"10:22.305 ","End":"10:25.400","Text":"we\u0027re left with 1 over 1 squared,"},{"Start":"10:25.400 ","End":"10:30.530","Text":"with a minus, minus 1 over 1 squared is minus 1."},{"Start":"10:30.530 ","End":"10:35.045","Text":"Now we compute that expression D at the point 1, 2,"},{"Start":"10:35.045 ","End":"10:41.240","Text":"D is this times this minus this squared is the product of the 2 pure second derivative,"},{"Start":"10:41.240 ","End":"10:44.740","Text":"so to speak, minus the mixed 1 squared."},{"Start":"10:44.740 ","End":"10:52.800","Text":"It\u0027s 0 times 0 less minus 1 squared,"},{"Start":"10:52.800 ","End":"10:58.000","Text":"this gives minus 1."},{"Start":"10:59.390 ","End":"11:04.530","Text":"We have a saddle point,"},{"Start":"11:04.530 ","End":"11:09.370","Text":"at the point, let me just write that."},{"Start":"11:11.030 ","End":"11:21.990","Text":"This point becomes a saddle for the function z of x, y."},{"Start":"11:21.990 ","End":"11:28.325","Text":"If you want, you could also put the z and you could say that 1, 2, 0."},{"Start":"11:28.325 ","End":"11:37.960","Text":"I\u0027ll just say that 1, 2 is this saddle point of the implicit function z and we are done."}],"ID":9030},{"Watched":false,"Name":"Exercise 4","Duration":"11m 13s","ChapterTopicVideoID":8725,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"Since we have a function of 2 variables,"},{"Start":"00:02.370 ","End":"00:06.090","Text":"we have to find its critical points and classify them."},{"Start":"00:06.090 ","End":"00:10.500","Text":"Meaning to say whether each one is a minimum,"},{"Start":"00:10.500 ","End":"00:13.965","Text":"maximum, or saddle point."},{"Start":"00:13.965 ","End":"00:19.035","Text":"Let\u0027s start out by differentiating,"},{"Start":"00:19.035 ","End":"00:23.010","Text":"f with respect to x is,"},{"Start":"00:23.010 ","End":"00:30.120","Text":"3x squared, and then minus 6x."},{"Start":"00:30.120 ","End":"00:34.110","Text":"From here we get plus 3, and that\u0027s it,"},{"Start":"00:34.110 ","End":"00:41.040","Text":"and fy is, let\u0027s see,"},{"Start":"00:41.040 ","End":"00:46.260","Text":"minus 3y squared, plus 12y,"},{"Start":"00:46.260 ","End":"00:49.900","Text":"and then minus 12."},{"Start":"00:52.160 ","End":"00:55.715","Text":"Before I even do the second-order derivative,"},{"Start":"00:55.715 ","End":"00:58.750","Text":"let\u0027s see what the critical points are."},{"Start":"00:58.750 ","End":"01:02.325","Text":"Let\u0027s set these 2 both to be 0,"},{"Start":"01:02.325 ","End":"01:06.505","Text":"but also divide, like this one I\u0027ll divide by 3."},{"Start":"01:06.505 ","End":"01:08.350","Text":"If I divide it by 3,"},{"Start":"01:08.350 ","End":"01:10.720","Text":"I get x squared,"},{"Start":"01:10.720 ","End":"01:15.930","Text":"minus 2x, plus 1 equals 0."},{"Start":"01:15.930 ","End":"01:20.240","Text":"If I take this one and here I also divide it by 3."},{"Start":"01:20.240 ","End":"01:26.820","Text":"In fact, I\u0027ll divide it by minus 3 and get y squared,"},{"Start":"01:26.820 ","End":"01:32.640","Text":"minus 4y, plus 4, equals 0."},{"Start":"01:32.640 ","End":"01:35.710","Text":"Now, this should be familiar to you."},{"Start":"01:35.710 ","End":"01:40.370","Text":"This one is just x minus 1 squared."},{"Start":"01:40.660 ","End":"01:45.680","Text":"If this is 0, that only gives one possibility that x equals 1."},{"Start":"01:45.680 ","End":"01:46.985","Text":"Of course you could do it with"},{"Start":"01:46.985 ","End":"01:52.585","Text":"the quadratic equation and you\u0027d still only get just one solution."},{"Start":"01:52.585 ","End":"02:02.420","Text":"This one happens to be y minus 2 all squared using the special binomial expansions."},{"Start":"02:02.420 ","End":"02:06.395","Text":"You can always check by multiplying this out and getting this."},{"Start":"02:06.395 ","End":"02:11.360","Text":"But should have seen these enough times already to recognize them."},{"Start":"02:11.360 ","End":"02:13.280","Text":"This is y minus 2 squared is 0,"},{"Start":"02:13.280 ","End":"02:15.080","Text":"and if something squared is 0,"},{"Start":"02:15.080 ","End":"02:18.980","Text":"then it is 0, so y minus 2 is 0 and y equals 2."},{"Start":"02:18.980 ","End":"02:21.005","Text":"In other words, the point 1,"},{"Start":"02:21.005 ","End":"02:23.600","Text":"2 is our critical point,"},{"Start":"02:23.600 ","End":"02:26.075","Text":"but we don\u0027t know what kind it is."},{"Start":"02:26.075 ","End":"02:29.510","Text":"The main method we have is to compute that funny thing"},{"Start":"02:29.510 ","End":"02:32.615","Text":"called D which uses the second derivative."},{"Start":"02:32.615 ","End":"02:36.665","Text":"Let\u0027s continue with second-order partial derivatives."},{"Start":"02:36.665 ","End":"02:42.860","Text":"We get fxx, which is this with respect to x,"},{"Start":"02:42.860 ","End":"02:47.780","Text":"is 6x, minus 6,"},{"Start":"02:47.780 ","End":"02:53.475","Text":"and f. I meant to say xy."},{"Start":"02:53.475 ","End":"02:55.620","Text":"Well, you know what? Let\u0027s do yy first,"},{"Start":"02:55.620 ","End":"03:00.195","Text":"fyy is minus 6y,"},{"Start":"03:00.195 ","End":"03:05.320","Text":"plus 12 and fxy,"},{"Start":"03:05.430 ","End":"03:08.290","Text":"which is actually the same as fyx."},{"Start":"03:08.290 ","End":"03:17.720","Text":"Doesn\u0027t matter if you differentiate this one with respect to x, we\u0027ll get 6."},{"Start":"03:17.720 ","End":"03:23.770","Text":"No. This one with respect to y, I\u0027m sorry."},{"Start":"03:23.770 ","End":"03:28.465","Text":"If we differentiate this with respect to y, we get 0."},{"Start":"03:28.465 ","End":"03:33.535","Text":"Similarly, if we differentiate this with respect to x,"},{"Start":"03:33.535 ","End":"03:37.750","Text":"then we also get 0 so that\u0027s a double-check on that."},{"Start":"03:37.750 ","End":"03:42.415","Text":"If we substitute what these are at the point,"},{"Start":"03:42.415 ","End":"03:46.855","Text":"if I take these and substitute at the point 1,2,"},{"Start":"03:46.855 ","End":"03:50.845","Text":"then what we get is if x is 1, this is 0."},{"Start":"03:50.845 ","End":"03:54.170","Text":"If y is 2 minus 12,"},{"Start":"03:54.170 ","End":"03:56.990","Text":"so this is 0 and this already is 0."},{"Start":"03:56.990 ","End":"03:59.570","Text":"The quantity d that we compute,"},{"Start":"03:59.570 ","End":"04:02.615","Text":"which is this times this minus this squared,"},{"Start":"04:02.615 ","End":"04:06.020","Text":"is 0 times 0 minus 0 squared."},{"Start":"04:06.020 ","End":"04:14.760","Text":"Anyway, it\u0027s 0. Strictly speaking,"},{"Start":"04:14.760 ","End":"04:16.990","Text":"this is d at the point 1,2."},{"Start":"04:16.990 ","End":"04:20.254","Text":"When this is 0, we don\u0027t know."},{"Start":"04:20.254 ","End":"04:26.000","Text":"This test doesn\u0027t give us any information as to the nature of this critical point,"},{"Start":"04:26.000 ","End":"04:27.170","Text":"what kind it is."},{"Start":"04:27.170 ","End":"04:32.165","Text":"We\u0027ll have to use some round about method, some tricks."},{"Start":"04:32.165 ","End":"04:36.530","Text":"Notice that there are no terms containing x and y both."},{"Start":"04:36.530 ","End":"04:42.680","Text":"I can separate f of xy and write just the x bits and just the y bits."},{"Start":"04:42.680 ","End":"04:48.780","Text":"I can write x cubed and then minus 3x squared,"},{"Start":"04:48.780 ","End":"04:52.200","Text":"and then plus 3x."},{"Start":"04:52.200 ","End":"04:56.680","Text":"Let me just leave a space here, you\u0027ll see why."},{"Start":"04:58.370 ","End":"05:00.440","Text":"They\u0027re going to put a number here,"},{"Start":"05:00.440 ","End":"05:02.180","Text":"but I\u0027m not sure yet what."},{"Start":"05:02.180 ","End":"05:06.800","Text":"Then for the y\u0027s, that\u0027s a minus."},{"Start":"05:06.800 ","End":"05:12.280","Text":"Why don\u0027t I put the minus already here?"},{"Start":"05:12.280 ","End":"05:15.000","Text":"I\u0027m going to reverse all the signs of the y\u0027s,"},{"Start":"05:15.000 ","End":"05:17.640","Text":"so it\u0027s going to be plus y cubed,"},{"Start":"05:17.640 ","End":"05:22.900","Text":"minus 6y squared, plus 12y."},{"Start":"05:22.900 ","End":"05:25.395","Text":"Again, I want to leave a blank here."},{"Start":"05:25.395 ","End":"05:29.410","Text":"Then we have plus 8."},{"Start":"05:31.010 ","End":"05:35.300","Text":"I see that the coefficients 1,"},{"Start":"05:35.300 ","End":"05:38.930","Text":"3, 3 something called 1 minus 3,3 something,"},{"Start":"05:38.930 ","End":"05:49.300","Text":"and I\u0027m reminded of a formula from algebra in general that says a minus b cubed,"},{"Start":"05:49.300 ","End":"05:54.465","Text":"is a cubed, minus 3a squared b,"},{"Start":"05:54.465 ","End":"06:00.240","Text":"plus 3ab squared, minus b cubed."},{"Start":"06:00.240 ","End":"06:03.075","Text":"I look at this and I think,"},{"Start":"06:03.075 ","End":"06:07.270","Text":"if I take my a as x,"},{"Start":"06:07.370 ","End":"06:14.555","Text":"and I take the b as 1x minus 1 cubed according to this,"},{"Start":"06:14.555 ","End":"06:16.700","Text":"would be x cubed,"},{"Start":"06:16.700 ","End":"06:21.510","Text":"minus 3x squared 1, which I don\u0027t write,"},{"Start":"06:21.510 ","End":"06:24.690","Text":"plus 3x 1 squared,"},{"Start":"06:24.690 ","End":"06:26.025","Text":"I don\u0027t need to write,"},{"Start":"06:26.025 ","End":"06:29.640","Text":"minus b cubed is minus 1."},{"Start":"06:29.640 ","End":"06:34.165","Text":"If I go back here and put a minus 1,"},{"Start":"06:34.165 ","End":"06:40.010","Text":"then I know that this will be just x minus 1 cubed."},{"Start":"06:40.010 ","End":"06:43.595","Text":"But I can\u0027t just put a minus 1 here."},{"Start":"06:43.595 ","End":"06:48.510","Text":"I\u0027ll change that 8, to a 9,"},{"Start":"06:48.510 ","End":"06:53.500","Text":"because I subtracted 1 and I must change the total."},{"Start":"06:53.500 ","End":"07:00.859","Text":"We can do the same thing with y. I will get that y minus 2 cubed from this formula,"},{"Start":"07:00.859 ","End":"07:04.525","Text":"is y cubed, minus 3."},{"Start":"07:04.525 ","End":"07:07.885","Text":"A squared is y squared times 2,"},{"Start":"07:07.885 ","End":"07:11.890","Text":"is minus 6y squared,"},{"Start":"07:11.890 ","End":"07:15.645","Text":"plus 3y 2 squared,"},{"Start":"07:15.645 ","End":"07:20.430","Text":"3 times 2 squared is 12y."},{"Start":"07:20.430 ","End":"07:22.530","Text":"Then here, the minus b cubed,"},{"Start":"07:22.530 ","End":"07:25.990","Text":"will be minus 8."},{"Start":"07:27.530 ","End":"07:30.725","Text":"I can put a minus 8 here,"},{"Start":"07:30.725 ","End":"07:35.270","Text":"which means that I\u0027ve added 8 altogether because it\u0027s minus minus 8,"},{"Start":"07:35.270 ","End":"07:37.025","Text":"so I have to subtract 8."},{"Start":"07:37.025 ","End":"07:38.750","Text":"I make a second adjustment,"},{"Start":"07:38.750 ","End":"07:40.955","Text":"and this is now 1."},{"Start":"07:40.955 ","End":"07:46.685","Text":"What I get is x minus 1 cubed minus,"},{"Start":"07:46.685 ","End":"07:50.420","Text":"and this bit is y minus 2 cubed,"},{"Start":"07:50.420 ","End":"07:52.505","Text":"and then plus 1."},{"Start":"07:52.505 ","End":"07:54.710","Text":"Let me draw a little sketch of what I\u0027m going to do."},{"Start":"07:54.710 ","End":"08:00.970","Text":"I\u0027m going to sketch the axes, the y-axis, x-axis."},{"Start":"08:00.970 ","End":"08:05.600","Text":"Then the point I\u0027m interested in is 1,2."},{"Start":"08:05.600 ","End":"08:08.850","Text":"Let\u0027s say 1,2 is here,"},{"Start":"08:09.040 ","End":"08:12.590","Text":"so this is 1,2."},{"Start":"08:12.590 ","End":"08:16.055","Text":"This is the y-axis,"},{"Start":"08:16.055 ","End":"08:20.065","Text":"and this is the x-axis."},{"Start":"08:20.065 ","End":"08:27.229","Text":"Let me take a horizontal line through this point like this."},{"Start":"08:27.229 ","End":"08:34.610","Text":"This will be the line where y equals 2 and just x changes."},{"Start":"08:34.610 ","End":"08:40.080","Text":"The value of the function at this point, in other words,"},{"Start":"08:40.080 ","End":"08:42.720","Text":"f of 1, 2,"},{"Start":"08:42.720 ","End":"08:46.400","Text":"is obtained by letting x equals 1 and y equals 2."},{"Start":"08:46.400 ","End":"08:48.830","Text":"This becomes 0, this becomes 0."},{"Start":"08:48.830 ","End":"08:51.665","Text":"All we\u0027re left with is 1."},{"Start":"08:51.665 ","End":"08:56.640","Text":"At this point, f is equal to 1."},{"Start":"08:56.710 ","End":"09:02.020","Text":"Suppose I take 2 nearby points on this line on either side."},{"Start":"09:02.020 ","End":"09:03.980","Text":"Let\u0027s take a point here,"},{"Start":"09:03.980 ","End":"09:06.500","Text":"and a point here."},{"Start":"09:06.500 ","End":"09:14.625","Text":"Both of these points are going to be of the form x, 2."},{"Start":"09:14.625 ","End":"09:18.480","Text":"Only here, the x will be bigger than one,"},{"Start":"09:18.480 ","End":"09:20.790","Text":"and here, the x will be less than 1,"},{"Start":"09:20.790 ","End":"09:22.770","Text":"but they\u0027re both x, 2."},{"Start":"09:22.770 ","End":"09:24.865","Text":"The y is [inaudible] 2."},{"Start":"09:24.865 ","End":"09:29.645","Text":"Let\u0027s figure out what the function is at each of these points."},{"Start":"09:29.645 ","End":"09:34.560","Text":"Well, in general, f of x, 2, will equal."},{"Start":"09:34.560 ","End":"09:37.230","Text":"Because y is 2, this part is still 0."},{"Start":"09:37.230 ","End":"09:39.320","Text":"Here we still get one."},{"Start":"09:39.320 ","End":"09:43.805","Text":"But we get here the x minus 1 cubed."},{"Start":"09:43.805 ","End":"09:48.770","Text":"We get x minus 1 cubed, plus 1."},{"Start":"09:48.770 ","End":"09:51.840","Text":"Let\u0027s take 2 cases."},{"Start":"09:55.240 ","End":"09:59.420","Text":"Well, let\u0027s write the 2 cases or x less than 1."},{"Start":"09:59.420 ","End":"10:01.130","Text":"If x is bigger than 1,"},{"Start":"10:01.130 ","End":"10:03.170","Text":"x minus 1 is positive,"},{"Start":"10:03.170 ","End":"10:05.435","Text":"positive cubed is positive,"},{"Start":"10:05.435 ","End":"10:12.610","Text":"so I get that f is bigger than 1 in this case."},{"Start":"10:12.610 ","End":"10:15.410","Text":"If x is less than 1,"},{"Start":"10:15.410 ","End":"10:17.195","Text":"x minus 1 is negative,"},{"Start":"10:17.195 ","End":"10:21.025","Text":"negative cubed is negative because 3 is an odd number,"},{"Start":"10:21.025 ","End":"10:26.170","Text":"negative plus 1 means that we have f which is less than 1."},{"Start":"10:26.170 ","End":"10:30.620","Text":"At this point, f equals 1."},{"Start":"10:30.620 ","End":"10:34.430","Text":"At this point, f is bigger than 1,"},{"Start":"10:34.430 ","End":"10:35.450","Text":"and at this point,"},{"Start":"10:35.450 ","End":"10:36.875","Text":"f is less than 1."},{"Start":"10:36.875 ","End":"10:38.450","Text":"A little bit small."},{"Start":"10:38.450 ","End":"10:42.500","Text":"The point is, I can find nearby points where the value of"},{"Start":"10:42.500 ","End":"10:47.570","Text":"the function f is both larger and smaller than the value at the given point."},{"Start":"10:47.570 ","End":"10:50.360","Text":"If it\u0027s neither a maximum nor a minimum,"},{"Start":"10:50.360 ","End":"10:52.450","Text":"that makes it a saddle."},{"Start":"10:52.450 ","End":"10:54.605","Text":"We\u0027ve now classified this,"},{"Start":"10:54.605 ","End":"10:59.075","Text":"as being a saddle point."},{"Start":"10:59.075 ","End":"11:01.820","Text":"We could have used the going along in"},{"Start":"11:01.820 ","End":"11:06.590","Text":"a vertical line where y is bigger than 2 or less than 2."},{"Start":"11:06.590 ","End":"11:07.700","Text":"Same trick would have worked,"},{"Start":"11:07.700 ","End":"11:09.930","Text":"but we only needed one of them."},{"Start":"11:10.190 ","End":"11:13.630","Text":"For this example, we\u0027re done."}],"ID":9031},{"Watched":false,"Name":"Exercise 5","Duration":"22m 21s","ChapterTopicVideoID":8726,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we\u0027re given a function f of x, y, and z."},{"Start":"00:05.130 ","End":"00:09.570","Text":"This is one of the differences from the previous exercises where we totally"},{"Start":"00:09.570 ","End":"00:14.475","Text":"worked with just functions of 2 variables, x and y."},{"Start":"00:14.475 ","End":"00:18.060","Text":"Now we\u0027re increasing the number of variables."},{"Start":"00:18.060 ","End":"00:21.930","Text":"We have a function of 3 variables given by this formula,"},{"Start":"00:21.930 ","End":"00:24.870","Text":"and we\u0027re also told that x,"},{"Start":"00:24.870 ","End":"00:26.760","Text":"y, and z are positive."},{"Start":"00:26.760 ","End":"00:32.430","Text":"For one thing, this means that the denominators are not 0, so everything\u0027s defined."},{"Start":"00:32.430 ","End":"00:37.500","Text":"We want to find the critical points and classify them as"},{"Start":"00:37.500 ","End":"00:44.475","Text":"to maximum, minimum, or saddle."},{"Start":"00:44.475 ","End":"00:48.740","Text":"In 3-dimensions, we also say that anything that\u0027s not a maximum or a minimum,"},{"Start":"00:48.740 ","End":"00:51.425","Text":"but it\u0027s a critical point, then it\u0027s a saddle."},{"Start":"00:51.425 ","End":"00:57.305","Text":"As usual, we start off by finding the partial derivatives,"},{"Start":"00:57.305 ","End":"01:00.785","Text":"f with respect to x."},{"Start":"01:00.785 ","End":"01:04.440","Text":"From here, we get 1."},{"Start":"01:05.480 ","End":"01:11.195","Text":"If the derivative of 1 over x is minus 1 over x squared,"},{"Start":"01:11.195 ","End":"01:18.605","Text":"then here we\u0027ll get minus the 1 times y squared is y squared and instead of x,"},{"Start":"01:18.605 ","End":"01:20.704","Text":"we have x squared."},{"Start":"01:20.704 ","End":"01:25.985","Text":"But the y squared over 4 stays because it\u0027s constant as far as x goes."},{"Start":"01:25.985 ","End":"01:28.585","Text":"Then this doesn\u0027t have x at all,"},{"Start":"01:28.585 ","End":"01:31.805","Text":"I don\u0027t know what is this so we can stop here."},{"Start":"01:31.805 ","End":"01:37.010","Text":"Now f with respect to y is, let\u0027s see."},{"Start":"01:37.010 ","End":"01:43.280","Text":"This is nothing with respect to y. Y squared gives us 2y,"},{"Start":"01:43.280 ","End":"01:47.780","Text":"so we get 2y over 4x."},{"Start":"01:47.780 ","End":"01:50.455","Text":"The denominator is a constant."},{"Start":"01:50.455 ","End":"01:56.284","Text":"From here, based on 1 over y giving minus 1 over y squared,"},{"Start":"01:56.284 ","End":"02:01.010","Text":"we have minus z squared over y squared and nothing else."},{"Start":"02:01.010 ","End":"02:07.590","Text":"I\u0027ll just cancel 2 from here and 2 from here, that leaves 2."},{"Start":"02:07.590 ","End":"02:11.510","Text":"Of course, we have more work in 3 variables,"},{"Start":"02:11.510 ","End":"02:13.925","Text":"there\u0027s more partial derivatives."},{"Start":"02:13.925 ","End":"02:18.450","Text":"Now, this is nothing, this is nothing."},{"Start":"02:18.450 ","End":"02:20.715","Text":"Here, z squared gives 2z,"},{"Start":"02:20.715 ","End":"02:24.825","Text":"so it\u0027s 2z over y."},{"Start":"02:24.825 ","End":"02:26.850","Text":"From here, once again,"},{"Start":"02:26.850 ","End":"02:30.030","Text":"1 over z would give us minus 1 over z squared,"},{"Start":"02:30.030 ","End":"02:34.910","Text":"only here it\u0027s minus 2 over z squared because we have the constant 2."},{"Start":"02:35.140 ","End":"02:40.115","Text":"At this point, we can either continue finding partial derivatives,"},{"Start":"02:40.115 ","End":"02:45.620","Text":"or we can look for critical points here. Either is okay."},{"Start":"02:45.620 ","End":"02:49.025","Text":"I like to look for critical points right away because if I don\u0027t find any,"},{"Start":"02:49.025 ","End":"02:52.370","Text":"then there\u0027s no point in finding second-order partial derivatives."},{"Start":"02:52.370 ","End":"02:56.885","Text":"It depends if you\u0027re optimistic or pessimistic about finding critical points out,"},{"Start":"02:56.885 ","End":"02:58.805","Text":"go and do it now."},{"Start":"02:58.805 ","End":"03:01.580","Text":"I\u0027m not going to copy this again,"},{"Start":"03:01.580 ","End":"03:06.115","Text":"but we just want these right-hand sides and we want to set them equal to 0."},{"Start":"03:06.115 ","End":"03:09.150","Text":"Here is 0, here is 0, here is 0."},{"Start":"03:09.150 ","End":"03:11.720","Text":"I\u0027ll just put a curly brace here to show that we\u0027re"},{"Start":"03:11.720 ","End":"03:15.140","Text":"doing 3 equations and 3 unknowns, x, y,"},{"Start":"03:15.140 ","End":"03:22.170","Text":"and z. I just noticed we can cancel the 2 here and we can cancel the 2 here,"},{"Start":"03:22.170 ","End":"03:25.420","Text":"but we\u0027ll have to leave a 1, can\u0027t leave it blank."},{"Start":"03:26.660 ","End":"03:29.570","Text":"There are a lot of quantities squared here,"},{"Start":"03:29.570 ","End":"03:34.400","Text":"but I want to remind you that we have this restriction that x,"},{"Start":"03:34.400 ","End":"03:36.830","Text":"y, and z are all positive."},{"Start":"03:36.830 ","End":"03:39.620","Text":"If we have to take square roots of something to find x,"},{"Start":"03:39.620 ","End":"03:41.900","Text":"y, or z, we take the positive."},{"Start":"03:41.900 ","End":"03:44.795","Text":"There\u0027s many ways to go about this."},{"Start":"03:44.795 ","End":"03:46.775","Text":"After staring at it a bit,"},{"Start":"03:46.775 ","End":"03:52.745","Text":"I decided that we\u0027ll go with the y over x and finding it because there\u0027s a y over x here,"},{"Start":"03:52.745 ","End":"03:59.040","Text":"admittedly squared, and there\u0027s also a y over x here."},{"Start":"03:59.270 ","End":"04:04.955","Text":"Also, I noticed that this contains z over y."},{"Start":"04:04.955 ","End":"04:09.620","Text":"If I can find y over x from here and z over y from here,"},{"Start":"04:09.620 ","End":"04:11.320","Text":"that might lead to something."},{"Start":"04:11.320 ","End":"04:20.360","Text":"Let\u0027s take the first one and I can bring this to the other side and multiply by 4,"},{"Start":"04:20.360 ","End":"04:28.605","Text":"so I get that y squared over x squared equals 4 from here."},{"Start":"04:28.605 ","End":"04:34.630","Text":"This will give me that y over x is,"},{"Start":"04:34.630 ","End":"04:40.475","Text":"normally I would say plus or minus 2 but because of what I said earlier,"},{"Start":"04:40.475 ","End":"04:43.370","Text":"if y and x are positive so is y over x,"},{"Start":"04:43.370 ","End":"04:46.680","Text":"so I can erase the plus or minus."},{"Start":"04:46.930 ","End":"04:49.280","Text":"Now from this one,"},{"Start":"04:49.280 ","End":"04:52.215","Text":"I can get z over y."},{"Start":"04:52.215 ","End":"04:58.270","Text":"Z over y, I just bring this to the other side is 1 over z squared."},{"Start":"04:58.270 ","End":"05:00.020","Text":"Now look at this middle equation."},{"Start":"05:00.020 ","End":"05:04.150","Text":"I have y over x and I have z over y, it\u0027s squared."},{"Start":"05:04.150 ","End":"05:07.985","Text":"How about I substitute both of these in here?"},{"Start":"05:07.985 ","End":"05:14.335","Text":"I\u0027m just saying I\u0027m substituting this bit and this bit,"},{"Start":"05:14.335 ","End":"05:16.760","Text":"both of these into this equation."},{"Start":"05:16.760 ","End":"05:18.380","Text":"If y over x is 2,"},{"Start":"05:18.380 ","End":"05:22.720","Text":"what I have here is 2 over 2 because there\u0027s a 2 in the denominator."},{"Start":"05:22.720 ","End":"05:26.390","Text":"So 2 over 2 minus,"},{"Start":"05:26.720 ","End":"05:30.830","Text":"I put little brackets here just to show you that z squared"},{"Start":"05:30.830 ","End":"05:35.045","Text":"over y squared is z over y all squared."},{"Start":"05:35.045 ","End":"05:37.520","Text":"I can put from here,"},{"Start":"05:37.520 ","End":"05:42.615","Text":"1 over z squared, all squared."},{"Start":"05:42.615 ","End":"05:46.875","Text":"Now this is 1, this is 1 over z to the 4."},{"Start":"05:46.875 ","End":"05:50.775","Text":"If 1 equals 1 over z to the 4,"},{"Start":"05:50.775 ","End":"05:53.520","Text":"we just get that z to the 4 equals 1."},{"Start":"05:53.520 ","End":"05:56.565","Text":"Maybe I skipped a step, but this is easy algebra."},{"Start":"05:56.565 ","End":"06:01.410","Text":"Then z has to be the fourth root of 1."},{"Start":"06:01.410 ","End":"06:03.255","Text":"Normally I would say,"},{"Start":"06:03.255 ","End":"06:06.990","Text":"z is plus or minus 1,"},{"Start":"06:06.990 ","End":"06:08.485","Text":"because the fourth root of 1 is 1."},{"Start":"06:08.485 ","End":"06:11.135","Text":"But again, because of this restriction,"},{"Start":"06:11.135 ","End":"06:13.430","Text":"then z has to be plus 1."},{"Start":"06:13.430 ","End":"06:16.795","Text":"So I just erased the plus or minus."},{"Start":"06:16.795 ","End":"06:19.940","Text":"We have one of the variables, z is 1."},{"Start":"06:19.940 ","End":"06:27.510","Text":"Now what should we do? Suppose we substituted it in here."},{"Start":"06:27.860 ","End":"06:34.800","Text":"We get 1 over y equals 1 over 1 squared."},{"Start":"06:34.800 ","End":"06:38.955","Text":"Obviously from this, y is equal to 1,"},{"Start":"06:38.955 ","End":"06:41.385","Text":"and we\u0027ll highlight that."},{"Start":"06:41.385 ","End":"06:44.675","Text":"Now, if we put y equals 1 in here,"},{"Start":"06:44.675 ","End":"06:48.470","Text":"we get that 1 over x equals 2,"},{"Start":"06:48.470 ","End":"06:52.580","Text":"which gives us that x equals 1/2,"},{"Start":"06:52.580 ","End":"06:55.730","Text":"and I\u0027ll highlight that."},{"Start":"06:55.730 ","End":"06:58.550","Text":"We have now x, y, and z."},{"Start":"06:58.550 ","End":"07:02.910","Text":"This gives us the point 1/2, 1,"},{"Start":"07:02.910 ","End":"07:07.170","Text":"1, x, y, z."},{"Start":"07:07.170 ","End":"07:10.380","Text":"We have exactly 1 critical point."},{"Start":"07:10.380 ","End":"07:15.710","Text":"Now that we see that there is a point to continuing with the second order derivatives,"},{"Start":"07:15.710 ","End":"07:20.570","Text":"and then we\u0027ll find out whether this is minimum, maximum, or saddle."},{"Start":"07:20.570 ","End":"07:24.110","Text":"Let\u0027s take partial derivative of f with respect to"},{"Start":"07:24.110 ","End":"07:27.320","Text":"x and find its partial derivatives with respect to x,"},{"Start":"07:27.320 ","End":"07:29.390","Text":"y, and z, 3 of them."},{"Start":"07:29.390 ","End":"07:35.095","Text":"We have f_xx is, the 1 disappears."},{"Start":"07:35.095 ","End":"07:40.430","Text":"The derivative of 1 over x squared is minus 2 over x cubed,"},{"Start":"07:40.430 ","End":"07:44.785","Text":"so we get minus,"},{"Start":"07:44.785 ","End":"07:51.555","Text":"minus 2 y squared over 4x cubed."},{"Start":"07:51.555 ","End":"07:56.330","Text":"The minus and the 4 stayed because they were constants and so did the y squared."},{"Start":"07:56.330 ","End":"08:00.085","Text":"From x squared, we got minus 2 over x cubed."},{"Start":"08:00.085 ","End":"08:02.250","Text":"I can cancel, minus,"},{"Start":"08:02.250 ","End":"08:04.035","Text":"minus 2 is 2,"},{"Start":"08:04.035 ","End":"08:07.290","Text":"2 goes into 4 twice."},{"Start":"08:07.290 ","End":"08:11.610","Text":"Here I just have y squared over 2x cubed."},{"Start":"08:13.860 ","End":"08:16.989","Text":"What I feel like doing this time is substituting"},{"Start":"08:16.989 ","End":"08:19.780","Text":"each one as it comes and then I won\u0027t forget."},{"Start":"08:19.780 ","End":"08:24.415","Text":"So fxx at the critical points,"},{"Start":"08:24.415 ","End":"08:28.420","Text":"1/2, 1, 1 is equal to,"},{"Start":"08:28.420 ","End":"08:32.020","Text":"see, x is 1/2,"},{"Start":"08:32.020 ","End":"08:35.050","Text":"so the denominator is 2 times 1/8,"},{"Start":"08:35.050 ","End":"08:38.395","Text":"which is 1/4, the numerator is 1,"},{"Start":"08:38.395 ","End":"08:41.530","Text":"1 over 1/4 is 4."},{"Start":"08:41.530 ","End":"08:48.269","Text":"Next, fxy is, this with respect to y,"},{"Start":"08:48.269 ","End":"08:51.450","Text":"derivative of y squared is 2y,"},{"Start":"08:51.450 ","End":"08:58.410","Text":"so it\u0027s minus 2y over 4x squared,"},{"Start":"08:58.410 ","End":"09:01.805","Text":"2 goes into 4 twice,"},{"Start":"09:01.805 ","End":"09:06.114","Text":"and then fxy at the same point,"},{"Start":"09:06.114 ","End":"09:10.690","Text":"I\u0027ll write it, is equal to,"},{"Start":"09:10.690 ","End":"09:14.740","Text":"y is 1, x is 1/2,"},{"Start":"09:14.740 ","End":"09:18.010","Text":"1/2 squared is 1/4 times 2 is 1/2,"},{"Start":"09:18.010 ","End":"09:22.615","Text":"minus 1 over 1/2, is minus 2."},{"Start":"09:22.615 ","End":"09:28.300","Text":"Then fxz is, well,"},{"Start":"09:28.300 ","End":"09:29.635","Text":"there is no z here,"},{"Start":"09:29.635 ","End":"09:39.805","Text":"so that\u0027s 0, so specifically fxz at the same point is also 0, I should write it."},{"Start":"09:39.805 ","End":"09:42.295","Text":"That\u0027s that."},{"Start":"09:42.295 ","End":"09:47.560","Text":"I\u0027ll continue and put a separator,"},{"Start":"09:47.560 ","End":"09:48.880","Text":"I\u0027ll continue over here."},{"Start":"09:48.880 ","End":"09:53.635","Text":"The next one is fy,"},{"Start":"09:53.635 ","End":"10:01.540","Text":"I don\u0027t need to do it with respect to x because fxy is the same as fyx,"},{"Start":"10:01.540 ","End":"10:05.090","Text":"so I go straight away to fyy."},{"Start":"10:05.640 ","End":"10:09.145","Text":"Maybe I should make a note of that."},{"Start":"10:09.145 ","End":"10:13.510","Text":"This is the same as if it was fyx,"},{"Start":"10:13.510 ","End":"10:16.550","Text":"this is the same as if it was fzx,"},{"Start":"10:17.070 ","End":"10:20.710","Text":"so we already have fyx,"},{"Start":"10:20.710 ","End":"10:24.010","Text":"now fyy, is equal to,"},{"Start":"10:24.010 ","End":"10:28.525","Text":"let\u0027s see, this one with respect to y,"},{"Start":"10:28.525 ","End":"10:33.265","Text":"from the first, I just get 1 over"},{"Start":"10:33.265 ","End":"10:40.225","Text":"2x because the y disappears and it\u0027s a constant times y,"},{"Start":"10:40.225 ","End":"10:42.310","Text":"the constant is 1 over 2x,"},{"Start":"10:42.310 ","End":"10:44.740","Text":"but that\u0027s just the first term here,"},{"Start":"10:44.740 ","End":"10:47.935","Text":"there\u0027s also a minus z squared over y squared,"},{"Start":"10:47.935 ","End":"10:51.130","Text":"and so we get minus,"},{"Start":"10:51.130 ","End":"10:55.090","Text":"now 1 over y squared gives us minus 2 over y cubed,"},{"Start":"10:55.090 ","End":"10:58.975","Text":"so the minus becomes a plus and I get a 2 here,"},{"Start":"10:58.975 ","End":"11:02.035","Text":"and I get a y cubed here."},{"Start":"11:02.035 ","End":"11:10.825","Text":"Now this means that fyy at the given point is equal to."},{"Start":"11:10.825 ","End":"11:12.580","Text":"If x is 1/2,"},{"Start":"11:12.580 ","End":"11:16.105","Text":"2 times 1/2 is 1, so this first term is 1,"},{"Start":"11:16.105 ","End":"11:19.420","Text":"and then z and y are both 1,"},{"Start":"11:19.420 ","End":"11:21.235","Text":"so here I get 2,"},{"Start":"11:21.235 ","End":"11:25.330","Text":"1 plus 2 equals 3."},{"Start":"11:25.330 ","End":"11:32.530","Text":"Next, fyz is equal to this with respect to z."},{"Start":"11:32.530 ","End":"11:35.470","Text":"Well, this doesn\u0027t contain z, so that\u0027s nothing."},{"Start":"11:35.470 ","End":"11:38.320","Text":"Z squared gives me minus 2z,"},{"Start":"11:38.320 ","End":"11:43.570","Text":"so I have minus 2z over y squared,"},{"Start":"11:43.570 ","End":"11:49.600","Text":"and therefore fyz at the point 1/2, 1,"},{"Start":"11:49.600 ","End":"11:54.490","Text":"1 is equal to z and y are both 1,"},{"Start":"11:54.490 ","End":"11:58.450","Text":"and so I just get minus 2."},{"Start":"11:58.450 ","End":"12:03.670","Text":"I\u0027m just mentioning this as the same as fzy."},{"Start":"12:03.670 ","End":"12:06.040","Text":"Now I realize I did something misleading before when"},{"Start":"12:06.040 ","End":"12:09.010","Text":"I multiplied the 2 sides of the equation by 2,"},{"Start":"12:09.010 ","End":"12:11.020","Text":"that\u0027s very well for solving the equation,"},{"Start":"12:11.020 ","End":"12:14.140","Text":"but I changed the partial derivatives,"},{"Start":"12:14.140 ","End":"12:19.990","Text":"so I\u0027d rather undo that because I\u0027m using this for the partial derivative."},{"Start":"12:19.990 ","End":"12:25.225","Text":"I was about to check if fyz is the same as fzy,"},{"Start":"12:25.225 ","End":"12:29.740","Text":"and I was just looking, if I take f of z with respect to y, this is nothing,"},{"Start":"12:29.740 ","End":"12:31.480","Text":"it doesn\u0027t contain y,"},{"Start":"12:31.480 ","End":"12:33.820","Text":"and this with respect to y,"},{"Start":"12:33.820 ","End":"12:37.630","Text":"means that it becomes minus 2z over y squared,"},{"Start":"12:37.630 ","End":"12:40.330","Text":"which is the same as this."},{"Start":"12:40.330 ","End":"12:44.530","Text":"Similarly, you could check that fyx is really what\u0027s written here,"},{"Start":"12:44.530 ","End":"12:48.355","Text":"and fxz is 0."},{"Start":"12:48.355 ","End":"12:53.650","Text":"The partial derivatives that are mixed really are equal if I reverse the order,"},{"Start":"12:53.650 ","End":"12:55.780","Text":"I do that as I\u0027ve check."},{"Start":"12:55.780 ","End":"12:58.165","Text":"That\u0027s that."},{"Start":"12:58.165 ","End":"13:03.850","Text":"The final one is fzz,"},{"Start":"13:03.850 ","End":"13:05.935","Text":"and this is equal to,"},{"Start":"13:05.935 ","End":"13:11.850","Text":"from here I get 2 over y, and from here,"},{"Start":"13:11.850 ","End":"13:15.360","Text":"1 over z squared is minus 2 over z cubed,"},{"Start":"13:15.360 ","End":"13:18.315","Text":"so I get the minus with the minus makes a plus,"},{"Start":"13:18.315 ","End":"13:21.050","Text":"the 2 with the 2 gives me a 4,"},{"Start":"13:21.050 ","End":"13:24.475","Text":"and it\u0027s over z cubed."},{"Start":"13:24.475 ","End":"13:29.410","Text":"Fzz at our particular point,"},{"Start":"13:29.410 ","End":"13:31.975","Text":"this is equal to,"},{"Start":"13:31.975 ","End":"13:36.190","Text":"I put in y equals 1 and z equals 1,"},{"Start":"13:36.190 ","End":"13:40.390","Text":"so I get 2 plus 4 is 6."},{"Start":"13:40.390 ","End":"13:42.640","Text":"Now I\u0027m going to remind you what I do with"},{"Start":"13:42.640 ","End":"13:46.690","Text":"all these partial derivatives in the case of 3 variables."},{"Start":"13:46.690 ","End":"13:52.780","Text":"There\u0027s a matrix, and I\u0027ll call it H of x and y,"},{"Start":"13:52.780 ","End":"13:58.689","Text":"there\u0027s a reason H it\u0027s in literature and the Wikipedia had called the Hessian,"},{"Start":"13:58.689 ","End":"14:04.525","Text":"and that is the matrix that contains all the partial derivatives in order."},{"Start":"14:04.525 ","End":"14:06.115","Text":"I\u0027ll show you what order do I mean."},{"Start":"14:06.115 ","End":"14:09.115","Text":"Here, on the first row, I put fx,"},{"Start":"14:09.115 ","End":"14:13.700","Text":"its partial derivatives, fxx, fxy, fxz."},{"Start":"14:15.780 ","End":"14:19.060","Text":"The second row, I have fy."},{"Start":"14:19.060 ","End":"14:21.730","Text":"I\u0027ll start off for just writing that 3 times,"},{"Start":"14:21.730 ","End":"14:25.360","Text":"and then here I put xyz."},{"Start":"14:25.360 ","End":"14:32.320","Text":"Here I put fz 3 times, and here xyz."},{"Start":"14:32.320 ","End":"14:41.410","Text":"But I want to remind you that this one is the same as this one."},{"Start":"14:41.410 ","End":"14:43.195","Text":"The 3 of them they\u0027re duplicates."},{"Start":"14:43.195 ","End":"14:47.545","Text":"The zy is the same as the yz,"},{"Start":"14:47.545 ","End":"14:51.280","Text":"and the yx, is the same as the xy."},{"Start":"14:51.280 ","End":"14:57.685","Text":"These 3 I can get from the mirror images on the other side of this diagonal."},{"Start":"14:57.685 ","End":"15:00.070","Text":"Close the matrix."},{"Start":"15:00.070 ","End":"15:03.220","Text":"What do we do with this 3 by 3 matrix?"},{"Start":"15:03.220 ","End":"15:06.820","Text":"Well, first of all,"},{"Start":"15:06.820 ","End":"15:14.230","Text":"let me write its value at the point 1/2, 1, 1."},{"Start":"15:14.230 ","End":"15:16.150","Text":"This is equal 2."},{"Start":"15:16.150 ","End":"15:20.020","Text":"Now, let\u0027s look at our values, f x,"},{"Start":"15:20.020 ","End":"15:25.870","Text":"x at the point is 4,"},{"Start":"15:25.870 ","End":"15:30.160","Text":"f x y is minus 2,"},{"Start":"15:30.160 ","End":"15:33.295","Text":"f x z is 0,"},{"Start":"15:33.295 ","End":"15:36.520","Text":"f y, y is 3."},{"Start":"15:36.520 ","End":"15:41.905","Text":"That goes here, f y z is minus 2,"},{"Start":"15:41.905 ","End":"15:44.005","Text":"f z, z is 6,"},{"Start":"15:44.005 ","End":"15:46.480","Text":"and because of this duplication,"},{"Start":"15:46.480 ","End":"15:48.850","Text":"the minus 2 here is the same here."},{"Start":"15:48.850 ","End":"15:51.160","Text":"The minus 2 here is the same here,"},{"Start":"15:51.160 ","End":"15:54.470","Text":"and the 0 here is the 0 here."},{"Start":"15:56.070 ","End":"16:01.870","Text":"So for example, the f z x is the same as the f x z."},{"Start":"16:01.870 ","End":"16:03.790","Text":"That gives these 2 0s."},{"Start":"16:03.790 ","End":"16:09.700","Text":"Now, what we have to do is just for organization,"},{"Start":"16:09.700 ","End":"16:13.150","Text":"I\u0027m going to put some lines here,"},{"Start":"16:13.150 ","End":"16:21.250","Text":"and here, and what we need to do is compute 3 determinants."},{"Start":"16:21.250 ","End":"16:26.410","Text":"The first determinant is a 1 by 1 of just this matrix,"},{"Start":"16:26.410 ","End":"16:27.715","Text":"the 1 by 1 matrix."},{"Start":"16:27.715 ","End":"16:29.440","Text":"Then we need to take the determinant of"},{"Start":"16:29.440 ","End":"16:32.095","Text":"this and then the determinants of the whole thing."},{"Start":"16:32.095 ","End":"16:33.845","Text":"Let\u0027s give them names."},{"Start":"16:33.845 ","End":"16:42.045","Text":"Let\u0027s call them H_1 is the determinant of the matrix that just contains 1 number 4,"},{"Start":"16:42.045 ","End":"16:47.260","Text":"and then, okay, I\u0027ll scroll a bit more."},{"Start":"16:47.970 ","End":"16:51.760","Text":"Then we need H_2,"},{"Start":"16:51.760 ","End":"16:55.240","Text":"which will be the determinant of this 2 by 2,"},{"Start":"16:55.240 ","End":"17:00.835","Text":"which is 4 minus 2 minus 2 3,"},{"Start":"17:00.835 ","End":"17:03.460","Text":"and then the 3rd determinant,"},{"Start":"17:03.460 ","End":"17:08.080","Text":"or call it H_3 is the determinant of this whole thing."},{"Start":"17:08.080 ","End":"17:12.385","Text":"4 minus 2, 0 minus 2,"},{"Start":"17:12.385 ","End":"17:20.120","Text":"3 minus 2, 0 minus 2, 6."},{"Start":"17:20.460 ","End":"17:25.735","Text":"Now, the determinant of a 1 by 1 matrix is just the entry itself."},{"Start":"17:25.735 ","End":"17:29.200","Text":"So this is equal to 4."},{"Start":"17:29.200 ","End":"17:35.840","Text":"For a 2 by 2, we take this diagonal multiplied minus this diagonal,"},{"Start":"17:35.900 ","End":"17:43.934","Text":"and so it\u0027s 12 minus 4 because the product of these 2 is 4,"},{"Start":"17:43.934 ","End":"17:47.115","Text":"and that gives us 8,"},{"Start":"17:47.115 ","End":"17:52.705","Text":"and this 1, let me just give you the answer."},{"Start":"17:52.705 ","End":"17:59.470","Text":"It\u0027s equal to 32 but I\u0027ll put an asterisk here and I\u0027ll show you at"},{"Start":"17:59.470 ","End":"18:01.825","Text":"the end how I reached this computation if you"},{"Start":"18:01.825 ","End":"18:06.955","Text":"care but I don\u0027t want it to stop the flow of things."},{"Start":"18:06.955 ","End":"18:09.310","Text":"Now, here\u0027s the thing we do."},{"Start":"18:09.310 ","End":"18:11.230","Text":"We look at the signs of these 3,"},{"Start":"18:11.230 ","End":"18:12.985","Text":"of H_1, H_2, H_3."},{"Start":"18:12.985 ","End":"18:14.920","Text":"This 1 is positive."},{"Start":"18:14.920 ","End":"18:17.860","Text":"Should I write a plus or should I write bigger than 0?"},{"Start":"18:17.860 ","End":"18:19.675","Text":"I\u0027ll go with bigger than 0."},{"Start":"18:19.675 ","End":"18:23.260","Text":"This is positive, bigger than 0."},{"Start":"18:23.260 ","End":"18:26.410","Text":"This is positive bigger than 0."},{"Start":"18:26.410 ","End":"18:29.125","Text":"There\u0027s 2 interesting cases."},{"Start":"18:29.125 ","End":"18:32.660","Text":"If they\u0027re all bigger than 0,"},{"Start":"18:33.120 ","End":"18:38.230","Text":"this is the condition when they\u0027re all bigger than 0,"},{"Start":"18:38.230 ","End":"18:44.569","Text":"that we can conclude that it is a minimum point."},{"Start":"18:44.790 ","End":"18:49.750","Text":"The other interesting case is as follows."},{"Start":"18:49.750 ","End":"18:52.195","Text":"If this were less than 0,"},{"Start":"18:52.195 ","End":"18:55.405","Text":"and this were greater than 0,"},{"Start":"18:55.405 ","End":"18:58.345","Text":"and this again is less than 0,"},{"Start":"18:58.345 ","End":"19:02.665","Text":"that would give us a maximum,"},{"Start":"19:02.665 ","End":"19:05.650","Text":"and we don\u0027t know in the other cases,"},{"Start":"19:05.650 ","End":"19:09.310","Text":"but this first case holds,"},{"Start":"19:09.310 ","End":"19:14.650","Text":"and so our particular point which was 1/2, 1,"},{"Start":"19:14.650 ","End":"19:17.665","Text":"1 is a minimum,"},{"Start":"19:17.665 ","End":"19:23.725","Text":"and I\u0027ll highlight that because that\u0027s the answer but I still owe you"},{"Start":"19:23.725 ","End":"19:29.500","Text":"the asterisk to show you in case you want why this determinant is this."},{"Start":"19:29.500 ","End":"19:32.665","Text":"If you don\u0027t to want to know or you know how to do this,"},{"Start":"19:32.665 ","End":"19:35.260","Text":"you can skip the remainder of this clip."},{"Start":"19:35.260 ","End":"19:39.020","Text":"I\u0027m going to just do the computation now."},{"Start":"19:39.360 ","End":"19:42.920","Text":"Now, just write it again."},{"Start":"19:44.090 ","End":"19:48.015","Text":"As a thing called expanding by a row or a column,"},{"Start":"19:48.015 ","End":"19:53.655","Text":"best to look for a row with a 0 or even more than 1 0."},{"Start":"19:53.655 ","End":"19:58.435","Text":"Let\u0027s say the first row contains a 0,"},{"Start":"19:58.435 ","End":"20:00.925","Text":"so I expand by the first row,"},{"Start":"20:00.925 ","End":"20:05.350","Text":"and I\u0027ll just remind you that you"},{"Start":"20:05.350 ","End":"20:09.910","Text":"can think of like a checkerboard pattern that each entry has like a plus or minus,"},{"Start":"20:09.910 ","End":"20:17.140","Text":"what we do is if we take each entry in this case it\u0027s the top row,"},{"Start":"20:17.140 ","End":"20:19.660","Text":"and we take the 4 with the plus,"},{"Start":"20:19.660 ","End":"20:23.290","Text":"I even going to write plus 4 just to emphasize,"},{"Start":"20:23.290 ","End":"20:28.600","Text":"and then we multiply it by its cofactor."},{"Start":"20:28.600 ","End":"20:31.450","Text":"The cofactor is what\u0027s left."},{"Start":"20:31.450 ","End":"20:34.060","Text":"If I\u0027m just imagining this,"},{"Start":"20:34.060 ","End":"20:40.720","Text":"I delete the row and column containing this for what I\u0027m left with is this here,"},{"Start":"20:40.720 ","End":"20:45.520","Text":"and we take the determinant of 3 minus 2,"},{"Start":"20:45.520 ","End":"20:51.175","Text":"minus 2, 6, and then we go to the next entry in this row."},{"Start":"20:51.175 ","End":"20:53.690","Text":"It\u0027s a minus 2,"},{"Start":"20:54.030 ","End":"20:56.500","Text":"but it\u0027s also has a minus."},{"Start":"20:56.500 ","End":"21:00.250","Text":"So I take the sign, the minus,"},{"Start":"21:00.250 ","End":"21:05.770","Text":"and then I take the minus 2,"},{"Start":"21:05.770 ","End":"21:10.270","Text":"and then I erase the row and column containing the minus 2,"},{"Start":"21:10.270 ","End":"21:13.105","Text":"what I\u0027m left with is these 4 entries."},{"Start":"21:13.105 ","End":"21:16.644","Text":"So it comes out minus 2,"},{"Start":"21:16.644 ","End":"21:20.980","Text":"minus 2, and then 0, 6,"},{"Start":"21:20.980 ","End":"21:25.570","Text":"and then finally the 0 with a plus, well,"},{"Start":"21:25.570 ","End":"21:30.700","Text":"it doesn\u0027t matter what it is because it\u0027s got a 0,"},{"Start":"21:30.700 ","End":"21:32.665","Text":"so I don\u0027t have to bother with it."},{"Start":"21:32.665 ","End":"21:36.550","Text":"So let\u0027s see. This is equal to 4 times."},{"Start":"21:36.550 ","End":"21:40.390","Text":"Now, this determinant is diagonal minus diagonal."},{"Start":"21:40.390 ","End":"21:46.900","Text":"18 minus 4 is 14. 4 times 14."},{"Start":"21:46.900 ","End":"21:48.040","Text":"Here, we have minus,"},{"Start":"21:48.040 ","End":"21:52.255","Text":"minus 2 is plus 2 times this minus this,"},{"Start":"21:52.255 ","End":"21:58.585","Text":"is minus 12, minus 0 is just minus 12,"},{"Start":"21:58.585 ","End":"22:00.790","Text":"and the last 1 is 0."},{"Start":"22:00.790 ","End":"22:08.065","Text":"4 times 14 is 56 minus 24,"},{"Start":"22:08.065 ","End":"22:11.785","Text":"and that is equal to 32,"},{"Start":"22:11.785 ","End":"22:13.525","Text":"and that\u0027s what I wrote here."},{"Start":"22:13.525 ","End":"22:16.285","Text":"That confirms this 32,"},{"Start":"22:16.285 ","End":"22:18.340","Text":"and so we are d1,"},{"Start":"22:18.340 ","End":"22:21.529","Text":"and this is the result."}],"ID":9032},{"Watched":false,"Name":"Exedrcise 6","Duration":"17m 17s","ChapterTopicVideoID":9768,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.210","Text":"In this exercise, we have to find the least distance between 2 parabolas."},{"Start":"00:06.210 ","End":"00:09.750","Text":"This 1, y equals x squared plus 1, and the other 1,"},{"Start":"00:09.750 ","End":"00:13.680","Text":"y equals minus x squared plus 2x."},{"Start":"00:13.680 ","End":"00:19.335","Text":"Perhaps I\u0027ll give a rough sketch though it\u0027s not strictly necessary to take an axis."},{"Start":"00:19.335 ","End":"00:23.530","Text":"That\u0027s the y-axis on another axis."},{"Start":"00:24.080 ","End":"00:29.805","Text":"This 1, we get something like this."},{"Start":"00:29.805 ","End":"00:32.510","Text":"This 1, I\u0027m just looking at it."},{"Start":"00:32.510 ","End":"00:34.010","Text":"It\u0027s an upside down,"},{"Start":"00:34.010 ","End":"00:36.409","Text":"face down parabola."},{"Start":"00:36.409 ","End":"00:41.105","Text":"Something like this."},{"Start":"00:41.105 ","End":"00:44.540","Text":"Let\u0027s just take a typical point here,"},{"Start":"00:44.540 ","End":"00:48.120","Text":"and a typical point here,"},{"Start":"00:48.440 ","End":"00:53.435","Text":"and draw a line to indicate the distance."},{"Start":"00:53.435 ","End":"00:57.290","Text":"Let\u0027s say the x of this point is a,"},{"Start":"00:57.290 ","End":"01:01.010","Text":"so this would be the point a comma,"},{"Start":"01:01.010 ","End":"01:03.770","Text":"and then y is x squared plus 1,"},{"Start":"01:03.770 ","End":"01:08.100","Text":"so this would be a squared plus 1."},{"Start":"01:08.120 ","End":"01:12.045","Text":"If the x of this point is b,"},{"Start":"01:12.045 ","End":"01:16.594","Text":"then this would be the point b comma."},{"Start":"01:16.594 ","End":"01:18.380","Text":"Looking at this equation,"},{"Start":"01:18.380 ","End":"01:23.135","Text":"minus b squared plus 2b,"},{"Start":"01:23.135 ","End":"01:32.180","Text":"and let\u0027s call this distance here d. Using the formula for the distance between 2 points,"},{"Start":"01:32.180 ","End":"01:34.250","Text":"let\u0027s write that d equals,"},{"Start":"01:34.250 ","End":"01:36.170","Text":"we have a square root sign."},{"Start":"01:36.170 ","End":"01:41.080","Text":"Then we need the difference between the Xs squared."},{"Start":"01:41.080 ","End":"01:43.920","Text":"Doesn\u0027t matter in which order we subtract,"},{"Start":"01:43.920 ","End":"01:46.160","Text":"let\u0027s say this minus this,"},{"Start":"01:46.160 ","End":"01:49.430","Text":"a minus b squared."},{"Start":"01:49.430 ","End":"01:54.500","Text":"Then the difference between the y squared,"},{"Start":"01:54.500 ","End":"01:57.740","Text":"so it\u0027s a squared plus 1."},{"Start":"01:57.740 ","End":"02:08.655","Text":"Subtracting this makes it plus b squared and minus to b or this squared."},{"Start":"02:08.655 ","End":"02:14.485","Text":"Actually, d is a function of a and b."},{"Start":"02:14.485 ","End":"02:19.350","Text":"Let\u0027s just write that. That\u0027s d of a and b."},{"Start":"02:19.520 ","End":"02:23.780","Text":"We\u0027re looking for the minimum of d as a function of a and b,"},{"Start":"02:23.780 ","End":"02:25.880","Text":"which values of a and b give the least."},{"Start":"02:25.880 ","End":"02:27.830","Text":"Trick here that we\u0027ve used before,"},{"Start":"02:27.830 ","End":"02:32.810","Text":"that when you have a square root for finding the a b,"},{"Start":"02:32.810 ","End":"02:34.729","Text":"which gives the minimum distance,"},{"Start":"02:34.729 ","End":"02:36.950","Text":"we can just throw out the square root"},{"Start":"02:36.950 ","End":"02:40.190","Text":"because we\u0027re really looking at the distance squared."},{"Start":"02:40.190 ","End":"02:44.120","Text":"F equals d squared."},{"Start":"02:44.120 ","End":"02:52.460","Text":"Then we have f of a and b equals this thing without the square root."},{"Start":"02:52.460 ","End":"02:56.360","Text":"This trick is usually used for square roots,"},{"Start":"02:56.360 ","End":"03:02.060","Text":"but it actually works for any function that\u0027s positive or non-negative."},{"Start":"03:02.060 ","End":"03:06.770","Text":"The least distance will occur when the squared distance is also least."},{"Start":"03:06.770 ","End":"03:11.390","Text":"We just have to remember afterwards when we find the smallest value f,"},{"Start":"03:11.390 ","End":"03:14.870","Text":"we have to take the square root to get back to d. I mean,"},{"Start":"03:14.870 ","End":"03:20.430","Text":"d is the square root of f. But the a and b will be the same."},{"Start":"03:21.470 ","End":"03:24.620","Text":"Function in 2 variables."},{"Start":"03:24.620 ","End":"03:28.550","Text":"Let\u0027s start by looking for critical points."},{"Start":"03:28.550 ","End":"03:34.940","Text":"Critical points, we get when fa is 0"},{"Start":"03:34.940 ","End":"03:42.480","Text":"and fb equals 0 simultaneously."},{"Start":"03:42.480 ","End":"03:46.390","Text":"What that gives us is,"},{"Start":"03:46.460 ","End":"03:51.395","Text":"let\u0027s see, the derivative with respect to a."},{"Start":"03:51.395 ","End":"03:56.150","Text":"From here, we get twice a minus b,"},{"Start":"03:56.150 ","End":"04:00.745","Text":"and then the inner derivative is 1. I\u0027ll write it."},{"Start":"04:00.745 ","End":"04:04.730","Text":"Plus twice this thing,"},{"Start":"04:04.730 ","End":"04:09.260","Text":"a squared plus 1 plus b squared minus 2b."},{"Start":"04:09.260 ","End":"04:20.700","Text":"Then the inner derivative with respect to a can just come from here, it\u0027s 2a."},{"Start":"04:23.450 ","End":"04:26.780","Text":"We want this to equal 0."},{"Start":"04:26.780 ","End":"04:29.540","Text":"For the other 1, with respect to b,"},{"Start":"04:29.540 ","End":"04:32.660","Text":"we get twice a minus b,"},{"Start":"04:32.660 ","End":"04:37.425","Text":"but this time, minus 1 for the inner derivative."},{"Start":"04:37.425 ","End":"04:40.610","Text":"Then here, the same starts out the same,"},{"Start":"04:40.610 ","End":"04:44.735","Text":"a squared plus 1 plus b squared minus 2b."},{"Start":"04:44.735 ","End":"04:48.945","Text":"But the inner derivative with respect to b, this time,"},{"Start":"04:48.945 ","End":"04:55.799","Text":"is 2b minus 2 just from these terms,"},{"Start":"04:55.799 ","End":"04:58.815","Text":"and that\u0027s also equal to 0."},{"Start":"04:58.815 ","End":"05:01.845","Text":"We can cancel by 2,"},{"Start":"05:01.845 ","End":"05:04.725","Text":"it will make it a bit nicer,"},{"Start":"05:04.725 ","End":"05:09.010","Text":"and also here and here."},{"Start":"05:09.140 ","End":"05:11.710","Text":"Now, here\u0027s what I propose."},{"Start":"05:11.710 ","End":"05:15.655","Text":"We have a minus b here and we have a minus b here."},{"Start":"05:15.655 ","End":"05:18.265","Text":"Well, yeah, here with a minus sign,"},{"Start":"05:18.265 ","End":"05:22.610","Text":"let\u0027s extract a minus b from each 1 of them."},{"Start":"05:22.820 ","End":"05:28.390","Text":"Now, we get that a minus b equals,"},{"Start":"05:28.390 ","End":"05:31.435","Text":"and again, I\u0027ll write a minus b equals."},{"Start":"05:31.435 ","End":"05:36.610","Text":"From the top 1, I just need to bring this to the other side."},{"Start":"05:36.610 ","End":"05:41.920","Text":"So I have minus and I\u0027ll just change the order 2a"},{"Start":"05:41.920 ","End":"05:50.090","Text":"times a squared plus 1 plus b squared minus 2b."},{"Start":"05:51.480 ","End":"05:57.290","Text":"From here, if I bring the a minus b to the other side,"},{"Start":"05:57.290 ","End":"05:58.970","Text":"it\u0027ll be plus a minus b."},{"Start":"05:58.970 ","End":"06:02.675","Text":"We just get this expression. This is equal."},{"Start":"06:02.675 ","End":"06:07.590","Text":"Again, I\u0027ll change the order, 2b minus 2."},{"Start":"06:07.590 ","End":"06:09.680","Text":"Then this same factor,"},{"Start":"06:09.680 ","End":"06:15.780","Text":"a squared plus 1 plus b squared minus 2b."},{"Start":"06:16.120 ","End":"06:22.160","Text":"Now, if I compare the a minus bs, or in other words,"},{"Start":"06:22.160 ","End":"06:26.210","Text":"both these right-hand sides are equal to the same left-hand side,"},{"Start":"06:26.210 ","End":"06:28.625","Text":"so they\u0027re equal amongst themselves."},{"Start":"06:28.625 ","End":"06:36.050","Text":"From here, I can get that minus 2a times this bracket"},{"Start":"06:36.050 ","End":"06:47.730","Text":"a squared plus 1 plus b squared minus 2b is equal to,"},{"Start":"06:50.440 ","End":"06:56.250","Text":"copying from here, 2b minus 2."},{"Start":"06:56.250 ","End":"06:58.220","Text":"This same thing in the brackets,"},{"Start":"06:58.220 ","End":"07:05.610","Text":"a squared plus 1 plus b squared minus 2b."},{"Start":"07:06.410 ","End":"07:12.095","Text":"Now, what I would like to do would be to cancel this common factor."},{"Start":"07:12.095 ","End":"07:17.250","Text":"But here, we have to be careful because it\u0027s possible that this thing is 0."},{"Start":"07:17.250 ","End":"07:20.895","Text":"You are not allowed to divide both sides by 0."},{"Start":"07:20.895 ","End":"07:26.390","Text":"Let me just show you at the side why this thing is not 0."},{"Start":"07:26.390 ","End":"07:30.080","Text":"Let\u0027s just suppose on the contrary that it is equal to 0."},{"Start":"07:30.080 ","End":"07:32.405","Text":"We get a squared."},{"Start":"07:32.405 ","End":"07:35.090","Text":"Let me just change the order here."},{"Start":"07:35.090 ","End":"07:41.235","Text":"Plus b squared minus 2b plus 1,"},{"Start":"07:41.235 ","End":"07:42.725","Text":"let\u0027s put the 1 at the end."},{"Start":"07:42.725 ","End":"07:46.625","Text":"Suppose that this does equal to 0."},{"Start":"07:46.625 ","End":"07:54.105","Text":"Notice that I can rewrite this as a squared,"},{"Start":"07:54.105 ","End":"07:59.360","Text":"and this bit here is just b minus 1 squared."},{"Start":"07:59.360 ","End":"08:04.820","Text":"We\u0027re familiar with this kind of special binomial expansion, this would be equal to 0."},{"Start":"08:04.820 ","End":"08:08.625","Text":"Now, if something squared plus something squared is 0,"},{"Start":"08:08.625 ","End":"08:14.250","Text":"the only way that can happen is that each of these has to be 0."},{"Start":"08:14.290 ","End":"08:16.700","Text":"Otherwise, it\u0027s going to be positive."},{"Start":"08:16.700 ","End":"08:19.920","Text":"If a is not 0, this thing is positive."},{"Start":"08:21.890 ","End":"08:24.180","Text":"If p minus 1 is not 0,"},{"Start":"08:24.180 ","End":"08:26.230","Text":"this is also positive."},{"Start":"08:28.110 ","End":"08:36.325","Text":"This can only come from a equals 0 and b minus 1 equals 0,"},{"Start":"08:36.325 ","End":"08:40.120","Text":"so b equals 1."},{"Start":"08:40.120 ","End":"08:44.080","Text":"That\u0027s the only way to get this thing to become 0."},{"Start":"08:44.080 ","End":"08:47.590","Text":"But if a is 0 and b is 1,"},{"Start":"08:47.590 ","End":"08:52.750","Text":"and we plug this in to 1 of these equations,"},{"Start":"08:52.750 ","End":"08:54.595","Text":"let\u0027s say this 1."},{"Start":"08:54.595 ","End":"08:56.935","Text":"If we plug it into here,"},{"Start":"08:56.935 ","End":"09:05.065","Text":"then we\u0027ll get a minus b would be minus 1 times"},{"Start":"09:05.065 ","End":"09:10.800","Text":"1 plus this thing we know is"},{"Start":"09:10.800 ","End":"09:17.310","Text":"0 because we assumed plus 0 times whatever,"},{"Start":"09:17.310 ","End":"09:21.580","Text":"it\u0027s 2_a but it doesn\u0027t matter it\u0027s also 0."},{"Start":"09:22.980 ","End":"09:30.310","Text":"Plus 0 has got to equal the other side, which is 0."},{"Start":"09:30.310 ","End":"09:33.055","Text":"In other words, we get that minus 1 is 0,"},{"Start":"09:33.055 ","End":"09:35.855","Text":"and this is a contradiction."},{"Start":"09:35.855 ","End":"09:38.415","Text":"Assuming this is equal to 0,"},{"Start":"09:38.415 ","End":"09:42.045","Text":"we get the minus 1 equals 0 that\u0027s not possible."},{"Start":"09:42.045 ","End":"09:45.480","Text":"This contradiction means that this was not possible."},{"Start":"09:45.480 ","End":"09:50.075","Text":"So this is justified and now we\u0027re back here,"},{"Start":"09:50.075 ","End":"09:57.535","Text":"and we get that minus 2_a equals 2_b minus 2."},{"Start":"09:57.535 ","End":"10:03.940","Text":"I can divide both sides by minus"},{"Start":"10:03.940 ","End":"10:10.765","Text":"2 and then I\u0027ll get that a equals minus b plus 1."},{"Start":"10:10.765 ","End":"10:14.170","Text":"I\u0027d rather write that as 1 minus b."},{"Start":"10:14.170 ","End":"10:16.075","Text":"Once I have this,"},{"Start":"10:16.075 ","End":"10:18.460","Text":"I can plug into 1 of these 2."},{"Start":"10:18.460 ","End":"10:20.965","Text":"Let\u0027s say the first 1."},{"Start":"10:20.965 ","End":"10:23.035","Text":"Into this 1, it\u0027s,"},{"Start":"10:23.035 ","End":"10:25.880","Text":"looks a bit easier,"},{"Start":"10:26.100 ","End":"10:29.815","Text":"a is 1 minus b in this 1."},{"Start":"10:29.815 ","End":"10:36.430","Text":"Here we have 1 minus b minus b is 1 minus 2_b,"},{"Start":"10:36.430 ","End":"10:41.810","Text":"and this equals, minus 2_a."},{"Start":"10:45.180 ","End":"10:47.800","Text":"We have here minus 2_a,"},{"Start":"10:47.800 ","End":"10:50.750","Text":"it\u0027s 2_b minus 2."},{"Start":"10:52.410 ","End":"11:03.410","Text":"Here we get a squared is 1 minus b squared."},{"Start":"11:03.720 ","End":"11:06.130","Text":"We actually saw over here,"},{"Start":"11:06.130 ","End":"11:14.740","Text":"the last part is b minus 1 squared brackets."},{"Start":"11:14.740 ","End":"11:17.950","Text":"You know what? I\u0027ll change these to square brackets."},{"Start":"11:17.950 ","End":"11:26.125","Text":"Now this right-hand side can be simplified."},{"Start":"11:26.125 ","End":"11:33.290","Text":"I mean, 1 minus b square and b minus 1 squared are the same."},{"Start":"11:34.830 ","End":"11:38.690","Text":"From here I\u0027ve got twice"},{"Start":"11:39.120 ","End":"11:46.960","Text":"b minus 1 squared and from here I\u0027ve got twice b minus 1,"},{"Start":"11:46.960 ","End":"11:50.600","Text":"if I take outside the brackets."},{"Start":"11:51.000 ","End":"11:55.690","Text":"What I get is that"},{"Start":"11:55.690 ","End":"12:04.970","Text":"1 minus 2_b is 4 times b minus 1 cubed."},{"Start":"12:05.340 ","End":"12:08.320","Text":"I see this is going to get messy,"},{"Start":"12:08.320 ","End":"12:10.600","Text":"and now I realize I made the wrong choice."},{"Start":"12:10.600 ","End":"12:13.240","Text":"Instead of writing a in terms of b,"},{"Start":"12:13.240 ","End":"12:20.484","Text":"I should have written b in terms of a minus a is b minus 1,"},{"Start":"12:20.484 ","End":"12:22.705","Text":"and that will take care of this."},{"Start":"12:22.705 ","End":"12:26.035","Text":"B is actually 1 minus a,"},{"Start":"12:26.035 ","End":"12:29.230","Text":"also because I can bring this here and this here."},{"Start":"12:29.230 ","End":"12:39.295","Text":"That means that 1 minus 2_b is 1 minus twice 1 minus a,"},{"Start":"12:39.295 ","End":"12:48.385","Text":"which comes out to be 1 minus 2 plus 2_a,"},{"Start":"12:48.385 ","End":"12:52.000","Text":"so it\u0027s 2_a minus 1."},{"Start":"12:52.000 ","End":"12:55.180","Text":"If I put instead of b minus 1,"},{"Start":"12:55.180 ","End":"12:57.010","Text":"I\u0027ll put minus a."},{"Start":"12:57.010 ","End":"13:00.295","Text":"On this side instead of 1 minus 2_b,"},{"Start":"13:00.295 ","End":"13:04.270","Text":"I put 2_a minus 1."},{"Start":"13:04.270 ","End":"13:13.735","Text":"Then I get the equation 2_a minus 1 is minus a cubed is a minus,"},{"Start":"13:13.735 ","End":"13:20.155","Text":"it\u0027s an odd power, I get minus 4_a cubed."},{"Start":"13:20.155 ","End":"13:25.315","Text":"Then just bringing everything to 1 side to have a cubic equation"},{"Start":"13:25.315 ","End":"13:34.935","Text":"4_a cubed plus 2_a minus 1 equals 0."},{"Start":"13:34.935 ","End":"13:37.395","Text":"Now this is a cubic equation,"},{"Start":"13:37.395 ","End":"13:41.225","Text":"and we don\u0027t know how to solve cubic equations."},{"Start":"13:41.225 ","End":"13:43.930","Text":"I suggest 1 of 2 things."},{"Start":"13:43.930 ","End":"13:46.930","Text":"We either use a numerical approximation method,"},{"Start":"13:46.930 ","End":"13:49.405","Text":"such as the Newton-Raphson method."},{"Start":"13:49.405 ","End":"13:54.440","Text":"I\u0027ll just put an asterisk here and I\u0027ll show you that at the end."},{"Start":"13:55.680 ","End":"13:58.420","Text":"Or, and I don\u0027t recommend this,"},{"Start":"13:58.420 ","End":"14:04.885","Text":"you can cheat because on the internet there are calculators that solve cubic equations."},{"Start":"14:04.885 ","End":"14:10.255","Text":"I\u0027ll just show you what the answer is using 1 such calculator."},{"Start":"14:10.255 ","End":"14:14.065","Text":"This is what I got and obviously this still continues."},{"Start":"14:14.065 ","End":"14:22.970","Text":"Let\u0027s just say to 3 decimal places that a equals 0.385."},{"Start":"14:23.970 ","End":"14:31.659","Text":"From here I\u0027ve got that b equal because b is 1 minus a,"},{"Start":"14:31.659 ","End":"14:36.640","Text":"and so 1 minus this is 0.615."},{"Start":"14:36.640 ","End":"14:43.180","Text":"I want to make some comments."},{"Start":"14:43.180 ","End":"14:46.675","Text":"First of all, how do I know that\u0027s the only solution?"},{"Start":"14:46.675 ","End":"14:49.600","Text":"Well, there is only 1 solution and I\u0027ll tell you"},{"Start":"14:49.600 ","End":"14:54.430","Text":"a simple way of telling how or rather why."},{"Start":"14:54.430 ","End":"14:56.710","Text":"If you differentiate this,"},{"Start":"14:56.710 ","End":"14:59.695","Text":"you get 12_a squared plus 2."},{"Start":"14:59.695 ","End":"15:03.460","Text":"Now 12_a squared plus 2 is always positive,"},{"Start":"15:03.460 ","End":"15:06.175","Text":"so this is an increase in function always."},{"Start":"15:06.175 ","End":"15:09.040","Text":"We can only cross the x-axis once,"},{"Start":"15:09.040 ","End":"15:12.100","Text":"so this is the only solution for a."},{"Start":"15:12.100 ","End":"15:13.915","Text":"The second thing is,"},{"Start":"15:13.915 ","End":"15:16.930","Text":"why is this a minimum?"},{"Start":"15:16.930 ","End":"15:20.605","Text":"I don\u0027t want to do the second derivative test"},{"Start":"15:20.605 ","End":"15:23.740","Text":"and substitute approximate values and it\u0027s messy."},{"Start":"15:23.740 ","End":"15:25.479","Text":"The answer is just intuitive."},{"Start":"15:25.479 ","End":"15:30.565","Text":"When you have 2 graphs like these that go onto infinity,"},{"Start":"15:30.565 ","End":"15:33.925","Text":"this cannot possibly be a maximum."},{"Start":"15:33.925 ","End":"15:37.450","Text":"It\u0027s just clear pictorially that\u0027s good enough that there"},{"Start":"15:37.450 ","End":"15:40.960","Text":"is some place that\u0027s the closest than the others."},{"Start":"15:40.960 ","End":"15:44.450","Text":"I mean, you can literally see it."},{"Start":"15:44.520 ","End":"15:51.070","Text":"We\u0027re not going to go and rigorously prove that it\u0027s a minimum, it\u0027s intuitively obvious."},{"Start":"15:51.070 ","End":"15:53.365","Text":"Now we\u0027re almost done."},{"Start":"15:53.365 ","End":"15:55.780","Text":"But we just haven\u0027t answered the question."},{"Start":"15:55.780 ","End":"15:57.010","Text":"We found the a and the b,"},{"Start":"15:57.010 ","End":"15:59.005","Text":"but the question was the distance."},{"Start":"15:59.005 ","End":"16:02.380","Text":"We substitute not in the function here,"},{"Start":"16:02.380 ","End":"16:07.825","Text":"but in the formula for D, this formula here."},{"Start":"16:07.825 ","End":"16:13.765","Text":"We get that D is equal to,"},{"Start":"16:13.765 ","End":"16:16.690","Text":"well, it\u0027s a bit messy."},{"Start":"16:16.690 ","End":"16:22.105","Text":"It\u0027s the square root of a minus b is"},{"Start":"16:22.105 ","End":"16:32.210","Text":"minus 0.23 squared plus,"},{"Start":"16:33.870 ","End":"16:43.315","Text":"I computed it at the side to 0.29645 squared."},{"Start":"16:43.315 ","End":"16:52.195","Text":"This comes out to 3 decimal places, 0.375."},{"Start":"16:52.195 ","End":"16:54.100","Text":"Well, this is approximately,"},{"Start":"16:54.100 ","End":"16:59.049","Text":"I should really put approximately equal to signs and all of these."},{"Start":"16:59.049 ","End":"17:02.660","Text":"Because we can see, it\u0027s not exact."},{"Start":"17:04.590 ","End":"17:10.075","Text":"That\u0027s the answer, approximately."},{"Start":"17:10.075 ","End":"17:13.910","Text":"All that remains is for me to show you a numerical method of how to"},{"Start":"17:13.910 ","End":"17:18.660","Text":"reach this solution to this equation."}],"ID":9621},{"Watched":false,"Name":"Exercise 6 - Numerical Approximation","Duration":"5m 11s","ChapterTopicVideoID":8729,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.715","Text":"It remains the matter of solving this equation using numerical methods."},{"Start":"00:05.715 ","End":"00:10.210","Text":"What I\u0027d like to do is just change the a to x."},{"Start":"00:10.210 ","End":"00:12.000","Text":"I\u0027ll write it as"},{"Start":"00:12.000 ","End":"00:18.540","Text":"4x^3 plus 2x minus 1 equals 0."},{"Start":"00:18.540 ","End":"00:25.110","Text":"Let\u0027s go to a clean page to scroll all the way."},{"Start":"00:25.110 ","End":"00:30.030","Text":"In general, the Newton-Raphson method talks about solving an equation of"},{"Start":"00:30.030 ","End":"00:34.800","Text":"the form f of x equals 0 with numerical methods."},{"Start":"00:34.800 ","End":"00:41.017","Text":"This is not the same f as the one we had above with a and b just be using the letter f."},{"Start":"00:41.017 ","End":"00:48.760","Text":"What it involves is making an initial guess called x_1,"},{"Start":"00:48.760 ","End":"00:50.855","Text":"and we\u0027ll come to that in a minute."},{"Start":"00:50.855 ","End":"00:53.330","Text":"Then getting x_2, x_3, x_4"},{"Start":"00:53.330 ","End":"01:04.185","Text":"or however many iterations we need using the formula that x_n plus 1."},{"Start":"01:04.185 ","End":"01:10.424","Text":"The successive guess when we know x_n is equal to x_n minus"},{"Start":"01:10.424 ","End":"01:17.415","Text":"f of x_n over f prime of x_n."},{"Start":"01:17.415 ","End":"01:19.740","Text":"For example, if n is 1,"},{"Start":"01:19.740 ","End":"01:26.610","Text":"this will give me x_2 in terms of x_1 and f of x_1 and f prime of x_1."},{"Start":"01:26.610 ","End":"01:32.115","Text":"If n is 2, it\u0027ll give me x_3 in terms of x_2 and so on."},{"Start":"01:32.115 ","End":"01:35.100","Text":"It uses the previous value,"},{"Start":"01:35.100 ","End":"01:36.580","Text":"the function at that value,"},{"Start":"01:36.580 ","End":"01:38.905","Text":"and the derivative function at that value."},{"Start":"01:38.905 ","End":"01:42.500","Text":"In our case, of course,"},{"Start":"01:42.500 ","End":"01:46.375","Text":"f of x is whatever\u0027s on the left-hand side here,"},{"Start":"01:46.375 ","End":"01:51.420","Text":"4x^3 plus 2x minus 1."},{"Start":"01:51.420 ","End":"01:57.355","Text":"f prime of x is the derivative is 12x^2 plus 2."},{"Start":"01:57.355 ","End":"02:00.779","Text":"Now, what about the initial guess?"},{"Start":"02:00.779 ","End":"02:04.640","Text":"There\u0027s no exact science in this,"},{"Start":"02:04.640 ","End":"02:07.279","Text":"but I\u0027d like to try some whole numbers."},{"Start":"02:07.279 ","End":"02:11.330","Text":"For example, if I try x equals 0,"},{"Start":"02:11.330 ","End":"02:17.925","Text":"f of 0 would be 0 plus 0 minus 1 is minus 1."},{"Start":"02:17.925 ","End":"02:20.744","Text":"If I try 1,"},{"Start":"02:20.744 ","End":"02:24.340","Text":"I\u0027ll get 4 plus 2 minus 1,"},{"Start":"02:24.340 ","End":"02:27.630","Text":"6 minus 1 is 5."},{"Start":"02:27.670 ","End":"02:31.610","Text":"I know that somewhere in between these two,"},{"Start":"02:31.610 ","End":"02:34.229","Text":"there is a root of f."},{"Start":"02:34.229 ","End":"02:37.670","Text":"Let\u0027s try x_1 between 0 and 1."},{"Start":"02:37.670 ","End":"02:44.135","Text":"Let\u0027s take x_1 to be equal to 0.5 or 1/2."},{"Start":"02:44.135 ","End":"02:46.510","Text":"I\u0027ll work with decimals."},{"Start":"02:46.510 ","End":"02:58.005","Text":"Using this formula, we would get that x_2 is equal to,"},{"Start":"02:58.005 ","End":"03:08.715","Text":"in general, it\u0027s x_1 minus f of x_1 over f prime of x_1."},{"Start":"03:08.715 ","End":"03:12.450","Text":"In our case, with x_1 being 0.5,"},{"Start":"03:12.450 ","End":"03:21.832","Text":"it\u0027s 0.5 minus f of 0.5 is"},{"Start":"03:21.832 ","End":"03:33.109","Text":"4 times 0.5 cubed plus twice 0.5 minus 1 over,"},{"Start":"03:33.109 ","End":"03:34.935","Text":"and from here,"},{"Start":"03:34.935 ","End":"03:40.140","Text":"12 times 0.5 squared plus 2,"},{"Start":"03:40.140 ","End":"03:46.395","Text":"and this comes out to be 0.4 exactly."},{"Start":"03:46.395 ","End":"03:50.295","Text":"I\u0027ll spare you the computations."},{"Start":"03:50.295 ","End":"03:54.735","Text":"Then using this, similarly,"},{"Start":"03:54.735 ","End":"03:58.990","Text":"I can get x_3 equals"},{"Start":"03:59.690 ","End":"04:07.875","Text":"x_2 minus f of x_2 divided by f prime of x_2."},{"Start":"04:07.875 ","End":"04:13.430","Text":"This equals, just like this,"},{"Start":"04:13.430 ","End":"04:16.040","Text":"only with 0.4 instead of 0.50,"},{"Start":"04:16.040 ","End":"04:26.630","Text":"0.4 minus 4 times 0.4 cubed plus twice 0.4 minus 1 over"},{"Start":"04:26.630 ","End":"04:31.930","Text":"12 times 0.4 squared plus 2."},{"Start":"04:31.930 ","End":"04:34.580","Text":"Again, I\u0027ll spare you the calculations."},{"Start":"04:34.580 ","End":"04:40.705","Text":"This comes out to be 0.385."},{"Start":"04:40.705 ","End":"04:45.050","Text":"Usually, it\u0027s safe to stop after 3 iterations unless"},{"Start":"04:45.050 ","End":"04:48.740","Text":"you started off with some wildly inaccurate first guess,"},{"Start":"04:48.740 ","End":"04:52.950","Text":"if I took x_1 as 20 or something, maybe not."},{"Start":"04:53.200 ","End":"05:02.115","Text":"Just for comparison, I did keep the value from the Internet calculator, which was this."},{"Start":"05:02.115 ","End":"05:05.145","Text":"We see it starts with 0.385."},{"Start":"05:05.145 ","End":"05:07.755","Text":"That\u0027s the numerical part."},{"Start":"05:07.755 ","End":"05:10.870","Text":"We\u0027re done with this exercise."}],"ID":9035},{"Watched":false,"Name":"Exercise 7 part a","Duration":"12m 51s","ChapterTopicVideoID":8730,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, we\u0027re given some data points and we have"},{"Start":"00:05.160 ","End":"00:10.425","Text":"to find a line of the form y equals ax plus b,"},{"Start":"00:10.425 ","End":"00:17.100","Text":"which is a best fit in the sense that it minimizes the sum of the squared errors."},{"Start":"00:17.100 ","End":"00:19.859","Text":"Now, I want to do it in general,"},{"Start":"00:19.859 ","End":"00:21.570","Text":"or at least to start out in general,"},{"Start":"00:21.570 ","End":"00:29.235","Text":"not to relate to these specific points that will take points x1 y1,"},{"Start":"00:29.235 ","End":"00:33.990","Text":"x2 y2, and so on,"},{"Start":"00:33.990 ","End":"00:39.870","Text":"up to xn, yn."},{"Start":"00:39.870 ","End":"00:45.035","Text":"Later on we\u0027ll plug in these particular points and others."},{"Start":"00:45.035 ","End":"00:48.890","Text":"There are 4 of them, so we\u0027ll have x1 y1 to x4 y4,"},{"Start":"00:48.890 ","End":"00:51.130","Text":"but we\u0027ll begin in general."},{"Start":"00:51.130 ","End":"00:56.160","Text":"Let\u0027s say a typical point from 1 to n is index i."},{"Start":"00:56.160 ","End":"01:00.090","Text":"So xi, yi will be one of these."},{"Start":"01:00.090 ","End":"01:07.625","Text":"Now, the approximation, this straight line means that instead of yi,"},{"Start":"01:07.625 ","End":"01:14.805","Text":"we\u0027re going to approximate it by ax plus b."},{"Start":"01:14.805 ","End":"01:20.040","Text":"For this particular x will be axi plus b."},{"Start":"01:20.040 ","End":"01:23.360","Text":"Now, the approximation means that there is an error."},{"Start":"01:23.360 ","End":"01:26.120","Text":"The squared error will be the difference."},{"Start":"01:26.120 ","End":"01:29.810","Text":"Let\u0027s take the right hand side minus the left-hand side."},{"Start":"01:29.810 ","End":"01:39.950","Text":"So axi plus b estimated minus actual squared."},{"Start":"01:39.950 ","End":"01:44.515","Text":"Now, the sum of the squared errors, sum means Sigma,"},{"Start":"01:44.515 ","End":"01:50.420","Text":"and i goes from 1 to n. The sum of the squared errors is this thing here,"},{"Start":"01:50.420 ","End":"01:57.700","Text":"axi plus b minus yi squared."},{"Start":"01:57.700 ","End":"02:02.215","Text":"I just moved this over here because I needed room to write that"},{"Start":"02:02.215 ","End":"02:06.350","Text":"this is a function of a and b."},{"Start":"02:06.350 ","End":"02:09.990","Text":"The xi, yi, the points will be given."},{"Start":"02:09.990 ","End":"02:14.550","Text":"Just like here, we will take them to be these specific numbers, but they\u0027re given."},{"Start":"02:14.550 ","End":"02:16.800","Text":"Only a and b are the variables."},{"Start":"02:16.800 ","End":"02:21.295","Text":"What we want now is to find the minimum."},{"Start":"02:21.295 ","End":"02:24.070","Text":"We\u0027re going to assume that we get a minimum."},{"Start":"02:24.070 ","End":"02:26.515","Text":"We just have to find the critical point,"},{"Start":"02:26.515 ","End":"02:29.410","Text":"and in the exercise book,"},{"Start":"02:29.410 ","End":"02:33.010","Text":"it said to assume that we really do get a minimum,"},{"Start":"02:33.010 ","End":"02:40.395","Text":"so we just have to find it not to prove that it is the least square error."},{"Start":"02:40.395 ","End":"02:44.850","Text":"We do this by taking the 2 partial derivatives."},{"Start":"02:44.850 ","End":"02:49.760","Text":"Let\u0027s start with f with respect to a and see what this equals."},{"Start":"02:49.760 ","End":"02:51.440","Text":"I want to make 2 comments."},{"Start":"02:51.440 ","End":"02:53.270","Text":"First of all, here I have a sum."},{"Start":"02:53.270 ","End":"02:55.415","Text":"When you take the derivative of a sum,"},{"Start":"02:55.415 ","End":"02:58.220","Text":"it\u0027s okay to take a derivative of each term separately."},{"Start":"02:58.220 ","End":"03:01.610","Text":"So the derivative can go inside the Sigma."},{"Start":"03:01.610 ","End":"03:05.820","Text":"The second is that when you differentiate something squared,"},{"Start":"03:05.890 ","End":"03:09.580","Text":"could be with respect to a or with respect to anything,"},{"Start":"03:09.580 ","End":"03:14.225","Text":"if I have this squared and I take its derivative with respect to whatever,"},{"Start":"03:14.225 ","End":"03:18.100","Text":"then you take twice this,"},{"Start":"03:18.100 ","End":"03:20.405","Text":"just like if it was x squared it would be 2x,"},{"Start":"03:20.405 ","End":"03:25.730","Text":"but then we need to multiply by the inner derivative where box is whatever."},{"Start":"03:25.730 ","End":"03:31.540","Text":"Here we will get the sum from 1 to n and the derivative of this,"},{"Start":"03:31.540 ","End":"03:33.050","Text":"but this thing is squared,"},{"Start":"03:33.050 ","End":"03:35.780","Text":"so it\u0027s twice times this thing,"},{"Start":"03:35.780 ","End":"03:40.320","Text":"which is axi plus b minus yi."},{"Start":"03:40.320 ","End":"03:48.390","Text":"Then the inner derivative with respect to a is just the coefficient of a, which is xi."},{"Start":"03:48.890 ","End":"03:52.640","Text":"Now, a constant can come in front of the Sigma,"},{"Start":"03:52.640 ","End":"03:55.700","Text":"so I can erase it here and write it here."},{"Start":"03:55.700 ","End":"03:58.260","Text":"Now let\u0027s go on to fb."},{"Start":"03:59.170 ","End":"04:01.865","Text":"This will be very similar."},{"Start":"04:01.865 ","End":"04:03.950","Text":"This will be the sum,"},{"Start":"04:03.950 ","End":"04:08.660","Text":"i goes from 1 to n. Now,"},{"Start":"04:08.660 ","End":"04:12.210","Text":"the 2 here, already,"},{"Start":"04:12.210 ","End":"04:15.185","Text":"we\u0027ll put it out front because you can see that I did that here."},{"Start":"04:15.185 ","End":"04:16.790","Text":"So it\u0027s twice."},{"Start":"04:16.790 ","End":"04:20.030","Text":"Then we have this thing here,"},{"Start":"04:20.030 ","End":"04:26.545","Text":"axi plus b minus yi."},{"Start":"04:26.545 ","End":"04:30.050","Text":"But the inner derivative this time,"},{"Start":"04:30.050 ","End":"04:34.205","Text":"because it\u0027s with respect to b, is just 1."},{"Start":"04:34.205 ","End":"04:37.280","Text":"So I\u0027ll write it as 1 here."},{"Start":"04:37.280 ","End":"04:40.380","Text":"Now, for a critical point,"},{"Start":"04:40.380 ","End":"04:46.110","Text":"fa and fb are both equal to 0."},{"Start":"04:46.110 ","End":"04:50.650","Text":"For suspected minimum or critical point, this is what we will do."},{"Start":"04:50.650 ","End":"04:55.560","Text":"From these 2 we will get a set of equations,"},{"Start":"04:55.560 ","End":"04:57.960","Text":"2 equations and 2 unknowns, a and b."},{"Start":"04:57.960 ","End":"05:02.090","Text":"The 2 won\u0027t make any difference when I\u0027m equating to 0,"},{"Start":"05:02.090 ","End":"05:03.785","Text":"but we\u0027ve got the sum,"},{"Start":"05:03.785 ","End":"05:08.235","Text":"i goes from 1 to n of"},{"Start":"05:08.235 ","End":"05:15.210","Text":"axi plus b minus yi times xi."},{"Start":"05:15.210 ","End":"05:16.635","Text":"This is equal to 0."},{"Start":"05:16.635 ","End":"05:19.200","Text":"From here, again, forgetting the 2,"},{"Start":"05:19.200 ","End":"05:22.385","Text":"because if twice something is 0, it is 0."},{"Start":"05:22.385 ","End":"05:27.155","Text":"I equals 1 to n of"},{"Start":"05:27.155 ","End":"05:33.375","Text":"just axi plus b minus yi."},{"Start":"05:33.375 ","End":"05:36.504","Text":"This is also equal to 0."},{"Start":"05:36.504 ","End":"05:38.240","Text":"I want to save the steps."},{"Start":"05:38.240 ","End":"05:41.360","Text":"Let me just note that this general term in this sum,"},{"Start":"05:41.360 ","End":"05:43.430","Text":"the xi I can multiply,"},{"Start":"05:43.430 ","End":"05:47.735","Text":"so it would be axi squared plus"},{"Start":"05:47.735 ","End":"05:54.505","Text":"bxi minus yi, xi."},{"Start":"05:54.505 ","End":"05:58.460","Text":"That will be the general term for the sum here."},{"Start":"05:58.460 ","End":"06:03.950","Text":"Next, I\u0027m going to use the property that for Sigma,"},{"Start":"06:03.950 ","End":"06:06.920","Text":"not only can you take a constant out,"},{"Start":"06:06.920 ","End":"06:10.955","Text":"but you can also break up an inner sum."},{"Start":"06:10.955 ","End":"06:15.015","Text":"You can apply Sigma to each piece separately."},{"Start":"06:15.015 ","End":"06:17.220","Text":"From the first one,"},{"Start":"06:17.220 ","End":"06:20.460","Text":"we\u0027ll get, first of all,"},{"Start":"06:20.460 ","End":"06:23.730","Text":"the sum of axi squared."},{"Start":"06:23.730 ","End":"06:31.220","Text":"I can take the a out and get a times the sum of xi squared from this one."},{"Start":"06:31.220 ","End":"06:34.790","Text":"I goes from 1 to n. The next one,"},{"Start":"06:34.790 ","End":"06:40.215","Text":"I\u0027ll just take b out the brackets and I\u0027ll get the sum."},{"Start":"06:40.215 ","End":"06:45.070","Text":"I goes from 1 to n of xi."},{"Start":"06:45.260 ","End":"06:47.805","Text":"From the last term,"},{"Start":"06:47.805 ","End":"06:50.870","Text":"I get minus the sum,"},{"Start":"06:50.870 ","End":"06:52.360","Text":"i goes from 1 to n,"},{"Start":"06:52.360 ","End":"06:53.740","Text":"and I\u0027ll just reverse the order,"},{"Start":"06:53.740 ","End":"06:56.810","Text":"write it as xi, yi."},{"Start":"06:56.810 ","End":"06:58.770","Text":"This equals 0."},{"Start":"06:58.770 ","End":"07:00.985","Text":"From the second equation,"},{"Start":"07:00.985 ","End":"07:03.430","Text":"similarly, we just take the a out."},{"Start":"07:03.430 ","End":"07:13.750","Text":"Sum, i goes from 1 to n xi plus the sum,"},{"Start":"07:13.750 ","End":"07:22.334","Text":"i goes from 1 to n of b minus the sum,"},{"Start":"07:22.334 ","End":"07:26.340","Text":"i goes from 1 to n of yi."},{"Start":"07:26.340 ","End":"07:32.080","Text":"I want you to notice that this can be simplified."},{"Start":"07:32.080 ","End":"07:39.705","Text":"The sum from 1 to n of a constant b is like saying b plus b plus,"},{"Start":"07:39.705 ","End":"07:42.150","Text":"and so on, plus b,"},{"Start":"07:42.150 ","End":"07:44.680","Text":"is going to be n of these,"},{"Start":"07:44.680 ","End":"07:47.920","Text":"so it\u0027s just n times b."},{"Start":"07:47.920 ","End":"07:58.615","Text":"Instead of this, I can write n times b. I forgot to write equals 0 here."},{"Start":"07:58.615 ","End":"08:06.125","Text":"This is the point where I now want to substitute the actual points in."},{"Start":"08:06.125 ","End":"08:08.555","Text":"Then everything will be a constant,"},{"Start":"08:08.555 ","End":"08:10.225","Text":"except for a and b."},{"Start":"08:10.225 ","End":"08:14.390","Text":"Then we\u0027re going to solve 2 equations and 2 unknowns, a and b."},{"Start":"08:14.390 ","End":"08:17.390","Text":"Then at the end, we have to remember to write"},{"Start":"08:17.390 ","End":"08:20.885","Text":"y equals ax plus b with the a and b that we found."},{"Start":"08:20.885 ","End":"08:22.310","Text":"I\u0027m going to scroll down."},{"Start":"08:22.310 ","End":"08:28.020","Text":"Let me first copy these so I don\u0027t lose them 2, 2.5."},{"Start":"08:28.020 ","End":"08:31.920","Text":"The next one is 1, 0.8."},{"Start":"08:31.920 ","End":"08:35.970","Text":"The next one is 3, 3.23,"},{"Start":"08:35.970 ","End":"08:38.760","Text":"and the 4th one is 4,"},{"Start":"08:38.760 ","End":"08:42.480","Text":"3.5 and 1, 2,"},{"Start":"08:42.480 ","End":"08:46.065","Text":"3, 4, n is going to be 4 in this case."},{"Start":"08:46.065 ","End":"08:48.880","Text":"Now I\u0027m going to scroll down."},{"Start":"08:51.440 ","End":"08:57.030","Text":"What I suggest for computing all these Sigmas is to make a table."},{"Start":"08:57.030 ","End":"09:02.839","Text":"I start with my first column,"},{"Start":"09:02.839 ","End":"09:05.465","Text":"which will be just the values of i,"},{"Start":"09:05.465 ","End":"09:09.385","Text":"1, 2, 3, and 4."},{"Start":"09:09.385 ","End":"09:17.750","Text":"Then also I\u0027ll want the sum from i goes from 1-4."},{"Start":"09:19.790 ","End":"09:23.770","Text":"I\u0027ll need a horizontal line."},{"Start":"09:23.990 ","End":"09:29.490","Text":"The next column will be xi."},{"Start":"09:29.490 ","End":"09:33.680","Text":"Then I\u0027ll need yi."},{"Start":"09:33.680 ","End":"09:34.980","Text":"Let\u0027s see what else I need."},{"Start":"09:34.980 ","End":"09:40.530","Text":"I\u0027ll need xi squared and I\u0027ll need xi,"},{"Start":"09:40.530 ","End":"09:48.270","Text":"yi, and added some extra grid lines and I fill in the data."},{"Start":"09:48.270 ","End":"09:52.450","Text":"The xi are going to be 2, 1, 3, 4."},{"Start":"09:54.470 ","End":"09:58.920","Text":"Then the yi, 2.5,"},{"Start":"09:58.920 ","End":"10:06.060","Text":"0.8, 3.2, and 3.5."},{"Start":"10:06.060 ","End":"10:10.695","Text":"The rest are computed xi squared,"},{"Start":"10:10.695 ","End":"10:12.510","Text":"just the square of this column,"},{"Start":"10:12.510 ","End":"10:18.195","Text":"4, 1, 9, 16."},{"Start":"10:18.195 ","End":"10:20.490","Text":"The product xi times yi,"},{"Start":"10:20.490 ","End":"10:22.920","Text":"2 times 2.5 is 5."},{"Start":"10:22.920 ","End":"10:28.050","Text":"Here 0.8, here 9.6,"},{"Start":"10:28.050 ","End":"10:31.950","Text":"and here 4 times 3 1/2 is 14."},{"Start":"10:31.950 ","End":"10:35.160","Text":"Now we need the last row for the sums."},{"Start":"10:35.160 ","End":"10:37.945","Text":"2 plus 1 plus 3 plus 4."},{"Start":"10:37.945 ","End":"10:41.760","Text":"That c is 10."},{"Start":"10:43.150 ","End":"10:46.954","Text":"Let\u0027s see 0.8 plus 3.2 is 4,"},{"Start":"10:46.954 ","End":"10:49.460","Text":"and these 2 together give me 6."},{"Start":"10:49.460 ","End":"10:51.609","Text":"Altogether, I get 10."},{"Start":"10:51.609 ","End":"10:54.000","Text":"This column, 109 is 10,"},{"Start":"10:54.000 ","End":"10:57.375","Text":"4 and 16 is 20, so I get 30."},{"Start":"10:57.375 ","End":"11:00.990","Text":"Let\u0027s see here, 5 and 4 is 19,"},{"Start":"11:00.990 ","End":"11:04.335","Text":"and this and this is 10.4,"},{"Start":"11:04.335 ","End":"11:10.445","Text":"so I get 29.4."},{"Start":"11:10.445 ","End":"11:16.000","Text":"Again, I can now go back here and have each of these quantities."},{"Start":"11:16.000 ","End":"11:18.010","Text":"For example, in the first equation,"},{"Start":"11:18.010 ","End":"11:19.510","Text":"the sum of the squares,"},{"Start":"11:19.510 ","End":"11:21.220","Text":"xi squared is 30."},{"Start":"11:21.220 ","End":"11:24.550","Text":"So this first equation becomes 30a."},{"Start":"11:24.550 ","End":"11:28.845","Text":"I\u0027ll leave room for the second."},{"Start":"11:28.845 ","End":"11:34.350","Text":"Then the sum of the x\u0027s is this column, is 10."},{"Start":"11:34.350 ","End":"11:37.720","Text":"So it\u0027s plus 10b."},{"Start":"11:38.480 ","End":"11:41.925","Text":"This sum is 29.4."},{"Start":"11:41.925 ","End":"11:46.130","Text":"I like to put the numbers on the right rather than minus something equals 0."},{"Start":"11:46.130 ","End":"11:48.425","Text":"So 29.4."},{"Start":"11:48.425 ","End":"11:53.760","Text":"The second equation, this part\u0027s the same, 30a."},{"Start":"11:54.170 ","End":"11:59.440","Text":"Here we have n is 4 times b."},{"Start":"11:59.930 ","End":"12:03.065","Text":"This, if I put it on the other side,"},{"Start":"12:03.065 ","End":"12:07.505","Text":"the sum of yi from here is 10,"},{"Start":"12:07.505 ","End":"12:13.545","Text":"so these are the 2 equations in a and b."},{"Start":"12:13.545 ","End":"12:16.910","Text":"Now, I\u0027ll spare you the details of solving"},{"Start":"12:16.910 ","End":"12:20.000","Text":"2 equations and 2 unknowns and I\u0027ll just tell you that"},{"Start":"12:20.000 ","End":"12:32.255","Text":"we get that a equals 0.88 and b equals 0.3."},{"Start":"12:32.255 ","End":"12:38.975","Text":"The line with the least sums of squared errors is y"},{"Start":"12:38.975 ","End":"12:45.230","Text":"equals 0.88x plus 0.3."},{"Start":"12:45.230 ","End":"12:47.315","Text":"Ax plus b."},{"Start":"12:47.315 ","End":"12:50.820","Text":"That is our answer."}],"ID":9036},{"Watched":false,"Name":"Exercise 7 part b","Duration":"11m 19s","ChapterTopicVideoID":8731,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"We\u0027ve seen exercises like this before,"},{"Start":"00:03.480 ","End":"00:07.485","Text":"the thing that changes is the type of function."},{"Start":"00:07.485 ","End":"00:11.730","Text":"I think previously we had one with linear ax plus b."},{"Start":"00:11.730 ","End":"00:16.395","Text":"Linear event, here we have a function of this form where a and b,"},{"Start":"00:16.395 ","End":"00:20.610","Text":"we have to customize them or"},{"Start":"00:20.610 ","End":"00:27.460","Text":"discover what they are to give the least sum of squared errors."},{"Start":"00:28.880 ","End":"00:32.625","Text":"As before, we\u0027re going to start out in general,"},{"Start":"00:32.625 ","End":"00:36.315","Text":"having points x_1, y_1,"},{"Start":"00:36.315 ","End":"00:46.050","Text":"x_2, y_2, and up to end points x_n, y_n."},{"Start":"00:46.050 ","End":"00:49.815","Text":"Later we\u0027ll apply our specific points."},{"Start":"00:49.815 ","End":"00:56.090","Text":"What we do is we estimate that y is"},{"Start":"00:56.090 ","End":"01:03.260","Text":"equal to ax squared plus bx for each of these points,"},{"Start":"01:03.260 ","End":"01:07.415","Text":"so we\u0027re going to let y_i,"},{"Start":"01:07.415 ","End":"01:10.330","Text":"the general point where i goes from 1 to n,"},{"Start":"01:10.330 ","End":"01:14.354","Text":"is going to be approximately equal to"},{"Start":"01:14.354 ","End":"01:23.160","Text":"ax_i squared plus bx_i."},{"Start":"01:23.160 ","End":"01:25.030","Text":"This is what my estimate is,"},{"Start":"01:25.030 ","End":"01:29.615","Text":"I\u0027m replacing the actual value by the predicted using this curve."},{"Start":"01:29.615 ","End":"01:33.065","Text":"So there\u0027s an error, discrepancy between these 2,"},{"Start":"01:33.065 ","End":"01:42.990","Text":"and the error, or rather I want the squared error,"},{"Start":"01:42.990 ","End":"01:47.350","Text":"and then it doesn\u0027t matter which way I subtract."},{"Start":"01:47.970 ","End":"01:52.465","Text":"I\u0027ll take the right-hand side minus the left-hand side,"},{"Start":"01:52.465 ","End":"01:58.170","Text":"ax_i squared plus bx_i,"},{"Start":"01:58.170 ","End":"02:04.200","Text":"subtract the y_i, that\u0027s the error squared."},{"Start":"02:04.200 ","End":"02:07.485","Text":"Now I want the sum of the squared errors,"},{"Start":"02:07.485 ","End":"02:11.904","Text":"so the sum of the squared errors is the sum,"},{"Start":"02:11.904 ","End":"02:15.550","Text":"and i goes from 1 to n of these squared errors,"},{"Start":"02:15.550 ","End":"02:16.750","Text":"so just copy that,"},{"Start":"02:16.750 ","End":"02:24.800","Text":"ax_i squared plus bx_i minus y_i squared."},{"Start":"02:24.800 ","End":"02:29.055","Text":"Now, these x_i and y_i will all be known,"},{"Start":"02:29.055 ","End":"02:31.350","Text":"for example, will be these points here."},{"Start":"02:31.350 ","End":"02:34.610","Text":"The only variables are a and b."},{"Start":"02:35.780 ","End":"02:39.635","Text":"This now becomes a function of a and b."},{"Start":"02:39.635 ","End":"02:46.265","Text":"So I just shifted it to the right so I can write f of a and b equals."},{"Start":"02:46.265 ","End":"02:51.230","Text":"Now I\u0027m going to use the techniques for finding a minimum point, well,"},{"Start":"02:51.230 ","End":"02:54.620","Text":"we\u0027re going to look for a critical point and I believe that in"},{"Start":"02:54.620 ","End":"02:58.100","Text":"the phrasing of the question in the exercise book,"},{"Start":"02:58.100 ","End":"03:03.205","Text":"it said to just assume that we\u0027re going to get a minimum when we find a critical point."},{"Start":"03:03.205 ","End":"03:05.660","Text":"So let\u0027s look for a critical point."},{"Start":"03:05.660 ","End":"03:09.365","Text":"Critical point is given by the derivative with respect to a,"},{"Start":"03:09.365 ","End":"03:13.510","Text":"and the derivative with respect to b are both 0."},{"Start":"03:13.510 ","End":"03:20.735","Text":"Let\u0027s differentiate this first with respect to a and then we get."},{"Start":"03:20.735 ","End":"03:23.795","Text":"Now we\u0027ve done this thing before,"},{"Start":"03:23.795 ","End":"03:26.465","Text":"what you can do is you can put the derivative"},{"Start":"03:26.465 ","End":"03:30.395","Text":"inside the sum and do it for each term separately."},{"Start":"03:30.395 ","End":"03:33.140","Text":"So the derivative of this with respect to x,"},{"Start":"03:33.140 ","End":"03:36.995","Text":"it\u0027s something squared so it\u0027s twice that something,"},{"Start":"03:36.995 ","End":"03:42.510","Text":"which is ax_i squared plus bx_i minus y_i."},{"Start":"03:44.510 ","End":"03:47.300","Text":"Now we\u0027re differentiating with respect to a,"},{"Start":"03:47.300 ","End":"03:49.970","Text":"so the derivative is just the coefficient of a,"},{"Start":"03:49.970 ","End":"03:51.785","Text":"everything else doesn\u0027t contain a,"},{"Start":"03:51.785 ","End":"03:59.890","Text":"so it\u0027s x_i squared, and that\u0027s that."},{"Start":"04:01.370 ","End":"04:06.225","Text":"I\u0027ll just mention that the constant 2,"},{"Start":"04:06.225 ","End":"04:08.130","Text":"which is a multiplier,"},{"Start":"04:08.130 ","End":"04:12.695","Text":"we can actually take it out and put it in front of the Sigma."},{"Start":"04:12.695 ","End":"04:15.644","Text":"Now f with respect to b is similar,"},{"Start":"04:15.644 ","End":"04:19.440","Text":"it\u0027s also going to be the sum from 1 to n,"},{"Start":"04:19.440 ","End":"04:22.780","Text":"and there already can see that I\u0027m going to get a 2 here,"},{"Start":"04:22.780 ","End":"04:25.565","Text":"so I\u0027ll bring it out front right away."},{"Start":"04:25.565 ","End":"04:32.820","Text":"Now we also have this ax_i squared plus bx_i minus y_i,"},{"Start":"04:32.820 ","End":"04:39.350","Text":"the difference is the inner derivative with respect to b is just x_i."},{"Start":"04:40.670 ","End":"04:46.265","Text":"That\u0027s the difference. Now we want to set both of these to be 0,"},{"Start":"04:46.265 ","End":"04:49.310","Text":"we\u0027ll get a pair of equations."},{"Start":"04:49.310 ","End":"04:56.285","Text":"The 2, it doesn\u0027t matter when we\u0027re comparing to 0 because it\u0027s going to be equal to 0/2,"},{"Start":"04:56.285 ","End":"04:58.025","Text":"so I\u0027ll just take this."},{"Start":"04:58.025 ","End":"05:03.045","Text":"I\u0027m going to multiply out already,"},{"Start":"05:03.045 ","End":"05:05.970","Text":"so we get the sum,"},{"Start":"05:05.970 ","End":"05:08.790","Text":"i goes from 1 to n,"},{"Start":"05:08.790 ","End":"05:15.300","Text":"I\u0027m going to put this inside, a, x_i squared,"},{"Start":"05:15.300 ","End":"05:24.335","Text":"x_i squared is ax_i^4 plus"},{"Start":"05:24.335 ","End":"05:33.170","Text":"bx_i cubed minus x_i"},{"Start":"05:33.170 ","End":"05:38.300","Text":"squared y_i that equals 0."},{"Start":"05:38.300 ","End":"05:40.659","Text":"In the other one,"},{"Start":"05:40.659 ","End":"05:47.730","Text":"we will get pretty much the same thing with 1 power less in the x\u0027s,"},{"Start":"05:47.730 ","End":"05:55.890","Text":"so it\u0027s x_i cubed plus bx_i squared minus x_i,"},{"Start":"05:55.890 ","End":"06:03.160","Text":"y_i equals 0, I forgot the brackets, it\u0027s important,"},{"Start":"06:05.930 ","End":"06:10.050","Text":"i equals 1 to n. At this point,"},{"Start":"06:10.050 ","End":"06:13.730","Text":"I\u0027m going to put the Sigma for each one separately and also take"},{"Start":"06:13.730 ","End":"06:18.520","Text":"constants out so we end up with a pair of equations,"},{"Start":"06:18.520 ","End":"06:22.965","Text":"a times the sum of"},{"Start":"06:22.965 ","End":"06:30.310","Text":"x_i^4 plus b times the sum of x_i cubed."},{"Start":"06:31.670 ","End":"06:34.684","Text":"I will put this on the other side,"},{"Start":"06:34.684 ","End":"06:40.070","Text":"equals the sum of x_i squared y_i,"},{"Start":"06:40.070 ","End":"06:44.254","Text":"all the sums from 1 to n,"},{"Start":"06:44.254 ","End":"06:46.390","Text":"from 1 to n,"},{"Start":"06:46.390 ","End":"06:50.040","Text":"from 1 to n. Here,"},{"Start":"06:50.040 ","End":"06:53.715","Text":"we\u0027ll get the same thing with 1 power less,"},{"Start":"06:53.715 ","End":"07:00.075","Text":"a sum from 1 to n of x_i cubed"},{"Start":"07:00.075 ","End":"07:07.275","Text":"plus b sum from 1 to n of x_i squared equals sum,"},{"Start":"07:07.275 ","End":"07:13.300","Text":"i goes from 1 to n, of x_i, y_i."},{"Start":"07:13.640 ","End":"07:22.460","Text":"At this point, I want to stop being abstract and put in our particular points."},{"Start":"07:22.460 ","End":"07:26.120","Text":"I\u0027m going to copy them so I don\u0027t lose them when I scroll."},{"Start":"07:26.120 ","End":"07:31.605","Text":"I have 3 points minus 1, 2, 2, 0,"},{"Start":"07:31.605 ","End":"07:33.810","Text":"and 0, minus 2,"},{"Start":"07:33.810 ","End":"07:37.925","Text":"so of course that our n is equal to 3 in this case."},{"Start":"07:37.925 ","End":"07:43.475","Text":"Now let me organize a table."},{"Start":"07:43.475 ","End":"07:49.970","Text":"I just modified the table from the previous exercise."},{"Start":"07:49.970 ","End":"07:52.280","Text":"Here, we have n is 3,"},{"Start":"07:52.280 ","End":"07:58.355","Text":"so I need 3 rows and the row for the total."},{"Start":"07:58.355 ","End":"08:01.820","Text":"Let\u0027s put in the x_i, y_i first."},{"Start":"08:01.820 ","End":"08:06.100","Text":"The x_i are minus 1, 2, and 0,"},{"Start":"08:06.100 ","End":"08:09.225","Text":"and the y_i are going to be 2,"},{"Start":"08:09.225 ","End":"08:11.855","Text":"0, and minus 2."},{"Start":"08:11.855 ","End":"08:14.270","Text":"Now the columns I\u0027m going to need,"},{"Start":"08:14.270 ","End":"08:20.605","Text":"looking here I\u0027m going to need the sum of x_i^4."},{"Start":"08:20.605 ","End":"08:24.030","Text":"I\u0027m going to need x_i cubed,"},{"Start":"08:24.030 ","End":"08:29.010","Text":"I\u0027m going to need x_i squared,"},{"Start":"08:29.010 ","End":"08:33.360","Text":"I\u0027m going to need x_i squared y_i,"},{"Start":"08:33.360 ","End":"08:41.310","Text":"and I\u0027m going to need x_i, y_i."},{"Start":"08:41.310 ","End":"08:44.970","Text":"Now, if x_i is minus 1 squared,"},{"Start":"08:44.970 ","End":"08:47.550","Text":"it\u0027s 1 cubed, it\u0027s minus 1,"},{"Start":"08:47.550 ","End":"08:49.950","Text":"and to the 4th, it\u0027s plus 1."},{"Start":"08:49.950 ","End":"08:52.770","Text":"Next one is 2,"},{"Start":"08:52.770 ","End":"08:56.115","Text":"squared is 4, cubed is 8,"},{"Start":"08:56.115 ","End":"08:57.990","Text":"to the 4th is 16,"},{"Start":"08:57.990 ","End":"09:02.010","Text":"powers of 2, 0, 0, 0."},{"Start":"09:02.010 ","End":"09:06.170","Text":"In fact these are all going to be 0 because it\u0027s going to have a 0 in the product."},{"Start":"09:06.170 ","End":"09:13.209","Text":"Now, let\u0027s see, x_i squared times y is this column with this column,"},{"Start":"09:13.209 ","End":"09:17.580","Text":"1 times 2 is 2,"},{"Start":"09:17.580 ","End":"09:20.640","Text":"4 times 0, 0."},{"Start":"09:20.640 ","End":"09:24.930","Text":"In fact, this is also going to be 0 because the y is 0 here."},{"Start":"09:25.720 ","End":"09:37.255","Text":"Then x_i, y_i here is minus 1 times 2 is minus 2."},{"Start":"09:37.255 ","End":"09:39.330","Text":"Next, totals."},{"Start":"09:39.330 ","End":"09:41.970","Text":"Minus 1 and 2 and 0 is 1,"},{"Start":"09:41.970 ","End":"09:45.180","Text":"2 and minus 2 and 0 is 0,"},{"Start":"09:45.180 ","End":"09:54.710","Text":"17, 7, 5, 2, minus 2."},{"Start":"09:54.710 ","End":"10:03.560","Text":"So our equations become sum of the 4th powers of x, 17a."},{"Start":"10:04.320 ","End":"10:09.890","Text":"For b, we have this here, 7b."},{"Start":"10:09.890 ","End":"10:15.695","Text":"This column is 2."},{"Start":"10:15.695 ","End":"10:19.680","Text":"Next one, the sum of x_i cubed is 7a,"},{"Start":"10:19.680 ","End":"10:25.220","Text":"and b goes with the sum of the squares of x,"},{"Start":"10:25.220 ","End":"10:32.570","Text":"is 5, and sum of this is minus 2."},{"Start":"10:32.570 ","End":"10:34.490","Text":"I\u0027ll tell you the answer,"},{"Start":"10:34.490 ","End":"10:36.935","Text":"a comes out to 2/3,"},{"Start":"10:36.935 ","End":"10:40.040","Text":"and b comes out minus 4/3."},{"Start":"10:40.040 ","End":"10:45.965","Text":"Let\u0027s substitute at least in one of them just as a partial check,"},{"Start":"10:45.965 ","End":"10:53.465","Text":"7a is 14/3, 5b is minus 20/3,"},{"Start":"10:53.465 ","End":"10:57.130","Text":"14 minus 20 is minus 6/3."},{"Start":"10:57.130 ","End":"10:59.080","Text":"Yeah, that won\u0027t work,"},{"Start":"10:59.080 ","End":"11:01.130","Text":"and I\u0027ll leave you to check the other one."},{"Start":"11:01.130 ","End":"11:02.390","Text":"Now that we have a and b,"},{"Start":"11:02.390 ","End":"11:09.675","Text":"all that we have to do is write the equation of the curve which was y equals ax squared,"},{"Start":"11:09.675 ","End":"11:11.865","Text":"which is 2/3x squared,"},{"Start":"11:11.865 ","End":"11:16.425","Text":"plus bx is minus 4/3x,"},{"Start":"11:16.425 ","End":"11:19.630","Text":"and we are done."}],"ID":9037},{"Watched":false,"Name":"Exercise 7 part c","Duration":"15m 21s","ChapterTopicVideoID":8732,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.870","Text":"In this exercise, we have a set of data points."},{"Start":"00:05.660 ","End":"00:09.150","Text":"We believe that they should have come from"},{"Start":"00:09.150 ","End":"00:13.305","Text":"a function of the form y equals ax plus b over x,"},{"Start":"00:13.305 ","End":"00:15.125","Text":"but there are some errors."},{"Start":"00:15.125 ","End":"00:18.610","Text":"We have to find the a and the b, i.e,"},{"Start":"00:18.610 ","End":"00:24.410","Text":"this curve, which minimizes the sum of squared errors for these."},{"Start":"00:24.410 ","End":"00:28.535","Text":"Now, I\u0027m going to start off in general,"},{"Start":"00:28.535 ","End":"00:30.710","Text":"not with these specific points,"},{"Start":"00:30.710 ","End":"00:35.325","Text":"but with say, n points, x_1,"},{"Start":"00:35.325 ","End":"00:45.360","Text":"y_1, x_2, y_2, and so on and so on up to x_n,"},{"Start":"00:45.360 ","End":"00:49.760","Text":"y_n because in this example we have n equals 4 and these are given,"},{"Start":"00:49.760 ","End":"00:52.624","Text":"but let\u0027s just first of all, be general."},{"Start":"00:52.624 ","End":"00:57.790","Text":"Now, when we estimate,"},{"Start":"00:58.210 ","End":"01:03.710","Text":"we typically have a discrepancy or error because y_1 will not equal,"},{"Start":"01:03.710 ","End":"01:11.140","Text":"it will only be approximately equal to ax."},{"Start":"01:13.820 ","End":"01:18.225","Text":"If we take a typical 1 of these, let\u0027s say x_i,"},{"Start":"01:18.225 ","End":"01:23.405","Text":"y_i, it will not satisfy this equation."},{"Start":"01:23.405 ","End":"01:28.330","Text":"y_i will just approximate"},{"Start":"01:28.850 ","End":"01:37.890","Text":"ax_i plus b over x_i and there\u0027ll be an error, discrepancy."},{"Start":"01:37.890 ","End":"01:42.965","Text":"We want to minimize the sum of the squared errors."},{"Start":"01:42.965 ","End":"01:50.090","Text":"Now the squared error is not the difference,"},{"Start":"01:50.090 ","End":"01:51.230","Text":"but the difference squared,"},{"Start":"01:51.230 ","End":"01:54.180","Text":"doesn\u0027t matter the order of subtraction."},{"Start":"01:54.800 ","End":"02:06.430","Text":"Let\u0027s say the right minus the left ax_i plus b over x_i minus y_i and squared."},{"Start":"02:06.640 ","End":"02:12.260","Text":"Now, the sum of the squared errors which we have to minimize."},{"Start":"02:12.260 ","End":"02:14.315","Text":"Let me just write it over here,"},{"Start":"02:14.315 ","End":"02:21.269","Text":"will be the sum where i goes from 1 to n of all of these,"},{"Start":"02:21.269 ","End":"02:28.200","Text":"ax_i plus b over x_i minus y_i squared."},{"Start":"02:28.200 ","End":"02:32.720","Text":"This is going to be a function of a and b"},{"Start":"02:32.720 ","End":"02:37.985","Text":"because all the rest will be known just like these x_i,"},{"Start":"02:37.985 ","End":"02:40.490","Text":"y_i might be from here or from anywhere,"},{"Start":"02:40.490 ","End":"02:49.485","Text":"but a and b are the variables and we want to minimize this sum of squared errors."},{"Start":"02:49.485 ","End":"02:52.155","Text":"We\u0027re going to look for a critical point"},{"Start":"02:52.155 ","End":"02:57.530","Text":"and we\u0027re going to assume that critical point is a minimum."},{"Start":"02:57.530 ","End":"03:02.460","Text":"In the exercise book,"},{"Start":"03:02.460 ","End":"03:04.800","Text":"it says to make that assumption,"},{"Start":"03:04.800 ","End":"03:08.000","Text":"so we\u0027re just going to look for the critical point and assume that it\u0027s a minimum."},{"Start":"03:08.000 ","End":"03:14.360","Text":"We do that by the usual technique of saying the partial derivatives,"},{"Start":"03:14.360 ","End":"03:21.175","Text":"in this case with respect to a and with respect to b are both 0."},{"Start":"03:21.175 ","End":"03:27.285","Text":"Let\u0027s see. f with respect to a will equal,"},{"Start":"03:27.285 ","End":"03:31.220","Text":"I differentiate this with respect to"},{"Start":"03:31.220 ","End":"03:36.770","Text":"a. I can differentiate each of the terms in the sum separately."},{"Start":"03:36.770 ","End":"03:38.090","Text":"We\u0027ve done these before,"},{"Start":"03:38.090 ","End":"03:40.550","Text":"so I\u0027m going to take a bit of a shortcut."},{"Start":"03:40.550 ","End":"03:46.015","Text":"We get 2 times something and the 2 I\u0027m going to pull out in front of the Sigma."},{"Start":"03:46.015 ","End":"03:50.900","Text":"Already I\u0027m starting off with twice the Sigma from the sum from 1"},{"Start":"03:50.900 ","End":"03:56.580","Text":"to n. Derivative of this is twice this, 2 is here."},{"Start":"03:56.780 ","End":"04:02.935","Text":"ax_i plus b over x_i minus y_i"},{"Start":"04:02.935 ","End":"04:10.530","Text":"multiplied by the inner derivative of this in a derivative with respect to a."},{"Start":"04:10.820 ","End":"04:15.000","Text":"Now, a doesn\u0027t appear in these 2, so it\u0027s just from here,"},{"Start":"04:15.000 ","End":"04:19.440","Text":"so it\u0027s the coefficient of a, which is x_i."},{"Start":"04:20.050 ","End":"04:24.215","Text":"Now the derivative with respect to b."},{"Start":"04:24.215 ","End":"04:27.620","Text":"Also we\u0027re going to take the 2 out because each term"},{"Start":"04:27.620 ","End":"04:31.090","Text":"is going to be twice some things so I pull the 2 out,"},{"Start":"04:31.090 ","End":"04:35.300","Text":"sum from 1 to n. This part\u0027s the same."},{"Start":"04:35.300 ","End":"04:40.325","Text":"The something squared gives us twice that something. Just copy it."},{"Start":"04:40.325 ","End":"04:47.665","Text":"ax_i plus b over x_i minus y_i,"},{"Start":"04:47.665 ","End":"04:51.140","Text":"but the derivative here,"},{"Start":"04:51.140 ","End":"04:53.600","Text":"since it\u0027s with respect to b,"},{"Start":"04:53.600 ","End":"04:55.370","Text":"is the coefficient of b,"},{"Start":"04:55.370 ","End":"04:59.000","Text":"which is 1 over x_i."},{"Start":"05:01.610 ","End":"05:08.770","Text":"This condition will give us a set of 2 equations and 2 unknowns, a and b."},{"Start":"05:08.770 ","End":"05:11.965","Text":"This is going to equal 0."},{"Start":"05:11.965 ","End":"05:19.875","Text":"Let\u0027s write it, the sum from 1 to n and l already open the brackets,"},{"Start":"05:19.875 ","End":"05:26.469","Text":"so ax_i over x_i is just a,"},{"Start":"05:26.469 ","End":"05:30.810","Text":"sorry, it\u0027s ax_i squared."},{"Start":"05:32.060 ","End":"05:38.010","Text":"Then plus b over x_i times x_i is just b,"},{"Start":"05:38.010 ","End":"05:43.965","Text":"and then minus now write x_i, y_i."},{"Start":"05:43.965 ","End":"05:47.980","Text":"All this is equal to 0."},{"Start":"05:48.130 ","End":"05:55.680","Text":"I guess we\u0027re assuming that all the x_i\u0027s will be non 0,"},{"Start":"05:55.680 ","End":"05:56.925","Text":"otherwise this won\u0027t make sense."},{"Start":"05:56.925 ","End":"05:58.280","Text":"Maybe I\u0027ll just write that down."},{"Start":"05:58.280 ","End":"06:03.110","Text":"We\u0027re assuming that x_i is not equal to 0,"},{"Start":"06:03.110 ","End":"06:08.965","Text":"where i goes from 1 to n. Next 1,"},{"Start":"06:08.965 ","End":"06:18.905","Text":"f_b equals 0, gives me the sum from 1 to n. This time I\u0027m multiplying by 1 over x_i,"},{"Start":"06:18.905 ","End":"06:28.905","Text":"so I get ax_i over x_i is just a. b gets a denominator of x_i"},{"Start":"06:28.905 ","End":"06:33.825","Text":"squared now and here we have"},{"Start":"06:33.825 ","End":"06:41.309","Text":"minus y_i over x_i and this equals 0."},{"Start":"06:41.309 ","End":"06:44.070","Text":"2 equations and 2 unknowns,"},{"Start":"06:44.070 ","End":"06:48.800","Text":"a and b. I\u0027m going to copy these points because I need to scroll more."},{"Start":"06:48.800 ","End":"06:51.319","Text":"Let\u0027s see, we have the first 1 is 10,"},{"Start":"06:51.319 ","End":"06:58.020","Text":"20.2, then we have 6 comma 12.9,"},{"Start":"06:58.020 ","End":"07:01.830","Text":"then we have 8.5,"},{"Start":"07:01.830 ","End":"07:08.510","Text":"and lastly 0.5 comma 4 and now I can safely scroll and lose those."},{"Start":"07:08.510 ","End":"07:12.530","Text":"Yeah."},{"Start":"07:12.530 ","End":"07:20.185","Text":"This should do. Now continue over here."},{"Start":"07:20.185 ","End":"07:23.514","Text":"The first 1, as usual,"},{"Start":"07:23.514 ","End":"07:31.345","Text":"we can put the Sigma in front of each of these,"},{"Start":"07:31.345 ","End":"07:35.020","Text":"and we can also take constants in front of a sigma."},{"Start":"07:35.020 ","End":"07:38.395","Text":"We get from here from the first 1,"},{"Start":"07:38.395 ","End":"07:44.440","Text":"a times the sum of x_i squared"},{"Start":"07:44.440 ","End":"07:52.270","Text":"plus the sum of b minus the sum,"},{"Start":"07:52.270 ","End":"07:55.960","Text":"you know what, instead of minus equals 0,"},{"Start":"07:55.960 ","End":"07:57.550","Text":"I\u0027ll make it equals,"},{"Start":"07:57.550 ","End":"08:01.030","Text":"put this over the other side, x_i, y_i,"},{"Start":"08:01.030 ","End":"08:05.260","Text":"I did several steps in 1 and I don\u0027t want to get too lazy."},{"Start":"08:05.260 ","End":"08:08.200","Text":"I\u0027ll write the i 1 to n,"},{"Start":"08:08.200 ","End":"08:10.360","Text":"i 1 to n,"},{"Start":"08:10.360 ","End":"08:12.550","Text":"and that\u0027s the first of 2."},{"Start":"08:12.550 ","End":"08:19.075","Text":"The second 1 becomes the sum of"},{"Start":"08:19.075 ","End":"08:28.559","Text":"a plus b times the sum of 1 over x_i squared equals,"},{"Start":"08:28.559 ","End":"08:34.980","Text":"again I put in the minus over the sum of y_i over x_i,"},{"Start":"08:34.980 ","End":"08:38.520","Text":"and each of these is from 1 to n,"},{"Start":"08:38.520 ","End":"08:39.990","Text":"from 1 to n,"},{"Start":"08:39.990 ","End":"08:49.180","Text":"i goes from 1 to n. There\u0027s just 1 more little shortcut simplification I can do."},{"Start":"08:49.180 ","End":"08:51.310","Text":"We\u0027ve seen this before."},{"Start":"08:51.310 ","End":"08:54.235","Text":"When we take the sum of a constant,"},{"Start":"08:54.235 ","End":"08:57.415","Text":"we have the sum of n terms and all of them b,"},{"Start":"08:57.415 ","End":"09:00.410","Text":"so this is n times b."},{"Start":"09:01.650 ","End":"09:07.135","Text":"Similarly here, we can replace this by n times a,"},{"Start":"09:07.135 ","End":"09:08.575","Text":"plus a plus a plus a,"},{"Start":"09:08.575 ","End":"09:11.000","Text":"n times just na."},{"Start":"09:14.850 ","End":"09:22.070","Text":"Now let\u0027s introduce our particular points."},{"Start":"09:22.230 ","End":"09:27.445","Text":"I want to compute some of the quantities."},{"Start":"09:27.445 ","End":"09:29.874","Text":"I\u0027m going to compute this,"},{"Start":"09:29.874 ","End":"09:40.510","Text":"and I\u0027m going to compute this sum and this sum and this sum,"},{"Start":"09:40.510 ","End":"09:42.985","Text":"but for our particular points,"},{"Start":"09:42.985 ","End":"09:47.170","Text":"so I\u0027ll do it in a table form."},{"Start":"09:47.170 ","End":"09:51.400","Text":"I just move this to the side to make room for the table,"},{"Start":"09:51.400 ","End":"09:54.610","Text":"and I copied 1 from the previous exercise."},{"Start":"09:54.610 ","End":"10:02.845","Text":"I just need to extend it a bit because I\u0027m going to need a column for 1"},{"Start":"10:02.845 ","End":"10:12.100","Text":"over x_i squared and a column for y_i over x_i."},{"Start":"10:12.100 ","End":"10:14.650","Text":"Now, I\u0027ll just fill in the xs and ys."},{"Start":"10:14.650 ","End":"10:20.875","Text":"We have 10, 20.2 and then 6,"},{"Start":"10:20.875 ","End":"10:30.655","Text":"12.9, then 4, 8.5, and 0.5, 4."},{"Start":"10:30.655 ","End":"10:34.765","Text":"This is the x_i squared column."},{"Start":"10:34.765 ","End":"10:38.260","Text":"10 gives me a 100, here I get 36,"},{"Start":"10:38.260 ","End":"10:43.150","Text":"4 squared is 16, 1/2 squared is 1/4."},{"Start":"10:43.150 ","End":"10:47.980","Text":"Next, I need the product x_i times y_i,"},{"Start":"10:47.980 ","End":"10:56.410","Text":"10 times 20.2 is 202, here, 77.4,"},{"Start":"10:56.410 ","End":"11:02.980","Text":"here I need 4 times 8.5 is 34,"},{"Start":"11:02.980 ","End":"11:07.225","Text":"and here 1/2 times 4 is 2,"},{"Start":"11:07.225 ","End":"11:11.080","Text":"then 1 over x_i squared,"},{"Start":"11:11.080 ","End":"11:15.440","Text":"so it\u0027s 1 over this, 0.01."},{"Start":"11:17.160 ","End":"11:22.970","Text":"The easy 1s, 1 over a 1/4 is 4,"},{"Start":"11:23.400 ","End":"11:28.390","Text":"1/16 is 0.0625,"},{"Start":"11:28.390 ","End":"11:34.780","Text":"and 1/36 is 0.02."},{"Start":"11:34.780 ","End":"11:42.640","Text":"Then 7 is indefinitely a rounded up to an 8 to make it 4 places."},{"Start":"11:43.610 ","End":"11:48.460","Text":"Y_i over x_i, this over this,"},{"Start":"11:53.290 ","End":"12:04.150","Text":"make that 2.02, this over this will be 2.15,"},{"Start":"12:04.150 ","End":"12:12.925","Text":"this over this will be 2.125,"},{"Start":"12:12.925 ","End":"12:22.130","Text":"and 4 over 1/2 will be 8."},{"Start":"12:22.130 ","End":"12:24.910","Text":"Then the last row is for the Sigma,"},{"Start":"12:24.910 ","End":"12:26.815","Text":"for the sum of them."},{"Start":"12:26.815 ","End":"12:29.320","Text":"I need this sum."},{"Start":"12:29.320 ","End":"12:34.670","Text":"That\u0027s for this column, 136, 152.25."},{"Start":"12:37.140 ","End":"12:41.980","Text":"Here, we\u0027ll get 315.4,"},{"Start":"12:41.980 ","End":"12:49.195","Text":"4.1003, and this total"},{"Start":"12:49.195 ","End":"12:54.670","Text":"is 14 from the 8 and 2 and 2 and 2."},{"Start":"12:54.670 ","End":"12:56.590","Text":"Then look at this,"},{"Start":"12:56.590 ","End":"12:58.600","Text":"there\u0027s a 5 here,"},{"Start":"12:58.600 ","End":"13:06.415","Text":"2 and 5 and 2 is 9, 0111, 14.295."},{"Start":"13:06.415 ","End":"13:08.365","Text":"These are the only 1s we need,"},{"Start":"13:08.365 ","End":"13:09.880","Text":"we don\u0027t need these 2."},{"Start":"13:09.880 ","End":"13:16.930","Text":"Now just substitute them in here and we get."},{"Start":"13:16.930 ","End":"13:18.940","Text":"This 1 is this 1."},{"Start":"13:18.940 ","End":"13:27.910","Text":"152.25 a plus n is 4."},{"Start":"13:27.910 ","End":"13:31.270","Text":"Let\u0027s write that again, n equals 4,"},{"Start":"13:31.270 ","End":"13:41.000","Text":"that\u0027s 4b equals, this sum is here, 315.4."},{"Start":"13:42.060 ","End":"13:44.740","Text":"Here once again, n is 4,"},{"Start":"13:44.740 ","End":"13:50.810","Text":"so it\u0027s 4a plus"},{"Start":"13:51.540 ","End":"13:55.180","Text":"4.1003 times b."},{"Start":"13:55.180 ","End":"14:06.070","Text":"B is this 14.295."},{"Start":"14:06.070 ","End":"14:09.730","Text":"Everything\u0027s exactly except for this was a slight approximation"},{"Start":"14:09.730 ","End":"14:14.305","Text":"because I made an approximation here."},{"Start":"14:14.305 ","End":"14:19.690","Text":"In any event I\u0027m not going to bore you with"},{"Start":"14:19.690 ","End":"14:25.600","Text":"the details of solving 2 equations in 2 unknowns,"},{"Start":"14:25.600 ","End":"14:27.265","Text":"and I\u0027m going to give you,"},{"Start":"14:27.265 ","End":"14:33.055","Text":"the results a is 2.032,"},{"Start":"14:33.055 ","End":"14:38.425","Text":"maybe approximately, everything\u0027s approximate."},{"Start":"14:38.425 ","End":"14:47.320","Text":"B came out as 1.5039."},{"Start":"14:47.320 ","End":"14:57.190","Text":"Lastly, we just have to answer the question that the equation y equals ax"},{"Start":"14:57.190 ","End":"15:03.115","Text":"plus b over x means that this is equal to"},{"Start":"15:03.115 ","End":"15:12.010","Text":"2.032x plus"},{"Start":"15:12.010 ","End":"15:16.570","Text":"1.5039 over x,"},{"Start":"15:16.570 ","End":"15:21.980","Text":"and that\u0027s the answer. We\u0027re d1."}],"ID":9038},{"Watched":false,"Name":"Exercise 7 part d","Duration":"19m 59s","ChapterTopicVideoID":8733,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.305","Text":"In this exercise, we have a set of data points,"},{"Start":"00:04.305 ","End":"00:06.405","Text":"x and y in each."},{"Start":"00:06.405 ","End":"00:15.180","Text":"We want to try and approximate these as coming from a curve, a function,"},{"Start":"00:15.180 ","End":"00:19.280","Text":"y equals ax squared plus b over x squared in such a way that it\u0027s"},{"Start":"00:19.280 ","End":"00:25.700","Text":"the best fit in the sense that it minimizes the sum of squared errors."},{"Start":"00:25.700 ","End":"00:29.360","Text":"I\u0027ll start out by taking points in general,"},{"Start":"00:29.360 ","End":"00:31.370","Text":"not this specific data."},{"Start":"00:31.370 ","End":"00:34.640","Text":"In other words, we\u0027ll take points x_1,"},{"Start":"00:34.640 ","End":"00:39.235","Text":"y_1, and so on, up to x_n,"},{"Start":"00:39.235 ","End":"00:49.325","Text":"y_n points, and let\u0027s develop it in general then we\u0027ll take the specific example."},{"Start":"00:49.325 ","End":"00:56.165","Text":"Now, we\u0027re going to take for a particular x_i, y_i,"},{"Start":"00:56.165 ","End":"01:00.140","Text":"we\u0027re going to assume that y_i is being approximated"},{"Start":"01:00.140 ","End":"01:10.075","Text":"by ax_i squared plus b over x_i squared,"},{"Start":"01:10.075 ","End":"01:15.225","Text":"which means that there\u0027s an error."},{"Start":"01:15.225 ","End":"01:20.415","Text":"What I want to do is see where the sum of squared errors is."},{"Start":"01:20.415 ","End":"01:26.240","Text":"Now the squared error and 1 of"},{"Start":"01:26.240 ","End":"01:28.670","Text":"the good things about squaring is it doesn\u0027t matter"},{"Start":"01:28.670 ","End":"01:31.940","Text":"the order of subtraction would be to take the difference,"},{"Start":"01:31.940 ","End":"01:34.810","Text":"lets\u0027s say the right minus the left,"},{"Start":"01:34.810 ","End":"01:41.680","Text":"ax_i squared plus b over x_i squared minus,"},{"Start":"01:41.680 ","End":"01:43.190","Text":"this is the estimated,"},{"Start":"01:43.190 ","End":"01:47.240","Text":"this is the actual and squared."},{"Start":"01:47.240 ","End":"01:53.014","Text":"Now, the sum of the squared errors will be the sum,"},{"Start":"01:53.014 ","End":"01:57.445","Text":"i goes from 1 to n. I is a typical,"},{"Start":"01:57.445 ","End":"02:02.295","Text":"all of these up this same expression,"},{"Start":"02:02.295 ","End":"02:11.400","Text":"ax_i squared plus b over x_i squared minus y_i."},{"Start":"02:11.400 ","End":"02:19.440","Text":"This is just a function of a and b because the x_is and the y_is are given quantities."},{"Start":"02:19.440 ","End":"02:21.230","Text":"In our case, there\u0027ll be these,"},{"Start":"02:21.230 ","End":"02:25.125","Text":"but in general they\u0027re assumed to be fixed."},{"Start":"02:25.125 ","End":"02:30.110","Text":"We want to vary a and b in such a way that this comes out to be a minimum."},{"Start":"02:30.110 ","End":"02:31.565","Text":"If it\u0027s a minimum,"},{"Start":"02:31.565 ","End":"02:33.875","Text":"then it\u0027s going to be a critical point."},{"Start":"02:33.875 ","End":"02:42.180","Text":"We\u0027re going to look for a critical point by setting f_a and f_b to equal 0."},{"Start":"02:42.180 ","End":"02:46.310","Text":"First of all, let\u0027s compute them and by the way,"},{"Start":"02:46.310 ","End":"02:47.915","Text":"in the exercise book,"},{"Start":"02:47.915 ","End":"02:51.560","Text":"we\u0027re not going to have to show that the critical point is actually a minimum,"},{"Start":"02:51.560 ","End":"02:53.600","Text":"we\u0027re just going to take that for granted."},{"Start":"02:53.600 ","End":"02:55.760","Text":"We\u0027re just looking for a critical point."},{"Start":"02:55.760 ","End":"02:58.355","Text":"F with respect to a,"},{"Start":"02:58.355 ","End":"03:04.150","Text":"now the derivative with respect to a can go inside the Sigma."},{"Start":"03:04.700 ","End":"03:08.830","Text":"I forgot the 2 here sorry."},{"Start":"03:09.140 ","End":"03:15.150","Text":"It\u0027s going to be twice this and the 2 can be put out front."},{"Start":"03:15.150 ","End":"03:22.110","Text":"Twice this thing, ax_i squared plus b over"},{"Start":"03:22.110 ","End":"03:28.790","Text":"x_i squared minus y_i times the inner derivative with respect to a."},{"Start":"03:28.790 ","End":"03:34.680","Text":"A only appears here and its coefficient is x_i squared;"},{"Start":"03:36.670 ","End":"03:39.805","Text":"i goes from 1 to n, sorry."},{"Start":"03:39.805 ","End":"03:43.185","Text":"Similarly for f be the 2 is also going to come out"},{"Start":"03:43.185 ","End":"03:49.020","Text":"front and we just going to get a different thing here."},{"Start":"03:49.020 ","End":"03:50.960","Text":"There\u0027s this part\u0027s going to be the same,"},{"Start":"03:50.960 ","End":"03:57.140","Text":"going to be ax_i squared plus b over x_i squared minus y_i."},{"Start":"03:57.140 ","End":"04:01.450","Text":"But the inner derivative will be different because we\u0027re now with respect to b,"},{"Start":"04:01.450 ","End":"04:06.050","Text":"the coefficient is 1 over x_i squared."},{"Start":"04:06.050 ","End":"04:09.515","Text":"All the rest of it is a constant as far as b goes."},{"Start":"04:09.515 ","End":"04:18.020","Text":"Now what we want for a critical point is for f_a and f_b to both be 0."},{"Start":"04:18.020 ","End":"04:24.570","Text":"We\u0027ll get 2 equations from assigning these to be 0."},{"Start":"04:24.570 ","End":"04:31.570","Text":"The first 1 will be the sum,"},{"Start":"04:32.240 ","End":"04:37.845","Text":"i goes from 1 to n. I\u0027m going to"},{"Start":"04:37.845 ","End":"04:42.065","Text":"expand this Sigma can be placed in"},{"Start":"04:42.065 ","End":"04:46.100","Text":"to each 1 of them and I can also take constants out to the Sigma,"},{"Start":"04:46.100 ","End":"04:48.035","Text":"so I\u0027ll get a,"},{"Start":"04:48.035 ","End":"04:55.685","Text":"the sum from 1 to n of x_i squared."},{"Start":"04:55.685 ","End":"04:57.380","Text":"Maybe I forgot to mention."},{"Start":"04:57.380 ","End":"04:59.690","Text":"When I\u0027m assigning something to be equal to 0,"},{"Start":"04:59.690 ","End":"05:01.775","Text":"I don\u0027t care about the 2."},{"Start":"05:01.775 ","End":"05:05.250","Text":"0 divided by 2 is still 0."},{"Start":"05:06.820 ","End":"05:11.915","Text":"Just already mentioned that I\u0027m going to make something equals 0."},{"Start":"05:11.915 ","End":"05:21.605","Text":"Next 1 would give me plus b times the sum of 1 over x_i"},{"Start":"05:21.605 ","End":"05:26.930","Text":"squared from 1 to n and the last 1 will be"},{"Start":"05:26.930 ","End":"05:33.650","Text":"just the sum of y_i,"},{"Start":"05:33.650 ","End":"05:38.490","Text":"i goes from 1 to n;"},{"Start":"05:42.010 ","End":"05:46.680","Text":"derivative with respect to b equation."},{"Start":"05:46.720 ","End":"05:50.330","Text":"Again, I\u0027ve forgotten i goes from 1 to n,"},{"Start":"05:50.330 ","End":"05:56.550","Text":"although 1 tends to sometimes omit that when it\u0027s always the same."},{"Start":"06:00.250 ","End":"06:03.680","Text":"Not only am I going to expand like before,"},{"Start":"06:03.680 ","End":"06:13.100","Text":"but I\u0027m going to make each of these equals 0."},{"Start":"06:13.100 ","End":"06:16.700","Text":"Obviously I can drop the 2 if I\u0027m making something equals"},{"Start":"06:16.700 ","End":"06:21.785","Text":"0 and I\u0027m also going to expand the brackets,"},{"Start":"06:21.785 ","End":"06:28.960","Text":"so we get the sum of,"},{"Start":"06:32.330 ","End":"06:34.440","Text":"but pretty adapt to this,"},{"Start":"06:34.440 ","End":"06:39.325","Text":"I\u0027m going to just do several steps in 1."},{"Start":"06:39.325 ","End":"06:43.600","Text":"I\u0027m going to put the Sigma in front of each,"},{"Start":"06:43.600 ","End":"06:51.930","Text":"so I would get x_i squared and I would take the a outside."},{"Start":"06:52.190 ","End":"06:55.685","Text":"I goes from 1 to n,"},{"Start":"06:55.685 ","End":"06:59.870","Text":"but I\u0027m also going to multiply by x_i squared,"},{"Start":"06:59.870 ","End":"07:04.300","Text":"so actually the 2 becomes a 4."},{"Start":"07:04.300 ","End":"07:08.690","Text":"Next, I\u0027m taking the b"},{"Start":"07:08.690 ","End":"07:19.450","Text":"out and the x_i squared with the x_i squared will just cancel."},{"Start":"07:19.730 ","End":"07:24.520","Text":"So I\u0027ll get the sum,"},{"Start":"07:24.650 ","End":"07:33.645","Text":"i goes from 1 to n of 1 and"},{"Start":"07:33.645 ","End":"07:37.185","Text":"here I will get"},{"Start":"07:37.185 ","End":"07:44.220","Text":"minus the sum of just y_i times x_i squared."},{"Start":"07:44.220 ","End":"07:48.820","Text":"I\u0027ll just write it as x_i squared times y_i."},{"Start":"07:49.160 ","End":"07:53.295","Text":"Also from 1 to n,"},{"Start":"07:53.295 ","End":"07:57.400","Text":"and this equals 0."},{"Start":"07:57.890 ","End":"08:00.815","Text":"Just to summarize again what I did here,"},{"Start":"08:00.815 ","End":"08:04.039","Text":"multiplied each of these by x_i squared,"},{"Start":"08:04.039 ","End":"08:10.050","Text":"then split the sum of the 3 parts into 3 separate Sigmas."},{"Start":"08:10.050 ","End":"08:17.940","Text":"I also took constants a and b outside the Sigma,"},{"Start":"08:17.940 ","End":"08:20.385","Text":"that\u0027s the first 1."},{"Start":"08:20.385 ","End":"08:30.730","Text":"Now the second 1, once again the 2 disappears and what I\u0027ll get here is this time I\u0027ll"},{"Start":"08:30.730 ","End":"08:35.890","Text":"have a times the sum i goes from 1"},{"Start":"08:35.890 ","End":"08:42.460","Text":"to n of x_i squared over x_i squared."},{"Start":"08:42.770 ","End":"08:49.215","Text":"X is 1 plus,"},{"Start":"08:49.215 ","End":"08:55.410","Text":"here I\u0027ll get b times the sum 1 to n."},{"Start":"08:55.410 ","End":"09:02.080","Text":"The x_i squared with the x_i squared will give me x_i to the 4."},{"Start":"09:02.990 ","End":"09:09.660","Text":"The last Sigma will be from 1 to n,"},{"Start":"09:09.660 ","End":"09:17.430","Text":"y_i over x_i squared, and equals 0."},{"Start":"09:17.430 ","End":"09:23.500","Text":"So that\u0027s 2 equations in 2 unknowns, a and b."},{"Start":"09:23.630 ","End":"09:27.310","Text":"Now what I have to compute,"},{"Start":"09:27.590 ","End":"09:29.700","Text":"when I say compute, I mean,"},{"Start":"09:29.700 ","End":"09:32.565","Text":"I\u0027m going to do this for our particular points."},{"Start":"09:32.565 ","End":"09:36.045","Text":"I\u0027m going to have this quantity,"},{"Start":"09:36.045 ","End":"09:40.245","Text":"the sum of the x_i to the fourth."},{"Start":"09:40.245 ","End":"09:44.069","Text":"I\u0027m going to need this sum."},{"Start":"09:44.069 ","End":"09:47.955","Text":"I\u0027m going to need this,"},{"Start":"09:47.955 ","End":"09:51.150","Text":"and I\u0027m going to need this."},{"Start":"09:51.150 ","End":"10:00.300","Text":"As for these 2, the sum 1 plus 1 plus 1 plus 1 n times is just n,"},{"Start":"10:00.300 ","End":"10:05.115","Text":"so I can replace this Sigma by n and this Sigma"},{"Start":"10:05.115 ","End":"10:12.550","Text":"also by n. Like here it\u0027s b_n and here it\u0027s a_n."},{"Start":"10:13.490 ","End":"10:24.270","Text":"Now it\u0027s time to introduce our particular points and I\u0027ll copy them."},{"Start":"10:24.270 ","End":"10:28.270","Text":"So I can scroll further."},{"Start":"10:28.640 ","End":"10:33.820","Text":"Here we are, they\u0027re copied just so I can scroll."},{"Start":"10:36.140 ","End":"10:43.515","Text":"Now I want to make a table so I can compute all these quantities."},{"Start":"10:43.515 ","End":"10:46.290","Text":"Note that we have 1, 2, 3, 4,"},{"Start":"10:46.290 ","End":"10:47.430","Text":"we have 5 points,"},{"Start":"10:47.430 ","End":"10:50.625","Text":"so our n is equal to 5."},{"Start":"10:50.625 ","End":"10:58.640","Text":"Here I copied the part of a table from previous clip."},{"Start":"10:58.640 ","End":"11:04.220","Text":"Just need to fill some things in, i goes from 1-5."},{"Start":"11:04.220 ","End":"11:08.900","Text":"So 1, 2, 3, 4, 5."},{"Start":"11:08.900 ","End":"11:15.270","Text":"Here I want to put in the points 4,"},{"Start":"11:15.270 ","End":"11:23.445","Text":"33 and so on, 2, 8.50, 0.5,"},{"Start":"11:23.445 ","End":"11:32.520","Text":"2.3, 1, 4.5, 0.1, 90."},{"Start":"11:32.520 ","End":"11:37.964","Text":"The last row will be for Sigma sum,"},{"Start":"11:37.964 ","End":"11:40.200","Text":"i goes from 1 to n,"},{"Start":"11:40.200 ","End":"11:43.605","Text":"which in our case is 5."},{"Start":"11:43.605 ","End":"11:51.610","Text":"The quantities we need to sum are, x_i to the 4."},{"Start":"11:51.890 ","End":"11:58.720","Text":"I\u0027m going to need x_i squared y_i."},{"Start":"12:01.610 ","End":"12:08.410","Text":"I\u0027m going to need 1 over x_i to the 4."},{"Start":"12:10.820 ","End":"12:19.245","Text":"I\u0027m going to need y_i over x_i squared."},{"Start":"12:19.245 ","End":"12:24.210","Text":"I think since I\u0027ve got x_i squared here and here,"},{"Start":"12:24.210 ","End":"12:32.530","Text":"maybe I\u0027ll put in an extra column for x_i squared,"},{"Start":"12:32.720 ","End":"12:35.460","Text":"even though we don\u0027t need it sum,"},{"Start":"12:35.460 ","End":"12:36.600","Text":"but we don\u0027t need this,"},{"Start":"12:36.600 ","End":"12:37.620","Text":"we don\u0027t need this,"},{"Start":"12:37.620 ","End":"12:38.790","Text":"we don\u0027t need this sum,"},{"Start":"12:38.790 ","End":"12:41.145","Text":"but we need 4 sums here,"},{"Start":"12:41.145 ","End":"12:43.095","Text":"like I highlighted here."},{"Start":"12:43.095 ","End":"12:45.045","Text":"Why don\u0027t I start with this one,"},{"Start":"12:45.045 ","End":"12:47.160","Text":"just take this column and square it."},{"Start":"12:47.160 ","End":"12:49.140","Text":"So I\u0027ve got 16,"},{"Start":"12:49.140 ","End":"12:51.900","Text":"which is 4 squared and then you go 4,"},{"Start":"12:51.900 ","End":"12:55.605","Text":"0.25, 1 squared is 1,"},{"Start":"12:55.605 ","End":"13:00.580","Text":"0.1 squared is 0.01."},{"Start":"13:01.040 ","End":"13:07.275","Text":"Now I could get this column by squaring this column,"},{"Start":"13:07.275 ","End":"13:09.120","Text":"because x_i squared squared."},{"Start":"13:09.120 ","End":"13:13.140","Text":"I\u0027ve got 256 from here,"},{"Start":"13:13.140 ","End":"13:16.185","Text":"4 squared is 16."},{"Start":"13:16.185 ","End":"13:22.780","Text":"A 1/4 squared is a 16th, which is 0.0625."},{"Start":"13:27.710 ","End":"13:30.555","Text":"This squared is 1,"},{"Start":"13:30.555 ","End":"13:37.060","Text":"and this squared is 0.0001."},{"Start":"13:37.060 ","End":"13:43.450","Text":"This column is the reciprocal of this column."},{"Start":"13:44.450 ","End":"13:47.040","Text":"I start with the easier ones."},{"Start":"13:47.040 ","End":"13:51.390","Text":"1 over 1 is 1."},{"Start":"13:51.390 ","End":"13:54.255","Text":"This is a 16th."},{"Start":"13:54.255 ","End":"14:03.705","Text":"So this reciprocal is 16 and this one goes here, 0.0625."},{"Start":"14:03.705 ","End":"14:06.120","Text":"This is 1 over 10,000."},{"Start":"14:06.120 ","End":"14:09.310","Text":"So here I want 10,000,"},{"Start":"14:09.310 ","End":"14:16.760","Text":"and 1 over 256 is 0.0039."},{"Start":"14:16.760 ","End":"14:19.040","Text":"I just took it to 4 decimal places,"},{"Start":"14:19.040 ","End":"14:21.210","Text":"there is a bit more."},{"Start":"14:21.740 ","End":"14:26.670","Text":"Next, let\u0027s do this one,"},{"Start":"14:26.670 ","End":"14:32.910","Text":"which is what I get when I take this column and multiply it by this column."},{"Start":"14:32.910 ","End":"14:40.305","Text":"33 times 16 is 528."},{"Start":"14:40.305 ","End":"14:47.890","Text":"8 and 1/2 times 4 is 34."},{"Start":"14:48.260 ","End":"14:51.014","Text":"For the easy ones,"},{"Start":"14:51.014 ","End":"14:57.225","Text":"1 times 4.5 is 4.5,"},{"Start":"14:57.225 ","End":"15:01.230","Text":"0.01 is 100, times 90,"},{"Start":"15:01.230 ","End":"15:05.685","Text":"so that would give us 90 over a 100, 0.9."},{"Start":"15:05.685 ","End":"15:09.390","Text":"Then, let\u0027s see, 2.3 times a 1/4,"},{"Start":"15:09.390 ","End":"15:12.300","Text":"just have to divide this by 4."},{"Start":"15:12.300 ","End":"15:17.710","Text":"So I\u0027ll get 0.575."},{"Start":"15:22.850 ","End":"15:25.350","Text":"Now I need this column,"},{"Start":"15:25.350 ","End":"15:28.800","Text":"which is this divided by this,"},{"Start":"15:28.800 ","End":"15:33.660","Text":"33 over 16 is 2 and a 16th."},{"Start":"15:33.660 ","End":"15:44.430","Text":"Then our 16th is 0.0625."},{"Start":"15:44.430 ","End":"15:57.690","Text":"Then 8.5 over 4 is 2.125,"},{"Start":"15:57.690 ","End":"16:03.915","Text":"2.3 over this is like 2.3 times 4,"},{"Start":"16:03.915 ","End":"16:05.640","Text":"because it\u0027s a 1/4."},{"Start":"16:05.640 ","End":"16:09.180","Text":"It\u0027s going to be 9.2,"},{"Start":"16:09.180 ","End":"16:14.310","Text":"4.5 over 1 is 4.5,"},{"Start":"16:14.310 ","End":"16:22.690","Text":"and 90 over 100th is 90 times a 100 is, 9,000."},{"Start":"16:23.780 ","End":"16:27.855","Text":"Now let\u0027s do some of the sums, the Sigmas."},{"Start":"16:27.855 ","End":"16:35.985","Text":"256 plus 16 is 272, that\u0027s 273."},{"Start":"16:35.985 ","End":"16:39.960","Text":"Then we have plus 0.0625."},{"Start":"16:39.960 ","End":"16:45.630","Text":"So it\u0027s 0.0626 because of that."},{"Start":"16:45.630 ","End":"16:51.150","Text":"Next one 567."},{"Start":"16:51.150 ","End":"16:53.920","Text":"I\u0027m just going to quote the result."},{"Start":"16:54.440 ","End":"17:00.850","Text":"Here the whole number per 10,017,"},{"Start":"17:03.950 ","End":"17:13.380","Text":"and the decimal part will give 0.0664,"},{"Start":"17:13.380 ","End":"17:16.665","Text":"I\u0027ll write it over here, 0.0664,"},{"Start":"17:16.665 ","End":"17:21.010","Text":"that\u0027s from the 39 plus 625."},{"Start":"17:23.270 ","End":"17:36.300","Text":"Here I get 9,017.8875."},{"Start":"17:36.300 ","End":"17:39.680","Text":"Now that we have all these quantities,"},{"Start":"17:39.680 ","End":"17:46.320","Text":"we can now substitute them in here."},{"Start":"17:47.180 ","End":"17:53.115","Text":"This will give us 2 equations in 2 unknowns, a and b."},{"Start":"17:53.115 ","End":"18:06.734","Text":"This bit is this one here, 273.0626a plus,"},{"Start":"18:06.734 ","End":"18:10.470","Text":"here we have n,"},{"Start":"18:10.470 ","End":"18:13.830","Text":"which is 5 times b,"},{"Start":"18:13.830 ","End":"18:16.595","Text":"this I\u0027ll just put on the other side."},{"Start":"18:16.595 ","End":"18:21.710","Text":"I prefer the 3 term on the right."},{"Start":"18:21.710 ","End":"18:28.470","Text":"This is from this column 567.975."},{"Start":"18:29.360 ","End":"18:34.520","Text":"The next equation is going to be again,"},{"Start":"18:34.520 ","End":"18:40.295","Text":"n is 5, so it\u0027s 5a plus this,"},{"Start":"18:40.295 ","End":"18:45.810","Text":"I take from here, 10,017.0664b,"},{"Start":"18:50.840 ","End":"18:53.660","Text":"is equal to, putting this on"},{"Start":"18:53.660 ","End":"19:07.235","Text":"the other side also, 9,017.8875."},{"Start":"19:07.235 ","End":"19:12.040","Text":"I\u0027m not going to show you the work of solving 2 equations and 2 unknowns,"},{"Start":"19:12.040 ","End":"19:13.689","Text":"I\u0027ll just quote the results,"},{"Start":"19:13.689 ","End":"19:16.315","Text":"I\u0027ll tell you what a and b are."},{"Start":"19:16.315 ","End":"19:22.515","Text":"2.06 and 0.9 Now,"},{"Start":"19:22.515 ","End":"19:30.040","Text":"the question asked us for the curve y equals,"},{"Start":"19:31.540 ","End":"19:35.930","Text":"it was, well, you can tell it from here without scrolling up,"},{"Start":"19:35.930 ","End":"19:39.425","Text":"it was ax squared plus b over x squared."},{"Start":"19:39.425 ","End":"19:43.850","Text":"So it\u0027s 2.06 x"},{"Start":"19:43.850 ","End":"19:52.935","Text":"squared plus 0.9 over x squared."},{"Start":"19:52.935 ","End":"19:58.740","Text":"This is the answer that we\u0027re looking for. Done."}],"ID":9039},{"Watched":false,"Name":"Exercise 7 part e","Duration":"17m 59s","ChapterTopicVideoID":8734,"CourseChapterTopicPlaylistID":4979,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.520","Text":"This is another one of those exercises where we have"},{"Start":"00:03.520 ","End":"00:09.620","Text":"to fit a curve to a set of data points."},{"Start":"00:09.680 ","End":"00:13.330","Text":"In some sense, this is going to be the best curve"},{"Start":"00:13.330 ","End":"00:16.705","Text":"that minimizes the sum of squared errors."},{"Start":"00:16.705 ","End":"00:21.490","Text":"The difference between this one on the previous ones is that here we have 3 parameters,"},{"Start":"00:21.490 ","End":"00:26.645","Text":"a, b, and c and until now we\u0027ve only had 2 parameters."},{"Start":"00:26.645 ","End":"00:30.100","Text":"But the idea is very similar."},{"Start":"00:30.100 ","End":"00:33.355","Text":"We start with a general set of points."},{"Start":"00:33.355 ","End":"00:39.805","Text":"Not our point, but general set x_1, y_1, etc."},{"Start":"00:39.805 ","End":"00:43.130","Text":"General term x_i, y_i."},{"Start":"00:44.370 ","End":"00:49.950","Text":"The last term x_n, y_n."},{"Start":"00:49.950 ","End":"00:51.420","Text":"In our particular case,"},{"Start":"00:51.420 ","End":"00:53.580","Text":"n will be 1, 2, 3, 4, 5."},{"Start":"00:53.580 ","End":"00:55.830","Text":"But we start off in general."},{"Start":"00:55.830 ","End":"01:01.170","Text":"We say that I\u0027m going to take each y_i,"},{"Start":"01:01.170 ","End":"01:03.400","Text":"each one of these y\u0027s,"},{"Start":"01:03.400 ","End":"01:08.630","Text":"and approximate it by a curve of the form this,"},{"Start":"01:08.630 ","End":"01:13.130","Text":"which means that it\u0027s going to be a times x_i squared"},{"Start":"01:13.130 ","End":"01:18.840","Text":"plus bx_i plus c. Now,"},{"Start":"01:18.840 ","End":"01:21.020","Text":"when I\u0027m estimating or approximating or"},{"Start":"01:21.020 ","End":"01:23.915","Text":"curve fitting or whatever there\u0027s going to be an error."},{"Start":"01:23.915 ","End":"01:26.435","Text":"This and this are not the same."},{"Start":"01:26.435 ","End":"01:30.695","Text":"Their difference might be positive in any event,"},{"Start":"01:30.695 ","End":"01:34.055","Text":"what we\u0027re interested in is the square of the error."},{"Start":"01:34.055 ","End":"01:42.005","Text":"The square error for the particular point i is going to be,"},{"Start":"01:42.005 ","End":"01:43.700","Text":"we can subtract them in any order."},{"Start":"01:43.700 ","End":"01:46.820","Text":"I prefer to take the right-hand side,"},{"Start":"01:46.820 ","End":"01:53.455","Text":"ax squared plus bx plus c for our x_i and subtract the y_i,"},{"Start":"01:53.455 ","End":"01:55.934","Text":"and that error squared."},{"Start":"01:55.934 ","End":"01:59.315","Text":"Now, we want to minimize the sum of the squared errors."},{"Start":"01:59.315 ","End":"02:08.040","Text":"The sum of the squared errors is going to be the sum from 1 to n. Of this,"},{"Start":"02:08.040 ","End":"02:11.100","Text":"I just copy-pasted it here."},{"Start":"02:11.100 ","End":"02:15.590","Text":"Now, this is a function of 3 variables, a, b,"},{"Start":"02:15.590 ","End":"02:19.175","Text":"and c. I\u0027m going to call it f of a,"},{"Start":"02:19.175 ","End":"02:23.030","Text":"b, and c because only other quantities will be known."},{"Start":"02:23.030 ","End":"02:24.740","Text":"These will be the particular point,"},{"Start":"02:24.740 ","End":"02:26.165","Text":"in our case these,"},{"Start":"02:26.165 ","End":"02:27.680","Text":"and so a, b,"},{"Start":"02:27.680 ","End":"02:35.699","Text":"and c are the variables and we want to minimize this sum squared errors, abbreviated SSE."},{"Start":"02:36.020 ","End":"02:40.670","Text":"As usual, when you want to minimize something,"},{"Start":"02:40.670 ","End":"02:43.930","Text":"you find its critical point."},{"Start":"02:43.930 ","End":"02:46.420","Text":"In our case, that\u0027s all we\u0027re going to do."},{"Start":"02:46.420 ","End":"02:48.500","Text":"We\u0027re going to assume that the critical point that we"},{"Start":"02:48.500 ","End":"02:51.380","Text":"find is in fact the minimum in the exercise book."},{"Start":"02:51.380 ","End":"02:54.170","Text":"I believe there was an instruction to make that assumption."},{"Start":"02:54.170 ","End":"02:59.660","Text":"Let\u0027s just take the partial derivatives and set them all to 0."},{"Start":"02:59.660 ","End":"03:03.800","Text":"In other words, fa and fb and fc."},{"Start":"03:03.800 ","End":"03:07.175","Text":"Partial derivatives are going to all be 0."},{"Start":"03:07.175 ","End":"03:10.070","Text":"Let\u0027s see what that gives us."},{"Start":"03:10.070 ","End":"03:13.430","Text":"I moved that bit over here to get more space here."},{"Start":"03:13.430 ","End":"03:18.300","Text":"Let me just add for critical point or more specifically for a minimum."},{"Start":"03:20.360 ","End":"03:24.340","Text":"Let\u0027s see what is f with respect to a we\u0027ll do"},{"Start":"03:24.340 ","End":"03:29.215","Text":"each of the 3 partial derivatives separately."},{"Start":"03:29.215 ","End":"03:32.685","Text":"Here, we have the sum,"},{"Start":"03:32.685 ","End":"03:35.050","Text":"so we can put the derivative inside"},{"Start":"03:35.050 ","End":"03:43.630","Text":"the sum and go from 1 to n. The derivative of this with respect to a."},{"Start":"03:43.630 ","End":"03:48.550","Text":"Now, something squared means twice that something but the 2 as we did before,"},{"Start":"03:48.550 ","End":"03:51.835","Text":"we pull out in front of the summation."},{"Start":"03:51.835 ","End":"03:55.570","Text":"We get twice of this thing,"},{"Start":"03:55.570 ","End":"03:56.995","Text":"I\u0027ll just copy it,"},{"Start":"03:56.995 ","End":"04:02.635","Text":"ax_i squared plus bx_i plus c minus y_i,"},{"Start":"04:02.635 ","End":"04:04.790","Text":"and then times the inner derivative."},{"Start":"04:04.790 ","End":"04:07.575","Text":"But inner derivative with respect to a,"},{"Start":"04:07.575 ","End":"04:14.925","Text":"a only appears here so it\u0027s the coefficient of a which is x_i squared."},{"Start":"04:14.925 ","End":"04:18.770","Text":"That\u0027s one partial derivative."},{"Start":"04:18.770 ","End":"04:22.680","Text":"The second one with respect to b."},{"Start":"04:22.780 ","End":"04:27.085","Text":"The beginning is going to be practically the same,"},{"Start":"04:27.085 ","End":"04:29.925","Text":"and in fact, it will be the same for fc,"},{"Start":"04:29.925 ","End":"04:39.435","Text":"I just did a copy-paste from here."},{"Start":"04:39.435 ","End":"04:43.185","Text":"Because the twice this thing,"},{"Start":"04:43.185 ","End":"04:47.150","Text":"the error, it\u0027s going to be the same."},{"Start":"04:47.150 ","End":"04:52.215","Text":"The only difference is that the inner derivative will be different."},{"Start":"04:52.215 ","End":"04:56.865","Text":"But with respect to a, I get x_i squared with respect to b,"},{"Start":"04:56.865 ","End":"05:02.590","Text":"here\u0027s b, the inner derivative is the coefficient of b is x_i."},{"Start":"05:02.600 ","End":"05:07.970","Text":"In the third one, the derivative with respect to c is just 1,"},{"Start":"05:07.970 ","End":"05:10.970","Text":"I\u0027ll write times 1, even though I don\u0027t need to."},{"Start":"05:10.970 ","End":"05:16.105","Text":"Now, we want to set each of these to be 0."},{"Start":"05:16.105 ","End":"05:18.300","Text":"Now, I\u0027m going to do 3 things at once."},{"Start":"05:18.300 ","End":"05:20.150","Text":"First of all, throw out the 2,"},{"Start":"05:20.150 ","End":"05:21.965","Text":"because when I\u0027m comparing to 0,"},{"Start":"05:21.965 ","End":"05:24.170","Text":"a factor of 2 makes no difference."},{"Start":"05:24.170 ","End":"05:31.155","Text":"Secondly, I\u0027m going to put the Sigma on each of the terms separately."},{"Start":"05:31.155 ","End":"05:37.080","Text":"Before that I\u0027m going to multiply by x_i squared here,"},{"Start":"05:37.080 ","End":"05:39.385","Text":"by x_i here, and so on."},{"Start":"05:39.385 ","End":"05:45.380","Text":"What we get when we do all that is we get 4 separate Sigmas, and without the 2,"},{"Start":"05:45.380 ","End":"05:53.050","Text":"the first Sigma will be ax_i^4."},{"Start":"05:53.050 ","End":"05:55.085","Text":"Because if I multiply this by ax_i squared,"},{"Start":"05:55.085 ","End":"06:01.460","Text":"the a comes out in front and the x_i^4 is here,"},{"Start":"06:01.460 ","End":"06:08.850","Text":"and i goes from 1 to n. The next one will be x_i times x_i squared is x_i cubed."},{"Start":"06:08.850 ","End":"06:15.985","Text":"The b comes out in front and we\u0027ve got the sum from 1 to n of x_i cubed."},{"Start":"06:15.985 ","End":"06:19.274","Text":"The next one we\u0027ll get cx_i squared,"},{"Start":"06:19.274 ","End":"06:26.340","Text":"but the c comes out in front of the Sigma and we just have x_i squared."},{"Start":"06:26.340 ","End":"06:36.330","Text":"The last one will be y_i times x_i squared, also Sigma."},{"Start":"06:36.330 ","End":"06:41.250","Text":"I\u0027ll just write the x_i squared before the y_i,"},{"Start":"06:41.250 ","End":"06:42.635","Text":"i goes from 1 to the n,"},{"Start":"06:42.635 ","End":"06:45.845","Text":"and this of course equals 0."},{"Start":"06:45.845 ","End":"06:48.185","Text":"Let\u0027s see what happens with the next."},{"Start":"06:48.185 ","End":"06:51.635","Text":"Once again, we throw out the 2 because we\u0027re comparing to 0."},{"Start":"06:51.635 ","End":"06:55.030","Text":"We apply the Sigma to each bit separately."},{"Start":"06:55.030 ","End":"06:59.705","Text":"We get, it\u0027s going to be very similar to this one,"},{"Start":"06:59.705 ","End":"07:02.525","Text":"but because it\u0027s x_i instead of x_i squared,"},{"Start":"07:02.525 ","End":"07:04.685","Text":"the powers are going to be 1 lower."},{"Start":"07:04.685 ","End":"07:09.890","Text":"It\u0027s going to have a times the sum of x_i cubed the only,"},{"Start":"07:09.890 ","End":"07:14.430","Text":"b with x_i squared sum,"},{"Start":"07:14.430 ","End":"07:24.810","Text":"c with the sum of x_i and minus the sum of x_i, y_i equals 0."},{"Start":"07:24.810 ","End":"07:30.885","Text":"I just have to write the index range from 1 to n,"},{"Start":"07:30.885 ","End":"07:37.505","Text":"i from 1 to n. The last one,"},{"Start":"07:37.505 ","End":"07:40.100","Text":"we just take each term as is,"},{"Start":"07:40.100 ","End":"07:45.035","Text":"but we put a Sigma on each and take any constant out front."},{"Start":"07:45.035 ","End":"07:54.360","Text":"We get a times the sum of x_i squared b times the sum of x_i."},{"Start":"07:54.360 ","End":"07:58.020","Text":"If I take c out,"},{"Start":"07:58.020 ","End":"08:00.870","Text":"I\u0027ll just get the sum of 1,"},{"Start":"08:00.870 ","End":"08:10.575","Text":"and then minus the sum of y_i equals 0."},{"Start":"08:10.575 ","End":"08:20.650","Text":"Again, we just have to write the index i goes from 1 to n on each one of them."},{"Start":"08:23.010 ","End":"08:29.695","Text":"In a moment, I\u0027ll be substituting the x_i\u0027s and the y_i\u0027s from our data,"},{"Start":"08:29.695 ","End":"08:33.370","Text":"but let\u0027s see which quantities we need to compute. Some of them repeat."},{"Start":"08:33.370 ","End":"08:36.940","Text":"We\u0027ll need to compute the sum of x_i to the 4th,"},{"Start":"08:36.940 ","End":"08:43.030","Text":"we\u0027ll also need to compute the sum of x_i cubed and the sum of x_i squared."},{"Start":"08:43.030 ","End":"08:46.790","Text":"We\u0027ll also need to compute this sum."},{"Start":"08:47.610 ","End":"08:51.505","Text":"From here, this is already the same as this,"},{"Start":"08:51.505 ","End":"08:54.040","Text":"and this is the same as this,"},{"Start":"08:54.040 ","End":"08:58.315","Text":"but we\u0027ll need the sum of the x_i,"},{"Start":"08:58.315 ","End":"09:04.240","Text":"we\u0027ll also need the sum of x_i, y_i."},{"Start":"09:04.240 ","End":"09:08.080","Text":"This is already covered here."},{"Start":"09:08.080 ","End":"09:10.300","Text":"This is covered here."},{"Start":"09:10.300 ","End":"09:13.720","Text":"We\u0027ll need the sum of y_i."},{"Start":"09:13.720 ","End":"09:17.600","Text":"There is one more, this one here."},{"Start":"09:17.760 ","End":"09:22.720","Text":"The sum from 1 to n of the constant 1 is just 1 plus 1,"},{"Start":"09:22.720 ","End":"09:25.330","Text":"plus 1 and so on n times."},{"Start":"09:25.330 ","End":"09:30.475","Text":"This thing is exactly equal to n. Now,"},{"Start":"09:30.475 ","End":"09:35.410","Text":"let\u0027s see what happens with our set of points."},{"Start":"09:35.410 ","End":"09:37.630","Text":"I don\u0027t want to lose them when I scroll,"},{"Start":"09:37.630 ","End":"09:40.375","Text":"I\u0027m going to copy them down here 1,"},{"Start":"09:40.375 ","End":"09:45.390","Text":"4.5 and all the rest of them."},{"Start":"09:45.390 ","End":"09:48.195","Text":"I also want to remember so I don\u0027t have to scroll back up,"},{"Start":"09:48.195 ","End":"09:57.010","Text":"that we\u0027re looking for y equals ax squared plus bx plus c. After we found a, b,"},{"Start":"09:57.010 ","End":"10:02.830","Text":"and c that minimizes the sum of squared errors,"},{"Start":"10:02.830 ","End":"10:06.055","Text":"then we can substitute them here and get the answer."},{"Start":"10:06.055 ","End":"10:08.960","Text":"Now, I\u0027m going to scroll a bit."},{"Start":"10:11.610 ","End":"10:15.290","Text":"I\u0027m going to put in a table."},{"Start":"10:15.540 ","End":"10:23.469","Text":"Here we are. I borrowed a table from a previous exercise and just erased the entries."},{"Start":"10:23.469 ","End":"10:25.540","Text":"Remember in our case,"},{"Start":"10:25.540 ","End":"10:28.270","Text":"n was equal to 5."},{"Start":"10:28.270 ","End":"10:30.445","Text":"1, 2, 3, 4, 5,"},{"Start":"10:30.445 ","End":"10:34.480","Text":"that\u0027s why I\u0027ve got 5 rows here."},{"Start":"10:34.480 ","End":"10:37.000","Text":"Now, let\u0027s get to the columns."},{"Start":"10:37.000 ","End":"10:42.265","Text":"I already have the x_i and the y_i,"},{"Start":"10:42.265 ","End":"10:47.335","Text":"what I\u0027ll need is x_i squared."},{"Start":"10:47.335 ","End":"10:50.460","Text":"I\u0027ll need x_i cubed,"},{"Start":"10:50.460 ","End":"10:53.730","Text":"we\u0027ll need x_i to the 4th."},{"Start":"10:53.730 ","End":"11:00.340","Text":"We need x_i, y_i,"},{"Start":"11:00.800 ","End":"11:07.880","Text":"and we need x_i squared times y_i."},{"Start":"11:07.880 ","End":"11:11.920","Text":"The points themselves are the first thing I put in."},{"Start":"11:11.920 ","End":"11:20.900","Text":"I have 1,4.5 and then 0.5,2.3,"},{"Start":"11:21.180 ","End":"11:25.990","Text":"0,0.8 minus 1, 0.1,"},{"Start":"11:25.990 ","End":"11:29.840","Text":"and finally minus 0.5,12."},{"Start":"11:30.090 ","End":"11:35.360","Text":"The last row is for the sums of the columns."},{"Start":"11:35.760 ","End":"11:40.405","Text":"X _i squared that go along 1 squared,"},{"Start":"11:40.405 ","End":"11:44.995","Text":"0.5 squared, 0 squared minus 1 squared is 1,"},{"Start":"11:44.995 ","End":"11:48.385","Text":"and this is also 0.25 when squared."},{"Start":"11:48.385 ","End":"11:52.510","Text":"Now, cubed, 1 cubed is 1,"},{"Start":"11:52.510 ","End":"11:54.655","Text":"1/2 cubed is an 8th,"},{"Start":"11:54.655 ","End":"12:05.320","Text":"in decimal, 0 cubed minus 1 cubed minus 1/2 cubed is just going to be minus 0.125."},{"Start":"12:05.320 ","End":"12:10.750","Text":"Now, to the 4th, all I have to do is square this column,"},{"Start":"12:10.750 ","End":"12:12.925","Text":"for example, squared, squared."},{"Start":"12:12.925 ","End":"12:15.370","Text":"1 squared is 1, 1/4,"},{"Start":"12:15.370 ","End":"12:17.035","Text":"squared is a 16th,"},{"Start":"12:17.035 ","End":"12:28.720","Text":"that is 0.062501 and also 0.0625."},{"Start":"12:28.720 ","End":"12:32.545","Text":"X_i, y_i is the product of this and this,"},{"Start":"12:32.545 ","End":"12:39.410","Text":"1 times 4.5,1/2 times 2.3 would make it 1.15,"},{"Start":"12:40.440 ","End":"12:48.115","Text":"0 times something is 0 minus 1 times this is minus 0.1,"},{"Start":"12:48.115 ","End":"12:51.895","Text":"minus 1/2 times 12 is minus 6."},{"Start":"12:51.895 ","End":"13:00.475","Text":"Then x_i squared y_i is product of these 2 columns, 4.5 here."},{"Start":"13:00.475 ","End":"13:05.680","Text":"This times this would give me,"},{"Start":"13:05.680 ","End":"13:09.505","Text":"it\u0027s a quarter of 2.3, quarter of 2,"},{"Start":"13:09.505 ","End":"13:12.370","Text":"and a quarter of 3 is 0.575,"},{"Start":"13:12.370 ","End":"13:13.960","Text":"this is what I make it."},{"Start":"13:13.960 ","End":"13:17.095","Text":"This times this is 0,"},{"Start":"13:17.095 ","End":"13:24.710","Text":"this times this is 0.1,12 times a 1/4 is 3."},{"Start":"13:24.810 ","End":"13:27.235","Text":"That\u0027s the columns."},{"Start":"13:27.235 ","End":"13:30.035","Text":"We just have to sum them now."},{"Start":"13:30.035 ","End":"13:34.050","Text":"This column 1 cancels with minus 1.5,"},{"Start":"13:34.050 ","End":"13:37.854","Text":"with minus 0.5, it just comes out to be 0."},{"Start":"13:37.854 ","End":"13:40.650","Text":"Next column."},{"Start":"13:41.110 ","End":"13:44.200","Text":"I just noticed I made a copying error."},{"Start":"13:44.200 ","End":"13:48.460","Text":"This is 0.12, I miscopied."},{"Start":"13:48.460 ","End":"13:50.245","Text":"I\u0027m sorry. Not too late though."},{"Start":"13:50.245 ","End":"13:59.980","Text":"This sum comes out to be 7.82 this column 1 and 1 is 2,"},{"Start":"13:59.980 ","End":"14:03.400","Text":"this and this is 1/2, so a 2.5."},{"Start":"14:03.400 ","End":"14:06.340","Text":"Next column, this cancels out with this,"},{"Start":"14:06.340 ","End":"14:10.420","Text":"this cancels out with this, that\u0027s 0."},{"Start":"14:10.420 ","End":"14:13.225","Text":"1 and 1 is 2."},{"Start":"14:13.225 ","End":"14:20.870","Text":"This, and this is 0.125, that\u0027s this sum."},{"Start":"14:21.900 ","End":"14:24.895","Text":"When I corrected this number,"},{"Start":"14:24.895 ","End":"14:28.885","Text":"I also have to correct this and this because I\u0027m using them."},{"Start":"14:28.885 ","End":"14:36.560","Text":"This one is minus 1/2 times 0.12 is minus 0.06."},{"Start":"14:38.760 ","End":"14:46.010","Text":"This one times this one is 0.03."},{"Start":"14:46.620 ","End":"14:49.975","Text":"The total of this,"},{"Start":"14:49.975 ","End":"14:54.730","Text":"comes out to 5.49."},{"Start":"14:54.730 ","End":"15:03.430","Text":"Finally, this is 5.205 when added up."},{"Start":"15:03.430 ","End":"15:07.135","Text":"Now, we have all these numbers here."},{"Start":"15:07.135 ","End":"15:10.060","Text":"Now, I substitute in here,"},{"Start":"15:10.060 ","End":"15:14.260","Text":"and we get 3 equations and 3 unknowns,"},{"Start":"15:14.260 ","End":"15:21.655","Text":"a, b, and c. This value I take from here."},{"Start":"15:21.655 ","End":"15:30.490","Text":"It\u0027s 2.125a, next one from this column, it\u0027s a 0,"},{"Start":"15:30.490 ","End":"15:37.419","Text":"but I\u0027ll write it in 0b and then for c,"},{"Start":"15:37.419 ","End":"15:45.470","Text":"I want this column 2.5c."},{"Start":"15:46.500 ","End":"15:49.930","Text":"Instead of a minus equals 0,"},{"Start":"15:49.930 ","End":"15:52.945","Text":"I\u0027d rather bring it over to the other side as a plus."},{"Start":"15:52.945 ","End":"15:58.760","Text":"This one is here, it\u0027s 5.205."},{"Start":"16:01.470 ","End":"16:04.225","Text":"Next one."},{"Start":"16:04.225 ","End":"16:06.580","Text":"Let\u0027s get the right columns."},{"Start":"16:06.580 ","End":"16:09.920","Text":"The x_i cubed column is 0a,"},{"Start":"16:11.100 ","End":"16:16.100","Text":"for b it\u0027s 2.5,"},{"Start":"16:16.170 ","End":"16:21.010","Text":"c is from this column,"},{"Start":"16:21.010 ","End":"16:30.110","Text":"it\u0027s 0c and it equals 5.49."},{"Start":"16:31.350 ","End":"16:35.930","Text":"The last one is 2.5a,"},{"Start":"16:36.240 ","End":"16:39.940","Text":"sum of x_i is 0,"},{"Start":"16:39.940 ","End":"16:45.640","Text":"b_n is 5, so this is 5."},{"Start":"16:45.640 ","End":"16:48.460","Text":"See this n here is 5."},{"Start":"16:48.460 ","End":"16:54.290","Text":"Then it equals sum of y_i, 7.82."},{"Start":"16:55.470 ","End":"17:00.210","Text":"I\u0027m not going to show you the calculations for this."},{"Start":"17:01.510 ","End":"17:04.475","Text":"It doesn\u0027t belong here."},{"Start":"17:04.475 ","End":"17:07.175","Text":"I\u0027m assuming you know how to solve such systems."},{"Start":"17:07.175 ","End":"17:13.560","Text":"I\u0027ll just give you the answer that a comes out to be 1.48,"},{"Start":"17:13.560 ","End":"17:18.520","Text":"b comes out 2.196,"},{"Start":"17:18.520 ","End":"17:22.880","Text":"and c comes out 0.824."},{"Start":"17:25.260 ","End":"17:28.550","Text":"Just don\u0027t forget to answer the question."},{"Start":"17:28.550 ","End":"17:32.100","Text":"Remember the question wanted a curve of this form."},{"Start":"17:32.100 ","End":"17:41.590","Text":"The answer is the parabola y equals 1.48x"},{"Start":"17:41.590 ","End":"17:57.400","Text":"squared plus 2.196x plus 0.824."},{"Start":"17:57.400 ","End":"18:00.800","Text":"This is our answer and we\u0027re done."}],"ID":9040}],"Thumbnail":null,"ID":4979}]

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1.1

1