[{"Name":"Practice Questions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"14m 23s","ChapterTopicVideoID":9781,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9781.jpeg","UploadDate":"2022-01-13T07:09:43.5070000","DurationForVideoObject":"PT14M23S","Description":null,"MetaTitle":"Exercise 1 - Practice Questions: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on Practice Questions practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/constrained-extrema/practice-questions/vid9666","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this clip, I\u0027m going to solve an example of a constrained extrema problem,"},{"Start":"00:06.000 ","End":"00:10.335","Text":"called it number 1, because I think there will be more examples after this."},{"Start":"00:10.335 ","End":"00:13.004","Text":"My example is as follows."},{"Start":"00:13.004 ","End":"00:16.215","Text":"We want to find the maximum."},{"Start":"00:16.215 ","End":"00:19.740","Text":"It\u0027s either maximum or minimum, always,"},{"Start":"00:19.740 ","End":"00:25.755","Text":"this time of the function natural log of x plus natural log of y."},{"Start":"00:25.755 ","End":"00:27.900","Text":"We need a constraint,"},{"Start":"00:27.900 ","End":"00:31.800","Text":"we need a subject to or such that,"},{"Start":"00:31.800 ","End":"00:39.780","Text":"and we have the y plus 2x is equal to 8,"},{"Start":"00:39.780 ","End":"00:42.085","Text":"this is the constraint."},{"Start":"00:42.085 ","End":"00:49.100","Text":"They\u0027ve also given us a bit of an extra to solve, and that\u0027s here."},{"Start":"00:49.100 ","End":"00:52.160","Text":"That is to show that the actual maximum value that we"},{"Start":"00:52.160 ","End":"00:57.720","Text":"obtain at that point is natural log of 8."},{"Start":"00:58.130 ","End":"01:00.755","Text":"Let\u0027s get started."},{"Start":"01:00.755 ","End":"01:04.280","Text":"First thing we do is identify 2 functions."},{"Start":"01:04.280 ","End":"01:09.065","Text":"We have an objective function and a constraint function."},{"Start":"01:09.065 ","End":"01:14.660","Text":"The objective is the one who\u0027s extremum we have to find, maximum or minimum."},{"Start":"01:14.660 ","End":"01:19.490","Text":"In this case, what\u0027s in the curly brackets here is our objective function."},{"Start":"01:19.490 ","End":"01:22.710","Text":"We\u0027ll use the letter f for it."},{"Start":"01:22.710 ","End":"01:25.535","Text":"F and some 2 variables, x, y,"},{"Start":"01:25.535 ","End":"01:31.265","Text":"that\u0027s going to be the natural log of x plus natural log of y,"},{"Start":"01:31.265 ","End":"01:36.545","Text":"objective or target, this is what it is."},{"Start":"01:36.545 ","End":"01:39.620","Text":"The constraint function is pretty"},{"Start":"01:39.620 ","End":"01:43.820","Text":"much what\u0027s written here except that we want it to be a function."},{"Start":"01:43.820 ","End":"01:49.100","Text":"We just put everything on the left-hand side and leave 0 on the right-hand side."},{"Start":"01:49.100 ","End":"01:51.350","Text":"We call that the constraint function,"},{"Start":"01:51.350 ","End":"01:57.310","Text":"are use letter G. This is going to be y plus 2x minus 8."},{"Start":"01:58.880 ","End":"02:02.069","Text":"That\u0027s like step 0."},{"Start":"02:02.069 ","End":"02:05.785","Text":"Step 1, what we do is find"},{"Start":"02:05.785 ","End":"02:13.525","Text":"all partial derivatives of both these functions of the first-order and second-order."},{"Start":"02:13.525 ","End":"02:15.640","Text":"There should be 5 each."},{"Start":"02:15.640 ","End":"02:22.170","Text":"There will be 2 first-order derivatives and 3 second-order derivatives."},{"Start":"02:22.170 ","End":"02:30.785","Text":"Let\u0027s start, begin with f. F with respect to x is equal to,"},{"Start":"02:30.785 ","End":"02:36.205","Text":"y is a constant and derivative of natural log is 1 over,"},{"Start":"02:36.205 ","End":"02:44.570","Text":"so it\u0027s 1 over x and f with respect to y."},{"Start":"02:46.410 ","End":"02:54.700","Text":"Same thing, 1 over y. Second-order derivatives."},{"Start":"02:54.710 ","End":"02:59.475","Text":"We have twice with respect to x,"},{"Start":"02:59.475 ","End":"03:05.185","Text":"we just take this 1 and take the partial with respect to x."},{"Start":"03:05.185 ","End":"03:08.960","Text":"We get minus 1 over x squared."},{"Start":"03:10.080 ","End":"03:16.285","Text":"F_yy. Second derivative with respect to y twice,"},{"Start":"03:16.285 ","End":"03:22.280","Text":"similarly minus 1 over y squared."},{"Start":"03:22.820 ","End":"03:25.110","Text":"Theoretically there\u0027s 2 more,"},{"Start":"03:25.110 ","End":"03:27.330","Text":"there\u0027s f_xy and f_yx."},{"Start":"03:27.330 ","End":"03:30.580","Text":"But we already mentioned that there\u0027s a theory on that,"},{"Start":"03:30.580 ","End":"03:32.155","Text":"in almost all cases,"},{"Start":"03:32.155 ","End":"03:34.000","Text":"at least in the cases that we encounter,"},{"Start":"03:34.000 ","End":"03:35.320","Text":"there will be the same thing."},{"Start":"03:35.320 ","End":"03:38.785","Text":"I\u0027ll call it f_xy and it will equal f_ yx."},{"Start":"03:38.785 ","End":"03:41.620","Text":"You can check for yourself if I differentiate this with"},{"Start":"03:41.620 ","End":"03:44.500","Text":"respect to x or this with respect to y,"},{"Start":"03:44.500 ","End":"03:47.140","Text":"I\u0027ll get the same thing and it happens to be 0,"},{"Start":"03:47.140 ","End":"03:52.840","Text":"because here x is a constant if I do with respect to y and similarly,"},{"Start":"03:52.840 ","End":"03:54.190","Text":"here x is a constant,"},{"Start":"03:54.190 ","End":"03:56.575","Text":"this is just equal to 0."},{"Start":"03:56.575 ","End":"04:06.489","Text":"That\u0027s just 1/2 of it because I also need the same thing for g. G with respect to x,"},{"Start":"04:07.190 ","End":"04:10.485","Text":"only x is not a constant."},{"Start":"04:10.485 ","End":"04:17.795","Text":"This is just 2, and g with respect to y,"},{"Start":"04:17.795 ","End":"04:22.590","Text":"that\u0027s just going to be 1 because the x is constant."},{"Start":"04:22.610 ","End":"04:27.210","Text":"As for the second-order derivatives,"},{"Start":"04:27.210 ","End":"04:29.760","Text":"0 will be going to be 0 because this is a constant."},{"Start":"04:29.760 ","End":"04:31.950","Text":"I\u0027ll just make a note of that,"},{"Start":"04:31.950 ","End":"04:37.620","Text":"that g_xx and g_yy"},{"Start":"04:37.620 ","End":"04:44.160","Text":"and g_xy are all equal to 0."},{"Start":"04:44.160 ","End":"04:47.080","Text":"That\u0027s the first step."},{"Start":"04:47.450 ","End":"04:51.600","Text":"On to step 2,"},{"Start":"04:51.600 ","End":"04:54.660","Text":"also I have some more space here."},{"Start":"04:54.660 ","End":"04:59.460","Text":"Step 2, it\u0027s always the same."},{"Start":"04:59.460 ","End":"05:03.150","Text":"It\u0027s always 3 equations and 3 unknowns,"},{"Start":"05:03.150 ","End":"05:05.730","Text":"x y, and a new unknown Lambda."},{"Start":"05:05.730 ","End":"05:11.780","Text":"The first 2 equations that the first derivative f with respect to"},{"Start":"05:11.780 ","End":"05:17.960","Text":"x is Lambda times first derivative of g with respect to x."},{"Start":"05:17.960 ","End":"05:21.200","Text":"The second equation is the same thing,"},{"Start":"05:21.200 ","End":"05:23.510","Text":"but with y instead of x."},{"Start":"05:23.510 ","End":"05:31.385","Text":"So we have f with respect to y is Lambda g with respect to y."},{"Start":"05:31.385 ","End":"05:37.835","Text":"The third equation is just the constraint function equaling 0."},{"Start":"05:37.835 ","End":"05:42.705","Text":"We have that, the constraint function was,"},{"Start":"05:42.705 ","End":"05:48.320","Text":"if we forget and we go back up and check and see that the constraint function was this."},{"Start":"05:48.320 ","End":"05:50.300","Text":"We let this equal 0,"},{"Start":"05:50.300 ","End":"05:51.440","Text":"or if we prefer,"},{"Start":"05:51.440 ","End":"05:54.890","Text":"we can even put the constraint in its original form."},{"Start":"05:54.890 ","End":"05:57.140","Text":"I generally like to use this original form,"},{"Start":"05:57.140 ","End":"05:59.090","Text":"y plus 2x is 8,"},{"Start":"05:59.090 ","End":"06:02.820","Text":"or this equals 0 either way."},{"Start":"06:03.200 ","End":"06:09.690","Text":"I\u0027ll just write it as y plus 2x equals 8."},{"Start":"06:09.690 ","End":"06:15.150","Text":"That gives us 3 equations and 3 unknowns,"},{"Start":"06:15.150 ","End":"06:18.495","Text":"x y, and lambda."},{"Start":"06:18.495 ","End":"06:20.960","Text":"We always solve them the same way."},{"Start":"06:20.960 ","End":"06:26.750","Text":"That way is, that we take the first 2 equations and we divide one by the other,"},{"Start":"06:26.750 ","End":"06:28.900","Text":"and that gets rid of Lambda."},{"Start":"06:28.900 ","End":"06:32.055","Text":"We have 2 equations and 2 unknowns, x and y."},{"Start":"06:32.055 ","End":"06:35.205","Text":"But I first of all plug in what I know."},{"Start":"06:35.205 ","End":"06:42.690","Text":"These become f with respect to x is 1 over x is equal to Lambda,"},{"Start":"06:42.690 ","End":"06:47.440","Text":"and then g_x is 2."},{"Start":"06:49.820 ","End":"06:53.055","Text":"The next 1, f with respect to y,"},{"Start":"06:53.055 ","End":"07:02.435","Text":"1 over y equals lambda times g_y is 1."},{"Start":"07:02.435 ","End":"07:06.655","Text":"The last 1 just as is."},{"Start":"07:06.655 ","End":"07:09.550","Text":"What I was saying, we divide the first 2,"},{"Start":"07:09.550 ","End":"07:11.920","Text":"1 by the other, it doesn\u0027t matter which way."},{"Start":"07:11.920 ","End":"07:13.780","Text":"I\u0027ll take that first over the second."},{"Start":"07:13.780 ","End":"07:19.615","Text":"1 over x over 1 over y is equal"},{"Start":"07:19.615 ","End":"07:27.135","Text":"to Lambda times 2 over Lambda times 1."},{"Start":"07:27.135 ","End":"07:36.880","Text":"The lambdas cancel, 2 over 1 is 2 and 1 over x over 1 over y is just y over x."},{"Start":"07:36.880 ","End":"07:44.800","Text":"If you think about it, you\u0027ll see that this gives us that y over x is equal to 2."},{"Start":"07:44.800 ","End":"07:49.345","Text":"So we have 1 equation that y over x is 2,"},{"Start":"07:49.345 ","End":"07:53.035","Text":"and the other equation is simply the constraint."},{"Start":"07:53.035 ","End":"07:55.300","Text":"So this is 2 equations in 2 unknowns,"},{"Start":"07:55.300 ","End":"07:58.090","Text":"x and y, we know how to solve that."},{"Start":"07:58.090 ","End":"08:04.900","Text":"I think I should put x to the other side and get that y equals 2x,"},{"Start":"08:04.900 ","End":"08:07.795","Text":"and this looks good because 2x here."},{"Start":"08:07.795 ","End":"08:14.350","Text":"So I could just say that y plus y equals 8,"},{"Start":"08:14.350 ","End":"08:19.450","Text":"or I could have put 2x plus 2x equals 8, it doesn\u0027t matter."},{"Start":"08:19.450 ","End":"08:21.160","Text":"From here 2y is 8,"},{"Start":"08:21.160 ","End":"08:24.550","Text":"so y equals 4,"},{"Start":"08:24.550 ","End":"08:28.885","Text":"and if y equals 4 and y is 2x,"},{"Start":"08:28.885 ","End":"08:31.405","Text":"so 2x is equal to 4,"},{"Start":"08:31.405 ","End":"08:33.835","Text":"so x is equal to 2."},{"Start":"08:33.835 ","End":"08:36.310","Text":"Y equals 4, x equals 2,"},{"Start":"08:36.310 ","End":"08:40.870","Text":"so that gives us both x and y."},{"Start":"08:40.870 ","End":"08:43.915","Text":"We will need Lambda later on."},{"Start":"08:43.915 ","End":"08:46.885","Text":"Let\u0027s do that now, you could take 1 of these."},{"Start":"08:46.885 ","End":"08:51.535","Text":"Here\u0027s a good 1. Here just says Lambda\u0027s equal to 1 over y."},{"Start":"08:51.535 ","End":"08:55.315","Text":"If Lambda is 1 over y and y is 4,"},{"Start":"08:55.315 ","End":"08:59.634","Text":"then Lambda equals 1 quarter."},{"Start":"08:59.634 ","End":"09:02.380","Text":"Sometimes the x, y,"},{"Start":"09:02.380 ","End":"09:07.360","Text":"and Lambda from the extremum I denote them with an asterisk so not just any old x,"},{"Start":"09:07.360 ","End":"09:09.190","Text":"y, and Lambda, they\u0027re v,"},{"Start":"09:09.190 ","End":"09:11.290","Text":"x, y, and Lambda that we want."},{"Start":"09:11.290 ","End":"09:13.970","Text":"I\u0027ll just put asterisks on them."},{"Start":"09:14.280 ","End":"09:18.445","Text":"That actually completes step 2."},{"Start":"09:18.445 ","End":"09:22.225","Text":"Step 3 will be to tell us if our suspect."},{"Start":"09:22.225 ","End":"09:28.165","Text":"The suspect is where x is 2 and y is 4."},{"Start":"09:28.165 ","End":"09:34.500","Text":"So far this is just a suspect for an extremum and I\u0027ll highlight this 1 as well."},{"Start":"09:34.500 ","End":"09:37.170","Text":"What we have to find out, and let\u0027s go back up,"},{"Start":"09:37.170 ","End":"09:41.130","Text":"and what we see here is that we need a maximum."},{"Start":"09:41.130 ","End":"09:44.150","Text":"Now I have a suspect for an extremum,"},{"Start":"09:44.150 ","End":"09:48.729","Text":"and I don\u0027t know if it\u0027s a maximum or minimum or possibly neither,"},{"Start":"09:48.729 ","End":"09:52.375","Text":"and step 3 is going to help us determine this."},{"Start":"09:52.375 ","End":"09:55.975","Text":"Remember we had the strange expression H,"},{"Start":"09:55.975 ","End":"09:58.370","Text":"well I\u0027ll write it again."},{"Start":"09:58.470 ","End":"10:02.170","Text":"Here\u0027s the expression H,"},{"Start":"10:02.170 ","End":"10:04.495","Text":"a very strange expression,"},{"Start":"10:04.495 ","End":"10:09.280","Text":"and we use this for finding out what kind of an extremum,"},{"Start":"10:09.280 ","End":"10:10.870","Text":"if any, we have."},{"Start":"10:10.870 ","End":"10:16.390","Text":"What we do is we substitute Lambda from here,"},{"Start":"10:16.390 ","End":"10:19.855","Text":"and we substitute also x and y."},{"Start":"10:19.855 ","End":"10:23.530","Text":"It will turn out that if H is positive,"},{"Start":"10:23.530 ","End":"10:25.645","Text":"our extremum is a minimum,"},{"Start":"10:25.645 ","End":"10:27.520","Text":"and if H is negative,"},{"Start":"10:27.520 ","End":"10:29.485","Text":"then it\u0027s a maximum."},{"Start":"10:29.485 ","End":"10:33.610","Text":"Here I copied the partial derivatives,"},{"Start":"10:33.610 ","End":"10:37.810","Text":"and we don\u0027t actually have to compute"},{"Start":"10:37.810 ","End":"10:43.210","Text":"H. All we need to find out is whether it\u0027s positive or negative,"},{"Start":"10:43.210 ","End":"10:46.240","Text":"so that can save us on some of the calculations."},{"Start":"10:46.240 ","End":"10:52.120","Text":"For example, here I see fxx minus 1 over x squared,"},{"Start":"10:52.120 ","End":"10:54.415","Text":"so I know it\u0027s negative, whatever x is,"},{"Start":"10:54.415 ","End":"11:04.390","Text":"this piece is negative and gxx is here and it\u0027s 0,"},{"Start":"11:04.390 ","End":"11:11.420","Text":"and gy squared is positive because it\u0027s squared."},{"Start":"11:12.810 ","End":"11:18.565","Text":"Altogether I have a negative,"},{"Start":"11:18.565 ","End":"11:23.170","Text":"takeaway 0 is negative times positive is still negative."},{"Start":"11:23.170 ","End":"11:27.295","Text":"So this whole first term up to the plus sign, it\u0027s negative."},{"Start":"11:27.295 ","End":"11:30.290","Text":"Now the second term,"},{"Start":"11:30.750 ","End":"11:33.700","Text":"fyy is minus 1 over y squared,"},{"Start":"11:33.700 ","End":"11:36.955","Text":"so that\u0027s negative, gyy is 0,"},{"Start":"11:36.955 ","End":"11:39.520","Text":"this is 0, and like before,"},{"Start":"11:39.520 ","End":"11:41.410","Text":"this thing squared is positive."},{"Start":"11:41.410 ","End":"11:44.185","Text":"We have a negative, less 0 is a negative,"},{"Start":"11:44.185 ","End":"11:46.450","Text":"times a positive is still negative."},{"Start":"11:46.450 ","End":"11:49.990","Text":"So this term is a negative."},{"Start":"11:49.990 ","End":"11:53.665","Text":"The last term, look fxy,"},{"Start":"11:53.665 ","End":"11:58.195","Text":"the mixed second-order derivative of f is 0."},{"Start":"11:58.195 ","End":"12:00.280","Text":"This bit is 0,"},{"Start":"12:00.280 ","End":"12:02.440","Text":"and the g of x,"},{"Start":"12:02.440 ","End":"12:07.660","Text":"y, the mixed 1 for g is also 0."},{"Start":"12:07.660 ","End":"12:12.745","Text":"That\u0027s enough for us to know that this whole last term is also 0,"},{"Start":"12:12.745 ","End":"12:19.450","Text":"so we have a negative plus a negative less 0 altogether it\u0027s negative,"},{"Start":"12:19.450 ","End":"12:23.110","Text":"which means that this is less than 0,"},{"Start":"12:23.110 ","End":"12:27.220","Text":"this whole H. Because it\u0027s less than 0,"},{"Start":"12:27.220 ","End":"12:30.960","Text":"that means that we have a maximum,"},{"Start":"12:30.960 ","End":"12:33.825","Text":"our extremum is a maximum,"},{"Start":"12:33.825 ","End":"12:37.680","Text":"and that\u0027s a good job because that\u0027s what we were asked anyway."},{"Start":"12:37.680 ","End":"12:43.945","Text":"We have a maximum at x equals 2 and y equals 4."},{"Start":"12:43.945 ","End":"12:48.520","Text":"But there was that other little bit of a question we also had,"},{"Start":"12:48.520 ","End":"12:55.630","Text":"where was it, we also have to show that the maximum is natural log of 8."},{"Start":"12:55.630 ","End":"13:01.490","Text":"So we will have to substitute and just make sure of this and then we\u0027ll be done."},{"Start":"13:01.740 ","End":"13:04.645","Text":"Let\u0027s get back down again."},{"Start":"13:04.645 ","End":"13:08.395","Text":"Our function, f of x, y,"},{"Start":"13:08.395 ","End":"13:16.915","Text":"the objective function, this was equal to natural log x plus natural log of y."},{"Start":"13:16.915 ","End":"13:21.460","Text":"But what we want is for our particular x and y,"},{"Start":"13:21.460 ","End":"13:24.160","Text":"which is 2 and 4,"},{"Start":"13:24.160 ","End":"13:34.585","Text":"so we have to substitute x equals 2 and y equals 4 to see what happens in our case."},{"Start":"13:34.585 ","End":"13:42.085","Text":"We get a natural log of 2 plus natural log of 4,"},{"Start":"13:42.085 ","End":"13:45.850","Text":"and this doesn\u0027t exactly look like natural log of H,"},{"Start":"13:45.850 ","End":"13:46.930","Text":"which is what we wanted."},{"Start":"13:46.930 ","End":"13:51.115","Text":"But then we remember there are logarithm rules that"},{"Start":"13:51.115 ","End":"13:58.300","Text":"the natural log in general of a product is the sum of the logarithms."},{"Start":"13:58.300 ","End":"14:01.780","Text":"This equation, I can also read it from right to left."},{"Start":"14:01.780 ","End":"14:06.450","Text":"If I let a is 2 and b is 4,"},{"Start":"14:06.450 ","End":"14:13.310","Text":"then I get that this equals natural log of 2 times 4,"},{"Start":"14:13.310 ","End":"14:16.010","Text":"which is natural log of 8,"},{"Start":"14:16.010 ","End":"14:24.150","Text":"and that answers that little extra question that they asked of us, and we\u0027re done."}],"ID":9666},{"Watched":false,"Name":"Exercise 2","Duration":"12m 34s","ChapterTopicVideoID":9782,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"Here, we have the second solved example of constrained extrema."},{"Start":"00:04.110 ","End":"00:12.330","Text":"This time, we need to find the minimum of x plus y subject to x plus y plus xy equals 15."},{"Start":"00:12.330 ","End":"00:15.435","Text":"We also thrown in this condition."},{"Start":"00:15.435 ","End":"00:18.270","Text":"It looks like it may have come from a word problem where it"},{"Start":"00:18.270 ","End":"00:21.990","Text":"only makes sense when x and y are bigger or equal to 0,"},{"Start":"00:21.990 ","End":"00:23.565","Text":"so just note that."},{"Start":"00:23.565 ","End":"00:25.680","Text":"After we\u0027ve done that, we also have to show that"},{"Start":"00:25.680 ","End":"00:29.940","Text":"the actual minimum value of x plus y is 6."},{"Start":"00:29.940 ","End":"00:33.895","Text":"Let\u0027s begin by defining 2 functions."},{"Start":"00:33.895 ","End":"00:37.790","Text":"These are the objective function and the constraint function."},{"Start":"00:37.790 ","End":"00:41.390","Text":"The objective is the one that we\u0027re trying to minimize or maximize."},{"Start":"00:41.390 ","End":"00:50.325","Text":"In this case, it will be x plus y. I\u0027ll use the letter f. f of xy equals x plus y."},{"Start":"00:50.325 ","End":"00:52.575","Text":"The constraint is what?"},{"Start":"00:52.575 ","End":"00:55.160","Text":"It goes after the subject to part."},{"Start":"00:55.160 ","End":"01:00.599","Text":"So that\u0027s x plus y plus xy,"},{"Start":"01:00.599 ","End":"01:02.010","Text":"but not the equation."},{"Start":"01:02.010 ","End":"01:03.990","Text":"We want something to equal 0,"},{"Start":"01:03.990 ","End":"01:07.015","Text":"we subtract 15, and then it equals 0,"},{"Start":"01:07.015 ","End":"01:09.175","Text":"and this is called the constraint function."},{"Start":"01:09.175 ","End":"01:12.130","Text":"I forgot to say, this is g of xy."},{"Start":"01:12.400 ","End":"01:17.060","Text":"Yes, g of xy. Now that we have these 2 functions, f and g,"},{"Start":"01:17.060 ","End":"01:20.570","Text":"step 1 will be to find"},{"Start":"01:20.570 ","End":"01:26.465","Text":"all the partial derivatives of these 2 functions of the first, and second order."},{"Start":"01:26.465 ","End":"01:33.020","Text":"Let\u0027s begin with f. f with respect to x is equal"},{"Start":"01:33.020 ","End":"01:40.365","Text":"to y is a constant so this is equal to 1. f with respect to y,"},{"Start":"01:40.365 ","End":"01:44.205","Text":"there, x is a constant but it\u0027s still equal to 1."},{"Start":"01:44.205 ","End":"01:49.820","Text":"Now, f with respect to x with respect to x."},{"Start":"01:49.820 ","End":"01:53.580","Text":"So this with respect to x is 0."},{"Start":"01:53.660 ","End":"01:59.420","Text":"Also, the other 2 will be 0 clearly so I\u0027ll just write that as f"},{"Start":"01:59.420 ","End":"02:07.110","Text":"of xx equals fxy equals fyy equals 0."},{"Start":"02:07.110 ","End":"02:12.180","Text":"As you remember, there is no 4th one because fyx is the same as fxy."},{"Start":"02:12.350 ","End":"02:16.025","Text":"That\u0027s it for f. Now, for g,"},{"Start":"02:16.025 ","End":"02:24.720","Text":"g with respect to x is y is a constant so this gives me 1 plus y,"},{"Start":"02:25.460 ","End":"02:31.565","Text":"and g with respect to y is equal to x is a constant,"},{"Start":"02:31.565 ","End":"02:34.980","Text":"so this becomes 1 plus x."},{"Start":"02:35.330 ","End":"02:39.690","Text":"Now, second-order derivatives, gxx."},{"Start":"02:39.690 ","End":"02:45.900","Text":"This with respect to x is 0 because everything is a constant there,"},{"Start":"02:45.900 ","End":"02:52.089","Text":"and gyx or gxy,"},{"Start":"02:52.160 ","End":"02:56.600","Text":"doesn\u0027t matter if I differentiate this with respect to y and I get"},{"Start":"02:56.600 ","End":"03:00.875","Text":"1 or I differentiate this with respect to x and I still get 1,"},{"Start":"03:00.875 ","End":"03:08.190","Text":"and then gyy, 0."},{"Start":"03:08.190 ","End":"03:13.065","Text":"In step 2, we always write the same 3 equations,"},{"Start":"03:13.065 ","End":"03:22.500","Text":"and these equations are f with respect to x is equal to Lambda times g with respect to"},{"Start":"03:22.500 ","End":"03:32.280","Text":"x. f with respect to y is equal to Lambda times g with respect to y."},{"Start":"03:32.280 ","End":"03:38.610","Text":"The last one is always the constraint function equals 0."},{"Start":"03:38.610 ","End":"03:43.570","Text":"This gives us 3 equations in 3 unknowns, xy and Lambda."},{"Start":"03:43.570 ","End":"03:47.240","Text":"Lambda is just an auxiliary variable and it will help us to"},{"Start":"03:47.240 ","End":"03:52.070","Text":"determine whether what we get is a maximum or a minimum."},{"Start":"03:52.070 ","End":"03:57.290","Text":"The solution to these 3 equations or solutions if there\u0027s more than 1,"},{"Start":"03:57.290 ","End":"04:00.549","Text":"are the suspects for extrema."},{"Start":"04:00.549 ","End":"04:04.730","Text":"Afterwards, we check each one and see whether it\u0027s minimum or maximum or neither."},{"Start":"04:04.730 ","End":"04:08.630","Text":"At first, I need to rewrite them just in general."},{"Start":"04:08.630 ","End":"04:10.430","Text":"In our specific case,"},{"Start":"04:10.430 ","End":"04:15.200","Text":"what we have, f with respect to x,"},{"Start":"04:15.200 ","End":"04:16.520","Text":"I\u0027ve got that here,"},{"Start":"04:16.520 ","End":"04:24.635","Text":"is 1 equals Lambda g with respect to x is here times 1 plus y."},{"Start":"04:24.635 ","End":"04:28.920","Text":"2nd equation, fy is here is 1,"},{"Start":"04:28.920 ","End":"04:32.645","Text":"and gy is here."},{"Start":"04:32.645 ","End":"04:36.660","Text":"So I get Lambda times 1 plus x."},{"Start":"04:36.660 ","End":"04:40.850","Text":"The last one is the original constraint which says that x"},{"Start":"04:40.850 ","End":"04:46.445","Text":"plus y plus xy minus 15 equals 0,"},{"Start":"04:46.445 ","End":"04:48.710","Text":"or you could write it as it was originally,"},{"Start":"04:48.710 ","End":"04:52.255","Text":"this plus this plus this equals 15, doesn\u0027t matter."},{"Start":"04:52.255 ","End":"04:55.805","Text":"We always solve these the same way."},{"Start":"04:55.805 ","End":"05:00.620","Text":"We divide one of these equations by the other,"},{"Start":"05:00.620 ","End":"05:01.820","Text":"the 2 top ones, I mean,"},{"Start":"05:01.820 ","End":"05:03.305","Text":"and we get rid of Lambda,"},{"Start":"05:03.305 ","End":"05:05.720","Text":"and then we get 2 equations in 2 unknowns."},{"Start":"05:05.720 ","End":"05:08.125","Text":"So if I divide, let\u0027s say,"},{"Start":"05:08.125 ","End":"05:09.460","Text":"the, I don\u0027t know,"},{"Start":"05:09.460 ","End":"05:11.630","Text":"the top one minus the second one."},{"Start":"05:11.630 ","End":"05:17.475","Text":"So I\u0027ll get 1 over 1 equals"},{"Start":"05:17.475 ","End":"05:27.269","Text":"Lambda 1 plus y over Lambda 1 plus x."},{"Start":"05:27.269 ","End":"05:31.165","Text":"Now, the Lambda cancels,"},{"Start":"05:31.165 ","End":"05:40.185","Text":"and then we cross multiply so we get that 1 plus y,"},{"Start":"05:40.185 ","End":"05:44.830","Text":"this times this equals 1 plus x."},{"Start":"05:45.980 ","End":"05:49.605","Text":"So tempting. Yeah, this cancel right away."},{"Start":"05:49.605 ","End":"05:53.785","Text":"This will give me that y equals x."},{"Start":"05:53.785 ","End":"05:56.380","Text":"Let me highlight these 2 equations."},{"Start":"05:56.380 ","End":"06:00.515","Text":"Here\u0027s one equation in x and y,"},{"Start":"06:00.515 ","End":"06:02.960","Text":"and here\u0027s the other equation in x and y,"},{"Start":"06:02.960 ","End":"06:05.960","Text":"2 equations and 2 unknowns."},{"Start":"06:05.960 ","End":"06:12.290","Text":"To solve this, it\u0027s easiest just to substitute y equals x in this equation,"},{"Start":"06:12.290 ","End":"06:14.720","Text":"and we will get,"},{"Start":"06:14.720 ","End":"06:20.025","Text":"I\u0027ll write it over here, x plus x"},{"Start":"06:20.025 ","End":"06:26.520","Text":"plus xx minus 15 equals 0."},{"Start":"06:26.520 ","End":"06:29.325","Text":"In other words, I\u0027ll just organize it,"},{"Start":"06:29.325 ","End":"06:37.170","Text":"it\u0027s x squared plus 2 x minus 15 equals 0."},{"Start":"06:37.170 ","End":"06:41.420","Text":"Now, this is a quadratic equation and you will know"},{"Start":"06:41.420 ","End":"06:44.780","Text":"how to solve quadratic equations so I won\u0027t waste time with the solution,"},{"Start":"06:44.780 ","End":"06:48.095","Text":"and I\u0027ll just tell you that x has 2 possibilities,"},{"Start":"06:48.095 ","End":"06:54.055","Text":"x could be either 3 or minus 5."},{"Start":"06:54.055 ","End":"06:57.875","Text":"But remember that at the beginning,"},{"Start":"06:57.875 ","End":"07:00.350","Text":"where is it? The other way."},{"Start":"07:00.350 ","End":"07:05.405","Text":"We were given that x and y are both non-negative so we can rule out"},{"Start":"07:05.405 ","End":"07:10.805","Text":"one of the possibilities and we can say that x has to be equal to 3."},{"Start":"07:10.805 ","End":"07:17.150","Text":"So we have that x is equal to 3 in our solution,"},{"Start":"07:17.150 ","End":"07:18.680","Text":"and then when x is 3,"},{"Start":"07:18.680 ","End":"07:22.460","Text":"y is equal to x, so y equals 3."},{"Start":"07:22.460 ","End":"07:26.990","Text":"But let\u0027s not forget that we had a third variable,"},{"Start":"07:26.990 ","End":"07:30.005","Text":"Lambda, and we have to find out what this equals."},{"Start":"07:30.005 ","End":"07:36.020","Text":"It turns out that you can skip the Lambda phase if the second-order derivatives of g,"},{"Start":"07:36.020 ","End":"07:38.660","Text":"this, this, and this are all 0."},{"Start":"07:38.660 ","End":"07:43.610","Text":"But in our case, that\u0027s not the case because this one is not 0,"},{"Start":"07:43.610 ","End":"07:45.470","Text":"so we do have to compute Lambda."},{"Start":"07:45.470 ","End":"07:47.780","Text":"Now, let\u0027s see where shall we do it?"},{"Start":"07:47.780 ","End":"07:49.535","Text":"We have these 2 equations."},{"Start":"07:49.535 ","End":"07:52.700","Text":"Let\u0027s take, say, the middle one,"},{"Start":"07:52.700 ","End":"07:58.205","Text":"1 equals Lambda times 1 plus x,"},{"Start":"07:58.205 ","End":"08:01.039","Text":"but x is equal to 3."},{"Start":"08:01.039 ","End":"08:07.445","Text":"So that easily gives us that Lambda is equal to 1/4."},{"Start":"08:07.445 ","End":"08:14.990","Text":"Now, these are the xy Lambda for our suspect point so I put asterisks on them."},{"Start":"08:14.990 ","End":"08:22.305","Text":"This xy Lambda belongs to our suspect to be an extremum. That\u0027s step 2."},{"Start":"08:22.305 ","End":"08:27.985","Text":"In step 3, we find out whether this xy gives us a maximum or a minimum."},{"Start":"08:27.985 ","End":"08:31.745","Text":"In step 3, we write down this strange expression called"},{"Start":"08:31.745 ","End":"08:34.940","Text":"H which you should have on a formula sheet."},{"Start":"08:34.940 ","End":"08:36.050","Text":"I\u0027ll just write it for you,"},{"Start":"08:36.050 ","End":"08:38.540","Text":"I don\u0027t even remember it by heart."},{"Start":"08:38.930 ","End":"08:46.845","Text":"Here it is, this lovely expression for H which looks quite formidable."},{"Start":"08:46.845 ","End":"08:50.280","Text":"Anyway, let\u0027s get to work on it."},{"Start":"08:50.280 ","End":"08:52.190","Text":"Now, I\u0027ve copied again"},{"Start":"08:52.190 ","End":"08:56.935","Text":"the partial derivatives of f and g because we\u0027re going to need them,"},{"Start":"08:56.935 ","End":"09:00.025","Text":"I copied them, we computed them before."},{"Start":"09:00.025 ","End":"09:02.170","Text":"Let\u0027s see what we get."},{"Start":"09:02.170 ","End":"09:07.330","Text":"fxx is 0,"},{"Start":"09:07.330 ","End":"09:13.790","Text":"so this bit is 0. gxx is also 0."},{"Start":"09:13.790 ","End":"09:18.100","Text":"This whole first term up to the plus is 0."},{"Start":"09:18.100 ","End":"09:20.075","Text":"Let\u0027s go onto the second term."},{"Start":"09:20.075 ","End":"09:29.400","Text":"fyy is 0, gyy is 0."},{"Start":"09:29.400 ","End":"09:32.445","Text":"This second term is also 0."},{"Start":"09:32.445 ","End":"09:35.185","Text":"Now, let\u0027s get to the last one."},{"Start":"09:35.185 ","End":"09:38.960","Text":"I just like to remind you that if you have difficulty"},{"Start":"09:38.960 ","End":"09:42.470","Text":"computing the actual answer and we don\u0027t really need it,"},{"Start":"09:42.470 ","End":"09:46.010","Text":"all we have to do is find whether H is positive or negative."},{"Start":"09:46.010 ","End":"09:47.480","Text":"Positive is a minimum,"},{"Start":"09:47.480 ","End":"09:49.595","Text":"negative means the maximum."},{"Start":"09:49.595 ","End":"09:51.200","Text":"Back to the last term,"},{"Start":"09:51.200 ","End":"09:56.475","Text":"fxy is 0 from here,"},{"Start":"09:56.475 ","End":"09:58.620","Text":"gxy is not 0,"},{"Start":"09:58.620 ","End":"10:05.350","Text":"it\u0027s 1. gx from here is 1 plus y,"},{"Start":"10:05.660 ","End":"10:10.065","Text":"and gy is 1 plus x."},{"Start":"10:10.065 ","End":"10:14.540","Text":"If I write down H over here,"},{"Start":"10:14.540 ","End":"10:18.440","Text":"I\u0027ll get that H equals."},{"Start":"10:18.440 ","End":"10:24.500","Text":"This is 0, this is 0 minus 2."},{"Start":"10:24.500 ","End":"10:30.150","Text":"What I get inside the brackets is minus Lambda."},{"Start":"10:30.730 ","End":"10:38.990","Text":"Then here, 1 plus y, 1 plus x."},{"Start":"10:38.990 ","End":"10:43.010","Text":"Now, what I want is not really the y and the x,"},{"Start":"10:43.010 ","End":"10:45.770","Text":"I actually want the y and the x for the suspect."},{"Start":"10:45.770 ","End":"10:50.030","Text":"These are all from the suspect points so I need to put an asterisk here."},{"Start":"10:50.030 ","End":"10:52.610","Text":"Now, we have these values here."},{"Start":"10:52.610 ","End":"10:56.600","Text":"We have that Lambda is equal to,"},{"Start":"10:56.600 ","End":"10:59.650","Text":"what is it? A quarter."},{"Start":"10:59.650 ","End":"11:07.295","Text":"y is equal to 3 and x is equal to 3."},{"Start":"11:07.295 ","End":"11:11.674","Text":"I could compute the actual answer but since I only need the sign,"},{"Start":"11:11.674 ","End":"11:15.970","Text":"I can see this is positive times positive times negative,"},{"Start":"11:15.970 ","End":"11:17.270","Text":"this is minus a 1/4,"},{"Start":"11:17.270 ","End":"11:19.370","Text":"times negative, which is minus 2."},{"Start":"11:19.370 ","End":"11:23.075","Text":"So altogether, it\u0027s positive, bigger than 0."},{"Start":"11:23.075 ","End":"11:27.655","Text":"Bigger than 0 means that it is a minimum,"},{"Start":"11:27.655 ","End":"11:29.510","Text":"and this is what we wanted."},{"Start":"11:29.510 ","End":"11:31.730","Text":"So at x equals 3,"},{"Start":"11:31.730 ","End":"11:35.210","Text":"y equals 3, we have a minimum."},{"Start":"11:35.210 ","End":"11:39.380","Text":"I\u0027ll show you. We were asked for, where is it?"},{"Start":"11:39.380 ","End":"11:43.255","Text":"The minimum of x plus y."},{"Start":"11:43.255 ","End":"11:45.405","Text":"We also had an extra bit."},{"Start":"11:45.405 ","End":"11:51.050","Text":"Let\u0027s not forget, we had an extra task to show that the minimum is actually 6."},{"Start":"11:51.050 ","End":"11:53.940","Text":"Let\u0027s get back down there again."},{"Start":"11:55.160 ","End":"12:01.500","Text":"What we need is the target function, f of xy,"},{"Start":"12:01.500 ","End":"12:06.600","Text":"at the suspect point which is now no longer a suspect,"},{"Start":"12:06.600 ","End":"12:13.670","Text":"but this is the asterisk which is f of 3 and 3."},{"Start":"12:13.670 ","End":"12:18.005","Text":"Since f of x plus y was x plus y,"},{"Start":"12:18.005 ","End":"12:23.805","Text":"this becomes 3 plus 3 which equals 6,"},{"Start":"12:23.805 ","End":"12:26.070","Text":"and this is what we were asked to show."},{"Start":"12:26.070 ","End":"12:28.055","Text":"We showed that second bit,"},{"Start":"12:28.055 ","End":"12:31.145","Text":"and we also showed that this point is a minimum,"},{"Start":"12:31.145 ","End":"12:33.810","Text":"and so we are done."}],"ID":9667},{"Watched":false,"Name":"Exercise 3","Duration":"9m 29s","ChapterTopicVideoID":9778,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, we have to find the maximum and minimum points"},{"Start":"00:04.470 ","End":"00:10.210","Text":"of this function subject to this constraint."},{"Start":"00:10.550 ","End":"00:17.265","Text":"This type of problem is best attacked using the Lagrange multiplier method."},{"Start":"00:17.265 ","End":"00:19.170","Text":"We define 2 new functions."},{"Start":"00:19.170 ","End":"00:21.930","Text":"First of all, we define g of x,"},{"Start":"00:21.930 ","End":"00:27.330","Text":"y to be the constraint in the sense that I make it as something equals 0."},{"Start":"00:27.330 ","End":"00:33.030","Text":"That would be x squared plus y squared minus 13."},{"Start":"00:33.030 ","End":"00:37.270","Text":"Then the constraint is just g equals 0."},{"Start":"00:37.540 ","End":"00:44.480","Text":"Also note for later use that this constraint is actually a circle,"},{"Start":"00:44.480 ","End":"00:47.150","Text":"x squared plus y squared equals r squared,"},{"Start":"00:47.150 ","End":"00:49.280","Text":"so the radius is root 13,"},{"Start":"00:49.280 ","End":"00:50.660","Text":"and it\u0027s centered at the origin."},{"Start":"00:50.660 ","End":"00:52.415","Text":"We will need this later."},{"Start":"00:52.415 ","End":"00:56.270","Text":"The second function we define is the Lagrangian function,"},{"Start":"00:56.270 ","End":"00:59.210","Text":"but it takes 3 variables x,"},{"Start":"00:59.210 ","End":"01:03.080","Text":"y, and the third 1 we\u0027ll call Lambda."},{"Start":"01:03.080 ","End":"01:07.960","Text":"Before I say what it is here in general, for short,"},{"Start":"01:07.960 ","End":"01:13.930","Text":"Lambda is L. L"},{"Start":"01:13.930 ","End":"01:19.075","Text":"is f minus Lambda g. We get,"},{"Start":"01:19.075 ","End":"01:23.185","Text":"this is f of x, y,"},{"Start":"01:23.185 ","End":"01:32.180","Text":"which is 4x plus 6y minus Lambda times the g part,"},{"Start":"01:32.180 ","End":"01:38.260","Text":"which is x squared plus y squared minus 13."},{"Start":"01:38.780 ","End":"01:43.040","Text":"The theory is that the critical points are"},{"Start":"01:43.040 ","End":"01:46.790","Text":"found when we let all 3 partial derivatives be 0."},{"Start":"01:46.790 ","End":"01:51.350","Text":"In other words, we\u0027ll let this be 0 with respect to y,"},{"Start":"01:51.350 ","End":"01:52.585","Text":"with respect to lambda."},{"Start":"01:52.585 ","End":"01:55.545","Text":"All 3 partial derivatives are 0."},{"Start":"01:55.545 ","End":"02:00.970","Text":"This gives us 3 equations in 3 unknowns."},{"Start":"02:01.750 ","End":"02:05.150","Text":"Let\u0027s see, with respect to x,"},{"Start":"02:05.150 ","End":"02:09.620","Text":"we get from here, just 4."},{"Start":"02:09.620 ","End":"02:15.950","Text":"From here, I get minus Lambda because it\u0027s a multiplicative constant."},{"Start":"02:15.950 ","End":"02:18.230","Text":"It stays on the derivative of what\u0027s here."},{"Start":"02:18.230 ","End":"02:23.764","Text":"That\u0027s just 2x Lambda times 2x. This is equal to 0."},{"Start":"02:23.764 ","End":"02:32.455","Text":"With respect to y, I get 6 minus Lambda times 2y equals 0."},{"Start":"02:32.455 ","End":"02:43.205","Text":"The last one just says that this thing in brackets with a minus is 0."},{"Start":"02:43.205 ","End":"02:46.445","Text":"I don\u0027t need the minus because it\u0027s something equals 0."},{"Start":"02:46.445 ","End":"02:52.925","Text":"It\u0027s just x squared plus y squared minus 13 equals 0."},{"Start":"02:52.925 ","End":"02:58.400","Text":"It always is so that the last one is the original constraint."},{"Start":"02:58.400 ","End":"03:01.640","Text":"Now let me slightly rewrite these,"},{"Start":"03:01.640 ","End":"03:04.445","Text":"I\u0027ll bring this to the other side."},{"Start":"03:04.445 ","End":"03:07.540","Text":"While I\u0027m at it, we can also divide by 2."},{"Start":"03:07.540 ","End":"03:13.595","Text":"We\u0027ll get that 2 equals Lambda x."},{"Start":"03:13.595 ","End":"03:18.200","Text":"From here, we\u0027ll get that also dividing by 2,"},{"Start":"03:18.200 ","End":"03:22.105","Text":"3 equals Lambda y."},{"Start":"03:22.105 ","End":"03:25.775","Text":"The last one, I don\u0027t have to copy it again and might as well"},{"Start":"03:25.775 ","End":"03:29.660","Text":"just write it as x squared plus y squared equals 13."},{"Start":"03:29.660 ","End":"03:31.565","Text":"It\u0027s always the original constraint."},{"Start":"03:31.565 ","End":"03:37.925","Text":"Now we have this set of 3 equations."},{"Start":"03:37.925 ","End":"03:47.585","Text":"Now, I would like to divide this equation by this equation."},{"Start":"03:47.585 ","End":"03:54.395","Text":"But I want to make sure that I\u0027m not dividing by 0."},{"Start":"03:54.395 ","End":"03:58.535","Text":"Well, what I want to write is that"},{"Start":"03:58.535 ","End":"04:09.650","Text":"2/3 equals Lambda x over Lambda y for these 2."},{"Start":"04:09.650 ","End":"04:13.700","Text":"But I don\u0027t want to be dividing by 0,"},{"Start":"04:13.700 ","End":"04:17.360","Text":"so I have to check separately the possibilities."},{"Start":"04:17.360 ","End":"04:20.915","Text":"What would happen if Lambda equals 0?"},{"Start":"04:20.915 ","End":"04:25.559","Text":"What would happen if y equals 0?"},{"Start":"04:26.290 ","End":"04:34.700","Text":"I need to check the separately consist only holds true if Lambda is not 0 and y is not 0."},{"Start":"04:34.700 ","End":"04:37.085","Text":"Otherwise I\u0027ll be dividing by 0."},{"Start":"04:37.085 ","End":"04:40.320","Text":"Let me take care of the exceptional cases separately."},{"Start":"04:40.320 ","End":"04:42.845","Text":"Lambda 0, and I plug it in here,"},{"Start":"04:42.845 ","End":"04:45.510","Text":"I get 2 equals 0."},{"Start":"04:45.850 ","End":"04:51.635","Text":"That\u0027s not possible, so this is ruled out."},{"Start":"04:51.635 ","End":"04:54.395","Text":"Now, how about y equals 0?"},{"Start":"04:54.395 ","End":"04:56.030","Text":"Well, if I plug that here,"},{"Start":"04:56.030 ","End":"04:58.430","Text":"if y is 0 and it gives me that 3 equals 0,"},{"Start":"04:58.430 ","End":"05:00.080","Text":"it\u0027s also not possible."},{"Start":"05:00.080 ","End":"05:02.090","Text":"This is also ruled out."},{"Start":"05:02.090 ","End":"05:05.660","Text":"We know that these conditions do hold."},{"Start":"05:05.660 ","End":"05:08.225","Text":"Now we can work on this one."},{"Start":"05:08.225 ","End":"05:12.810","Text":"In fact, what we have now is this equation."},{"Start":"05:13.120 ","End":"05:15.140","Text":"If I cancel Lambda,"},{"Start":"05:15.140 ","End":"05:18.455","Text":"I have this equation in x and y and this equation in x and y."},{"Start":"05:18.455 ","End":"05:22.070","Text":"Now I have 2 equations in just x and y."},{"Start":"05:23.230 ","End":"05:30.055","Text":"From this 1, if I cross multiply,"},{"Start":"05:30.055 ","End":"05:33.890","Text":"then I will get that."},{"Start":"05:33.890 ","End":"05:35.495","Text":"Well, let me just say this."},{"Start":"05:35.495 ","End":"05:41.900","Text":"Let me extract y in terms of x. Y is going to be this times this over this,"},{"Start":"05:41.900 ","End":"05:47.660","Text":"y is going to be 3x/2."},{"Start":"05:47.660 ","End":"05:50.440","Text":"Yeah, y is 3x/2,"},{"Start":"05:50.440 ","End":"05:54.720","Text":"because 3x is 2y and then divide by 2."},{"Start":"05:54.720 ","End":"05:59.610","Text":"This is what we get and then substitute that in here,"},{"Start":"05:59.770 ","End":"06:07.000","Text":"and we get x squared."},{"Start":"06:07.000 ","End":"06:10.440","Text":"Y squared would be this thing squared."},{"Start":"06:10.440 ","End":"06:12.285","Text":"It\u0027s square everything."},{"Start":"06:12.285 ","End":"06:15.515","Text":"It\u0027s 9x squared over 4."},{"Start":"06:15.515 ","End":"06:17.240","Text":"3 squared is 9, 2 squared is 4,"},{"Start":"06:17.240 ","End":"06:21.380","Text":"and so on, equals 13."},{"Start":"06:21.380 ","End":"06:26.820","Text":"Now, if I multiply both sides by 4,"},{"Start":"06:26.820 ","End":"06:34.140","Text":"what I\u0027ll get is 4x squared plus 9x squared is 13x squared."},{"Start":"06:34.140 ","End":"06:37.730","Text":"This by 4 well I\u0027ll just leave it as 13 times"},{"Start":"06:37.730 ","End":"06:43.355","Text":"4 because the 13 is going to cancel here and here,"},{"Start":"06:43.355 ","End":"06:48.615","Text":"x squared is 4, so x is plus or minus 2."},{"Start":"06:48.615 ","End":"06:51.255","Text":"Let me see, x could be 2,"},{"Start":"06:51.255 ","End":"06:53.265","Text":"and then we\u0027ll see what its y is."},{"Start":"06:53.265 ","End":"06:56.085","Text":"X could be minus 2."},{"Start":"06:56.085 ","End":"06:58.445","Text":"We\u0027ll see what the y is."},{"Start":"06:58.445 ","End":"07:00.080","Text":"Maybe I should have written here."},{"Start":"07:00.080 ","End":"07:05.300","Text":"This gives me that x is plus or minus square root of 4 plus or minus 2."},{"Start":"07:05.300 ","End":"07:11.310","Text":"If x is 2 put that in here, y is 3."},{"Start":"07:11.310 ","End":"07:13.815","Text":"If x is minus 2, put it here,"},{"Start":"07:13.815 ","End":"07:18.220","Text":"y is minus 3."},{"Start":"07:19.750 ","End":"07:22.070","Text":"By the way, if we wanted to,"},{"Start":"07:22.070 ","End":"07:23.629","Text":"we could find Lambda."},{"Start":"07:23.629 ","End":"07:28.080","Text":"For example, if x is 2 and y is 3,"},{"Start":"07:28.080 ","End":"07:32.690","Text":"each of these gives us that Lambda is 1, so it\u0027s consistent."},{"Start":"07:32.690 ","End":"07:37.445","Text":"I get the same value of Lambda if I put x equals 2 or y equals 3."},{"Start":"07:37.445 ","End":"07:39.710","Text":"Here Lambda is equal to 1,"},{"Start":"07:39.710 ","End":"07:41.795","Text":"but we don\u0027t care about Lambda."},{"Start":"07:41.795 ","End":"07:45.215","Text":"Here, by the way, if I put minus 2 minus 3,"},{"Start":"07:45.215 ","End":"07:47.920","Text":"I would get Lambda is minus 1."},{"Start":"07:47.920 ","End":"07:50.540","Text":"But as I said, we don\u0027t need Lambda,"},{"Start":"07:50.540 ","End":"07:51.950","Text":"I\u0027m just throwing it away,"},{"Start":"07:51.950 ","End":"07:53.330","Text":"but I could find x, y,"},{"Start":"07:53.330 ","End":"07:55.730","Text":"and Lambda without contradiction."},{"Start":"07:55.730 ","End":"07:59.959","Text":"Now, these are the only 2 critical points possible."},{"Start":"07:59.959 ","End":"08:03.000","Text":"We\u0027ll call them, I don\u0027t know, A and B."},{"Start":"08:03.440 ","End":"08:12.780","Text":"Because our constraint function is on a circle and target function f is continuous,"},{"Start":"08:12.780 ","End":"08:16.595","Text":"when we have a continuous function on a closed bounded curve,"},{"Start":"08:16.595 ","End":"08:21.455","Text":"there\u0027s a theorem of Weierstrass that it always has a maximum and minimum."},{"Start":"08:21.455 ","End":"08:24.035","Text":"Let\u0027s just plug in."},{"Start":"08:24.035 ","End":"08:28.115","Text":"What is f of A and what is f of B?"},{"Start":"08:28.115 ","End":"08:30.140","Text":"The largest would be the maximum,"},{"Start":"08:30.140 ","End":"08:32.165","Text":"and the smallest will be the minimum."},{"Start":"08:32.165 ","End":"08:37.355","Text":"F of A is going to be 4"},{"Start":"08:37.355 ","End":"08:43.340","Text":"times 2 plus 6 times 3."},{"Start":"08:43.340 ","End":"08:49.945","Text":"That\u0027s equal to 8 plus 18 is 26."},{"Start":"08:49.945 ","End":"08:53.220","Text":"Here it\u0027s going to be minus 2 and minus 3,"},{"Start":"08:53.220 ","End":"08:57.580","Text":"so it\u0027s going to be minus 26."},{"Start":"08:57.910 ","End":"09:06.780","Text":"We can conclude that this point is the maximum type of extremum maximum."},{"Start":"09:06.780 ","End":"09:11.030","Text":"This is the minimum."},{"Start":"09:11.290 ","End":"09:17.065","Text":"Just to make it clear that both of them are under"},{"Start":"09:17.065 ","End":"09:20.090","Text":"the constraint."},{"Start":"09:28.150 ","End":"09:30.810","Text":"That is it."}],"ID":9668},{"Watched":false,"Name":"Exercise 4","Duration":"10m 27s","ChapterTopicVideoID":9779,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we have one of those maximum,"},{"Start":"00:03.060 ","End":"00:05.850","Text":"minimum of a function and the constraint."},{"Start":"00:05.850 ","End":"00:08.580","Text":"This is the function of 2 variables,"},{"Start":"00:08.580 ","End":"00:11.025","Text":"and this is the constraint."},{"Start":"00:11.025 ","End":"00:19.575","Text":"I just want to point out that this constraint is actually the equation of an ellipse."},{"Start":"00:19.575 ","End":"00:23.670","Text":"In general, an ellipse has an equation,"},{"Start":"00:23.670 ","End":"00:28.545","Text":"x squared over a squared plus y squared over b squared equals 1."},{"Start":"00:28.545 ","End":"00:30.510","Text":"There\u0027s no problem getting it into that form."},{"Start":"00:30.510 ","End":"00:35.505","Text":"If I divide by 6, I get x squared over 6 plus y squared;"},{"Start":"00:35.505 ","End":"00:37.890","Text":"2 over 6 is like a 3 in the bottom,"},{"Start":"00:37.890 ","End":"00:40.480","Text":"equals 1, and instead of 6,"},{"Start":"00:40.480 ","End":"00:43.235","Text":"I could write root 6 squared,"},{"Start":"00:43.235 ","End":"00:46.370","Text":"and instead of 3, I could write root 3 squared."},{"Start":"00:46.370 ","End":"00:49.290","Text":"So it is in this form."},{"Start":"00:49.480 ","End":"00:52.310","Text":"I\u0027ll just erase this."},{"Start":"00:52.310 ","End":"00:54.800","Text":"I just wanted to note that this is an ellipse,"},{"Start":"00:54.800 ","End":"00:56.825","Text":"and even that\u0027s not important."},{"Start":"00:56.825 ","End":"01:01.890","Text":"It\u0027s important that it\u0027s a closed bounded curve."},{"Start":"01:01.890 ","End":"01:03.930","Text":"We\u0027ll use that later."},{"Start":"01:03.930 ","End":"01:07.560","Text":"Now to the extremum under constraint,"},{"Start":"01:07.560 ","End":"01:10.785","Text":"we use the Lagrange method, it runs multipliers."},{"Start":"01:10.785 ","End":"01:14.035","Text":"We define a new function,"},{"Start":"01:14.035 ","End":"01:16.160","Text":"g of x and y,"},{"Start":"01:16.160 ","End":"01:21.170","Text":"to be the constraint after you put everything on one side, and to be equal to 0."},{"Start":"01:21.170 ","End":"01:27.125","Text":"So g of x, y is x squared plus 2y squared minus 6."},{"Start":"01:27.125 ","End":"01:33.180","Text":"Then this is like g equals 0 is the constraint function."},{"Start":"01:33.580 ","End":"01:40.670","Text":"The second thing we do is define the Lagrangian function L. It has 3 variables: x,"},{"Start":"01:40.670 ","End":"01:44.945","Text":"y, and the third one everyone calls Lambda."},{"Start":"01:44.945 ","End":"01:54.035","Text":"In general, we define L as f minus Lambda g, that\u0027s just shorthand."},{"Start":"01:54.035 ","End":"01:55.745","Text":"In practice, what this is,"},{"Start":"01:55.745 ","End":"01:58.805","Text":"and in our case, is f of x, y,"},{"Start":"01:58.805 ","End":"02:03.710","Text":"which is x squared y minus Lambda times this function,"},{"Start":"02:03.710 ","End":"02:09.110","Text":"the constraint x squared plus 2y squared minus 6."},{"Start":"02:09.110 ","End":"02:11.450","Text":"Now what we do with this according to the theory,"},{"Start":"02:11.450 ","End":"02:15.665","Text":"is we find the 3 partial derivatives with respect to x,"},{"Start":"02:15.665 ","End":"02:20.315","Text":"with respect to y, and with respect to Lambda,"},{"Start":"02:20.315 ","End":"02:23.555","Text":"and set them all equal to 0."},{"Start":"02:23.555 ","End":"02:27.515","Text":"So we get a system of 3 equations."},{"Start":"02:27.515 ","End":"02:30.305","Text":"What this will give us in our case,"},{"Start":"02:30.305 ","End":"02:32.600","Text":"derivative with respect to x."},{"Start":"02:32.600 ","End":"02:35.910","Text":"From here I get 2xy,"},{"Start":"02:36.590 ","End":"02:46.105","Text":"and from here I get minus Lambda times just 2x."},{"Start":"02:46.105 ","End":"02:49.595","Text":"This has to be 0. With respect to y,"},{"Start":"02:49.595 ","End":"02:58.845","Text":"we get x squared minus Lambda times 4y,"},{"Start":"02:58.845 ","End":"03:01.095","Text":"and that equals 0."},{"Start":"03:01.095 ","End":"03:05.460","Text":"The last equation, respect to Lambda is minus;"},{"Start":"03:05.460 ","End":"03:08.480","Text":"this is 0, I don\u0027t need the minus if it\u0027s equal to 0,"},{"Start":"03:08.480 ","End":"03:10.340","Text":"because minus 0 is the same."},{"Start":"03:10.340 ","End":"03:12.065","Text":"So it\u0027s just this bit,"},{"Start":"03:12.065 ","End":"03:16.675","Text":"x squared plus 2y squared minus 6 equals 0."},{"Start":"03:16.675 ","End":"03:18.110","Text":"Unless that should be minus,"},{"Start":"03:18.110 ","End":"03:20.735","Text":"but minus 0 is 0."},{"Start":"03:20.735 ","End":"03:22.925","Text":"So that\u0027s the next set."},{"Start":"03:22.925 ","End":"03:27.925","Text":"Then I can rewrite these."},{"Start":"03:27.925 ","End":"03:32.070","Text":"I\u0027d like to put the stuff with the Lambda on the right-hand side,"},{"Start":"03:32.070 ","End":"03:34.910","Text":"and while I\u0027m at it I can cancel by 2."},{"Start":"03:34.910 ","End":"03:41.555","Text":"The first one will give me that xy equals Lambda x."},{"Start":"03:41.555 ","End":"03:46.955","Text":"The second one will give me that x squared equals,"},{"Start":"03:46.955 ","End":"03:49.960","Text":"there\u0027s nothing to divide by here."},{"Start":"03:52.820 ","End":"03:56.685","Text":"I\u0027ll keep it as Lambda times 4y."},{"Start":"03:56.685 ","End":"04:00.015","Text":"The reason is I\u0027m going to divide by Lambda."},{"Start":"04:00.015 ","End":"04:02.855","Text":"The last one I can rewrite or not,"},{"Start":"04:02.855 ","End":"04:06.185","Text":"I could rewrite it as the original constraint,"},{"Start":"04:06.185 ","End":"04:11.580","Text":"x squared plus 2y squared equals 6."},{"Start":"04:12.790 ","End":"04:16.740","Text":"It always comes out that the last one is the constraint."},{"Start":"04:16.740 ","End":"04:20.740","Text":"Now the strategy is to divide these 2 equations,"},{"Start":"04:20.740 ","End":"04:25.015","Text":"one by another, whichever is more convenient."},{"Start":"04:25.015 ","End":"04:30.380","Text":"I don\u0027t see that one\u0027s particularly better than the other."},{"Start":"04:31.290 ","End":"04:40.150","Text":"Maybe I\u0027ll do the second divided by the first. I don\u0027t know why."},{"Start":"04:40.150 ","End":"04:46.630","Text":"I\u0027ll get that x squared over"},{"Start":"04:46.630 ","End":"04:55.380","Text":"xy is equal to Lambda 4y,"},{"Start":"04:55.380 ","End":"05:02.120","Text":"or 4 Lambda y, over Lambda x."},{"Start":"05:02.120 ","End":"05:06.545","Text":"But this, I can only do under certain conditions,"},{"Start":"05:06.545 ","End":"05:13.685","Text":"provided that the denominators can\u0027t be 0,"},{"Start":"05:13.685 ","End":"05:19.010","Text":"which means that we have to have that Lambda\u0027s not 0,"},{"Start":"05:19.010 ","End":"05:24.810","Text":"x is not 0 and y is not 0."},{"Start":"05:26.750 ","End":"05:31.490","Text":"I\u0027ll solve this and then we\u0027ll explore these possibilities separately,"},{"Start":"05:31.490 ","End":"05:34.745","Text":"because they may give additional critical points."},{"Start":"05:34.745 ","End":"05:38.800","Text":"But if I just take it like this and assume these 3,"},{"Start":"05:38.800 ","End":"05:42.640","Text":"then I can cancel the Lambda."},{"Start":"05:43.190 ","End":"05:48.680","Text":"Also here I can cancel top and bottom by x."},{"Start":"05:48.680 ","End":"05:50.390","Text":"Like this x cancels,"},{"Start":"05:50.390 ","End":"05:53.735","Text":"1 of the x\u0027s is like canceling the 2."},{"Start":"05:53.735 ","End":"05:57.960","Text":"Now I have this equation."},{"Start":"05:57.960 ","End":"06:00.870","Text":"If I cross multiply,"},{"Start":"06:00.870 ","End":"06:04.515","Text":"we get x times x is x squared,"},{"Start":"06:04.515 ","End":"06:08.730","Text":"and here 4y squared."},{"Start":"06:08.730 ","End":"06:14.675","Text":"From here it\u0027s clear that x has to be plus or minus 2y;"},{"Start":"06:14.675 ","End":"06:16.850","Text":"2 being the square root of 4."},{"Start":"06:16.850 ","End":"06:20.190","Text":"That\u0027s the only 2 possibilities."},{"Start":"06:21.350 ","End":"06:24.520","Text":"If that\u0027s the case,"},{"Start":"06:24.520 ","End":"06:28.610","Text":"I can substitute this in the constraint equation."},{"Start":"06:28.610 ","End":"06:32.090","Text":"So x squared would be 2y all squared,"},{"Start":"06:32.090 ","End":"06:36.295","Text":"which is 4y squared,"},{"Start":"06:36.295 ","End":"06:40.830","Text":"and then plus 2y squared equals 6."},{"Start":"06:40.830 ","End":"06:43.725","Text":"Here I have 6y squared equals 6."},{"Start":"06:43.725 ","End":"06:47.320","Text":"So y squared is 1."},{"Start":"06:47.690 ","End":"06:55.955","Text":"That gives me that y equals plus or minus 1."},{"Start":"06:55.955 ","End":"06:59.820","Text":"Now each of these can go with each of these."},{"Start":"07:01.370 ","End":"07:07.185","Text":"Sorry. What I mean is,"},{"Start":"07:07.185 ","End":"07:10.160","Text":"I\u0027m free to choose either one of these for y,"},{"Start":"07:10.160 ","End":"07:15.060","Text":"and then plug in here and get one of 2 possibilities for x,"},{"Start":"07:15.060 ","End":"07:18.365","Text":"all together, 4 possibilities is what I\u0027m saying."},{"Start":"07:18.365 ","End":"07:22.859","Text":"I\u0027m saying that if we take y equals 1,"},{"Start":"07:23.200 ","End":"07:28.270","Text":"and we can still get that x is plus or minus 2 times 1,"},{"Start":"07:28.270 ","End":"07:32.630","Text":"so it could be 2,1 or minus 2,1."},{"Start":"07:32.630 ","End":"07:35.150","Text":"If y is minus 1,"},{"Start":"07:35.150 ","End":"07:40.265","Text":"we can still get that x is plus or minus twice minus 1."},{"Start":"07:40.265 ","End":"07:46.800","Text":"So I would get here, minus 2,"},{"Start":"07:46.800 ","End":"07:48.615","Text":"if I took the plus,"},{"Start":"07:48.615 ","End":"07:53.895","Text":"and plus 2 if I took the minus 2y."},{"Start":"07:53.895 ","End":"07:56.865","Text":"I\u0027ve got 4 points basically."},{"Start":"07:56.865 ","End":"07:58.650","Text":"I could label them;"},{"Start":"07:58.650 ","End":"08:04.200","Text":"it doesn\u0027t really matter in what order: A, B, C,"},{"Start":"08:04.200 ","End":"08:10.745","Text":"D. Now all these 4 points satisfy the constraint,"},{"Start":"08:10.745 ","End":"08:13.415","Text":"they\u0027re on this constraint function,"},{"Start":"08:13.415 ","End":"08:15.440","Text":"they\u0027re on the ellipse."},{"Start":"08:15.440 ","End":"08:17.855","Text":"Because it\u0027s an ellipse,"},{"Start":"08:17.855 ","End":"08:19.950","Text":"and an ellipse is a,"},{"Start":"08:19.950 ","End":"08:22.460","Text":"as I mentioned before, a closed bounded curve,"},{"Start":"08:22.460 ","End":"08:29.345","Text":"there\u0027s a theorem due to Weierstrass that any continuous function,"},{"Start":"08:29.345 ","End":"08:33.440","Text":"and certainly this function is continuous,"},{"Start":"08:33.440 ","End":"08:36.500","Text":"will achieve a maximum and a minimum,"},{"Start":"08:36.500 ","End":"08:38.495","Text":"at least one of each,"},{"Start":"08:38.495 ","End":"08:41.885","Text":"on this ellipse, in this case."},{"Start":"08:41.885 ","End":"08:48.195","Text":"All we have to do is compute what is f of A,"},{"Start":"08:48.195 ","End":"08:51.719","Text":"what is f of B,"},{"Start":"08:51.719 ","End":"08:55.035","Text":"what is f of C,"},{"Start":"08:55.035 ","End":"08:56.820","Text":"and what is f of D,"},{"Start":"08:56.820 ","End":"08:59.920","Text":"and then look for the largest and the smallest."},{"Start":"09:00.620 ","End":"09:06.995","Text":"That\u0027s the f is x squared y. I need x squared y for each of these,"},{"Start":"09:06.995 ","End":"09:09.110","Text":"2 squared times 1,"},{"Start":"09:09.110 ","End":"09:10.700","Text":"I\u0027ll just write one of them in full,"},{"Start":"09:10.700 ","End":"09:13.795","Text":"2 squared times 1 is 4."},{"Start":"09:13.795 ","End":"09:19.120","Text":"Here, negative 2 squared times 1 is also 4."},{"Start":"09:19.220 ","End":"09:27.300","Text":"Here, negative 2 squared times minus 1 is minus 4,"},{"Start":"09:27.300 ","End":"09:31.380","Text":"and 2 squared times minus 1 is also minus 4."},{"Start":"09:31.380 ","End":"09:36.735","Text":"So both of these 2 are minimum,"},{"Start":"09:36.735 ","End":"09:40.050","Text":"and both of these 2 are maximum."},{"Start":"09:40.050 ","End":"09:42.020","Text":"Let me just write that here,"},{"Start":"09:42.020 ","End":"09:45.800","Text":"that this here is a,"},{"Start":"09:46.090 ","End":"09:52.885","Text":"let\u0027s see, the 2,1, was A, these 2."},{"Start":"09:52.885 ","End":"09:56.645","Text":"I\u0027ll just write max for short, for maximum."},{"Start":"09:56.645 ","End":"09:58.760","Text":"Everyone does that. That\u0027s max,"},{"Start":"09:58.760 ","End":"10:01.340","Text":"max, that\u0027s min, min."},{"Start":"10:01.340 ","End":"10:03.200","Text":"It\u0027s a tie for minimum,"},{"Start":"10:03.200 ","End":"10:05.030","Text":"there\u0027s a value here."},{"Start":"10:05.030 ","End":"10:06.440","Text":"Both of these is 4,"},{"Start":"10:06.440 ","End":"10:08.270","Text":"both of these is minus 4."},{"Start":"10:08.270 ","End":"10:11.855","Text":"These are the 4 extrema."},{"Start":"10:11.855 ","End":"10:15.020","Text":"Just to be complete,"},{"Start":"10:15.020 ","End":"10:25.080","Text":"I\u0027ll write that these are maxima but under constraint. Then we\u0027re done."}],"ID":9669},{"Watched":false,"Name":"Exercise 5 part a","Duration":"13m 27s","ChapterTopicVideoID":9780,"CourseChapterTopicPlaylistID":114745,"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":"This exercise has 2 parts."},{"Start":"00:02.370 ","End":"00:04.080","Text":"We\u0027ll start with part a."},{"Start":"00:04.080 ","End":"00:06.270","Text":"Part a is an extremum problem."},{"Start":"00:06.270 ","End":"00:08.625","Text":"Extremum means maximum or minimum,"},{"Start":"00:08.625 ","End":"00:11.925","Text":"and in this case, it\u0027s a maximum function,"},{"Start":"00:11.925 ","End":"00:15.315","Text":"a maximum of this function xy,"},{"Start":"00:15.315 ","End":"00:19.815","Text":"and subject to, this is the constraint."},{"Start":"00:19.815 ","End":"00:22.785","Text":"We can also assume that x and y are positive."},{"Start":"00:22.785 ","End":"00:26.070","Text":"What I want to do is define,"},{"Start":"00:26.070 ","End":"00:29.955","Text":"first of all, a function f of x, y."},{"Start":"00:29.955 ","End":"00:32.400","Text":"That\u0027s the target function."},{"Start":"00:32.400 ","End":"00:38.070","Text":"The one I want to maximize or minimize, that is xy."},{"Start":"00:38.070 ","End":"00:44.330","Text":"I also want to define a function g of x, y,"},{"Start":"00:44.330 ","End":"00:46.205","Text":"which is the constraint,"},{"Start":"00:46.205 ","End":"00:48.245","Text":"but not as an equation,"},{"Start":"00:48.245 ","End":"00:53.705","Text":"as the function I get if I bring everything to the left-hand side."},{"Start":"00:53.705 ","End":"01:00.510","Text":"That is equal to x plus 3y minus 12."},{"Start":"01:00.510 ","End":"01:02.445","Text":"Here\u0027s the function f,"},{"Start":"01:02.445 ","End":"01:08.190","Text":"and the constraint is g equals 0 because I bring the 12 to the other side."},{"Start":"01:08.190 ","End":"01:10.954","Text":"Now, what we\u0027re going to need,"},{"Start":"01:10.954 ","End":"01:13.370","Text":"and this is a preparation,"},{"Start":"01:13.370 ","End":"01:17.555","Text":"are the partial derivatives of f and g up to second order."},{"Start":"01:17.555 ","End":"01:19.130","Text":"We have the functions themselves."},{"Start":"01:19.130 ","End":"01:22.130","Text":"Let\u0027s just continue, f with respect to x,"},{"Start":"01:22.130 ","End":"01:24.560","Text":"I won\u0027t keep writing of x and y,"},{"Start":"01:24.560 ","End":"01:32.460","Text":"this is equal to y. f with respect to y is x."},{"Start":"01:32.460 ","End":"01:38.280","Text":"Now, second order is 3 of them, fxx is,"},{"Start":"01:38.280 ","End":"01:41.700","Text":"this with respect to x is 0,"},{"Start":"01:41.700 ","End":"01:49.370","Text":"fxy is equal to the derivative of this with respect to y is 1."},{"Start":"01:49.370 ","End":"01:51.920","Text":"You might think there\u0027s also an fyx,"},{"Start":"01:51.920 ","End":"01:55.100","Text":"and there is, but it turns out to be the same as fxy."},{"Start":"01:55.100 ","End":"01:58.685","Text":"There\u0027s only 3 second order partial derivatives,"},{"Start":"01:58.685 ","End":"02:01.340","Text":"fyy being the last one,"},{"Start":"02:01.340 ","End":"02:05.695","Text":"is this with respect to y and that\u0027s 0."},{"Start":"02:05.695 ","End":"02:14.505","Text":"Now, let\u0027s do the same thing for g. g with respect to x is equal to just 1,"},{"Start":"02:14.505 ","End":"02:22.395","Text":"g with respect to y is 3, gxx is this."},{"Start":"02:22.395 ","End":"02:24.330","Text":"Well, these are both constants."},{"Start":"02:24.330 ","End":"02:27.339","Text":"All these 3 are going to be 0,"},{"Start":"02:28.280 ","End":"02:34.055","Text":"and gyy is 0."},{"Start":"02:34.055 ","End":"02:38.120","Text":"Next step is to find critical points."},{"Start":"02:38.120 ","End":"02:41.095","Text":"There are 3 equations."},{"Start":"02:41.095 ","End":"02:49.225","Text":"In general, it\u0027s always that fx equals Lambda times gx,"},{"Start":"02:49.225 ","End":"02:51.635","Text":"partial derivatives with respect to x."},{"Start":"02:51.635 ","End":"02:56.700","Text":"Same thing with y Lambda gy,"},{"Start":"02:56.700 ","End":"03:01.805","Text":"and the third equation is always the constraint,"},{"Start":"03:01.805 ","End":"03:04.234","Text":"the g equals 0."},{"Start":"03:04.234 ","End":"03:09.035","Text":"Or if you like, the original constraint in its original form."},{"Start":"03:09.035 ","End":"03:18.340","Text":"In our case, what we\u0027ll get from these is the following: fx is y,"},{"Start":"03:18.340 ","End":"03:22.770","Text":"and this is equal to Lambda times,"},{"Start":"03:22.770 ","End":"03:26.155","Text":"lets see, gx is 1."},{"Start":"03:26.155 ","End":"03:30.950","Text":"The second equation says that this is Lambda times this,"},{"Start":"03:30.950 ","End":"03:36.240","Text":"so x is Lambda times 3."},{"Start":"03:36.620 ","End":"03:41.600","Text":"The last one, I can write it that this equals 0."},{"Start":"03:41.600 ","End":"03:44.795","Text":"Sometimes I like to just write it as it was originally,"},{"Start":"03:44.795 ","End":"03:48.240","Text":"x plus 3y equals 12."},{"Start":"03:48.640 ","End":"03:51.620","Text":"Now, the usual technique,"},{"Start":"03:51.620 ","End":"03:56.810","Text":"what we do is we divide these 2 equations and that gets rid of Lambda."},{"Start":"03:56.810 ","End":"04:04.100","Text":"So I get that y over x is equal"},{"Start":"04:04.100 ","End":"04:12.550","Text":"to Lambda times 1 over Lambda times 3."},{"Start":"04:12.550 ","End":"04:16.130","Text":"Now, because we\u0027re not allowed to divide by 0,"},{"Start":"04:16.130 ","End":"04:21.895","Text":"I have to check separately for the possibility that something in the denominator is 0."},{"Start":"04:21.895 ","End":"04:25.575","Text":"Let\u0027s see, is it possible that Lambda is 0?"},{"Start":"04:25.575 ","End":"04:28.805","Text":"Well, if Lambda equals 0,"},{"Start":"04:28.805 ","End":"04:33.760","Text":"then what we get if we look at these 2 equations."},{"Start":"04:33.760 ","End":"04:38.040","Text":"In fact, these are the 2 that gave birth to this one."},{"Start":"04:38.040 ","End":"04:39.990","Text":"If Lambda equals 0,"},{"Start":"04:39.990 ","End":"04:42.795","Text":"then it means that y is 0 from here,"},{"Start":"04:42.795 ","End":"04:45.285","Text":"and x is 0 from here,"},{"Start":"04:45.285 ","End":"04:52.400","Text":"and that gives us that 0 plus 3 times 0 equals 12."},{"Start":"04:52.400 ","End":"04:57.730","Text":"In other words, we get that 0 equals 12 which is impossible."},{"Start":"04:57.730 ","End":"05:06.070","Text":"This possibility is ruled out that Lambda is not equal to 0. That\u0027s okay."},{"Start":"05:06.070 ","End":"05:10.190","Text":"Now, let\u0027s see if it\u0027s possible that x is 0."},{"Start":"05:10.190 ","End":"05:13.415","Text":"Well, if x is 0,"},{"Start":"05:13.415 ","End":"05:21.175","Text":"then that means that if I divide this by 3 Lambda 0, then Lambda is 0."},{"Start":"05:21.175 ","End":"05:26.750","Text":"If Lambda is 0, then we already know that that\u0027s a contradiction."},{"Start":"05:26.750 ","End":"05:29.110","Text":"This implies that Lambda equals 0,"},{"Start":"05:29.110 ","End":"05:31.145","Text":"which I can continue over here,"},{"Start":"05:31.145 ","End":"05:33.620","Text":"implies that 0 equals 12."},{"Start":"05:33.620 ","End":"05:37.075","Text":"This is also ruled out."},{"Start":"05:37.075 ","End":"05:40.945","Text":"What we\u0027ve divided by is not 0."},{"Start":"05:40.945 ","End":"05:44.690","Text":"Sometimes these things are"},{"Start":"05:44.690 ","End":"05:47.840","Text":"not ruled out and then you get extra possibilities for critical points."},{"Start":"05:47.840 ","End":"05:49.295","Text":"But in this case not."},{"Start":"05:49.295 ","End":"05:52.310","Text":"This one, it\u0027s always the same."},{"Start":"05:52.310 ","End":"05:56.435","Text":"We cancel by Lambda and then cross multiply."},{"Start":"05:56.435 ","End":"05:59.015","Text":"If we cross multiply,"},{"Start":"05:59.015 ","End":"06:07.680","Text":"we get that this diagonal which is x is equal to the other diagonal which is 3y."},{"Start":"06:07.680 ","End":"06:10.580","Text":"Now we have 2 equations and 2 unknowns,"},{"Start":"06:10.580 ","End":"06:17.210","Text":"x and y. I can substitute this in here."},{"Start":"06:17.210 ","End":"06:20.015","Text":"If I substitute this in that equation,"},{"Start":"06:20.015 ","End":"06:24.370","Text":"we now get that 3y plus 3y equals 12,"},{"Start":"06:24.370 ","End":"06:27.484","Text":"or 6y equals 12,"},{"Start":"06:27.484 ","End":"06:30.635","Text":"or y equals 2."},{"Start":"06:30.635 ","End":"06:36.295","Text":"Once we have y, we can now find x by putting that in here."},{"Start":"06:36.295 ","End":"06:43.770","Text":"That gives us that x is 3 times 2 and this is 6."},{"Start":"06:43.770 ","End":"06:45.855","Text":"Now we can find Lambda."},{"Start":"06:45.855 ","End":"06:47.670","Text":"That\u0027s just something I want to mention,"},{"Start":"06:47.670 ","End":"06:48.915","Text":"a kind of a shortcut."},{"Start":"06:48.915 ","End":"06:52.685","Text":"If these 3 happened to all be 0,"},{"Start":"06:52.685 ","End":"06:54.800","Text":"we don\u0027t need to find Lambda."},{"Start":"06:54.800 ","End":"06:58.015","Text":"You\u0027ll see later and why."},{"Start":"06:58.015 ","End":"06:59.940","Text":"But it\u0027s good practice,"},{"Start":"06:59.940 ","End":"07:03.350","Text":"and that we won\u0027t need Lambda in this particular example,"},{"Start":"07:03.350 ","End":"07:05.705","Text":"in general, we do need Lambda."},{"Start":"07:05.705 ","End":"07:07.820","Text":"We\u0027ll need it later when we do"},{"Start":"07:07.820 ","End":"07:12.950","Text":"a second derivative test to determine what kind of extremum it is."},{"Start":"07:12.950 ","End":"07:15.080","Text":"To get Lambda, well,"},{"Start":"07:15.080 ","End":"07:16.730","Text":"I could use either one of these."},{"Start":"07:16.730 ","End":"07:22.295","Text":"For example, if I hit the top one is maybe the easiest, y equals Lambda."},{"Start":"07:22.295 ","End":"07:24.170","Text":"If y equals Lambda,"},{"Start":"07:24.170 ","End":"07:30.845","Text":"then that means that Lambda equals y and Lambda therefore equals 2."},{"Start":"07:30.845 ","End":"07:32.690","Text":"Could\u0027ve done it from the second,"},{"Start":"07:32.690 ","End":"07:34.940","Text":"6 equals 3 Lambda."},{"Start":"07:34.940 ","End":"07:38.990","Text":"That\u0027s just for check and also gives me Lambda equals 2,"},{"Start":"07:38.990 ","End":"07:41.000","Text":"so I\u0027ve double-checked that."},{"Start":"07:41.000 ","End":"07:44.314","Text":"Now I have my critical point."},{"Start":"07:44.314 ","End":"07:47.055","Text":"The critical point is 6,"},{"Start":"07:47.055 ","End":"07:50.440","Text":"2, and there\u0027s only 1 critical point."},{"Start":"07:50.510 ","End":"07:55.035","Text":"Let us write the word critical point,"},{"Start":"07:55.035 ","End":"08:00.180","Text":"and we also have our Lambda which will come in handy."},{"Start":"08:00.180 ","End":"08:02.875","Text":"Now we proceed."},{"Start":"08:02.875 ","End":"08:09.545","Text":"We have to compute a quantity called H. H stands for Hessian but doesn\u0027t matter,"},{"Start":"08:09.545 ","End":"08:11.840","Text":"and it\u0027s a horrible expression."},{"Start":"08:11.840 ","End":"08:16.670","Text":"I don\u0027t believe you\u0027ll be asked to remember it."},{"Start":"08:16.670 ","End":"08:21.005","Text":"It\u0027s usually given on the formula sheet and it\u0027s equal to,"},{"Start":"08:21.005 ","End":"08:24.275","Text":"and I\u0027m just going to write it in shorthand."},{"Start":"08:24.275 ","End":"08:28.760","Text":"I\u0027m not going to write fxx of x and y but just like this,"},{"Start":"08:28.760 ","End":"08:37.895","Text":"minus Lambda gxx times gy squared,"},{"Start":"08:37.895 ","End":"08:39.470","Text":"that\u0027s just the beginning."},{"Start":"08:39.470 ","End":"08:43.725","Text":"Next, we have the same thing with x and y reversed."},{"Start":"08:43.725 ","End":"08:51.420","Text":"It\u0027s fyy minus Lambda gyy."},{"Start":"08:53.300 ","End":"08:57.825","Text":"Then instead of this gx squared,"},{"Start":"08:57.825 ","End":"09:07.295","Text":"next comes a minus 2 and after the minus 2 we have a hybrid between these 2."},{"Start":"09:07.295 ","End":"09:09.335","Text":"Instead of fxx or fyy,"},{"Start":"09:09.335 ","End":"09:12.560","Text":"we have fxy mixed,"},{"Start":"09:12.560 ","End":"09:15.035","Text":"and here we have minus Lambda,"},{"Start":"09:15.035 ","End":"09:19.679","Text":"also gxy, a mixture."},{"Start":"09:19.679 ","End":"09:21.900","Text":"Instead of gy squared or gx squared,"},{"Start":"09:21.900 ","End":"09:25.765","Text":"we get 1 of each, we get gx gy."},{"Start":"09:25.765 ","End":"09:28.430","Text":"I\u0027m saying this as if you have to remember it,"},{"Start":"09:28.430 ","End":"09:35.775","Text":"but no reasonable professor would ask you to memorize this."},{"Start":"09:35.775 ","End":"09:42.625","Text":"What I want to do is find H at our given point."},{"Start":"09:42.625 ","End":"09:49.710","Text":"I want to compute H at the critical point 6, 2."},{"Start":"09:50.510 ","End":"09:53.870","Text":"If this comes out to be positive,"},{"Start":"09:53.870 ","End":"09:57.320","Text":"it\u0027s a minimum, and negative, it\u0027s a maximum."},{"Start":"09:57.320 ","End":"10:00.260","Text":"Also, I want to relate to something I said earlier."},{"Start":"10:00.260 ","End":"10:03.515","Text":"I said something about if all these 3 things are 0,"},{"Start":"10:03.515 ","End":"10:07.925","Text":"if these are 0, notice that these are all the things that go with Lambda,"},{"Start":"10:07.925 ","End":"10:11.024","Text":"that this is 0, this is 0,"},{"Start":"10:11.024 ","End":"10:14.815","Text":"and this is 0, and that\u0027s why I said that we don\u0027t need Lambda,"},{"Start":"10:14.815 ","End":"10:17.670","Text":"but I did it for practice anyway."},{"Start":"10:18.140 ","End":"10:25.160","Text":"Now I just have to substitute all these quantities for our particular point."},{"Start":"10:25.160 ","End":"10:31.420","Text":"Notice that all these 12 partial derivatives."},{"Start":"10:31.420 ","End":"10:34.080","Text":"Well, only 10 if you don\u0027t count."},{"Start":"10:34.080 ","End":"10:36.620","Text":"They\u0027re almost all constants."},{"Start":"10:36.620 ","End":"10:39.395","Text":"The only place it\u0027s not a constant is here,"},{"Start":"10:39.395 ","End":"10:44.760","Text":"so I just want to write what these are at the point 6, 2."},{"Start":"10:44.760 ","End":"10:51.935","Text":"At 6, 2, this is equal to 2 because it\u0027s x,"},{"Start":"10:51.935 ","End":"10:54.170","Text":"y. The y is 2."},{"Start":"10:54.170 ","End":"10:56.150","Text":"Here, at the point 6,"},{"Start":"10:56.150 ","End":"10:59.030","Text":"2, this is equal to 6."},{"Start":"10:59.030 ","End":"11:02.675","Text":"Maybe I\u0027ll just write that at the point 6,"},{"Start":"11:02.675 ","End":"11:05.745","Text":"2, where x is 6 and y is 2."},{"Start":"11:05.745 ","End":"11:07.365","Text":"All the rest are constants,"},{"Start":"11:07.365 ","End":"11:16.000","Text":"so I can say that this equals fxx is written here as 0."},{"Start":"11:17.320 ","End":"11:20.585","Text":"I\u0027ll write it minus 0,"},{"Start":"11:20.585 ","End":"11:30.555","Text":"gy squared times 3 squared plus fyy is also 0."},{"Start":"11:30.555 ","End":"11:37.850","Text":"That we said is 0 and g with respect to x is 1 squared."},{"Start":"11:37.850 ","End":"11:42.455","Text":"But it\u0027s not going to matter because it\u0027s 0 anyway, minus twice."},{"Start":"11:42.455 ","End":"11:47.205","Text":"Let\u0027s see, fxy is 1, it\u0027s not 0."},{"Start":"11:47.205 ","End":"11:54.780","Text":"This part is 0, and gx gy is this times this,"},{"Start":"11:54.780 ","End":"11:59.115","Text":"is 1 times 3."},{"Start":"11:59.115 ","End":"12:02.760","Text":"Now 0 minus 0 is 0, so all this is 0."},{"Start":"12:02.760 ","End":"12:05.680","Text":"Again, this is 0."},{"Start":"12:05.690 ","End":"12:11.235","Text":"All we\u0027re left with is minus 2 times 1 times 1 times 3,"},{"Start":"12:11.235 ","End":"12:13.995","Text":"what I get is minus 6."},{"Start":"12:13.995 ","End":"12:19.250","Text":"Minus 6 is most certainly negative, less than 0."},{"Start":"12:19.250 ","End":"12:21.785","Text":"When it\u0027s less than 0,"},{"Start":"12:21.785 ","End":"12:29.910","Text":"then that means that our critical point is a maximum."},{"Start":"12:32.620 ","End":"12:40.350","Text":"What is the actual maximum value of f, which is xy?"},{"Start":"12:40.350 ","End":"12:45.330","Text":"Is the value equals 12, 6 times 2."},{"Start":"12:45.330 ","End":"12:50.610","Text":"6, 2 is the maximum,"},{"Start":"12:50.610 ","End":"12:54.739","Text":"and the maximum value is actually 12."},{"Start":"12:54.739 ","End":"12:58.235","Text":"There\u0027s something small that I forgot to do is just to check"},{"Start":"12:58.235 ","End":"13:01.760","Text":"that this condition is satisfied."},{"Start":"13:01.760 ","End":"13:07.909","Text":"Certainly, 6 and 2 are both bigger than 0, so that\u0027s okay."},{"Start":"13:07.909 ","End":"13:09.890","Text":"Just as an extra check,"},{"Start":"13:09.890 ","End":"13:15.860","Text":"I\u0027d like to check that this does satisfy the constraint x plus 3y,"},{"Start":"13:15.860 ","End":"13:20.249","Text":"6 plus 3 times 2 is indeed 12,"},{"Start":"13:20.249 ","End":"13:22.875","Text":"so that\u0027s an extra verification,"},{"Start":"13:22.875 ","End":"13:27.400","Text":"and that concludes part a."}],"ID":9670},{"Watched":false,"Name":"Exercise 5 part b","Duration":"12m 16s","ChapterTopicVideoID":9772,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.029","Text":"Now let\u0027s move on to Part B. I\u0027m going to erase what I don\u0027t need."},{"Start":"00:06.029 ","End":"00:08.355","Text":"I just kept the answer,"},{"Start":"00:08.355 ","End":"00:10.390","Text":"let me write it up here."},{"Start":"00:10.390 ","End":"00:13.580","Text":"The xy, which give the maximum,"},{"Start":"00:13.580 ","End":"00:15.585","Text":"sometimes we write it with an asterisk,"},{"Start":"00:15.585 ","End":"00:17.250","Text":"special xy or 6,2,"},{"Start":"00:17.250 ","End":"00:22.830","Text":"and the actual value of that xy was 12."},{"Start":"00:22.830 ","End":"00:28.455","Text":"Also allow me to just change this word interpret to illustrate,"},{"Start":"00:28.455 ","End":"00:30.960","Text":"and here we are."},{"Start":"00:30.960 ","End":"00:38.870","Text":"The idea for the illustration is to use the concept of level curves."},{"Start":"00:38.870 ","End":"00:42.290","Text":"The level curves of this function,"},{"Start":"00:42.290 ","End":"00:45.810","Text":"the target function that we want to maximize."},{"Start":"00:46.030 ","End":"00:53.810","Text":"The level curves of this function f will be of the form f of xy."},{"Start":"00:53.810 ","End":"01:00.500","Text":"In other words, xy equals some constant k. But we know that k is going"},{"Start":"01:00.500 ","End":"01:07.370","Text":"to be bigger than 0 because we\u0027re talking about only the first quadrant,"},{"Start":"01:07.370 ","End":"01:08.540","Text":"x and y are positive,"},{"Start":"01:08.540 ","End":"01:10.670","Text":"so the product is going to be positive."},{"Start":"01:10.670 ","End":"01:12.290","Text":"For different values of k,"},{"Start":"01:12.290 ","End":"01:14.875","Text":"we get different level curves."},{"Start":"01:14.875 ","End":"01:18.710","Text":"We\u0027re going to want to sketch a few of these level curves."},{"Start":"01:18.710 ","End":"01:22.805","Text":"Let\u0027s investigate what a typical 1 looks like."},{"Start":"01:22.805 ","End":"01:30.235","Text":"I\u0027ll write it in the form y equals k/x."},{"Start":"01:30.235 ","End":"01:33.589","Text":"I\u0027ll do a quick investigation before sketching,"},{"Start":"01:33.589 ","End":"01:35.900","Text":"let\u0027s see what I want to do."},{"Start":"01:35.900 ","End":"01:45.590","Text":"I want to find the places where it\u0027s increasing or decreasing."},{"Start":"01:45.590 ","End":"01:47.810","Text":"What else do we do?"},{"Start":"01:47.810 ","End":"01:52.710","Text":"We want to find where it\u0027s concave up,"},{"Start":"01:54.020 ","End":"01:59.100","Text":"and concave down, or concave convex."},{"Start":"01:59.100 ","End":"02:07.165","Text":"I want intersection with the axes."},{"Start":"02:07.165 ","End":"02:10.235","Text":"If you\u0027ve learned about asymptotes,"},{"Start":"02:10.235 ","End":"02:14.130","Text":"maybe plot a couple of points."},{"Start":"02:14.600 ","End":"02:18.350","Text":"Well, we\u0027re looking at increasing and decreasing,"},{"Start":"02:18.350 ","End":"02:22.145","Text":"maybe also extrema, maximum and minimum."},{"Start":"02:22.145 ","End":"02:24.829","Text":"I\u0027m just going to do these quickly."},{"Start":"02:24.829 ","End":"02:27.140","Text":"You just want to get a general idea."},{"Start":"02:27.140 ","End":"02:30.740","Text":"We\u0027ll need the derivative,"},{"Start":"02:30.740 ","End":"02:33.740","Text":"first and second-order for investigation."},{"Start":"02:33.740 ","End":"02:42.140","Text":"Let\u0027s see, y prime is minus k over x squared."},{"Start":"02:42.140 ","End":"02:45.200","Text":"We can assume k is of some specific fixed number."},{"Start":"02:45.200 ","End":"02:46.910","Text":"I could take k equals 1,"},{"Start":"02:46.910 ","End":"02:54.655","Text":"but it\u0027s pretty much the same work for a general k. We also have the y double prime,"},{"Start":"02:54.655 ","End":"02:57.180","Text":"let\u0027s say it is minus k, x is the minus 2,"},{"Start":"02:57.180 ","End":"03:02.715","Text":"so it\u0027s minus 2k x to the minus 3."},{"Start":"03:02.715 ","End":"03:05.100","Text":"It\u0027s going to be plus."},{"Start":"03:05.100 ","End":"03:09.450","Text":"Now let\u0027s take this point-by-point for increasing, decreasing, and extrema,"},{"Start":"03:09.450 ","End":"03:16.835","Text":"I need to know where y prime is bigger than 0 or less than 0."},{"Start":"03:16.835 ","End":"03:22.435","Text":"Y prime is always negative because k is a positive number,"},{"Start":"03:22.435 ","End":"03:25.605","Text":"and x squared is positive."},{"Start":"03:25.605 ","End":"03:31.780","Text":"Actually y prime is less than 0 always."},{"Start":"03:32.390 ","End":"03:37.965","Text":"That means it\u0027s always decreasing."},{"Start":"03:37.965 ","End":"03:42.015","Text":"There\u0027s not going to be any maximum or minimum,"},{"Start":"03:42.015 ","End":"03:46.640","Text":"because maximum, minimum are the border between increasing and decreasing."},{"Start":"03:46.640 ","End":"03:50.020","Text":"Let\u0027s see as for concave up down,"},{"Start":"03:50.020 ","End":"03:55.489","Text":"we look at the second derivative, y double prime."},{"Start":"03:55.489 ","End":"03:57.575","Text":"Since x is positive,"},{"Start":"03:57.575 ","End":"04:01.360","Text":"this is always going to be bigger than 0."},{"Start":"04:01.360 ","End":"04:11.100","Text":"It\u0027s going to always be concave up, something like this."},{"Start":"04:11.100 ","End":"04:13.095","Text":"Intersection with the axes,"},{"Start":"04:13.095 ","End":"04:15.600","Text":"I claim that there are none."},{"Start":"04:15.600 ","End":"04:17.940","Text":"Intersection with the y-axis,"},{"Start":"04:17.940 ","End":"04:20.074","Text":"we would let x equal 0,"},{"Start":"04:20.074 ","End":"04:21.725","Text":"and that\u0027s not possible."},{"Start":"04:21.725 ","End":"04:24.320","Text":"In fact, neither x nor y can be 0,"},{"Start":"04:24.320 ","End":"04:26.360","Text":"because that would force k to be 0,"},{"Start":"04:26.360 ","End":"04:27.470","Text":"but k is positive,"},{"Start":"04:27.470 ","End":"04:29.660","Text":"so there are no intersections,"},{"Start":"04:29.660 ","End":"04:31.475","Text":"but there are asymptotes."},{"Start":"04:31.475 ","End":"04:36.410","Text":"The vertical asymptote is when you let x go to 0."},{"Start":"04:36.410 ","End":"04:38.600","Text":"If x goes to 0,"},{"Start":"04:38.600 ","End":"04:39.860","Text":"since x is positive,"},{"Start":"04:39.860 ","End":"04:47.840","Text":"it goes to 0 plus that implies that y goes to infinity,"},{"Start":"04:47.840 ","End":"04:50.515","Text":"and I\u0027ll just emphasize plus infinity."},{"Start":"04:50.515 ","End":"04:56.610","Text":"We have a vertical asymptote at 0 that the function goes to infinity."},{"Start":"04:56.810 ","End":"05:02.325","Text":"For horizontal, we let x go to infinity."},{"Start":"05:02.325 ","End":"05:05.140","Text":"In this case only to plus infinity,"},{"Start":"05:05.140 ","End":"05:06.835","Text":"because x is positive,"},{"Start":"05:06.835 ","End":"05:08.364","Text":"and if this happens,"},{"Start":"05:08.364 ","End":"05:13.785","Text":"then we get that y is 1 over plus infinity, so y is 0."},{"Start":"05:13.785 ","End":"05:18.610","Text":"That\u0027s the horizontal. In other words,"},{"Start":"05:18.610 ","End":"05:23.810","Text":"the axes are the asymptotes."},{"Start":"05:24.290 ","End":"05:27.329","Text":"As for plotting some points,"},{"Start":"05:27.329 ","End":"05:30.405","Text":"if I want product of 2 numbers to be k,"},{"Start":"05:30.405 ","End":"05:34.035","Text":"I could sketch the point 1,k,"},{"Start":"05:34.035 ","End":"05:38.070","Text":"and I could sketch the point k,1."},{"Start":"05:38.070 ","End":"05:41.700","Text":"Those would be easy to plug in."},{"Start":"05:41.700 ","End":"05:52.415","Text":"I take a vertical axis and the horizontal axis, the y-axis, x-axis."},{"Start":"05:52.415 ","End":"05:56.675","Text":"I know that the thing is decreasing in concave up."},{"Start":"05:56.675 ","End":"06:00.605","Text":"Let\u0027s say this is the point 1,k,"},{"Start":"06:00.605 ","End":"06:04.055","Text":"and this is the point k,1,"},{"Start":"06:04.055 ","End":"06:07.370","Text":"and it goes towards infinity."},{"Start":"06:07.370 ","End":"06:08.840","Text":"The y-axis is an asymptote,"},{"Start":"06:08.840 ","End":"06:10.130","Text":"so comes from here,"},{"Start":"06:10.130 ","End":"06:11.780","Text":"goes through here and here,"},{"Start":"06:11.780 ","End":"06:15.885","Text":"and then asymptotes down to 0."},{"Start":"06:15.885 ","End":"06:22.650","Text":"There are going to be several curves like this 1 for each k. In fact,"},{"Start":"06:22.650 ","End":"06:31.065","Text":"notice that k is f of x and y,"},{"Start":"06:31.065 ","End":"06:36.905","Text":"and we can tell in which direction k increases."},{"Start":"06:36.905 ","End":"06:43.850","Text":"The way we do this is we take the gradient vector and that\u0027s always"},{"Start":"06:43.850 ","End":"06:51.725","Text":"perpendicular to the curve and shows the direction in which the constant increases."},{"Start":"06:51.725 ","End":"06:55.970","Text":"We take gradient of f,"},{"Start":"06:55.970 ","End":"06:57.860","Text":"and that is in general,"},{"Start":"06:57.860 ","End":"07:00.440","Text":"the derivative of f with respect to x,"},{"Start":"07:00.440 ","End":"07:03.440","Text":"derivative of f with respect to y,"},{"Start":"07:03.440 ","End":"07:12.785","Text":"and if we look at f. It\u0027s easy to see that this is just with respect to x, it\u0027s just y."},{"Start":"07:12.785 ","End":"07:16.270","Text":"With respect to y, it\u0027s x,"},{"Start":"07:16.270 ","End":"07:19.450","Text":"and these are both positive."},{"Start":"07:20.900 ","End":"07:23.599","Text":"Let\u0027s just take, for example,"},{"Start":"07:23.599 ","End":"07:32.735","Text":"the gradient vector at the point k,1 is just going to be 1,k,"},{"Start":"07:32.735 ","End":"07:34.940","Text":"something in this direction."},{"Start":"07:34.940 ","End":"07:36.830","Text":"What I\u0027m saying is that there\u0027s going to"},{"Start":"07:36.830 ","End":"07:42.320","Text":"be several curves 1 for each k. There might be another 1 here,"},{"Start":"07:42.320 ","End":"07:44.060","Text":"might be another 1 here."},{"Start":"07:44.060 ","End":"07:50.285","Text":"The ones that are more outward are the ones with the larger value of k. Now,"},{"Start":"07:50.285 ","End":"07:51.830","Text":"this is a pretty messy picture."},{"Start":"07:51.830 ","End":"07:53.850","Text":"Let me go and bring a better 1."},{"Start":"07:53.850 ","End":"07:55.610","Text":"I found 1 on the Internet."},{"Start":"07:55.610 ","End":"07:59.640","Text":"Here\u0027s a picture that\u0027s a bit nicer."},{"Start":"07:59.680 ","End":"08:05.965","Text":"This curve is the curve where k equals 3."},{"Start":"08:05.965 ","End":"08:12.045","Text":"You can see the point 3,1 is here and the point 1,3 is here."},{"Start":"08:12.045 ","End":"08:19.945","Text":"This is the curve where k equals 6."},{"Start":"08:19.945 ","End":"08:25.105","Text":"This is the curve where k equals 12."},{"Start":"08:25.105 ","End":"08:30.640","Text":"This 1 is k equals 18."},{"Start":"08:30.710 ","End":"08:40.040","Text":"As we noticed here the ks increase as we go upwards and rightwards outwardly,"},{"Start":"08:40.040 ","End":"08:43.175","Text":"the values of k increase."},{"Start":"08:43.175 ","End":"08:46.015","Text":"I want to mark a special point,"},{"Start":"08:46.015 ","End":"08:50.270","Text":"remember in the first part of the exercise,"},{"Start":"08:50.270 ","End":"08:58.955","Text":"we found that our maximum point was 6,2 and k for it was equal to 12."},{"Start":"08:58.955 ","End":"09:03.605","Text":"Let\u0027s mark that point on our sketch."},{"Start":"09:03.605 ","End":"09:07.410","Text":"This would be the point, roughly 6,2."},{"Start":"09:10.340 ","End":"09:14.355","Text":"That\u0027s the answer we got,"},{"Start":"09:14.355 ","End":"09:20.640","Text":"and 6 times 2 equals 12, that\u0027s the xy."},{"Start":"09:20.640 ","End":"09:25.590","Text":"It\u0027s on this turquoise colored curve."},{"Start":"09:25.590 ","End":"09:27.530","Text":"The thing that\u0027s missing,"},{"Start":"09:27.530 ","End":"09:31.340","Text":"and probably I should have done it first was the constraint."},{"Start":"09:31.340 ","End":"09:36.110","Text":"The constraint was x plus 3y equals 12,"},{"Start":"09:36.110 ","End":"09:37.760","Text":"if you remember."},{"Start":"09:37.760 ","End":"09:40.415","Text":"Let\u0027s sketch this."},{"Start":"09:40.415 ","End":"09:44.540","Text":"Let\u0027s do it by the intersection with the axes."},{"Start":"09:44.540 ","End":"09:48.160","Text":"If x is 0,"},{"Start":"09:48.160 ","End":"09:49.900","Text":"then 3y is 12,"},{"Start":"09:49.900 ","End":"09:52.190","Text":"so y is 4."},{"Start":"09:52.190 ","End":"09:58.975","Text":"That\u0027s here, but I\u0027ll put a hollow circle because it\u0027s not included there."},{"Start":"09:58.975 ","End":"10:01.534","Text":"We want x and y to be strictly positive."},{"Start":"10:01.534 ","End":"10:04.700","Text":"When y is 0, x equals 12."},{"Start":"10:04.700 ","End":"10:06.300","Text":"I don\u0027t quite have enough space,"},{"Start":"10:06.300 ","End":"10:08.715","Text":"let me extend this line."},{"Start":"10:08.715 ","End":"10:12.135","Text":"Let\u0027s say this is 12."},{"Start":"10:12.135 ","End":"10:15.675","Text":"The point 12,0 is here,"},{"Start":"10:15.675 ","End":"10:20.305","Text":"and our line segment is from here to here."},{"Start":"10:20.305 ","End":"10:24.350","Text":"This here is our line,"},{"Start":"10:24.350 ","End":"10:28.680","Text":"which is the constraint x plus 3y equals 12."},{"Start":"10:28.880 ","End":"10:36.020","Text":"Tell from here, but actually this constraint line actually is tangent."},{"Start":"10:36.020 ","End":"10:42.335","Text":"If I continue this a bit to this level curve of 12,"},{"Start":"10:42.335 ","End":"10:46.430","Text":"this 1 would cut it also somewhere here."},{"Start":"10:46.430 ","End":"10:50.525","Text":"The k equals 3 would also cut it somewhere."},{"Start":"10:50.525 ","End":"10:55.980","Text":"This 1 wouldn\u0027t intersect with this line at all."},{"Start":"10:55.980 ","End":"11:00.585","Text":"Basically, what we want to do is keep increasing k,"},{"Start":"11:00.585 ","End":"11:05.540","Text":"that we still have an intersection with the line and to get k as large as possible,"},{"Start":"11:05.540 ","End":"11:09.230","Text":"and the value 12 is the largest possible."},{"Start":"11:09.230 ","End":"11:12.230","Text":"It just grazes it is tangent."},{"Start":"11:12.230 ","End":"11:15.740","Text":"This is the illustration of what it"},{"Start":"11:15.740 ","End":"11:23.100","Text":"means to find the level curve with the largest possible level,"},{"Start":"11:23.100 ","End":"11:29.170","Text":"it would be the 1 that would just touch the constraint."},{"Start":"11:29.510 ","End":"11:36.155","Text":"Hopefully this illustration explains what we discovered algebraically."},{"Start":"11:36.155 ","End":"11:41.795","Text":"In practice, you would draw the constraint line first,"},{"Start":"11:41.795 ","End":"11:44.810","Text":"and then you\u0027d keep drawing level curves."},{"Start":"11:44.810 ","End":"11:49.760","Text":"Maybe you start low and then keep increasing until you just graze,"},{"Start":"11:49.760 ","End":"11:52.280","Text":"meaning tangent, and that would be the best,"},{"Start":"11:52.280 ","End":"11:55.205","Text":"the largest possible k that you would get."},{"Start":"11:55.205 ","End":"11:57.755","Text":"This 12 is not the same as this 12."},{"Start":"11:57.755 ","End":"12:03.590","Text":"This 12 is from the value of xy,6 times 2 is 12."},{"Start":"12:03.590 ","End":"12:07.070","Text":"That\u0027s just happens to be a 12 here,"},{"Start":"12:07.070 ","End":"12:09.265","Text":"so that something else."},{"Start":"12:09.265 ","End":"12:15.750","Text":"That\u0027s it for the illustration and I hope it gives you some intuition."}],"ID":9671},{"Watched":false,"Name":"Exercise 6 part a","Duration":"13m 20s","ChapterTopicVideoID":9773,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"We have here an extremum problem,"},{"Start":"00:02.640 ","End":"00:04.350","Text":"that just means maximum or minimum,"},{"Start":"00:04.350 ","End":"00:06.150","Text":"and in this case, it\u0027s minimum."},{"Start":"00:06.150 ","End":"00:11.430","Text":"Minimum of this function subject to this constraint."},{"Start":"00:11.430 ","End":"00:15.450","Text":"That\u0027s Part a and later we\u0027ll see about Part b."},{"Start":"00:15.450 ","End":"00:18.270","Text":"Now for Part a, we have to define some functions."},{"Start":"00:18.270 ","End":"00:20.010","Text":"First of all, the target function,"},{"Start":"00:20.010 ","End":"00:21.360","Text":"the 1 we want the minimum of,"},{"Start":"00:21.360 ","End":"00:28.560","Text":"we\u0027ll call that f. We have f of x and y equals 2x plus y."},{"Start":"00:28.560 ","End":"00:31.890","Text":"The constraint, we also write as a function."},{"Start":"00:31.890 ","End":"00:34.545","Text":"If I write it as something equals 0,"},{"Start":"00:34.545 ","End":"00:40.035","Text":"I can write that g of xy equals root x plus"},{"Start":"00:40.035 ","End":"00:47.240","Text":"root y minus 9 and then the constraint is of the form g equals 0."},{"Start":"00:47.240 ","End":"00:49.445","Text":"I just put the 9 to the other side."},{"Start":"00:49.445 ","End":"00:52.235","Text":"Notice that x and y are positive."},{"Start":"00:52.235 ","End":"00:55.700","Text":"There\u0027s certainly no problem with taking square roots."},{"Start":"00:55.700 ","End":"00:57.440","Text":"Now in this type of problem,"},{"Start":"00:57.440 ","End":"01:00.830","Text":"we always need the partial derivatives up to second-order."},{"Start":"01:00.830 ","End":"01:03.590","Text":"Let\u0027s do that technical part now."},{"Start":"01:03.590 ","End":"01:08.150","Text":"In fact, why don\u0027t I rewrite the square root of"},{"Start":"01:08.150 ","End":"01:13.080","Text":"x and the square root of y as a fractional exponent."},{"Start":"01:13.080 ","End":"01:14.870","Text":"Instead of square root,"},{"Start":"01:14.870 ","End":"01:17.510","Text":"I\u0027ll take to the power of 1/2,"},{"Start":"01:17.510 ","End":"01:20.085","Text":"that will be easier for differentiating."},{"Start":"01:20.085 ","End":"01:21.780","Text":"Let\u0027s do f first."},{"Start":"01:21.780 ","End":"01:26.705","Text":"F with respect to x is just the constant 2,"},{"Start":"01:26.705 ","End":"01:31.000","Text":"f with respect to y will be 1,"},{"Start":"01:31.000 ","End":"01:34.760","Text":"and then f_xx, well,"},{"Start":"01:34.760 ","End":"01:38.435","Text":"that\u0027s 0 and so is f_xy,"},{"Start":"01:38.435 ","End":"01:42.250","Text":"that\u0027s 0 and f_yy,"},{"Start":"01:42.250 ","End":"01:46.430","Text":"all these 3 second-order derivatives are 0."},{"Start":"01:46.430 ","End":"01:54.050","Text":"Let\u0027s see. G with respect to x is equal to just 1/2."},{"Start":"01:54.050 ","End":"01:56.645","Text":"I don\u0027t know why I put the brackets here."},{"Start":"01:56.645 ","End":"02:01.420","Text":"That looks better. 1/2 x and then we reduce the power by 1,"},{"Start":"02:01.420 ","End":"02:04.970","Text":"so it\u0027s minus 1/2 and all the rest of it doesn\u0027t contain x,"},{"Start":"02:04.970 ","End":"02:06.470","Text":"so that\u0027s like a constant."},{"Start":"02:06.470 ","End":"02:08.930","Text":"Then g with respect to y,"},{"Start":"02:08.930 ","End":"02:13.025","Text":"similarly is 1/2 y^1/2."},{"Start":"02:13.025 ","End":"02:17.975","Text":"Now, second-order, this with respect to x,"},{"Start":"02:17.975 ","End":"02:22.930","Text":"minus 1/2 times a 1/2 is"},{"Start":"02:22.930 ","End":"02:29.835","Text":"minus 1/4 and we get x^1/1/2 or minus 3/2."},{"Start":"02:29.835 ","End":"02:31.460","Text":"The mixed order 1s are 0,"},{"Start":"02:31.460 ","End":"02:35.120","Text":"whether you differentiate this with respect to y or this with respect to x,"},{"Start":"02:35.120 ","End":"02:44.720","Text":"either way you get that this is 0 and g_yy is exactly the same like this,"},{"Start":"02:44.720 ","End":"02:50.870","Text":"except for the y, we get minus 1/4 y to the minus 3/2."},{"Start":"02:50.870 ","End":"02:54.710","Text":"These are the second-order."},{"Start":"02:54.710 ","End":"02:59.345","Text":"Next thing in the process is to find critical points."},{"Start":"02:59.345 ","End":"03:04.160","Text":"It involves solving 3 equations in 3 unknowns."},{"Start":"03:04.160 ","End":"03:06.590","Text":"Let me remind you in general,"},{"Start":"03:06.590 ","End":"03:08.120","Text":"not just in our case,"},{"Start":"03:08.120 ","End":"03:09.380","Text":"we get the 3 equations,"},{"Start":"03:09.380 ","End":"03:15.705","Text":"we have an extra variable artificial if you like, called Lambda."},{"Start":"03:15.705 ","End":"03:23.780","Text":"What we do is we say that the partial derivative with respect to x of f is Lambda"},{"Start":"03:23.780 ","End":"03:31.770","Text":"times partial derivative of g. Same thing with respect to y, Lambda times g_y."},{"Start":"03:31.770 ","End":"03:41.045","Text":"The last equation is the constraint where g is equal to 0."},{"Start":"03:41.045 ","End":"03:45.815","Text":"If I rewrite this for our particular case,"},{"Start":"03:45.815 ","End":"03:50.660","Text":"what we get is f with respect to x is 2 equals"},{"Start":"03:50.660 ","End":"03:56.465","Text":"Lambda and g with respect to x is 1/2, x^1/2."},{"Start":"03:56.465 ","End":"04:03.170","Text":"But I\u0027m going to rewrite the fractional power as with radicals,"},{"Start":"04:03.170 ","End":"04:08.290","Text":"so this comes out to be 1 over the square root of x."},{"Start":"04:08.290 ","End":"04:14.775","Text":"For this 1, f_y is 1 equals Lambda times this,"},{"Start":"04:14.775 ","End":"04:20.630","Text":"so it\u0027s 1/2 and the same thing with y^1/2 is 1 over the square root of y."},{"Start":"04:20.630 ","End":"04:24.000","Text":"The last 1, g equals 0,"},{"Start":"04:24.000 ","End":"04:27.860","Text":"you can either write that this equals 0 or this bit equals 9."},{"Start":"04:27.860 ","End":"04:30.290","Text":"I just like to copy the original constraint and that"},{"Start":"04:30.290 ","End":"04:36.710","Text":"usually I find it most pleasing to do it that way."},{"Start":"04:36.710 ","End":"04:39.710","Text":"But you could write minus 9 equals 0."},{"Start":"04:39.710 ","End":"04:41.510","Text":"No big difference anyway."},{"Start":"04:41.510 ","End":"04:47.360","Text":"The next step, this almost always the same,"},{"Start":"04:47.360 ","End":"04:54.570","Text":"is to divide these 2 equations 1 by another and then that gets rid of Lambda."},{"Start":"04:57.260 ","End":"05:03.220","Text":"Let me note that Lambda is not equal to 0."},{"Start":"05:03.380 ","End":"05:07.170","Text":"Because if Lambda were 0,"},{"Start":"05:07.170 ","End":"05:09.415","Text":"I\u0027d get 2 equals 0,"},{"Start":"05:09.415 ","End":"05:11.370","Text":"so when I divide,"},{"Start":"05:11.370 ","End":"05:13.260","Text":"there\u0027s no worry about that."},{"Start":"05:13.260 ","End":"05:18.205","Text":"Also notice that this is defined."},{"Start":"05:18.205 ","End":"05:21.050","Text":"I could have already said it at this stage that x to"},{"Start":"05:21.050 ","End":"05:23.960","Text":"the minus 1/2 or 1 over the square root of x is defined."},{"Start":"05:23.960 ","End":"05:28.235","Text":"Because x and y are strictly positive so 1 over a positive,"},{"Start":"05:28.235 ","End":"05:31.350","Text":"that\u0027s okay, as long as it\u0027s not 0."},{"Start":"05:32.120 ","End":"05:35.160","Text":"Let me put this out of the way."},{"Start":"05:35.160 ","End":"05:37.280","Text":"Now when I do the division,"},{"Start":"05:37.280 ","End":"05:40.790","Text":"let\u0027s say we do the first divided by the second."},{"Start":"05:40.790 ","End":"05:47.645","Text":"From these 2, I get that 2/1 is equal to,"},{"Start":"05:47.645 ","End":"05:49.160","Text":"and I\u0027ll put a dividing line,"},{"Start":"05:49.160 ","End":"05:56.880","Text":"Lambda times 1/2 times 1 over root x divided by this 1,"},{"Start":"05:56.880 ","End":"06:06.225","Text":"Lambda times 1/2 times 1 over root y. I can cancel the Lambda here and here."},{"Start":"06:06.225 ","End":"06:07.830","Text":"I already noted it\u0027s not 0."},{"Start":"06:07.830 ","End":"06:10.485","Text":"I can cancel the 1/2 with 1/2."},{"Start":"06:10.485 ","End":"06:17.685","Text":"What I\u0027m left with is that 2/1 equals,"},{"Start":"06:17.685 ","End":"06:20.615","Text":"now I hope you remember how to divide fractions."},{"Start":"06:20.615 ","End":"06:22.550","Text":"This is a fraction over a fraction."},{"Start":"06:22.550 ","End":"06:23.570","Text":"That if a division,"},{"Start":"06:23.570 ","End":"06:25.550","Text":"we multiply by the reciprocal,"},{"Start":"06:25.550 ","End":"06:29.520","Text":"so it\u0027s 1 over root x times root y over 1."},{"Start":"06:29.520 ","End":"06:37.235","Text":"In short, we get root y over root x, and just cross-multiplying,"},{"Start":"06:37.235 ","End":"06:47.410","Text":"this gives me another equation that 2 root x equals root y."},{"Start":"06:47.660 ","End":"06:51.440","Text":"If I look at this equation and this equation,"},{"Start":"06:51.440 ","End":"06:55.340","Text":"it\u0027s 2 equations in 2 unknowns, x and y,"},{"Start":"06:55.340 ","End":"06:59.300","Text":"but really it\u0027s 2 equations in root x and root y,"},{"Start":"06:59.300 ","End":"07:02.785","Text":"and it\u0027ll be easy for me to find root x and y,"},{"Start":"07:02.785 ","End":"07:06.805","Text":"and later on we\u0027ll square to find x and y."},{"Start":"07:06.805 ","End":"07:12.185","Text":"If I put root y from here into here,"},{"Start":"07:12.185 ","End":"07:20.200","Text":"then this equation becomes root x plus twice root x equals 9."},{"Start":"07:20.200 ","End":"07:23.310","Text":"This gives me 3 root x is 9,"},{"Start":"07:23.310 ","End":"07:30.030","Text":"so it gives me that root x equals 3."},{"Start":"07:30.030 ","End":"07:32.700","Text":"Root y is twice root x,"},{"Start":"07:32.700 ","End":"07:37.750","Text":"so root y is 6 and from these 2,"},{"Start":"07:37.750 ","End":"07:45.180","Text":"I can get x and y. I get that x equals 9 and"},{"Start":"07:45.180 ","End":"07:53.430","Text":"y equals 36 and I also need Lambda."},{"Start":"07:53.430 ","End":"07:56.965","Text":"Let\u0027s say I take this middle 1 here."},{"Start":"07:56.965 ","End":"08:07.930","Text":"From here, I can get the 1 equals Lambda times 1/2 and root y is 6 times 1/6th."},{"Start":"08:07.930 ","End":"08:17.580","Text":"Just multiply both sides by 2 times 6 and then we get that Lambda equals 12."},{"Start":"08:17.580 ","End":"08:20.360","Text":"I\u0027ll just change the color to highlight it."},{"Start":"08:20.360 ","End":"08:23.870","Text":"In fact, when we find the xy Lambda for the critical point,"},{"Start":"08:23.870 ","End":"08:26.825","Text":"often it\u0027s indicated with an asterisk."},{"Start":"08:26.825 ","End":"08:30.650","Text":"Not everyone, but some people like to indicate with an asterisk the"},{"Start":"08:30.650 ","End":"08:35.060","Text":"special xy Lambda that we found for the critical point."},{"Start":"08:35.060 ","End":"08:40.160","Text":"The next step is to figure out if this gives the maximum or minimum."},{"Start":"08:40.160 ","End":"08:44.615","Text":"We\u0027re hoping for a minimum and there is a test and the test involves"},{"Start":"08:44.615 ","End":"08:50.960","Text":"some expression called H. Here\u0027s this horrible expression."},{"Start":"08:50.960 ","End":"08:53.445","Text":"H stands for Hessian."},{"Start":"08:53.445 ","End":"08:59.300","Text":"We have to figure out the value of H at our particular point,"},{"Start":"08:59.800 ","End":"09:05.445","Text":"which is for these values of xy and Lambda."},{"Start":"09:05.445 ","End":"09:08.465","Text":"We don\u0027t even have to actually figure it out."},{"Start":"09:08.465 ","End":"09:12.200","Text":"We just need to know if it\u0027s bigger than 0 or less than 0."},{"Start":"09:12.200 ","End":"09:16.760","Text":"If it\u0027s bigger than 0, it\u0027s a minimum and if it comes out less than 0,"},{"Start":"09:16.760 ","End":"09:20.090","Text":"it\u0027s a maximum. Let\u0027s see."},{"Start":"09:20.090 ","End":"09:22.150","Text":"Well, some of the things we know already,"},{"Start":"09:22.150 ","End":"09:23.690","Text":"look at these 3 0s."},{"Start":"09:23.690 ","End":"09:26.405","Text":"That means this is a 0, this is 0,"},{"Start":"09:26.405 ","End":"09:30.450","Text":"and this is a 0 and what else?"},{"Start":"09:30.450 ","End":"09:34.290","Text":"We\u0027ve got this 1\u0027s a 0 so that\u0027s 0."},{"Start":"09:34.290 ","End":"09:38.110","Text":"Already the whole last term with the minus 2,"},{"Start":"09:38.110 ","End":"09:42.380","Text":"all this basically disappears."},{"Start":"09:42.380 ","End":"09:48.120","Text":"Now I can simplify this and what remains,"},{"Start":"09:48.120 ","End":"09:52.790","Text":"I can take out minus Lambda from here as well as from here,"},{"Start":"09:52.790 ","End":"09:56.610","Text":"so I\u0027ve got minus Lambda times this times this,"},{"Start":"09:56.610 ","End":"10:03.600","Text":"which is g_xx times g_y squared plus,"},{"Start":"10:03.600 ","End":"10:06.195","Text":"it\u0027s a plus because we\u0027ve taken the minus Lambda out,"},{"Start":"10:06.195 ","End":"10:12.730","Text":"g_yy times g_x squared."},{"Start":"10:14.870 ","End":"10:19.215","Text":"Now, g_y squared is positive,"},{"Start":"10:19.215 ","End":"10:22.585","Text":"this bit here is bigger than 0,"},{"Start":"10:22.585 ","End":"10:26.215","Text":"and this bit here is bigger than 0 because it\u0027s positive."},{"Start":"10:26.215 ","End":"10:30.290","Text":"Let\u0027s just look at this and at this."},{"Start":"10:30.290 ","End":"10:34.095","Text":"Now, g_xx in general,"},{"Start":"10:34.095 ","End":"10:36.300","Text":"not just at our x."},{"Start":"10:36.300 ","End":"10:40.440","Text":"Well, our particular x is 9,"},{"Start":"10:40.440 ","End":"10:48.580","Text":"but 9 to any power is going to be positive."},{"Start":"10:48.580 ","End":"10:54.870","Text":"This bit is positive and this minus 1/4 is negative,"},{"Start":"10:54.870 ","End":"10:58.085","Text":"so g_xx is negative."},{"Start":"10:58.085 ","End":"11:01.805","Text":"I don\u0027t have to plug in the x equals 9."},{"Start":"11:01.805 ","End":"11:06.020","Text":"You\u0027d get the square root of 9 is 3 and then raise it to the power of 3."},{"Start":"11:06.020 ","End":"11:10.125","Text":"You get 1/27, but doesn\u0027t matter, it\u0027s positive."},{"Start":"11:10.125 ","End":"11:16.610","Text":"Similarly here the g_yy, y is 36,"},{"Start":"11:16.610 ","End":"11:18.170","Text":"36 to any power,"},{"Start":"11:18.170 ","End":"11:20.330","Text":"whether you take the root or 1 over,"},{"Start":"11:20.330 ","End":"11:25.625","Text":"this bit is positive and the minus 1/4 is still negative."},{"Start":"11:25.625 ","End":"11:28.070","Text":"This times this is negative."},{"Start":"11:28.070 ","End":"11:30.010","Text":"So here we have,"},{"Start":"11:30.010 ","End":"11:38.980","Text":"this is negative and this is negative and Lambda is 12."},{"Start":"11:39.080 ","End":"11:45.080","Text":"If Lambda is 12, then this bit is also negative."},{"Start":"11:45.080 ","End":"11:49.895","Text":"If you think about this, this times this is negative,"},{"Start":"11:49.895 ","End":"11:51.290","Text":"this times this is negative,"},{"Start":"11:51.290 ","End":"11:55.145","Text":"negative and negative is negative and altogether times negative,"},{"Start":"11:55.145 ","End":"11:57.530","Text":"it altogether comes out positive."},{"Start":"11:57.530 ","End":"12:01.160","Text":"Because it\u0027s positive, then we are in"},{"Start":"12:01.160 ","End":"12:05.645","Text":"the case of a minimum and this is what we had to show. Let\u0027s just look."},{"Start":"12:05.645 ","End":"12:12.695","Text":"Yeah, we had the minimum and so we have found the x and the y for the minimum."},{"Start":"12:12.695 ","End":"12:14.615","Text":"But that\u0027s not all,"},{"Start":"12:14.615 ","End":"12:19.250","Text":"because we usually expected to find the actual minimum value,"},{"Start":"12:19.250 ","End":"12:21.455","Text":"so what we would say is,"},{"Start":"12:21.455 ","End":"12:30.675","Text":"at the minimum x is 9 and y is 36."},{"Start":"12:30.675 ","End":"12:35.400","Text":"The minimum value f of xy,"},{"Start":"12:35.400 ","End":"12:37.935","Text":"or in our case,"},{"Start":"12:37.935 ","End":"12:47.970","Text":"2x plus y for our particular asterisk values."},{"Start":"12:49.280 ","End":"12:51.660","Text":"Boy, this is a nuisance."},{"Start":"12:51.660 ","End":"12:54.195","Text":"Anyway, 2x plus y,"},{"Start":"12:54.195 ","End":"13:00.180","Text":"twice 9 is 18 plus 36 is 54,"},{"Start":"13:00.180 ","End":"13:04.770","Text":"so the 3 values that are important,"},{"Start":"13:04.770 ","End":"13:07.845","Text":"the x for the minimum is 9,"},{"Start":"13:07.845 ","End":"13:10.310","Text":"the y of the minimum is 36,"},{"Start":"13:10.310 ","End":"13:15.725","Text":"and the minimum value under this constraint is 54."},{"Start":"13:15.725 ","End":"13:20.700","Text":"That concludes Part a of the exercise."}],"ID":9672},{"Watched":false,"Name":"Exercise 6 part b","Duration":"6m 48s","ChapterTopicVideoID":9774,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.665","Text":"Now we\u0027re going to move on to Part b."},{"Start":"00:02.665 ","End":"00:05.955","Text":"Let me erase what I don\u0027t need from here."},{"Start":"00:05.955 ","End":"00:10.515","Text":"I\u0027ll copy what we really need over here."},{"Start":"00:10.515 ","End":"00:15.960","Text":"What we found in Part a was that the minimum occurs when x is 9,"},{"Start":"00:15.960 ","End":"00:19.815","Text":"y is 36, and the minimum value of this,"},{"Start":"00:19.815 ","End":"00:22.935","Text":"which we called f, was 54."},{"Start":"00:22.935 ","End":"00:26.010","Text":"Now let\u0027s get to the graphical part."},{"Start":"00:26.010 ","End":"00:30.270","Text":"Also, allow me to change the word interpret to illustrate."},{"Start":"00:30.270 ","End":"00:37.475","Text":"What I\u0027d like to do now is to sketch the constraint function, which is this."},{"Start":"00:37.475 ","End":"00:41.930","Text":"But I suddenly realized that this is going to get"},{"Start":"00:41.930 ","End":"00:47.035","Text":"very involved because it\u0027s given an implicit form and it gets messy."},{"Start":"00:47.035 ","End":"00:50.795","Text":"What we can do is just plot a few points."},{"Start":"00:50.795 ","End":"00:53.060","Text":"Or even what I did,"},{"Start":"00:53.060 ","End":"00:54.440","Text":"which may be cheating,"},{"Start":"00:54.440 ","End":"00:58.180","Text":"is to use an online graph plotter."},{"Start":"00:58.180 ","End":"01:04.290","Text":"I just brought a graphing from a graph plotter."},{"Start":"01:04.290 ","End":"01:11.130","Text":"There\u0027s a reason why I chose to have it marked off every 9 units. You\u0027ll see in a moment."},{"Start":"01:11.130 ","End":"01:16.870","Text":"Actually, the question originally was with greater or equal to 0 here."},{"Start":"01:16.870 ","End":"01:23.500","Text":"It\u0027s just that the derivatives don\u0027t exist when x or y was 0."},{"Start":"01:23.500 ","End":"01:28.190","Text":"But if you substitute x equals 0,"},{"Start":"01:28.190 ","End":"01:32.775","Text":"you\u0027ll get root y is 9 or y is 81."},{"Start":"01:32.775 ","End":"01:35.230","Text":"That\u0027s a point and likewise,"},{"Start":"01:35.230 ","End":"01:37.450","Text":"if y is 0,"},{"Start":"01:37.450 ","End":"01:41.470","Text":"then we get x is 81."},{"Start":"01:41.470 ","End":"01:45.430","Text":"We actually also have our extremum."},{"Start":"01:45.430 ","End":"01:47.770","Text":"X is 9, y is 36,"},{"Start":"01:47.770 ","End":"01:49.390","Text":"and that\u0027s this point here."},{"Start":"01:49.390 ","End":"01:51.400","Text":"I\u0027ll do that in red."},{"Start":"01:51.400 ","End":"01:55.265","Text":"Because it\u0027s symmetric with x and y,"},{"Start":"01:55.265 ","End":"01:58.950","Text":"the point 36, 9 is also on."},{"Start":"01:58.950 ","End":"02:01.855","Text":"You could have just plotted a few points."},{"Start":"02:01.855 ","End":"02:07.100","Text":"My idea is to illustrate rather than do a precise solving."},{"Start":"02:07.100 ","End":"02:09.155","Text":"This is our graph,"},{"Start":"02:09.155 ","End":"02:12.110","Text":"root x plus root y equals 9."},{"Start":"02:12.110 ","End":"02:16.565","Text":"Next, we want to plot some level curves for the"},{"Start":"02:16.565 ","End":"02:21.620","Text":"function f. That\u0027s the 1 we\u0027re trying to minimize or maximize."},{"Start":"02:21.620 ","End":"02:26.000","Text":"We let this equal k and take different values of"},{"Start":"02:26.000 ","End":"02:32.580","Text":"k. I\u0027m going to choose some values so that they come out nicely on this graph."},{"Start":"02:32.580 ","End":"02:36.080","Text":"For example, if I took k equals 18,"},{"Start":"02:36.080 ","End":"02:41.240","Text":"I\u0027d get 2x plus y equals 18."},{"Start":"02:41.240 ","End":"02:44.345","Text":"Then when x is 0,"},{"Start":"02:44.345 ","End":"02:53.044","Text":"y would be 18 and when y is 0,"},{"Start":"02:53.044 ","End":"02:56.320","Text":"2x is 18 so x would be 9."},{"Start":"02:56.320 ","End":"03:03.455","Text":"I joined the points and I\u0027m sticking to the first quadrant."},{"Start":"03:03.455 ","End":"03:09.720","Text":"But this doesn\u0027t cut the constraint curve,"},{"Start":"03:09.720 ","End":"03:11.850","Text":"so this is no good."},{"Start":"03:11.850 ","End":"03:18.870","Text":"Let\u0027s try a larger value of k. Let\u0027s try 2x plus y equals,"},{"Start":"03:18.870 ","End":"03:20.010","Text":"I want an even number,"},{"Start":"03:20.010 ","End":"03:29.110","Text":"36 and then when x is 0, y is 36."},{"Start":"03:29.900 ","End":"03:33.570","Text":"When y is 0, 2x is 36."},{"Start":"03:33.570 ","End":"03:36.150","Text":"So x is 18."},{"Start":"03:36.150 ","End":"03:41.145","Text":"We get this and it\u0027s still doesn\u0027t cut the constraint."},{"Start":"03:41.145 ","End":"03:43.910","Text":"Let\u0027s try something still larger."},{"Start":"03:43.910 ","End":"03:50.765","Text":"Let\u0027s try 2x plus y equals say,"},{"Start":"03:50.765 ","End":"03:56.970","Text":"72 and then when y is 0,"},{"Start":"03:58.580 ","End":"04:08.820","Text":"x is 1/2 of this, which is 36."},{"Start":"04:09.440 ","End":"04:14.090","Text":"This does cut the curve at 2 points here and another point here,"},{"Start":"04:14.090 ","End":"04:15.275","Text":"which is hard to see."},{"Start":"04:15.275 ","End":"04:17.330","Text":"Oh, and I should be labeling this,"},{"Start":"04:17.330 ","End":"04:20.300","Text":"the values of k. Here,"},{"Start":"04:20.300 ","End":"04:23.600","Text":"k is equal to 18."},{"Start":"04:23.600 ","End":"04:29.225","Text":"This was the curve where k equals 36."},{"Start":"04:29.225 ","End":"04:33.575","Text":"This is the curve where k equals 72."},{"Start":"04:33.575 ","End":"04:40.290","Text":"We noticed that the k is increasing in this direction."},{"Start":"04:41.120 ","End":"04:48.515","Text":"Actually, we could also do that using the gradient of f,"},{"Start":"04:48.515 ","End":"04:52.430","Text":"which is in general f with respect to x,"},{"Start":"04:52.430 ","End":"04:55.110","Text":"f with respect to y as a vector."},{"Start":"04:55.110 ","End":"04:57.710","Text":"In our case with respect to x,"},{"Start":"04:57.710 ","End":"05:01.760","Text":"it\u0027s the constant 2 and with respect to y,"},{"Start":"05:01.760 ","End":"05:03.305","Text":"it\u0027s the constant 1."},{"Start":"05:03.305 ","End":"05:05.495","Text":"If I took a vector 2,"},{"Start":"05:05.495 ","End":"05:10.190","Text":"1 that would show the direction of increase of k,"},{"Start":"05:10.190 ","End":"05:13.170","Text":"but we see its outward."},{"Start":"05:13.600 ","End":"05:17.315","Text":"What I\u0027m getting at is that if I take k too small,"},{"Start":"05:17.315 ","End":"05:21.770","Text":"I won\u0027t hit the constraint curve at all."},{"Start":"05:21.770 ","End":"05:24.260","Text":"If I take k too large,"},{"Start":"05:24.260 ","End":"05:26.059","Text":"then it won\u0027t be the minimum."},{"Start":"05:26.059 ","End":"05:28.455","Text":"It will actually cut at more than 1 point."},{"Start":"05:28.455 ","End":"05:33.135","Text":"It seems clear that there\u0027s an in-between value that\u0027s just right."},{"Start":"05:33.135 ","End":"05:36.230","Text":"That\u0027s going to be the 1 that just grazes"},{"Start":"05:36.230 ","End":"05:40.115","Text":"the curve is tangent to it and we already know the answer."},{"Start":"05:40.115 ","End":"05:42.610","Text":"It\u0027s going to go through here."},{"Start":"05:42.610 ","End":"05:50.200","Text":"The value will be 54 in fact and so let me just sketch that 1."},{"Start":"05:50.200 ","End":"05:57.500","Text":"If I take 2x plus y equals 54,"},{"Start":"05:57.500 ","End":"05:59.420","Text":"then when x is 0,"},{"Start":"05:59.420 ","End":"06:03.570","Text":"y is 54, so here,"},{"Start":"06:03.570 ","End":"06:06.284","Text":"and when y is 0,"},{"Start":"06:06.284 ","End":"06:10.545","Text":"x is 1/2 of that, which is 27."},{"Start":"06:10.545 ","End":"06:14.449","Text":"Let me join them and this line is just perfect."},{"Start":"06:14.449 ","End":"06:18.530","Text":"It\u0027s the level curve, k equals 54,"},{"Start":"06:18.530 ","End":"06:25.280","Text":"and that\u0027s the least value of k that will still cut the constraint curve."},{"Start":"06:25.280 ","End":"06:27.275","Text":"It cuts it here."},{"Start":"06:27.275 ","End":"06:33.560","Text":"Really this was just for illustration purposes to show how we play with"},{"Start":"06:33.560 ","End":"06:38.540","Text":"the level curves of the thing we\u0027re trying to minimize or"},{"Start":"06:38.540 ","End":"06:45.865","Text":"maximize and find the least or greatest value that cuts the constraint."},{"Start":"06:45.865 ","End":"06:49.150","Text":"That\u0027s it for illustration."}],"ID":9673},{"Watched":false,"Name":"Exercise 7","Duration":"5m 12s","ChapterTopicVideoID":9775,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"We have here a word problem,"},{"Start":"00:02.280 ","End":"00:04.650","Text":"and they\u0027re always the most difficult,"},{"Start":"00:04.650 ","End":"00:06.090","Text":"at least, for some people."},{"Start":"00:06.090 ","End":"00:08.925","Text":"Let\u0027s see if we can figure out what\u0027s going on here."},{"Start":"00:08.925 ","End":"00:14.690","Text":"It starts off by talking about points on the line and let me give a really rough sketch."},{"Start":"00:14.690 ","End":"00:17.435","Text":"It\u0027s not meant to be accurate."},{"Start":"00:17.435 ","End":"00:19.610","Text":"There are many points on the line."},{"Start":"00:19.610 ","End":"00:23.120","Text":"For example, if I let y equals 0,"},{"Start":"00:23.120 ","End":"00:25.895","Text":"then I would get that x equals 12,"},{"Start":"00:25.895 ","End":"00:29.030","Text":"so I would know that the point 12,"},{"Start":"00:29.030 ","End":"00:30.890","Text":"0 is on the line."},{"Start":"00:30.890 ","End":"00:39.330","Text":"Another example, well, I could take x equals 0 and then y would equal 4."},{"Start":"00:39.330 ","End":"00:42.265","Text":"Now it\u0027s not to scale, it\u0027s just schematic,"},{"Start":"00:42.265 ","End":"00:47.230","Text":"so the point 4, 0 would also be on the line."},{"Start":"00:47.270 ","End":"00:50.120","Text":"I could have done it as a table."},{"Start":"00:50.120 ","End":"00:55.920","Text":"In fact, I think a table might even be a better way of representing things."},{"Start":"00:56.950 ","End":"01:00.330","Text":"I might have 2 columns,"},{"Start":"01:00.330 ","End":"01:02.365","Text":"value of x, for example,"},{"Start":"01:02.365 ","End":"01:04.479","Text":"I know that when x is 12,"},{"Start":"01:04.479 ","End":"01:08.665","Text":"y is 0 on this line."},{"Start":"01:08.665 ","End":"01:11.765","Text":"If I know that if x is 4,"},{"Start":"01:11.765 ","End":"01:14.370","Text":"did I get something backwards?"},{"Start":"01:14.370 ","End":"01:18.030","Text":"Yes, silly me. This is 0, 4."},{"Start":"01:18.030 ","End":"01:23.140","Text":"Sorry. Here when x is 0, y is 4."},{"Start":"01:23.140 ","End":"01:25.345","Text":"I could take other examples."},{"Start":"01:25.345 ","End":"01:30.960","Text":"I could take x equals 6,"},{"Start":"01:30.960 ","End":"01:35.040","Text":"and then I\u0027d get 3y equals 12 minus 6 is 6,"},{"Start":"01:35.040 ","End":"01:40.620","Text":"so y would equal 2, and so on."},{"Start":"01:40.620 ","End":"01:47.380","Text":"I\u0027ll do 1 more. Suppose that y equals 1,"},{"Start":"01:47.380 ","End":"01:52.840","Text":"then x plus 3 is 12 so x is 9."},{"Start":"01:52.840 ","End":"01:54.909","Text":"Now for each point,"},{"Start":"01:54.909 ","End":"01:58.780","Text":"there are 2 coordinates: an x-coordinate and a y-coordinate."},{"Start":"01:58.780 ","End":"02:01.095","Text":"I could also put these on the line,"},{"Start":"02:01.095 ","End":"02:02.984","Text":"but it wouldn\u0027t be to scale."},{"Start":"02:02.984 ","End":"02:06.510","Text":"I could say 9, 1 is a point on the line."},{"Start":"02:06.510 ","End":"02:09.125","Text":"They might not be in the right order,"},{"Start":"02:09.125 ","End":"02:12.205","Text":"and I\u0027d know that somewhere there\u0027s a 6, 2 on the line."},{"Start":"02:12.205 ","End":"02:18.990","Text":"I\u0027m just illustrating a point that for each point on the line,"},{"Start":"02:18.990 ","End":"02:21.375","Text":"there are 2 coordinates x and y,"},{"Start":"02:21.375 ","End":"02:24.545","Text":"and then we can talk about the product of the coordinates."},{"Start":"02:24.545 ","End":"02:26.225","Text":"I\u0027ll need another column."},{"Start":"02:26.225 ","End":"02:32.270","Text":"I\u0027ll make some room and add an extra column for the product of the coordinates."},{"Start":"02:32.270 ","End":"02:35.885","Text":"Here I would say 12 times 0 is 0,"},{"Start":"02:35.885 ","End":"02:38.000","Text":"0 times 4 is 0,"},{"Start":"02:38.000 ","End":"02:40.235","Text":"6 times 2 is 12,"},{"Start":"02:40.235 ","End":"02:43.385","Text":"9 times 1 is 9, and so on."},{"Start":"02:43.385 ","End":"02:46.310","Text":"I could get for different x\u0027s and y\u0027s."},{"Start":"02:46.310 ","End":"02:49.325","Text":"I would get a different product of coordinates,"},{"Start":"02:49.325 ","End":"02:53.375","Text":"and we want to know which 1 has the greatest value here."},{"Start":"02:53.375 ","End":"02:56.000","Text":"So far the greatest we\u0027ve seen is 12,"},{"Start":"02:56.000 ","End":"02:59.320","Text":"but it could be greater."},{"Start":"02:59.320 ","End":"03:03.860","Text":"I can even label this column because what is the product of the coordinates?"},{"Start":"03:03.860 ","End":"03:07.700","Text":"It\u0027s just x times y, and this is what we want."},{"Start":"03:07.700 ","End":"03:09.335","Text":"The greatest would be maximum,"},{"Start":"03:09.335 ","End":"03:16.890","Text":"so I can now phrase this as a constrained extremum problem."},{"Start":"03:16.890 ","End":"03:24.540","Text":"What I want is the maximum of x,"},{"Start":"03:24.540 ","End":"03:28.690","Text":"y, and this would be the target function."},{"Start":"03:29.220 ","End":"03:32.200","Text":"The maximum of x, y,"},{"Start":"03:32.200 ","End":"03:37.330","Text":"subject to the constraint that it\u0027s on the line."},{"Start":"03:37.330 ","End":"03:44.900","Text":"In other words, subject to x plus 3y equals 12."},{"Start":"03:47.450 ","End":"03:52.740","Text":"Then we start solving it by letting f of x,"},{"Start":"03:52.740 ","End":"03:55.575","Text":"y be the target function xy."},{"Start":"03:55.575 ","End":"03:57.640","Text":"We define a g of x,"},{"Start":"03:57.640 ","End":"04:03.430","Text":"y equals the constraint which is usually written as something equals 0,"},{"Start":"04:03.430 ","End":"04:07.025","Text":"x plus 3y minus 12,"},{"Start":"04:07.025 ","End":"04:09.985","Text":"and then we use the Lagrange multiplier method."},{"Start":"04:09.985 ","End":"04:11.730","Text":"Now I\u0027m not going to continue,"},{"Start":"04:11.730 ","End":"04:13.960","Text":"and there\u0027s a very good reason for that."},{"Start":"04:13.960 ","End":"04:19.330","Text":"That this is exactly the same as 1 of the problems we\u0027ve done earlier."},{"Start":"04:19.330 ","End":"04:25.110","Text":"I refer you to Exercise 5,"},{"Start":"04:25.110 ","End":"04:27.380","Text":"at least, unless someone has renumbered them."},{"Start":"04:27.380 ","End":"04:36.534","Text":"In Exercise 5, this was exactly the constrained extremum problem that we had to solve,"},{"Start":"04:36.534 ","End":"04:40.020","Text":"and I can just quote the answer from there."},{"Start":"04:40.090 ","End":"04:44.300","Text":"There the answer was that the x,"},{"Start":"04:44.300 ","End":"04:52.230","Text":"y for the maximum which we also sometimes put an asterisk to indicate that special x,"},{"Start":"04:52.230 ","End":"04:55.940","Text":"y for the extremum turned out to be 6,"},{"Start":"04:55.940 ","End":"05:00.920","Text":"2 and the maximum value turned out to be 12."},{"Start":"05:00.920 ","End":"05:06.440","Text":"It\u0027s just coincidence that I happen to have chosen this point and that really 12"},{"Start":"05:06.440 ","End":"05:12.590","Text":"is the largest you could possibly get. We are done."}],"ID":9674},{"Watched":false,"Name":"Exercise 8","Duration":"10m 33s","ChapterTopicVideoID":9776,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"Let\u0027s see if we can understand what is asked of us in this exercise."},{"Start":"00:05.790 ","End":"00:09.570","Text":"We\u0027re given a curve and it\u0027s given an implicit form with"},{"Start":"00:09.570 ","End":"00:14.505","Text":"y and x mixed together as an equation."},{"Start":"00:14.505 ","End":"00:19.350","Text":"That gives us a curve in the plane and the curve consists of"},{"Start":"00:19.350 ","End":"00:25.790","Text":"many points and each point has a certain distance from the origin."},{"Start":"00:25.790 ","End":"00:30.410","Text":"We have to find the points respectively with"},{"Start":"00:30.410 ","End":"00:35.435","Text":"the greatest distance from the origin and with the shortest distance from the origin."},{"Start":"00:35.435 ","End":"00:40.190","Text":"Let\u0027s draw a sketch and that will explain it better."},{"Start":"00:40.190 ","End":"00:43.310","Text":"Now, I happen to know that this is an ellipse,"},{"Start":"00:43.310 ","End":"00:46.050","Text":"but you might not know that."},{"Start":"00:46.150 ","End":"00:51.080","Text":"I used an online calculator."},{"Start":"00:51.080 ","End":"00:56.885","Text":"There are plenty of websites that will sketch such equations for you."},{"Start":"00:56.885 ","End":"00:59.510","Text":"I came up with this curve,"},{"Start":"00:59.510 ","End":"01:03.690","Text":"I happened to include the grid lines, and doesn\u0027t matter."},{"Start":"01:04.360 ","End":"01:07.655","Text":"For any given point on this,"},{"Start":"01:07.655 ","End":"01:14.490","Text":"Let\u0027s say I took I this point here,"},{"Start":"01:15.290 ","End":"01:20.000","Text":"then it has a certain distance to the origin."},{"Start":"01:20.000 ","End":"01:21.485","Text":"The origin is here."},{"Start":"01:21.485 ","End":"01:23.270","Text":"This is the fixed point,"},{"Start":"01:23.270 ","End":"01:29.375","Text":"the origin and this point could be anywhere along this curve and between the two,"},{"Start":"01:29.375 ","End":"01:30.950","Text":"if I draw a line,"},{"Start":"01:30.950 ","End":"01:37.745","Text":"then I get a distance d. If this is the origin,"},{"Start":"01:37.745 ","End":"01:45.149","Text":"this would be the point 0, 0 the origin,"},{"Start":"01:45.149 ","End":"01:48.289","Text":"and this will be a variable point,"},{"Start":"01:48.289 ","End":"01:54.324","Text":"x, y, somewhere along this ellipse."},{"Start":"01:54.324 ","End":"01:56.060","Text":"At least to the eye,"},{"Start":"01:56.060 ","End":"01:58.790","Text":"it looks like maybe this point is the furthest"},{"Start":"01:58.790 ","End":"02:04.355","Text":"away or may be equal with this because there is a symmetry about this."},{"Start":"02:04.355 ","End":"02:08.645","Text":"Looks like this 45-degree line is a symmetry and so is this."},{"Start":"02:08.645 ","End":"02:10.925","Text":"If I go in the other direction,"},{"Start":"02:10.925 ","End":"02:21.020","Text":"I might end up here or here and these look to be the closest."},{"Start":"02:21.020 ","End":"02:26.975","Text":"Perhaps I\u0027ll just draw some faint lines in and at least to the eye,"},{"Start":"02:26.975 ","End":"02:31.954","Text":"this looks like it\u0027s going to be the closest distance and this is the longest distance."},{"Start":"02:31.954 ","End":"02:36.445","Text":"Now, how do we do this mathematically?"},{"Start":"02:36.445 ","End":"02:40.100","Text":"I want to convert this word problem into"},{"Start":"02:40.100 ","End":"02:45.365","Text":"a problem of minimum or maximum under constraint."},{"Start":"02:45.365 ","End":"02:47.285","Text":"In this case, we have both,"},{"Start":"02:47.285 ","End":"02:49.730","Text":"we have maximum and minimum."},{"Start":"02:49.730 ","End":"02:55.925","Text":"Note that by Pythagoras\u0027s theorem or from the distance formula between 2 points,"},{"Start":"02:55.925 ","End":"03:02.530","Text":"that d is the square root of x squared plus y squared"},{"Start":"03:02.530 ","End":"03:10.790","Text":"and normally the function of x and y that we\u0027re trying to maximize or minimize,"},{"Start":"03:10.790 ","End":"03:16.609","Text":"we call it f of x y but here we\u0027re going to do something different."},{"Start":"03:16.609 ","End":"03:20.090","Text":"There\u0027s a standard trick that gets rid of"},{"Start":"03:20.090 ","End":"03:25.325","Text":"square roots because square roots are messy to work with when you start differentiating."},{"Start":"03:25.325 ","End":"03:29.720","Text":"What we say is that the distance is greatest to least at"},{"Start":"03:29.720 ","End":"03:34.865","Text":"the same x y where the square of the distance is greatest or least."},{"Start":"03:34.865 ","End":"03:41.195","Text":"We actually take our target function not to be the distance,"},{"Start":"03:41.195 ","End":"03:45.830","Text":"but we take our target function f to be the distance squared."},{"Start":"03:45.830 ","End":"03:52.400","Text":"The distance squared will just be x squared plus y squared."},{"Start":"03:52.400 ","End":"03:57.965","Text":"All we have to do is to remember at the end that when we find the x and the y,"},{"Start":"03:57.965 ","End":"04:02.340","Text":"not to substitute in f but to substitute in d,"},{"Start":"04:02.340 ","End":"04:07.640","Text":"ie, to take the square root of this at the end and hopefully we won\u0027t forget."},{"Start":"04:07.640 ","End":"04:11.295","Text":"The other function is the constraint function."},{"Start":"04:11.295 ","End":"04:13.565","Text":"This is the constraint as an equation."},{"Start":"04:13.565 ","End":"04:16.460","Text":"Usually we want this to be in the form of"},{"Start":"04:16.460 ","End":"04:20.240","Text":"some function g equals 0 so we just bring everything,"},{"Start":"04:20.240 ","End":"04:23.670","Text":"say to the left-hand side and we can write g of x,"},{"Start":"04:23.670 ","End":"04:26.010","Text":"y is equal to"},{"Start":"04:26.010 ","End":"04:31.020","Text":"2x squared plus 3xy and then I\u0027ll put over"},{"Start":"04:31.020 ","End":"04:37.810","Text":"the plus 2y squared minus 1 equals 0."},{"Start":"04:37.870 ","End":"04:41.750","Text":"This was actually the equation that I fed into"},{"Start":"04:41.750 ","End":"04:50.105","Text":"the plotter of equations and it gave me this picture."},{"Start":"04:50.105 ","End":"04:57.920","Text":"Now if I want to write it in the language of extra moment the constraint,"},{"Start":"04:57.920 ","End":"05:04.685","Text":"I will say that I want to find the maximum"},{"Start":"05:04.685 ","End":"05:09.020","Text":"of when you put the curly braces x"},{"Start":"05:09.020 ","End":"05:14.510","Text":"squared plus y squared and then the constraints you write s.t,"},{"Start":"05:14.510 ","End":"05:17.160","Text":"which means subject to"},{"Start":"05:19.150 ","End":"05:28.560","Text":"2x squared plus 3xy plus 2y squared and you have"},{"Start":"05:28.560 ","End":"05:31.830","Text":"the choice you need that you either can say this equals"},{"Start":"05:31.830 ","End":"05:38.570","Text":"1 or we can just keep everything on the left and say equals 0."},{"Start":"05:38.570 ","End":"05:41.480","Text":"Only, not quite right because we"},{"Start":"05:41.480 ","End":"05:46.550","Text":"have both the greatest distance and the shortest distances,"},{"Start":"05:46.550 ","End":"05:48.875","Text":"so it\u0027s really 2 problems in 1,"},{"Start":"05:48.875 ","End":"05:54.475","Text":"we have a maximum problem and then we also have"},{"Start":"05:54.475 ","End":"06:01.600","Text":"the same thing with the minimum and because it\u0027s the same method for both,"},{"Start":"06:01.600 ","End":"06:04.450","Text":"we find critical points and then we feed it"},{"Start":"06:04.450 ","End":"06:07.720","Text":"into a formula and it tells us maximum or minimum."},{"Start":"06:07.720 ","End":"06:12.610","Text":"Really it\u0027s the same problem to find the extremum points of"},{"Start":"06:12.610 ","End":"06:18.875","Text":"this subject to this and then we test each extremum whether it\u0027s a maximum or a minimum."},{"Start":"06:18.875 ","End":"06:21.870","Text":"Now, as it turns out,"},{"Start":"06:21.870 ","End":"06:27.520","Text":"this problem of extremum and the constraint is actually"},{"Start":"06:27.520 ","End":"06:33.715","Text":"taken from a previous problem unless someone\u0027s changed the numbering,"},{"Start":"06:33.715 ","End":"06:38.055","Text":"this was exercise number"},{"Start":"06:38.055 ","End":"06:46.050","Text":"1 in this section on Lagrange multipliers."},{"Start":"06:46.050 ","End":"06:47.810","Text":"If we look at exercise 1,"},{"Start":"06:47.810 ","End":"06:50.060","Text":"it was stated a bit differently."},{"Start":"06:50.060 ","End":"06:52.580","Text":"First of all, it didn\u0027t say maximum or minimum,"},{"Start":"06:52.580 ","End":"06:56.550","Text":"it said find the extremum or extrema,"},{"Start":"06:56.550 ","End":"06:59.400","Text":"I believe it was stated as extrema which is the plural of"},{"Start":"06:59.400 ","End":"07:03.920","Text":"extremum and also it didn\u0027t give this constraint."},{"Start":"07:03.920 ","End":"07:07.445","Text":"It gave it in this form,"},{"Start":"07:07.445 ","End":"07:09.620","Text":"which is essentially the same thing."},{"Start":"07:09.620 ","End":"07:12.470","Text":"I mean this equation is the same as this equals"},{"Start":"07:12.470 ","End":"07:16.924","Text":"0 so other than minor variations in phrasing,"},{"Start":"07:16.924 ","End":"07:19.700","Text":"we\u0027ve done this exercise already."},{"Start":"07:19.700 ","End":"07:24.000","Text":"I can quote the solutions from there in"},{"Start":"07:24.000 ","End":"07:30.049","Text":"that exercise which I\u0027m referring to actually labeled the points are called this one A,"},{"Start":"07:30.049 ","End":"07:33.499","Text":"this one I labeled B, this one C,"},{"Start":"07:33.499 ","End":"07:41.910","Text":"and this one D and we got that A and C were minimum point,"},{"Start":"07:41.910 ","End":"07:46.260","Text":"I write the plural of minimum, where A,"},{"Start":"07:46.260 ","End":"07:54.840","Text":"which was the point 1 over square root of 7,"},{"Start":"07:54.840 ","End":"07:57.979","Text":"1 over square root of 7."},{"Start":"07:57.979 ","End":"08:04.069","Text":"This makes sense because we noticed that this 45-degree line is an axis of symmetry."},{"Start":"08:04.069 ","End":"08:06.745","Text":"The other one was just minus,"},{"Start":"08:06.745 ","End":"08:11.895","Text":"B was the point minus 1 over root 7,"},{"Start":"08:11.895 ","End":"08:17.490","Text":"minus 1 over root 7 and these to were a tie for minimum and that"},{"Start":"08:17.490 ","End":"08:25.190","Text":"the minimum f at the point A as well as f at the point B,"},{"Start":"08:25.190 ","End":"08:30.265","Text":"was equal to 2/7"},{"Start":"08:30.265 ","End":"08:36.800","Text":"and this is where we have to be careful because we didn\u0027t want the minimum for f,"},{"Start":"08:36.800 ","End":"08:44.905","Text":"we wanted the minimum distance and the distance is the square root so in our case,"},{"Start":"08:44.905 ","End":"08:47.480","Text":"we have the same two points,"},{"Start":"08:47.480 ","End":"08:53.580","Text":"but let\u0027s call it d,"},{"Start":"08:53.580 ","End":"08:55.520","Text":"the distance of the point A from"},{"Start":"08:55.520 ","End":"08:59.750","Text":"the origin equals the distance of the point B from the origin."},{"Start":"08:59.750 ","End":"09:04.895","Text":"That would be the square root of 2/7 because in that problem,"},{"Start":"09:04.895 ","End":"09:10.480","Text":"the target function was x squared plus y squared and here it\u0027s the square root."},{"Start":"09:10.480 ","End":"09:15.650","Text":"Reminding you again, we knew that they would occur at the same places,"},{"Start":"09:15.650 ","End":"09:17.330","Text":"but the values would be different."},{"Start":"09:17.330 ","End":"09:20.630","Text":"One would be with a square root and one would be without."},{"Start":"09:20.630 ","End":"09:22.505","Text":"Now, as for the other,"},{"Start":"09:22.505 ","End":"09:24.590","Text":"we also had two maxima."},{"Start":"09:24.590 ","End":"09:26.840","Text":"In that same exercise 1,"},{"Start":"09:26.840 ","End":"09:34.805","Text":"we found that we had to maximum points or maxima and they were B and D. B,"},{"Start":"09:34.805 ","End":"09:44.540","Text":"it turned out was minus 1,1 and C was the point 1,"},{"Start":"09:44.540 ","End":"09:48.815","Text":"minus 1 and the function f,"},{"Start":"09:48.815 ","End":"09:56.150","Text":"both at B and at C was equal to 2."},{"Start":"09:56.150 ","End":"10:05.645","Text":"In our case, the maximum for the distance is also B and D but we don\u0027t want f,"},{"Start":"10:05.645 ","End":"10:10.760","Text":"we want the distance of B from the origin and"},{"Start":"10:10.760 ","End":"10:17.300","Text":"the distance of C from the origin are a tie and they are each the square root of 2."},{"Start":"10:17.300 ","End":"10:20.890","Text":"Like I said, d is the square root of x-squared plus y-squared,"},{"Start":"10:20.890 ","End":"10:23.600","Text":"f is d squared or d is the square root of f,"},{"Start":"10:23.600 ","End":"10:25.190","Text":"whichever way you want to look at it."},{"Start":"10:25.190 ","End":"10:32.520","Text":"We need to take the square root of the answers from exercise 1 and we are done."}],"ID":9675},{"Watched":false,"Name":"Exercise 9","Duration":"17m 1s","ChapterTopicVideoID":9777,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this exercise, we\u0027re asked to find"},{"Start":"00:02.340 ","End":"00:08.310","Text":"the shortest distance from a given line to a given parabola."},{"Start":"00:08.310 ","End":"00:12.970","Text":"We recognize this as the equation of the line."},{"Start":"00:13.220 ","End":"00:18.585","Text":"Just take it on trust that this is actually a parabola."},{"Start":"00:18.585 ","End":"00:26.130","Text":"I\u0027ve provided a sketch just to help imagine what\u0027s going on."},{"Start":"00:26.130 ","End":"00:32.850","Text":"Somewhere or other, there\u0027s a place where this distance,"},{"Start":"00:32.850 ","End":"00:34.500","Text":"which means that if you take"},{"Start":"00:34.500 ","End":"00:39.480","Text":"a point on the parabola and you drop a perpendicular to the line,"},{"Start":"00:39.480 ","End":"00:42.280","Text":"that\u0027s the distance, somewhere it looks like here."},{"Start":"00:42.280 ","End":"00:44.375","Text":"This is a minimum. In general,"},{"Start":"00:44.375 ","End":"00:48.305","Text":"if you take a point x,y on the parabola,"},{"Start":"00:48.305 ","End":"00:53.255","Text":"the distance as I said is gotten by dropping a perpendicular,"},{"Start":"00:53.255 ","End":"00:59.345","Text":"but there is a formula for the distance between a point and a line."},{"Start":"00:59.345 ","End":"01:04.590","Text":"If this is the point, in general, x,"},{"Start":"01:04.590 ","End":"01:10.515","Text":"y, then there is a formula for the distance,"},{"Start":"01:10.515 ","End":"01:16.890","Text":"and the distance is a function of x and y from the point to the line."},{"Start":"01:16.890 ","End":"01:28.205","Text":"You can remember it by just taking the equation of the line but without the equal to 0,"},{"Start":"01:28.205 ","End":"01:34.615","Text":"so it\u0027s 3x minus 6y plus 4,"},{"Start":"01:34.615 ","End":"01:41.375","Text":"and then we divide that by the square root of"},{"Start":"01:41.375 ","End":"01:49.240","Text":"the coefficient of x squared plus the coefficient of y squared."},{"Start":"01:49.240 ","End":"01:51.800","Text":"Actually what we have to do is take"},{"Start":"01:51.800 ","End":"01:57.150","Text":"the absolute value of this because distance is positive."},{"Start":"01:57.190 ","End":"02:00.350","Text":"This is the target function,"},{"Start":"02:00.350 ","End":"02:03.080","Text":"the thing that we have to make maximum or minimum."},{"Start":"02:03.080 ","End":"02:06.500","Text":"I prefer the letter f, I\u0027m used to it."},{"Start":"02:06.500 ","End":"02:11.704","Text":"Let\u0027s change that d to an f. That\u0027s the target function,"},{"Start":"02:11.704 ","End":"02:15.140","Text":"and then there\u0027s also a constraint function,"},{"Start":"02:15.140 ","End":"02:17.960","Text":"which would be this one."},{"Start":"02:17.960 ","End":"02:25.530","Text":"That would be g of x,y is equal to x"},{"Start":"02:25.530 ","End":"02:33.645","Text":"squared plus 2xy plus y squared plus 4y."},{"Start":"02:33.645 ","End":"02:37.545","Text":"In general, we write"},{"Start":"02:37.545 ","End":"02:46.175","Text":"the extremum under constraint problem as max or min,"},{"Start":"02:46.175 ","End":"02:48.415","Text":"in this case min."},{"Start":"02:48.415 ","End":"02:54.800","Text":"Then curly braces the target function f of x,y which in this,"},{"Start":"02:54.800 ","End":"03:00.210","Text":"I\u0027ll just write it as f of x,y."},{"Start":"03:00.210 ","End":"03:04.249","Text":"Subject to, and then it\u0027s always,"},{"Start":"03:04.249 ","End":"03:05.750","Text":"I mean, I could just copy this,"},{"Start":"03:05.750 ","End":"03:11.835","Text":"but it\u0027s always subject to g of x,y equals 0."},{"Start":"03:11.835 ","End":"03:15.660","Text":"The target subject to the constraint,"},{"Start":"03:15.660 ","End":"03:18.180","Text":"and it\u0027s minimum or maximum,"},{"Start":"03:18.180 ","End":"03:20.440","Text":"in our case obviously minimum."},{"Start":"03:20.440 ","End":"03:24.350","Text":"What bothers me is this absolute value,"},{"Start":"03:24.350 ","End":"03:26.690","Text":"I want to try and get rid of that somehow."},{"Start":"03:26.690 ","End":"03:30.030","Text":"Let me say as follows."},{"Start":"03:32.510 ","End":"03:35.930","Text":"If the parabola intersected the line,"},{"Start":"03:35.930 ","End":"03:38.240","Text":"then the shortest distance would be 0."},{"Start":"03:38.240 ","End":"03:41.750","Text":"If you check algebraically, for example,"},{"Start":"03:41.750 ","End":"03:45.710","Text":"you could isolate y in terms of x here and substitute it here,"},{"Start":"03:45.710 ","End":"03:48.125","Text":"you\u0027ll find that there\u0027s no solution."},{"Start":"03:48.125 ","End":"03:52.519","Text":"They don\u0027t intersect, and in the picture indeed they don\u0027t intersect."},{"Start":"03:52.519 ","End":"03:58.040","Text":"Now, whenever you have an equation of a line,"},{"Start":"03:58.040 ","End":"04:04.099","Text":"something equals 0, then it divides the plane into 3 parts."},{"Start":"04:04.099 ","End":"04:05.540","Text":"One side of the line,"},{"Start":"04:05.540 ","End":"04:06.550","Text":"the other side of the line,"},{"Start":"04:06.550 ","End":"04:07.760","Text":"and the line itself."},{"Start":"04:07.760 ","End":"04:12.920","Text":"If I substitute x,y from the line into this left-hand side,"},{"Start":"04:12.920 ","End":"04:19.535","Text":"I\u0027ll get 0, but on one side I always get a positive and the other side negative."},{"Start":"04:19.535 ","End":"04:22.410","Text":"It\u0027s easy to check which is which."},{"Start":"04:23.050 ","End":"04:32.225","Text":"Here I can see that the 0,0 is on the parabola."},{"Start":"04:32.225 ","End":"04:34.070","Text":"I don\u0027t just mean looking from the picture,"},{"Start":"04:34.070 ","End":"04:38.465","Text":"you can see that if x is 0 and y is 0,"},{"Start":"04:38.465 ","End":"04:40.610","Text":"then it satisfies the equation."},{"Start":"04:40.610 ","End":"04:48.240","Text":"All we\u0027ll have to do is see what happens if I substitute 0,0 into this expression,"},{"Start":"04:48.240 ","End":"04:54.620","Text":"and it comes out to be 3 times 0 minus 6 times 0 plus 4, it\u0027s positive."},{"Start":"04:54.620 ","End":"04:57.155","Text":"What I get is that on the line,"},{"Start":"04:57.155 ","End":"05:01.880","Text":"this expression is equal to 0 on this side of the line,"},{"Start":"05:01.880 ","End":"05:05.980","Text":"it\u0027s bigger than 0, and on this side of the line it\u0027s less than 0."},{"Start":"05:05.980 ","End":"05:08.330","Text":"Because the parabola doesn\u0027t intersect,"},{"Start":"05:08.330 ","End":"05:10.670","Text":"it\u0027s always on one side of the line."},{"Start":"05:10.670 ","End":"05:15.110","Text":"It\u0027s always on this lower right parts of the line."},{"Start":"05:15.110 ","End":"05:20.035","Text":"I know that this expression is positive,"},{"Start":"05:20.035 ","End":"05:24.095","Text":"and so I can just erase the absolute value."},{"Start":"05:24.095 ","End":"05:26.630","Text":"Maybe I\u0027ll just rewrite it over here."},{"Start":"05:26.630 ","End":"05:30.160","Text":"f of x,y equals,"},{"Start":"05:30.160 ","End":"05:32.330","Text":"now without the absolute value,"},{"Start":"05:32.330 ","End":"05:37.310","Text":"it\u0027s just 3x minus 6y plus 4."},{"Start":"05:37.310 ","End":"05:42.990","Text":"The denominator, 6 squared plus 3 squared is 45."},{"Start":"05:43.010 ","End":"05:46.050","Text":"I could even write that in front,"},{"Start":"05:46.050 ","End":"05:50.835","Text":"1 over square root of 45."},{"Start":"05:50.835 ","End":"05:55.275","Text":"This is the function that I want to minimize."},{"Start":"05:55.275 ","End":"06:00.990","Text":"Now, the way we solve such a problem with an extra moment there,"},{"Start":"06:00.990 ","End":"06:05.080","Text":"a constraint, is to introduce a new variable Lambda,"},{"Start":"06:05.080 ","End":"06:09.160","Text":"and to write 3 equations in 3 unknowns."},{"Start":"06:09.160 ","End":"06:17.215","Text":"Derivative of f with respect to x is Lambda times the derivative of g with respect to x,"},{"Start":"06:17.215 ","End":"06:20.330","Text":"and similarly with y."},{"Start":"06:20.540 ","End":"06:29.990","Text":"The third equation is just the constraint that g equals 0."},{"Start":"06:29.990 ","End":"06:34.210","Text":"In our case, let\u0027s see what this gives us."},{"Start":"06:34.210 ","End":"06:41.845","Text":"The derivative of f with respect to x, which is here,"},{"Start":"06:41.845 ","End":"06:45.870","Text":"is 1 over root"},{"Start":"06:45.870 ","End":"06:54.100","Text":"45 times the derivative of this is just 3."},{"Start":"06:54.100 ","End":"07:00.765","Text":"I can just put that 3 here instead of the 1,and then Lambda."},{"Start":"07:00.765 ","End":"07:07.985","Text":"The derivative of g with respect to x is, let\u0027s see,"},{"Start":"07:07.985 ","End":"07:16.315","Text":"2x from here plus 2y from here, and that\u0027s it."},{"Start":"07:16.315 ","End":"07:23.730","Text":"The second equation, f with respect to y is similar,"},{"Start":"07:23.730 ","End":"07:26.115","Text":"instead of a 3 we get a minus 6,"},{"Start":"07:26.115 ","End":"07:32.415","Text":"so it\u0027s minus 6 over root 45 equals Lambda."},{"Start":"07:32.415 ","End":"07:37.680","Text":"The derivative with respect to y here comes out to be let\u0027s see,"},{"Start":"07:37.680 ","End":"07:40.080","Text":"from here we get a 2x,"},{"Start":"07:40.080 ","End":"07:43.260","Text":"from here we get a 2y,"},{"Start":"07:43.260 ","End":"07:46.515","Text":"and from here we get 4."},{"Start":"07:46.515 ","End":"07:52.305","Text":"The last equation is just the constraint,"},{"Start":"07:52.305 ","End":"07:57.560","Text":"but I\u0027d like to slightly rewrite it."},{"Start":"07:57.560 ","End":"08:07.290","Text":"I noticed that this part here is one of those special products,"},{"Start":"08:07.290 ","End":"08:10.035","Text":"square of a binomial,"},{"Start":"08:10.035 ","End":"08:14.940","Text":"it\u0027s just x plus y all squared."},{"Start":"08:14.940 ","End":"08:20.780","Text":"I\u0027ll just write the constraint as x plus"},{"Start":"08:20.780 ","End":"08:29.910","Text":"y squared plus 4y equals 0."},{"Start":"08:29.910 ","End":"08:35.170","Text":"Just rewriting the first 3 terms as a perfect square."},{"Start":"08:36.560 ","End":"08:40.310","Text":"The technique is always the same."},{"Start":"08:40.310 ","End":"08:43.625","Text":"We divide one of these equations by the other,"},{"Start":"08:43.625 ","End":"08:46.120","Text":"and that way we get rid of Lambda."},{"Start":"08:46.120 ","End":"08:48.110","Text":"It doesn\u0027t really matter,"},{"Start":"08:48.110 ","End":"08:51.590","Text":"but I like to divide by the simpler one,"},{"Start":"08:51.590 ","End":"08:55.220","Text":"somehow, I\u0027ll take the second over the first."},{"Start":"08:55.220 ","End":"08:59.555","Text":"What I get on the left-hand side,"},{"Start":"08:59.555 ","End":"09:01.370","Text":"this divided by this,"},{"Start":"09:01.370 ","End":"09:06.110","Text":"using division of fractions is multiplying by"},{"Start":"09:06.110 ","End":"09:11.810","Text":"the reciprocal minus 6 over root 45 times root 45 over 3,"},{"Start":"09:11.810 ","End":"09:15.020","Text":"just comes out to be minus 2,"},{"Start":"09:15.020 ","End":"09:21.070","Text":"and on the right-hand side I get this over this."},{"Start":"09:21.070 ","End":"09:22.210","Text":"I\u0027ll just copy it,"},{"Start":"09:22.210 ","End":"09:24.550","Text":"but I can take also a 2 out."},{"Start":"09:24.550 ","End":"09:26.830","Text":"From here, I can take a 2 out,"},{"Start":"09:26.830 ","End":"09:34.020","Text":"so it\u0027s 2 Lambda times x plus y plus 2,"},{"Start":"09:34.020 ","End":"09:36.630","Text":"after taking the 2 out, over,"},{"Start":"09:36.630 ","End":"09:39.644","Text":"and here taking the 2 out,2 Lambda,"},{"Start":"09:39.644 ","End":"09:43.930","Text":"just x plus y."},{"Start":"09:45.020 ","End":"09:49.005","Text":"Now we\u0027re not allowed to divide by 0."},{"Start":"09:49.005 ","End":"09:57.660","Text":"At the moment I\u0027m going to assume that Lambda is not equal to 0 and that x plus"},{"Start":"09:57.660 ","End":"10:02.040","Text":"y is not equal to 0 and later I\u0027ll return to"},{"Start":"10:02.040 ","End":"10:08.085","Text":"this point and see what happens if this is 0 or this is 0."},{"Start":"10:08.085 ","End":"10:10.350","Text":"Let\u0027s just take the path meanwhile,"},{"Start":"10:10.350 ","End":"10:13.770","Text":"assuming that these are not 0."},{"Start":"10:13.770 ","End":"10:18.720","Text":"I can cancel this 2 Lambda with this 2 Lambda and"},{"Start":"10:18.720 ","End":"10:24.440","Text":"then what\u0027s known exactly cross multiplication but if this was minus 2/1.Anyway,"},{"Start":"10:24.440 ","End":"10:27.720","Text":"this times this equals this."},{"Start":"10:28.370 ","End":"10:34.980","Text":"I get an equation that minus 2 times x plus"},{"Start":"10:34.980 ","End":"10:41.620","Text":"y is equal to x plus y plus 2."},{"Start":"10:44.180 ","End":"10:50.715","Text":"If I open brackets and bring this to this side,"},{"Start":"10:50.715 ","End":"10:53.670","Text":"here I\u0027ll get 3x plus 3y."},{"Start":"10:53.670 ","End":"10:56.550","Text":"On the other side, I can bring the 2."},{"Start":"10:56.550 ","End":"10:59.595","Text":"In short you can do this at the side."},{"Start":"10:59.595 ","End":"11:07.105","Text":"We can conclude that x plus y is minus 2/3."},{"Start":"11:07.105 ","End":"11:12.990","Text":"Actually, another way of seeing this is just looking at x plus y. I have minus 2 of them,"},{"Start":"11:12.990 ","End":"11:18.720","Text":"I\u0027ll bring this over, I have another minus 1 of them I have minus 3 times x plus y is 2."},{"Start":"11:18.720 ","End":"11:21.255","Text":"So dividing by minus 3."},{"Start":"11:21.255 ","End":"11:29.100","Text":"Anyway, this is what I get and this is good because we have x plus y here also."},{"Start":"11:29.100 ","End":"11:33.585","Text":"Essentially, we\u0027re concentrating on 2 equations and 2 unknowns,"},{"Start":"11:33.585 ","End":"11:35.475","Text":"this one and this one,"},{"Start":"11:35.475 ","End":"11:40.180","Text":"and if I substitute x plus y in here,"},{"Start":"11:42.020 ","End":"11:46.725","Text":"then what we\u0027ll get is"},{"Start":"11:46.725 ","End":"11:55.480","Text":"minus 2/3 squared plus 4y equals 0."},{"Start":"11:57.020 ","End":"11:58.395","Text":"This"},{"Start":"11:58.395 ","End":"12:08.635","Text":"squared is 4/9."},{"Start":"12:08.635 ","End":"12:17.895","Text":"4y plus"},{"Start":"12:17.895 ","End":"12:20.640","Text":"4/9 is equal to 0."},{"Start":"12:20.640 ","End":"12:31.020","Text":"This gives us that y is equal to minus 4/9 over 4 minus 1/9."},{"Start":"12:31.020 ","End":"12:37.545","Text":"Now if I substitute y equals minus 1/9 in here,"},{"Start":"12:37.545 ","End":"12:47.940","Text":"we get that x is equal to minus 2/3 plus 1/9 minus 6/9 plus a 1/9."},{"Start":"12:47.940 ","End":"12:52.480","Text":"We get that x equals minus 5/9."},{"Start":"12:53.660 ","End":"12:58.275","Text":"We only have 1 critical point."},{"Start":"12:58.275 ","End":"13:01.560","Text":"I\u0027ll call it x asterisk,"},{"Start":"13:01.560 ","End":"13:05.565","Text":"y asterisk and that is"},{"Start":"13:05.565 ","End":"13:11.970","Text":"minus 5/9 minus 1/9."},{"Start":"13:11.970 ","End":"13:15.810","Text":"In a minute, I\u0027ll get back to this and show you why this is true."},{"Start":"13:15.810 ","End":"13:21.045","Text":"There\u0027s only 1 critical point and I\u0027m not going to do"},{"Start":"13:21.045 ","End":"13:27.870","Text":"the second derivative test because clearly there\u0027s going to be a closest point,"},{"Start":"13:27.870 ","End":"13:30.030","Text":"it has to be a minimum."},{"Start":"13:30.030 ","End":"13:34.844","Text":"We\u0027ll just intuitively say that this is a minimum."},{"Start":"13:34.844 ","End":"13:37.470","Text":"When you have a parabola there has to be a point closest,"},{"Start":"13:37.470 ","End":"13:40.740","Text":"is not going to be a maximum because the maximum is infinity."},{"Start":"13:40.740 ","End":"13:43.950","Text":"I mean, you can get as far away as you like."},{"Start":"13:43.950 ","End":"13:49.275","Text":"Then I don\u0027t need Lambda because we\u0027re not doing the second derivative test."},{"Start":"13:49.275 ","End":"13:54.435","Text":"But I do have to explain why I can make these assumptions."},{"Start":"13:54.435 ","End":"13:58.454","Text":"Well, the first 1 is easy because if lambda is 0,"},{"Start":"13:58.454 ","End":"14:02.085","Text":"I get 3 over root 45 equals 0."},{"Start":"14:02.085 ","End":"14:05.100","Text":"So that leads to an impossibility."},{"Start":"14:05.100 ","End":"14:07.545","Text":"This assumption is true."},{"Start":"14:07.545 ","End":"14:12.375","Text":"Now why can I assume that x plus y is not 0?"},{"Start":"14:12.375 ","End":"14:15.570","Text":"Because if x plus y is 0,"},{"Start":"14:15.570 ","End":"14:19.965","Text":"suppose it is, then if I put that in here,"},{"Start":"14:19.965 ","End":"14:24.900","Text":"then I get 0 plus 4y is 0, that makes y,"},{"Start":"14:24.900 ","End":"14:30.660","Text":"0 and if y is 0 and x plus y is 0,"},{"Start":"14:30.660 ","End":"14:33.370","Text":"then x is also 0."},{"Start":"14:33.950 ","End":"14:36.270","Text":"If I put x equals 0,"},{"Start":"14:36.270 ","End":"14:38.280","Text":"y equals 0 in this equation,"},{"Start":"14:38.280 ","End":"14:43.215","Text":"again, I get this contradiction that 3 over root 45 is 0."},{"Start":"14:43.215 ","End":"14:45.705","Text":"This assumption also holds."},{"Start":"14:45.705 ","End":"14:56.640","Text":"Really this is the only critical point and therefore it\u0027s the minimum point."},{"Start":"14:56.640 ","End":"14:59.745","Text":"What we\u0027re missing now,"},{"Start":"14:59.745 ","End":"15:02.100","Text":"now that we\u0027ve found this point,"},{"Start":"15:02.100 ","End":"15:07.830","Text":"which is this is the actual minimum distance and for that,"},{"Start":"15:07.830 ","End":"15:13.335","Text":"we just substitute in here,"},{"Start":"15:13.335 ","End":"15:19.070","Text":"substitute the point minus 5/9, 1/9."},{"Start":"15:19.070 ","End":"15:25.035","Text":"So we get f of minus 5/9 minus"},{"Start":"15:25.035 ","End":"15:31.875","Text":"1/9 is equal to 1 over root 45 times,"},{"Start":"15:31.875 ","End":"15:41.760","Text":"let\u0027s see, 3 times minus 5/9,"},{"Start":"15:41.760 ","End":"15:49.550","Text":"minus 6 times minus 1/9,"},{"Start":"15:49.550 ","End":"15:52.830","Text":"plus 4."},{"Start":"15:52.830 ","End":"15:56.530","Text":"Let\u0027s see, we get"},{"Start":"15:58.280 ","End":"16:07.680","Text":"minus 15/9 plus 6/9."},{"Start":"16:07.680 ","End":"16:11.505","Text":"Maybe it better write this, 1 over root 45."},{"Start":"16:11.505 ","End":"16:17.985","Text":"I\u0027ve got minus 15/9 and then plus"},{"Start":"16:17.985 ","End":"16:23.040","Text":"6/ 9 and then plus"},{"Start":"16:23.040 ","End":"16:30.600","Text":"4 but minus 15 plus 6 is minus 9,"},{"Start":"16:30.600 ","End":"16:38.280","Text":"so together this is minus 1 plus the 4 makes it 3."},{"Start":"16:38.280 ","End":"16:47.050","Text":"The answer comes out to be 3 over the square root of 45."},{"Start":"16:47.570 ","End":"16:52.860","Text":"This is the answer we just asked to find the shortest distance."},{"Start":"16:52.860 ","End":"16:58.725","Text":"We weren\u0027t asked to find the point on the parabola which is closest."},{"Start":"16:58.725 ","End":"17:01.300","Text":"That\u0027s it. We\u0027re done."}],"ID":9676},{"Watched":false,"Name":"Exercise 10","Duration":"12m 41s","ChapterTopicVideoID":9786,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"Let\u0027s start by reading the exercise."},{"Start":"00:02.040 ","End":"00:06.525","Text":"From all the open boxes whose volume is 32 cubic centimeters,"},{"Start":"00:06.525 ","End":"00:10.770","Text":"compute the dimensions of the 1 with the least surface area."},{"Start":"00:10.770 ","End":"00:13.320","Text":"Certainly, a picture could help."},{"Start":"00:13.320 ","End":"00:17.340","Text":"Here\u0027s the box, X is the width, Y is the length,"},{"Start":"00:17.340 ","End":"00:18.660","Text":"Z is the height,"},{"Start":"00:18.660 ","End":"00:21.910","Text":"and they\u0027re all in centimeters."},{"Start":"00:22.400 ","End":"00:27.975","Text":"This is an optimization problem with constraints."},{"Start":"00:27.975 ","End":"00:31.875","Text":"It\u0027s going to be a minimum problem because of the word least."},{"Start":"00:31.875 ","End":"00:36.330","Text":"We want to identify the constraint function and the target function."},{"Start":"00:36.330 ","End":"00:39.060","Text":"Let\u0027s start with the constraint."},{"Start":"00:39.060 ","End":"00:45.345","Text":"The constraint says that the volume is 32."},{"Start":"00:45.345 ","End":"00:49.530","Text":"Volume is width times length times height,"},{"Start":"00:49.530 ","End":"00:56.530","Text":"so that says that xyz equals 32."},{"Start":"00:56.990 ","End":"00:59.370","Text":"What we want to optimize,"},{"Start":"00:59.370 ","End":"01:01.005","Text":"or in this case, minimize,"},{"Start":"01:01.005 ","End":"01:06.345","Text":"is the target function,"},{"Start":"01:06.345 ","End":"01:09.585","Text":"and that would be the surface area."},{"Start":"01:09.585 ","End":"01:20.100","Text":"Now, the word open means that it has a base and 4 sides but no top."},{"Start":"01:20.100 ","End":"01:22.950","Text":"Maybe like an aquarium."},{"Start":"01:22.950 ","End":"01:26.160","Text":"The surface area will be,"},{"Start":"01:26.160 ","End":"01:31.185","Text":"there\u0027s a base, and that\u0027s length times the width, xy,"},{"Start":"01:31.185 ","End":"01:36.090","Text":"there\u0027s 2 rectangles, x by z, so it\u0027s 2xz,"},{"Start":"01:36.090 ","End":"01:39.040","Text":"and then we have also 2yz,"},{"Start":"01:39.350 ","End":"01:42.810","Text":"that\u0027s the surface area."},{"Start":"01:42.810 ","End":"01:47.490","Text":"We can phrase this as a minimize problem,"},{"Start":"01:47.490 ","End":"01:57.340","Text":"the minimum curly braces of xy plus 2xz plus 2yz,"},{"Start":"01:58.670 ","End":"02:06.315","Text":"subject to the constraint xyz is 32,"},{"Start":"02:06.315 ","End":"02:12.735","Text":"or sometimes, I like to write it as something equals 0."},{"Start":"02:12.735 ","End":"02:18.360","Text":"This part is the target function,"},{"Start":"02:18.360 ","End":"02:22.155","Text":"we call it f of x, y, and z,"},{"Start":"02:22.155 ","End":"02:25.740","Text":"and the constraint to at least the part that\u0027s equal to 0,"},{"Start":"02:25.740 ","End":"02:28.290","Text":"I\u0027ll call that g of x,"},{"Start":"02:28.290 ","End":"02:32.205","Text":"y, and z. I forgot to mention something,"},{"Start":"02:32.205 ","End":"02:36.090","Text":"there are natural limitations because it\u0027s a real-world problem."},{"Start":"02:36.090 ","End":"02:37.710","Text":"The dimensions are positive."},{"Start":"02:37.710 ","End":"02:41.720","Text":"We should also add the conditions that x,"},{"Start":"02:41.720 ","End":"02:45.110","Text":"y, and z are all strictly positive."},{"Start":"02:45.110 ","End":"02:48.140","Text":"It makes no sense for it to be flat."},{"Start":"02:48.140 ","End":"02:51.865","Text":"Next step is to look for critical points."},{"Start":"02:51.865 ","End":"02:58.080","Text":"There\u0027s 3 equations in 1 which we can write as the gradient of the function f"},{"Start":"02:58.080 ","End":"03:01.290","Text":"equals Lambda times the gradient of"},{"Start":"03:01.290 ","End":"03:05.100","Text":"the function g. Lambda is going to be a 4th variable besides x,"},{"Start":"03:05.100 ","End":"03:08.025","Text":"y, and z, an auxiliary variable."},{"Start":"03:08.025 ","End":"03:11.700","Text":"Besides this is also the original constraint."},{"Start":"03:11.700 ","End":"03:13.965","Text":"Let\u0027s write them down."},{"Start":"03:13.965 ","End":"03:15.690","Text":"We\u0027re going to get 4 equation in all."},{"Start":"03:15.690 ","End":"03:22.035","Text":"From here, we\u0027ll get that f with respect to x is Lambda times g with respect to x,"},{"Start":"03:22.035 ","End":"03:24.660","Text":"and similarly for y,"},{"Start":"03:24.660 ","End":"03:28.035","Text":"and similarly for z,"},{"Start":"03:28.035 ","End":"03:30.780","Text":"and Lambda g of z."},{"Start":"03:30.780 ","End":"03:34.350","Text":"The 4th equation is the constraint."},{"Start":"03:34.350 ","End":"03:38.610","Text":"Well, in general, it\u0027s g equals 0,"},{"Start":"03:38.610 ","End":"03:43.440","Text":"but when I convert this to our particular case,"},{"Start":"03:43.440 ","End":"03:50.055","Text":"what we will get will be f with respect to x,"},{"Start":"03:50.055 ","End":"03:53.230","Text":"that will be y plus 2z,"},{"Start":"03:53.720 ","End":"04:01.005","Text":"that\u0027s equal to Lambda g with respect to x is yz."},{"Start":"04:01.005 ","End":"04:04.905","Text":"Next one, with respect to y here,"},{"Start":"04:04.905 ","End":"04:08.385","Text":"that gives us x plus 2z,"},{"Start":"04:08.385 ","End":"04:14.280","Text":"and here, we get Lambda times xz."},{"Start":"04:14.280 ","End":"04:23.960","Text":"With respect to z, we get here 2x plus 2y equals Lambda,"},{"Start":"04:23.960 ","End":"04:27.590","Text":"and then it will be xy."},{"Start":"04:27.590 ","End":"04:29.990","Text":"The original constraint, well,"},{"Start":"04:29.990 ","End":"04:36.780","Text":"I\u0027ll write it in this form, xyz equals 32."},{"Start":"04:37.460 ","End":"04:41.340","Text":"Just like in the case of 2 variables xy,"},{"Start":"04:41.340 ","End":"04:45.720","Text":"we can get rid of Lambda by dividing 1 equation by another."},{"Start":"04:45.720 ","End":"04:50.745","Text":"Let\u0027s say we divide the first by the second,"},{"Start":"04:50.745 ","End":"04:59.460","Text":"we would get y plus 2z over x plus 2z."},{"Start":"04:59.690 ","End":"05:10.570","Text":"You\u0027re dividing the first by the second equals Lambda yz over Lambda xz."},{"Start":"05:10.700 ","End":"05:16.020","Text":"Now, I claim that we\u0027re not dividing by 0 anywhere,"},{"Start":"05:16.020 ","End":"05:19.090","Text":"that there\u0027s no problem with this division."},{"Start":"05:20.810 ","End":"05:23.205","Text":"Let\u0027s take them one at a time."},{"Start":"05:23.205 ","End":"05:29.070","Text":"I say that Lambda is not equal to 0 because if Lambda was 0,"},{"Start":"05:29.070 ","End":"05:31.365","Text":"we get 0 on the right here,"},{"Start":"05:31.365 ","End":"05:33.750","Text":"and the left, we\u0027d get something positive, I mean,"},{"Start":"05:33.750 ","End":"05:37.260","Text":"y and z are positive numbers,"},{"Start":"05:37.260 ","End":"05:39.435","Text":"so y plus 2z is certainly positive."},{"Start":"05:39.435 ","End":"05:40.620","Text":"That rules that out."},{"Start":"05:40.620 ","End":"05:46.635","Text":"X and z, we already know are not 0 because they\u0027re positive."},{"Start":"05:46.635 ","End":"05:53.220","Text":"X is not 0 and z is not 0 from the given conditions."},{"Start":"05:53.220 ","End":"05:58.920","Text":"The only thing that could possibly be 0 and the denominator would be the x plus 2z."},{"Start":"05:58.920 ","End":"06:04.710","Text":"I say that x plus 2z is also not 0, because if it was 0,"},{"Start":"06:04.710 ","End":"06:07.410","Text":"then we\u0027d have 0 here,"},{"Start":"06:07.410 ","End":"06:14.190","Text":"and that\u0027s not possible because we\u0027ve already shown that Lambda x and z are all not 0."},{"Start":"06:14.190 ","End":"06:15.870","Text":"The product is not 0,"},{"Start":"06:15.870 ","End":"06:17.910","Text":"so this is also not 0."},{"Start":"06:17.910 ","End":"06:19.800","Text":"We\u0027re fine with this."},{"Start":"06:19.800 ","End":"06:23.130","Text":"Now, we can cross multiply out, but first,"},{"Start":"06:23.130 ","End":"06:29.160","Text":"we\u0027ll cancel Lambda with Lambda, z with z."},{"Start":"06:29.160 ","End":"06:31.455","Text":"Now, if we cross multiply,"},{"Start":"06:31.455 ","End":"06:37.890","Text":"we get x times this would be xy plus 2xz,"},{"Start":"06:37.890 ","End":"06:46.930","Text":"and this with this would be xy plus 2yz,"},{"Start":"06:48.920 ","End":"06:53.385","Text":"xy I can take away from both sides."},{"Start":"06:53.385 ","End":"06:58.050","Text":"We know that z is not 0,"},{"Start":"06:58.050 ","End":"07:01.410","Text":"so I can divide by z and then by 2."},{"Start":"07:01.410 ","End":"07:06.540","Text":"We end up with x equals y."},{"Start":"07:06.540 ","End":"07:07.980","Text":"That\u0027s a good start."},{"Start":"07:07.980 ","End":"07:14.640","Text":"Now, let\u0027s try this trick again on a different pair of equations."},{"Start":"07:14.640 ","End":"07:18.900","Text":"Let\u0027s try this 1 over this 1,"},{"Start":"07:18.900 ","End":"07:20.475","Text":"the second over the third."},{"Start":"07:20.475 ","End":"07:29.040","Text":"This time, we would get x plus 2z over 2x plus 2y,"},{"Start":"07:29.040 ","End":"07:32.730","Text":"I\u0027ll write that as twice x plus y,"},{"Start":"07:32.730 ","End":"07:42.930","Text":"equals Lambda xz over Lambda xy."},{"Start":"07:42.930 ","End":"07:48.540","Text":"Now, just as we checked before about all the denominators not being 0,"},{"Start":"07:48.540 ","End":"07:50.280","Text":"same thing works here."},{"Start":"07:50.280 ","End":"07:52.350","Text":"We already know Lambda is not 0."},{"Start":"07:52.350 ","End":"07:54.720","Text":"X and y are given to be positive."},{"Start":"07:54.720 ","End":"07:57.840","Text":"This is the left-hand side."},{"Start":"07:57.840 ","End":"08:02.175","Text":"Here, can\u0027t be 0 because if this was 0,"},{"Start":"08:02.175 ","End":"08:04.680","Text":"then 1 of these 3 would be 0,"},{"Start":"08:04.680 ","End":"08:09.045","Text":"which it isn\u0027t because these 3 already we showed were not 0."},{"Start":"08:09.045 ","End":"08:15.000","Text":"We can again cancel this time this with this,"},{"Start":"08:15.000 ","End":"08:18.945","Text":"and this with this, and then cross-multiply."},{"Start":"08:18.945 ","End":"08:24.510","Text":"I\u0027ll change this back to 2x plus 2y as we got no point in taking 2 out the brackets."},{"Start":"08:24.510 ","End":"08:32.085","Text":"Cross-multiplying this with this will give me that xy plus"},{"Start":"08:32.085 ","End":"08:43.060","Text":"2yz is equal to 2xz plus 2yz."},{"Start":"08:44.660 ","End":"08:47.865","Text":"This goes with this,"},{"Start":"08:47.865 ","End":"08:58.715","Text":"and divide both sides by x and we get that y equals 2z."},{"Start":"08:58.715 ","End":"09:04.160","Text":"Now, we have 3 equations and 3 unknowns without Lambda,"},{"Start":"09:04.160 ","End":"09:07.380","Text":"the ones that I\u0027ve just highlighted."},{"Start":"09:07.670 ","End":"09:11.360","Text":"I\u0027m going to get an equation in just z because look,"},{"Start":"09:11.360 ","End":"09:15.410","Text":"y is 2z and x is equal to y,"},{"Start":"09:15.410 ","End":"09:19.430","Text":"so therefore, it\u0027s equal to 2z also."},{"Start":"09:19.430 ","End":"09:22.220","Text":"If I put all that in here,"},{"Start":"09:22.220 ","End":"09:26.480","Text":"what I\u0027ll get is that from here,"},{"Start":"09:26.480 ","End":"09:32.740","Text":"x which is 2z times y,"},{"Start":"09:32.740 ","End":"09:41.650","Text":"which is also 2z times z is equal to 32."},{"Start":"09:41.650 ","End":"09:47.625","Text":"Now, this gives me that 4z cubed is 32,"},{"Start":"09:47.625 ","End":"09:54.855","Text":"so z cubed is 32 over 4, which is 8."},{"Start":"09:54.855 ","End":"09:56.685","Text":"If I take the cube root,"},{"Start":"09:56.685 ","End":"10:01.890","Text":"that gives me that z is equal to 2."},{"Start":"10:01.890 ","End":"10:11.840","Text":"Now y which is 2z is equal to 4 and x equals y,"},{"Start":"10:11.840 ","End":"10:16.100","Text":"so x equals 4 also,"},{"Start":"10:16.100 ","End":"10:22.215","Text":"and we often label the optimum with an asterisk."},{"Start":"10:22.215 ","End":"10:28.595","Text":"The question is, how do we know that this is a minimum and not a maximum, for example?"},{"Start":"10:28.595 ","End":"10:31.935","Text":"Well, let\u0027s first of all see what is the surface area."},{"Start":"10:31.935 ","End":"10:38.610","Text":"The surface area, which is the function f of x, y,"},{"Start":"10:38.610 ","End":"10:46.350","Text":"and z would be xy which"},{"Start":"10:46.350 ","End":"10:51.060","Text":"is 4 times 4 plus 2xz"},{"Start":"10:51.060 ","End":"10:57.810","Text":"plus 2 times 4 times 2 plus 2yz,"},{"Start":"10:57.810 ","End":"11:02.385","Text":"2 times 4 times 2,"},{"Start":"11:02.385 ","End":"11:10.320","Text":"that comes out to be 48 centimeters squared."},{"Start":"11:10.320 ","End":"11:14.459","Text":"I could give a specific example that gives more than 48,"},{"Start":"11:14.459 ","End":"11:16.290","Text":"but I\u0027d like to give you some intuition."},{"Start":"11:16.290 ","End":"11:18.225","Text":"There actually is no maximum,"},{"Start":"11:18.225 ","End":"11:19.680","Text":"because if you think about it,"},{"Start":"11:19.680 ","End":"11:23.250","Text":"I can take the base, the rectangular base,"},{"Start":"11:23.250 ","End":"11:25.125","Text":"as large as I want,"},{"Start":"11:25.125 ","End":"11:30.630","Text":"and always reduce the height small enough that the volume still comes out 32."},{"Start":"11:30.630 ","End":"11:33.570","Text":"Suppose I took x to be a large number,"},{"Start":"11:33.570 ","End":"11:37.155","Text":"100 could be 1,000 centimeters,"},{"Start":"11:37.155 ","End":"11:42.330","Text":"and I could take y to be also 100 centimeters."},{"Start":"11:42.330 ","End":"11:47.940","Text":"Then I could still get xyz to be 32 because if I take z to be,"},{"Start":"11:47.940 ","End":"11:52.545","Text":"let\u0027s see, 32 over 10,000,"},{"Start":"11:52.545 ","End":"11:54.240","Text":"what would that come out to?"},{"Start":"11:54.240 ","End":"12:03.870","Text":"0.0032, that is 32 over 10,000."},{"Start":"12:03.870 ","End":"12:09.570","Text":"Then the volume would still come out to be 32 cubic centimeters,"},{"Start":"12:09.570 ","End":"12:14.310","Text":"but the surface area would be more than 10,000 because just the base alone,"},{"Start":"12:14.310 ","End":"12:18.225","Text":"forgetting the sides is already x times y is 10,000,"},{"Start":"12:18.225 ","End":"12:19.785","Text":"so there is no maximum,"},{"Start":"12:19.785 ","End":"12:24.210","Text":"and intuitively, we feel that there is a minimum."},{"Start":"12:24.210 ","End":"12:28.570","Text":"This is really a minimum."},{"Start":"12:30.410 ","End":"12:33.570","Text":"The precise methods for proving that it\u0027s"},{"Start":"12:33.570 ","End":"12:37.635","Text":"a minimum is just beyond the scope of this course."},{"Start":"12:37.635 ","End":"12:40.605","Text":"We\u0027ll leave it at that,"},{"Start":"12:40.605 ","End":"12:42.940","Text":"and we are done."}],"ID":9677},{"Watched":false,"Name":"Exercise 11","Duration":"17m 40s","ChapterTopicVideoID":9787,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this exercise, we\u0027re given the equation of"},{"Start":"00:03.870 ","End":"00:07.515","Text":"a sphere and we have to consider the points on this sphere."},{"Start":"00:07.515 ","End":"00:11.310","Text":"We want the ones that are closest and"},{"Start":"00:11.310 ","End":"00:19.200","Text":"furthest away from the 1, 2, 2."},{"Start":"00:19.200 ","End":"00:22.920","Text":"You should be able to recognize the equation of a sphere."},{"Start":"00:22.920 ","End":"00:24.884","Text":"It\u0027s actually centered at the origin,"},{"Start":"00:24.884 ","End":"00:29.475","Text":"and the radius is 6 because 32 is 6 squared."},{"Start":"00:29.475 ","End":"00:31.885","Text":"Let me just give a quick sketch."},{"Start":"00:31.885 ","End":"00:35.510","Text":"I\u0027m not very good at drawing spheres."},{"Start":"00:35.510 ","End":"00:39.680","Text":"I\u0027ll draw a circle that we imagine this is a sphere,"},{"Start":"00:39.680 ","End":"00:44.420","Text":"and then there\u0027s a special 1, 2, 2."},{"Start":"00:44.420 ","End":"00:47.225","Text":"Let\u0027s say that\u0027s this one here,"},{"Start":"00:47.225 ","End":"00:51.470","Text":"it\u0027s actually inside the circle because this you substitute here,"},{"Start":"00:51.470 ","End":"00:56.340","Text":"1 squared plus 2 squared plus 2 squared is less than 36."},{"Start":"00:56.420 ","End":"01:03.050","Text":"Now, it seems clear that there\u0027s going to be a point on the sphere that\u0027s closest to it."},{"Start":"01:03.050 ","End":"01:04.940","Text":"It might be somewhere here,"},{"Start":"01:04.940 ","End":"01:08.180","Text":"and there\u0027ll be some point that\u0027s furthest away,"},{"Start":"01:08.180 ","End":"01:11.475","Text":"might be somewhere here,"},{"Start":"01:11.475 ","End":"01:16.350","Text":"and we have to find these points."},{"Start":"01:16.350 ","End":"01:20.900","Text":"We\u0027re not going to prove that what we find is actually minimum or maximum,"},{"Start":"01:20.900 ","End":"01:23.390","Text":"but intuitively it will be clear."},{"Start":"01:23.390 ","End":"01:28.830","Text":"Let\u0027s set this up as a problem of optimization on the constraint,"},{"Start":"01:28.830 ","End":"01:32.970","Text":"and let\u0027s go first for the constraint."},{"Start":"01:32.970 ","End":"01:38.615","Text":"That would just be this equation of the sphere,"},{"Start":"01:38.615 ","End":"01:48.925","Text":"which I\u0027d rather write as x squared plus y squared plus z squared minus 36 equals 0,"},{"Start":"01:48.925 ","End":"01:54.845","Text":"and then this here would be what we call the constraint function."},{"Start":"01:54.845 ","End":"02:01.030","Text":"We could call it g of x, y, and z,"},{"Start":"02:01.030 ","End":"02:06.860","Text":"and this refers to the left-hand side of the constraints,"},{"Start":"02:06.860 ","End":"02:08.540","Text":"so g equals 0."},{"Start":"02:08.540 ","End":"02:11.779","Text":"Then we also have a target function,"},{"Start":"02:11.779 ","End":"02:16.415","Text":"the one we\u0027re trying to maximize or minimize or both."},{"Start":"02:16.415 ","End":"02:20.705","Text":"In this case, it would be the distance."},{"Start":"02:20.705 ","End":"02:23.660","Text":"Let\u0027s just take any point on the sphere,"},{"Start":"02:23.660 ","End":"02:25.610","Text":"just some x, y,"},{"Start":"02:25.610 ","End":"02:28.235","Text":"z in general on the sphere."},{"Start":"02:28.235 ","End":"02:36.560","Text":"What we need is for the distance from the general point to a specific 1,"},{"Start":"02:36.560 ","End":"02:41.105","Text":"2, 2 for this d to be maximum or minimum."},{"Start":"02:41.105 ","End":"02:46.610","Text":"The target function, d of x, y,"},{"Start":"02:46.610 ","End":"02:54.005","Text":"and z is equal to the distance between 2 points is given by the standard formula,"},{"Start":"02:54.005 ","End":"03:01.070","Text":"x minus the x of the point squared plus y minus the y of the point squared plus"},{"Start":"03:01.070 ","End":"03:08.405","Text":"z minus the z of the point squared."},{"Start":"03:08.405 ","End":"03:11.250","Text":"I need to make this a bit longer."},{"Start":"03:11.440 ","End":"03:17.580","Text":"But this is not a very convenient function because it has a square root,"},{"Start":"03:17.580 ","End":"03:22.040","Text":"so we have our standard trick of instead of considering"},{"Start":"03:22.040 ","End":"03:28.055","Text":"the distance d. The trick is that because the square root is a positive function,"},{"Start":"03:28.055 ","End":"03:31.640","Text":"the distance is least when the square of the distance is least,"},{"Start":"03:31.640 ","End":"03:35.830","Text":"so we take f to be the distance squared,"},{"Start":"03:35.830 ","End":"03:43.940","Text":"and at the end you have to remember that when you find the greatest or least value for f,"},{"Start":"03:43.940 ","End":"03:47.870","Text":"that you go back and take the square root of f to get"},{"Start":"03:47.870 ","End":"03:51.860","Text":"back to d. But the points will be the same,"},{"Start":"03:51.860 ","End":"03:55.470","Text":"whether we take the distance or the distance squared."},{"Start":"03:55.960 ","End":"04:01.520","Text":"Let\u0027s take our targets as f of x, y, z,"},{"Start":"04:01.520 ","End":"04:06.620","Text":"and this will be this thing without the square root x minus 1"},{"Start":"04:06.620 ","End":"04:13.010","Text":"squared plus y minus 2 squared plus z minus 2 squared."},{"Start":"04:13.010 ","End":"04:19.965","Text":"Then we have our problem setup as we want, the maximum."},{"Start":"04:19.965 ","End":"04:23.150","Text":"Actually we want 2 problems."},{"Start":"04:23.150 ","End":"04:32.120","Text":"First of all, we want the maximum of put it in curly braces of our f of x,"},{"Start":"04:32.120 ","End":"04:36.620","Text":"y, z. I just write it as f of x, y,"},{"Start":"04:36.620 ","End":"04:44.430","Text":"z rather than copying this whole thing again, subject to."},{"Start":"04:44.430 ","End":"04:50.025","Text":"Then we have the equation g of x,"},{"Start":"04:50.025 ","End":"04:55.150","Text":"y, z equals 0."},{"Start":"04:57.470 ","End":"05:02.120","Text":"That\u0027s how it\u0027s setup except that we have 2 problems."},{"Start":"05:02.120 ","End":"05:07.130","Text":"We have maximum and then we also have the minimum of the same,"},{"Start":"05:07.130 ","End":"05:10.640","Text":"so I\u0027ll just write it twice,"},{"Start":"05:10.640 ","End":"05:17.790","Text":"but we have maximum and minimum of the same function subject to the same constraint."},{"Start":"05:18.220 ","End":"05:27.130","Text":"The standard technique for this is to find the critical points by introducing"},{"Start":"05:27.130 ","End":"05:33.320","Text":"a new variable Lambda and then writing the equation that the gradient"},{"Start":"05:33.320 ","End":"05:39.815","Text":"of f equals Lambda times the gradient of g. That gives 3 equations,"},{"Start":"05:39.815 ","End":"05:41.660","Text":"and together with the constraint,"},{"Start":"05:41.660 ","End":"05:44.355","Text":"that will be 4 equations."},{"Start":"05:44.355 ","End":"05:45.900","Text":"Let\u0027s write this out,"},{"Start":"05:45.900 ","End":"05:48.320","Text":"the gradient we\u0027ll just write it component-wise,"},{"Start":"05:48.320 ","End":"05:52.345","Text":"is the derivative of f with respect to x,"},{"Start":"05:52.345 ","End":"05:57.065","Text":"and that\u0027s going to equal Lambda derivative of g with respect to x."},{"Start":"05:57.065 ","End":"06:04.285","Text":"Similarly for y and similarly for z,"},{"Start":"06:04.285 ","End":"06:07.815","Text":"Lambda g with respect to z,"},{"Start":"06:07.815 ","End":"06:15.455","Text":"and we also get a fourth equation is that g equals 0,"},{"Start":"06:15.455 ","End":"06:17.950","Text":"which is the constraint."},{"Start":"06:17.950 ","End":"06:21.270","Text":"That\u0027s 4 equations and 4 unknowns,"},{"Start":"06:21.270 ","End":"06:23.795","Text":"x, y, z, Lambda, that\u0027s in general."},{"Start":"06:23.795 ","End":"06:26.435","Text":"Let\u0027s see what we get in our case."},{"Start":"06:26.435 ","End":"06:28.730","Text":"Just to make it easier to see,"},{"Start":"06:28.730 ","End":"06:32.645","Text":"I\u0027ll highlight f and g. This is our function f,"},{"Start":"06:32.645 ","End":"06:37.040","Text":"the target function, and this is our function g,"},{"Start":"06:37.040 ","End":"06:40.955","Text":"the constraint function when set to 0."},{"Start":"06:40.955 ","End":"06:51.020","Text":"So f with respect to x is just twice x minus 1 times 1,"},{"Start":"06:51.020 ","End":"06:54.685","Text":"the internal derivative, so I don\u0027t need to write times 1,"},{"Start":"06:54.685 ","End":"06:58.325","Text":"and all the rest of it is a constant as far as x goes."},{"Start":"06:58.325 ","End":"07:07.560","Text":"That will equal Lambda times g with respect to x,"},{"Start":"07:07.560 ","End":"07:11.475","Text":"g is this with respect to x is just 2x."},{"Start":"07:11.475 ","End":"07:14.180","Text":"Now with respect to y,"},{"Start":"07:14.180 ","End":"07:24.085","Text":"we get twice y minus 2 equals Lambda and then 2y,"},{"Start":"07:24.085 ","End":"07:30.830","Text":"and this 1 is twice z minus 2"},{"Start":"07:30.830 ","End":"07:37.830","Text":"from here equals Lambda times 2z."},{"Start":"07:37.970 ","End":"07:43.490","Text":"Finally, the last equation is g equals 0."},{"Start":"07:43.490 ","End":"07:49.145","Text":"I can either write it as this equals 0 or I can write it in the original form,"},{"Start":"07:49.145 ","End":"07:57.170","Text":"that x squared plus y squared plus z squared minus 36 equals 0."},{"Start":"07:57.170 ","End":"08:00.720","Text":"I sometimes refer to write it like this."},{"Start":"08:00.940 ","End":"08:04.355","Text":"4 equations, 4 unknowns."},{"Start":"08:04.355 ","End":"08:09.080","Text":"I can tidy up a bit because all these top 3 equations have a 2 in both sides,"},{"Start":"08:09.080 ","End":"08:14.210","Text":"so I can just cancel this 2 with this 2 and so on and so on,"},{"Start":"08:14.210 ","End":"08:16.685","Text":"This makes it a bit simpler."},{"Start":"08:16.685 ","End":"08:20.600","Text":"Now the standard trick we use is to take"},{"Start":"08:20.600 ","End":"08:25.285","Text":"a pair of these and divide 1 by the other and that gets rid of Lambda,"},{"Start":"08:25.285 ","End":"08:27.710","Text":"and we may do this more than once."},{"Start":"08:27.710 ","End":"08:29.990","Text":"Let\u0027s take the top 2,"},{"Start":"08:29.990 ","End":"08:33.530","Text":"so dividing this one by this one,"},{"Start":"08:33.530 ","End":"08:41.935","Text":"I get x minus 1 over y minus 2 equals"},{"Start":"08:41.935 ","End":"08:47.770","Text":"Lambda x over Lambda y. I"},{"Start":"08:47.770 ","End":"08:53.590","Text":"have to also take into account that these denominators could be zeros."},{"Start":"08:53.590 ","End":"08:58.630","Text":"I actually claim that none of these denominators is 0."},{"Start":"08:58.630 ","End":"09:04.329","Text":"For example, let\u0027s check what would happen if Lambda were equal to 0."},{"Start":"09:04.329 ","End":"09:07.090","Text":"Well, if Lambda were equal to 0,"},{"Start":"09:07.090 ","End":"09:10.465","Text":"then in all of the top 3 equations,"},{"Start":"09:10.465 ","End":"09:12.730","Text":"I get that x minus 1 is 0,"},{"Start":"09:12.730 ","End":"09:14.500","Text":"y minus 2 is 0,"},{"Start":"09:14.500 ","End":"09:16.570","Text":"z minus 2 is 0."},{"Start":"09:16.570 ","End":"09:19.640","Text":"That would give me that x, y,"},{"Start":"09:19.640 ","End":"09:23.850","Text":"z is 1, 2, 2."},{"Start":"09:23.930 ","End":"09:26.130","Text":"This point 1, 2,"},{"Start":"09:26.130 ","End":"09:28.030","Text":"2 doesn\u0027t satisfy the constraint,"},{"Start":"09:28.030 ","End":"09:29.520","Text":"it\u0027s not on the circle."},{"Start":"09:29.520 ","End":"09:31.940","Text":"Here\u0027s the original constraint."},{"Start":"09:31.940 ","End":"09:35.780","Text":"If I put in 1 squared plus 2 squared plus 2 squared,"},{"Start":"09:35.780 ","End":"09:38.280","Text":"I get something less than 36,"},{"Start":"09:38.280 ","End":"09:41.415","Text":"so this is not good."},{"Start":"09:41.415 ","End":"09:46.869","Text":"Therefore Lambda is not equal to 0,"},{"Start":"09:47.150 ","End":"09:51.500","Text":"and then I can also cancel the Lambda."},{"Start":"09:51.500 ","End":"09:56.125","Text":"Now what about the other things in the denominator, can y be 0?"},{"Start":"09:56.125 ","End":"10:02.060","Text":"Well, suppose that y were equal to 0."},{"Start":"10:02.060 ","End":"10:03.545","Text":"What would that give us?"},{"Start":"10:03.545 ","End":"10:05.825","Text":"If y is 0 here,"},{"Start":"10:05.825 ","End":"10:09.180","Text":"it would give us that."},{"Start":"10:09.450 ","End":"10:12.805","Text":"Plugging it into the second equation,"},{"Start":"10:12.805 ","End":"10:17.410","Text":"we would get minus 2 equals 0."},{"Start":"10:17.410 ","End":"10:19.780","Text":"That\u0027s also a full suit,"},{"Start":"10:19.780 ","End":"10:21.730","Text":"so y is not 0."},{"Start":"10:21.730 ","End":"10:24.085","Text":"What about this denominator?"},{"Start":"10:24.085 ","End":"10:27.385","Text":"What about y minus 2 equals 0?"},{"Start":"10:27.385 ","End":"10:30.295","Text":"Well, if y minus 2 is 0,"},{"Start":"10:30.295 ","End":"10:32.470","Text":"y would be equal to 2."},{"Start":"10:32.470 ","End":"10:36.490","Text":"But would get 0 equals 2 Lambda,"},{"Start":"10:36.490 ","End":"10:39.865","Text":"then that would give us that Lambda equals 0,"},{"Start":"10:39.865 ","End":"10:42.145","Text":"and that\u0027s already been ruled out."},{"Start":"10:42.145 ","End":"10:45.280","Text":"None of these denominators is 0,"},{"Start":"10:45.280 ","End":"10:48.100","Text":"and we can cross-multiply."},{"Start":"10:48.100 ","End":"10:49.450","Text":"I\u0027ll do that in a moment."},{"Start":"10:49.450 ","End":"10:56.470","Text":"I just want to also get another equation by dividing this one by this one,"},{"Start":"10:56.470 ","End":"10:59.050","Text":"and then we get this over this."},{"Start":"10:59.050 ","End":"11:05.425","Text":"We get y minus 2 over z minus 2"},{"Start":"11:05.425 ","End":"11:13.060","Text":"equals Lambda y over Lambda z."},{"Start":"11:13.060 ","End":"11:18.065","Text":"We\u0027ve already checked that Lambda is not equal to 0."},{"Start":"11:18.065 ","End":"11:20.970","Text":"This goes with this."},{"Start":"11:20.970 ","End":"11:23.370","Text":"The others can be ruled out similar to here."},{"Start":"11:23.370 ","End":"11:30.275","Text":"For example, if z equals 0 and we plug it into the last equation,"},{"Start":"11:30.275 ","End":"11:35.350","Text":"then we get that minus 2 equals 0,"},{"Start":"11:35.350 ","End":"11:37.585","Text":"again, and that\u0027s no good."},{"Start":"11:37.585 ","End":"11:39.310","Text":"Z can\u0027t be 0."},{"Start":"11:39.310 ","End":"11:43.075","Text":"If z minus 2 were equal to 0,"},{"Start":"11:43.075 ","End":"11:44.920","Text":"and we plugged it into here,"},{"Start":"11:44.920 ","End":"11:47.155","Text":"we\u0027d get that Lambda equals 0,"},{"Start":"11:47.155 ","End":"11:49.795","Text":"and this is already being ruled out."},{"Start":"11:49.795 ","End":"11:53.740","Text":"All these divisions are not by 0."},{"Start":"11:53.740 ","End":"11:57.010","Text":"Now we can just rewrite these in"},{"Start":"11:57.010 ","End":"12:01.975","Text":"a more convenient form by cross-multiplying from the first pair,"},{"Start":"12:01.975 ","End":"12:11.230","Text":"we\u0027d get that y times"},{"Start":"12:11.230 ","End":"12:17.650","Text":"x minus 1 is equal to the other diagonal,"},{"Start":"12:17.650 ","End":"12:21.140","Text":"x times y minus 2."},{"Start":"12:22.500 ","End":"12:29.740","Text":"The last one would give us that z times y"},{"Start":"12:29.740 ","End":"12:33.670","Text":"minus 2 is equal"},{"Start":"12:33.670 ","End":"12:40.465","Text":"to y times z minus 2."},{"Start":"12:40.465 ","End":"12:44.090","Text":"If I multiply this one out,"},{"Start":"12:44.550 ","End":"12:49.510","Text":"we get that yx minus"},{"Start":"12:49.510 ","End":"12:56.545","Text":"y equals xy minus 2x."},{"Start":"12:56.545 ","End":"13:02.245","Text":"Now, the yx cancels with the xy."},{"Start":"13:02.245 ","End":"13:05.500","Text":"I can get rid of the minus on both sides."},{"Start":"13:05.500 ","End":"13:12.170","Text":"What I get is that y equals to x."},{"Start":"13:12.270 ","End":"13:15.190","Text":"For this one, if I multiply out,"},{"Start":"13:15.190 ","End":"13:22.270","Text":"I get zy minus 2z equals yz minus 2y,"},{"Start":"13:22.270 ","End":"13:26.140","Text":"which gives me, zy and yz are the same."},{"Start":"13:26.140 ","End":"13:30.625","Text":"They cancel, divide both sides by minus 2,"},{"Start":"13:30.625 ","End":"13:34.735","Text":"and we get that z equals y."},{"Start":"13:34.735 ","End":"13:37.660","Text":"Now I\u0027m going to highlight 3 equations,"},{"Start":"13:37.660 ","End":"13:41.380","Text":"this one, this one, and this one."},{"Start":"13:41.380 ","End":"13:44.320","Text":"3 equations and 3 unknowns."},{"Start":"13:44.320 ","End":"13:48.745","Text":"I can do some substitutions to get this all in terms of x."},{"Start":"13:48.745 ","End":"13:51.100","Text":"Note that z equals y and y equals to 2x,"},{"Start":"13:51.100 ","End":"13:54.520","Text":"and I can get that z is also equal to 2x."},{"Start":"13:54.520 ","End":"13:57.115","Text":"We get x squared,"},{"Start":"13:57.115 ","End":"13:59.575","Text":"I\u0027m reading from here and doing a substitution,"},{"Start":"13:59.575 ","End":"14:06.130","Text":"plus y squared but y is 2x squared plus z,"},{"Start":"14:06.130 ","End":"14:10.090","Text":"which is equal also to x."},{"Start":"14:10.090 ","End":"14:15.860","Text":"So it\u0027s another 2x squared equals 36."},{"Start":"14:16.290 ","End":"14:19.495","Text":"Now if I think about it,"},{"Start":"14:19.495 ","End":"14:21.340","Text":"how many x squareds do I have?"},{"Start":"14:21.340 ","End":"14:25.090","Text":"1 plus 2 squared is 4 plus another 4,"},{"Start":"14:25.090 ","End":"14:30.860","Text":"9x squared equals 36."},{"Start":"14:30.860 ","End":"14:34.260","Text":"Divide both sides by 9,"},{"Start":"14:34.260 ","End":"14:41.255","Text":"and we get that x squared equals 4."},{"Start":"14:41.255 ","End":"14:44.990","Text":"This gives us 2 possibilities."},{"Start":"14:45.000 ","End":"14:53.140","Text":"We either get that x equals 2 or we get x equals minus 2."},{"Start":"14:53.140 ","End":"14:55.265","Text":"Now if x is 2,"},{"Start":"14:55.265 ","End":"14:59.745","Text":"then the xyz will be 2,"},{"Start":"14:59.745 ","End":"15:01.965","Text":"then y is 2x,"},{"Start":"15:01.965 ","End":"15:08.710","Text":"so that\u0027s 4, and z is equal to y, that\u0027s also 4."},{"Start":"15:08.710 ","End":"15:12.070","Text":"If I take x equals minus 2,"},{"Start":"15:12.070 ","End":"15:20.905","Text":"then the xyz becomes minus 2 for x, y is 2x."},{"Start":"15:20.905 ","End":"15:25.120","Text":"That\u0027s minus 4 and z is y is minus 4."},{"Start":"15:25.120 ","End":"15:27.985","Text":"These are both my x,"},{"Start":"15:27.985 ","End":"15:33.339","Text":"y, and z for the extrema."},{"Start":"15:33.339 ","End":"15:34.720","Text":"Let\u0027s give them names."},{"Start":"15:34.720 ","End":"15:39.160","Text":"Let\u0027s say this is the point A and this is the point B."},{"Start":"15:39.160 ","End":"15:42.220","Text":"How do I know which is the maximum and which is the minimum?"},{"Start":"15:42.220 ","End":"15:45.084","Text":"Well, we just substitute in the target function."},{"Start":"15:45.084 ","End":"15:49.780","Text":"I just went and copy-pasted the target function here because we couldn\u0027t see it."},{"Start":"15:49.780 ","End":"15:55.420","Text":"We want to check what is f of A and what is f of B."},{"Start":"15:55.420 ","End":"16:01.450","Text":"The larger will be the maximum and the smaller will be the minimum."},{"Start":"16:01.450 ","End":"16:04.820","Text":"Because I need a bit more room."},{"Start":"16:05.280 ","End":"16:07.600","Text":"I\u0027m plugging in 2,"},{"Start":"16:07.600 ","End":"16:09.085","Text":"4, 4 into here."},{"Start":"16:09.085 ","End":"16:13.900","Text":"So it\u0027s 2 minus 1 squared plus 4 minus 2"},{"Start":"16:13.900 ","End":"16:20.965","Text":"squared plus 4 minus 2 squared."},{"Start":"16:20.965 ","End":"16:25.390","Text":"This is equal to 1 squared plus 2 squared plus 2 squared,"},{"Start":"16:25.390 ","End":"16:27.370","Text":"that comes out to be 9."},{"Start":"16:27.370 ","End":"16:33.250","Text":"The other point will be minus 2 minus 1 squared,"},{"Start":"16:33.250 ","End":"16:36.234","Text":"minus 4 minus 2 squared,"},{"Start":"16:36.234 ","End":"16:43.975","Text":"and minus 4 minus 2 squared comes out to be 81."},{"Start":"16:43.975 ","End":"16:49.930","Text":"This point A is the minimum,"},{"Start":"16:49.930 ","End":"16:54.205","Text":"and this point B is the maximum."},{"Start":"16:54.205 ","End":"16:59.020","Text":"Although we weren\u0027t asked to find the actual distances,"},{"Start":"16:59.020 ","End":"17:02.320","Text":"just for completion, let\u0027s figure it out."},{"Start":"17:02.320 ","End":"17:05.770","Text":"Remember that the distance we\u0027re looking for is not f,"},{"Start":"17:05.770 ","End":"17:10.075","Text":"but the square root of f. I\u0027ll just write that the distance"},{"Start":"17:10.075 ","End":"17:17.440","Text":"of point A is equal to the square root of 9,"},{"Start":"17:17.440 ","End":"17:22.810","Text":"which is 3, and the maximum"},{"Start":"17:22.810 ","End":"17:31.015","Text":"relating to point B is equal to the square root of 81, which is 9."},{"Start":"17:31.015 ","End":"17:40.760","Text":"From here, these 2 I need to take the square root to get from f to d, and we are done."}],"ID":9678},{"Watched":false,"Name":"Exercise 12","Duration":"19m 25s","ChapterTopicVideoID":9788,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.325","Text":"In this exercise, we have to find the shortest distance from this point to this plane."},{"Start":"00:08.325 ","End":"00:14.160","Text":"We also have to find the point on this plane which is the closest."},{"Start":"00:14.160 ","End":"00:16.320","Text":"These 2 are very related."},{"Start":"00:16.320 ","End":"00:19.290","Text":"At the end, we also have to check our answer because there\u0027s"},{"Start":"00:19.290 ","End":"00:23.355","Text":"a standard formula for distance from point to a plane."},{"Start":"00:23.355 ","End":"00:27.165","Text":"Anyway, let\u0027s begin with a little sketch."},{"Start":"00:27.165 ","End":"00:30.165","Text":"Let\u0027s say that this is our plane,"},{"Start":"00:30.165 ","End":"00:37.110","Text":"minus 2x minus 2y plus z equals 0."},{"Start":"00:37.110 ","End":"00:41.625","Text":"It\u0027s not really relevant, but I noticed that it passes through the origin."},{"Start":"00:41.625 ","End":"00:44.270","Text":"Just might as well note that."},{"Start":"00:44.270 ","End":"00:47.619","Text":"But let\u0027s take a general point."},{"Start":"00:47.619 ","End":"00:50.235","Text":"Let\u0027s call this general point,"},{"Start":"00:50.235 ","End":"00:52.770","Text":"we\u0027ll just call it x, y, z."},{"Start":"00:52.770 ","End":"00:57.285","Text":"We have a special point somewhere here,"},{"Start":"00:57.285 ","End":"01:01.650","Text":"which is the point 1, 2, 3."},{"Start":"01:01.650 ","End":"01:04.565","Text":"It\u0027s not on the plane because if you substitute,"},{"Start":"01:04.565 ","End":"01:08.470","Text":"you\u0027ll see that it doesn\u0027t fit the plane equation."},{"Start":"01:08.470 ","End":"01:10.350","Text":"For each x, y, z,"},{"Start":"01:10.350 ","End":"01:16.855","Text":"it has a certain distance from our special point 1, 2, 3."},{"Start":"01:16.855 ","End":"01:19.640","Text":"We want the shortest distance,"},{"Start":"01:19.640 ","End":"01:21.290","Text":"meaning as we vary x, y,"},{"Start":"01:21.290 ","End":"01:23.200","Text":"z along the plane,"},{"Start":"01:23.200 ","End":"01:26.120","Text":"we\u0027re looking for a minimum value of d,"},{"Start":"01:26.120 ","End":"01:30.395","Text":"which is a function of x, y, and z."},{"Start":"01:30.395 ","End":"01:35.960","Text":"We\u0027re going to do it not with geometry and perpendiculars and all that,"},{"Start":"01:35.960 ","End":"01:39.830","Text":"but we\u0027re going to do it with optimization on the constraint."},{"Start":"01:39.830 ","End":"01:46.660","Text":"In this case, minimization on the constraint using the Lagrange multiplier method."},{"Start":"01:46.660 ","End":"01:48.620","Text":"We need 2 functions."},{"Start":"01:48.620 ","End":"01:51.800","Text":"We need a constraint function and we need a target function."},{"Start":"01:51.800 ","End":"01:56.115","Text":"Now, the constraint should be something equal to 0,"},{"Start":"01:56.115 ","End":"01:58.445","Text":"and actually, it is just the plane"},{"Start":"01:58.445 ","End":"02:01.190","Text":"because the constraint is that the point is on the plane."},{"Start":"02:01.190 ","End":"02:03.350","Text":"The constraint which is this,"},{"Start":"02:03.350 ","End":"02:07.040","Text":"but I\u0027ll write the function g of x, y,"},{"Start":"02:07.040 ","End":"02:14.775","Text":"z is equal to minus 2x minus 2y plus z."},{"Start":"02:14.775 ","End":"02:19.580","Text":"Then the constraint is written in the form g equals 0,"},{"Start":"02:19.580 ","End":"02:21.660","Text":"which is how we like it."},{"Start":"02:21.660 ","End":"02:24.300","Text":"Then we have the target function,"},{"Start":"02:24.300 ","End":"02:28.310","Text":"that\u0027s the thing we\u0027re trying to maximize or minimize."},{"Start":"02:28.310 ","End":"02:30.005","Text":"In this case, minimize."},{"Start":"02:30.005 ","End":"02:33.294","Text":"That would be this distance d,"},{"Start":"02:33.294 ","End":"02:39.545","Text":"so that would be d of x, y, z,"},{"Start":"02:39.545 ","End":"02:41.465","Text":"which is equal to,"},{"Start":"02:41.465 ","End":"02:44.780","Text":"using the formula for the distance between 2 points,"},{"Start":"02:44.780 ","End":"02:51.465","Text":"it will be the square root x minus 1 squared plus"},{"Start":"02:51.465 ","End":"03:02.050","Text":"y minus 2 squared plus z minus 3 squared."},{"Start":"03:02.050 ","End":"03:04.010","Text":"Because of the square root,"},{"Start":"03:04.010 ","End":"03:10.370","Text":"we have our usual standard trick of taking the square of the target,"},{"Start":"03:10.370 ","End":"03:14.390","Text":"and we\u0027ll call that f. We\u0027ll take f of x,"},{"Start":"03:14.390 ","End":"03:17.450","Text":"y, z, this to be our new target,"},{"Start":"03:17.450 ","End":"03:23.225","Text":"which will just be the same thing but without the square root sign,"},{"Start":"03:23.225 ","End":"03:27.070","Text":"z minus 3 squared."},{"Start":"03:27.070 ","End":"03:29.999","Text":"F is basically just d squared,"},{"Start":"03:29.999 ","End":"03:32.660","Text":"but we have to also remember at the end,"},{"Start":"03:32.660 ","End":"03:34.940","Text":"if we want the shortest distance,"},{"Start":"03:34.940 ","End":"03:36.650","Text":"it\u0027s not f, it\u0027s d,"},{"Start":"03:36.650 ","End":"03:41.555","Text":"and d is the square root of f. The point will be the same point."},{"Start":"03:41.555 ","End":"03:46.235","Text":"The square distance will be least when the distance is least,"},{"Start":"03:46.235 ","End":"03:50.390","Text":"but the actual value we have to take a square root at the end."},{"Start":"03:50.390 ","End":"03:59.930","Text":"Basically, our problem in standard form is we want the minimum of this f of x,"},{"Start":"03:59.930 ","End":"04:04.160","Text":"y, and z,"},{"Start":"04:04.160 ","End":"04:10.415","Text":"subject to the constraint that g of x,"},{"Start":"04:10.415 ","End":"04:14.885","Text":"y, z is equal to 0."},{"Start":"04:14.885 ","End":"04:17.480","Text":"In other words, the minimum of the square distance"},{"Start":"04:17.480 ","End":"04:21.240","Text":"subject to the point being on the plane."},{"Start":"04:21.940 ","End":"04:25.160","Text":"We use the usual technique."},{"Start":"04:25.160 ","End":"04:31.265","Text":"The usual technique with the Lagrange multipliers is to get 4 equations."},{"Start":"04:31.265 ","End":"04:35.375","Text":"The first equation comes from using the gradient."},{"Start":"04:35.375 ","End":"04:39.350","Text":"We say that the gradient of f equals Lambda,"},{"Start":"04:39.350 ","End":"04:42.340","Text":"Lambda\u0027s a new auxiliary variable,"},{"Start":"04:42.340 ","End":"04:47.370","Text":"times the gradient of g. That gives us 3 equations."},{"Start":"04:47.370 ","End":"04:48.795","Text":"Let\u0027s write those."},{"Start":"04:48.795 ","End":"04:57.915","Text":"It gives us that f with respect to x equals Lambda g with respect to x,"},{"Start":"04:57.915 ","End":"05:03.390","Text":"f with respect to y is Lambda g with respect to y,"},{"Start":"05:03.390 ","End":"05:09.819","Text":"f with respect to z is Lambda g with respect to z."},{"Start":"05:10.460 ","End":"05:14.600","Text":"The last equation is always the original constraint,"},{"Start":"05:14.600 ","End":"05:17.830","Text":"is that g is equal to 0."},{"Start":"05:17.830 ","End":"05:21.330","Text":"That gives us 4 equations and 4 unknowns,"},{"Start":"05:21.330 ","End":"05:22.800","Text":"x, y, z, Lambda."},{"Start":"05:22.800 ","End":"05:26.625","Text":"Let\u0027s see how that translates to our particular case."},{"Start":"05:26.625 ","End":"05:29.415","Text":"Here\u0027s our curly brace."},{"Start":"05:29.415 ","End":"05:32.625","Text":"Now, the first 1, f with respect to x,"},{"Start":"05:32.625 ","End":"05:39.020","Text":"it\u0027s just twice x minus 1 because the inner derivative"},{"Start":"05:39.020 ","End":"05:45.350","Text":"of x minus 1 is just 1 and the rest of it is constant as far as x goes."},{"Start":"05:45.350 ","End":"05:47.660","Text":"This equals Lambda."},{"Start":"05:47.660 ","End":"05:49.955","Text":"Now, g with respect to x,"},{"Start":"05:49.955 ","End":"05:53.160","Text":"clearly, it\u0027s just minus 2."},{"Start":"05:53.900 ","End":"05:57.885","Text":"Next equation, f with respect to y,"},{"Start":"05:57.885 ","End":"06:03.360","Text":"twice y minus 2 equals Lambda,"},{"Start":"06:03.360 ","End":"06:07.925","Text":"g with respect to y also minus 2."},{"Start":"06:07.925 ","End":"06:12.725","Text":"Next one, we get twice"},{"Start":"06:12.725 ","End":"06:21.960","Text":"z minus 3 equals Lambda times 1."},{"Start":"06:21.960 ","End":"06:28.740","Text":"Finally, g equal 0 is our original plane,"},{"Start":"06:28.740 ","End":"06:37.065","Text":"or this equal 0, minus 2x minus 2y plus z equal 0."},{"Start":"06:37.065 ","End":"06:41.000","Text":"Let\u0027s use our usual trick of"},{"Start":"06:41.000 ","End":"06:46.220","Text":"taking a couple of equations and dividing one by the other and getting rid of Lambda."},{"Start":"06:46.220 ","End":"06:49.700","Text":"Let\u0027s first of all do this divided by this,"},{"Start":"06:49.700 ","End":"06:51.935","Text":"1st divided by the 2nd."},{"Start":"06:51.935 ","End":"06:59.915","Text":"That\u0027s twice x minus 1 over twice y minus 2"},{"Start":"06:59.915 ","End":"07:08.865","Text":"is equal to Lambda times minus 2 over Lambda times minus 2."},{"Start":"07:08.865 ","End":"07:13.180","Text":"Now, of course, we have to make sure that we\u0027re not dividing by 0."},{"Start":"07:13.180 ","End":"07:15.990","Text":"Not saying that these can\u0027t be 0,"},{"Start":"07:15.990 ","End":"07:19.895","Text":"but if they\u0027re 0, we have to take care of them separately as a special case."},{"Start":"07:19.895 ","End":"07:22.475","Text":"Suppose Lambda were equal to 0,"},{"Start":"07:22.475 ","End":"07:24.260","Text":"what would happen then?"},{"Start":"07:24.260 ","End":"07:27.250","Text":"Well, if all these Lambdas are 0,"},{"Start":"07:27.250 ","End":"07:31.230","Text":"these 3 right-hand sides are 0 and the 2 doesn\u0027t matter,"},{"Start":"07:31.230 ","End":"07:40.455","Text":"so we\u0027d get that x would equal 1 in order for this to be 0,"},{"Start":"07:40.455 ","End":"07:42.810","Text":"then y would have to be 2,"},{"Start":"07:42.810 ","End":"07:45.615","Text":"and z would have to be 3."},{"Start":"07:45.615 ","End":"07:50.255","Text":"Then we\u0027d also have to satisfy this equation,"},{"Start":"07:50.255 ","End":"07:51.530","Text":"which means that 1, 2,"},{"Start":"07:51.530 ","End":"07:53.210","Text":"3 would be on this plane."},{"Start":"07:53.210 ","End":"07:56.810","Text":"We already mentioned that this point is not on the plane,"},{"Start":"07:56.810 ","End":"07:59.000","Text":"so this is not correct,"},{"Start":"07:59.000 ","End":"08:00.560","Text":"so that takes care of that."},{"Start":"08:00.560 ","End":"08:02.260","Text":"Now, what other worry do we have?"},{"Start":"08:02.260 ","End":"08:06.530","Text":"That y minus 2 could be 0."},{"Start":"08:06.530 ","End":"08:11.820","Text":"Well, if y minus 2 equal 0,"},{"Start":"08:12.590 ","End":"08:15.645","Text":"the left-hand side is 0,"},{"Start":"08:15.645 ","End":"08:18.700","Text":"so we get minus 2 Lambda is 0,"},{"Start":"08:18.700 ","End":"08:21.175","Text":"which would give us that Lambda is 0,"},{"Start":"08:21.175 ","End":"08:23.020","Text":"but we\u0027ve already ruled this out."},{"Start":"08:23.020 ","End":"08:25.300","Text":"This also is not true."},{"Start":"08:25.300 ","End":"08:27.160","Text":"Before I do all the simplification,"},{"Start":"08:27.160 ","End":"08:29.395","Text":"let me just write another one of these."},{"Start":"08:29.395 ","End":"08:31.060","Text":"I need another equation,"},{"Start":"08:31.060 ","End":"08:35.125","Text":"so let\u0027s take this time the 2nd divided by the 3rd."},{"Start":"08:35.125 ","End":"08:42.915","Text":"This over this is twice y minus 2 over twice"},{"Start":"08:42.915 ","End":"08:53.100","Text":"z minus 3 is equal to Lambda times minus 2 over Lambda times 1."},{"Start":"08:53.100 ","End":"08:55.525","Text":"Let us erase the 1."},{"Start":"08:55.525 ","End":"08:59.379","Text":"Now, we\u0027ve already seen that Lambda is not equal to 0."},{"Start":"08:59.379 ","End":"09:00.610","Text":"Now we have to worry,"},{"Start":"09:00.610 ","End":"09:04.825","Text":"is it possible that z minus 3 equals 0?"},{"Start":"09:04.825 ","End":"09:07.720","Text":"Well, just as above, if z minus 3 is 0,"},{"Start":"09:07.720 ","End":"09:09.515","Text":"left-hand side is 0,"},{"Start":"09:09.515 ","End":"09:13.060","Text":"and so the right-hand side Lambda will be equal to 0,"},{"Start":"09:13.060 ","End":"09:15.190","Text":"and this has been ruled out already."},{"Start":"09:15.190 ","End":"09:18.805","Text":"In both of these new equations,"},{"Start":"09:18.805 ","End":"09:21.275","Text":"we\u0027re not dividing by 0."},{"Start":"09:21.275 ","End":"09:23.985","Text":"We can do some cancellation."},{"Start":"09:23.985 ","End":"09:26.225","Text":"Lambda cancels with Lambda."},{"Start":"09:26.225 ","End":"09:28.750","Text":"In fact, minus 2 cancels with minus 2,"},{"Start":"09:28.750 ","End":"09:30.040","Text":"but we just can\u0027t leave it blank,"},{"Start":"09:30.040 ","End":"09:33.250","Text":"so it\u0027s 1 over 1, which is 1."},{"Start":"09:33.250 ","End":"09:35.990","Text":"This 2 cancels with this 2."},{"Start":"09:35.990 ","End":"09:38.355","Text":"From the first equation,"},{"Start":"09:38.355 ","End":"09:41.425","Text":"what we get is,"},{"Start":"09:41.425 ","End":"09:46.055","Text":"cross multiply, we just get that x minus 1 equals y minus 2."},{"Start":"09:46.055 ","End":"09:53.199","Text":"Let me write that, x minus 1 equals y minus 2."},{"Start":"09:56.750 ","End":"09:59.380","Text":"That\u0027s the 2nd equation."},{"Start":"09:59.380 ","End":"10:02.465","Text":"I\u0027m going to get 3 equations and 3 unknowns, x, y, and z."},{"Start":"10:02.465 ","End":"10:05.075","Text":"From the last one,"},{"Start":"10:05.075 ","End":"10:09.370","Text":"we can cancel Lambda with Lambda,"},{"Start":"10:09.370 ","End":"10:11.610","Text":"and I\u0027ll leave a 1 here."},{"Start":"10:11.610 ","End":"10:14.205","Text":"We can cancel the 2 with the 2."},{"Start":"10:14.205 ","End":"10:16.550","Text":"Now if we cross multiply,"},{"Start":"10:16.550 ","End":"10:23.030","Text":"we get from here y minus 2 equals,"},{"Start":"10:25.160 ","End":"10:29.775","Text":"minus 2 times z minus 3,"},{"Start":"10:29.775 ","End":"10:32.910","Text":"so it\u0027s minus 2z plus 6,"},{"Start":"10:32.910 ","End":"10:35.495","Text":"just multiplied this out by this."},{"Start":"10:35.495 ","End":"10:40.310","Text":"Notice that what I now have is 3 equations and 3 unknowns,"},{"Start":"10:40.310 ","End":"10:43.680","Text":"x, y, and z, and I\u0027ve gotten rid of Lambda."},{"Start":"10:43.680 ","End":"10:46.990","Text":"After looking at this a moment, here\u0027s what I suggest."},{"Start":"10:46.990 ","End":"10:48.625","Text":"I have y both here and here."},{"Start":"10:48.625 ","End":"10:49.960","Text":"Here\u0027s what I suggest."},{"Start":"10:49.960 ","End":"10:52.375","Text":"I see there is y in both of these."},{"Start":"10:52.375 ","End":"10:55.960","Text":"Let\u0027s put x and z,"},{"Start":"10:55.960 ","End":"11:01.285","Text":"both in terms of y and then we can substitute here and get an equation in y."},{"Start":"11:01.285 ","End":"11:05.440","Text":"What I get is that from here,"},{"Start":"11:05.440 ","End":"11:11.919","Text":"x is equal to y minus 2 plus 1. x is y minus 1."},{"Start":"11:11.919 ","End":"11:16.585","Text":"From here, put the 2z over to the left, it\u0027s 2z."},{"Start":"11:16.585 ","End":"11:18.610","Text":"But everything else on the right,"},{"Start":"11:18.610 ","End":"11:26.290","Text":"I\u0027ve got minus y and then plus 8 and so z is equal"},{"Start":"11:26.290 ","End":"11:37.820","Text":"to minus 1/2y plus 4."},{"Start":"11:37.820 ","End":"11:40.765","Text":"Now I substitute x and z,"},{"Start":"11:40.765 ","End":"11:45.820","Text":"this and this from here and I\u0027ll get everything in terms of y."},{"Start":"11:45.820 ","End":"11:52.585","Text":"Minus 2x is y minus 1."},{"Start":"11:52.585 ","End":"11:56.410","Text":"Minus 2y, y don\u0027t need to substitute,"},{"Start":"11:56.410 ","End":"12:00.174","Text":"plus z, which is,"},{"Start":"12:00.174 ","End":"12:03.670","Text":"well, it\u0027s just minus 1/2y plus 4."},{"Start":"12:03.670 ","End":"12:11.365","Text":"This is a minus and it\u0027s 1/2y plus 4 equals 0."},{"Start":"12:11.365 ","End":"12:17.590","Text":"Just multiply both sides by 2 and I\u0027ll open brackets up."},{"Start":"12:17.590 ","End":"12:20.005","Text":"Here I have minus 4."},{"Start":"12:20.005 ","End":"12:26.140","Text":"So it\u0027s minus 4y plus 2 would be plus 4."},{"Start":"12:26.140 ","End":"12:29.665","Text":"This multiplied by 2 is minus 4y."},{"Start":"12:29.665 ","End":"12:34.240","Text":"This multiplied by 2 is minus y and plus 8,"},{"Start":"12:34.240 ","End":"12:36.595","Text":"and 0 times 2 is just 0."},{"Start":"12:36.595 ","End":"12:39.200","Text":"Now let\u0027s collect."},{"Start":"12:39.390 ","End":"12:49.580","Text":"I\u0027ve got minus 4y minus 4y minus y is minus 9y."},{"Start":"12:51.060 ","End":"12:53.230","Text":"Numbers on the left,"},{"Start":"12:53.230 ","End":"13:00.505","Text":"I\u0027ve got minus 4 minus 8 is minus 12."},{"Start":"13:00.505 ","End":"13:06.985","Text":"That gives me that y equals minus 12 over 9,"},{"Start":"13:06.985 ","End":"13:11.059","Text":"comes out to be 4 over 3."},{"Start":"13:11.220 ","End":"13:14.890","Text":"Now that I have y, I can find x and z."},{"Start":"13:14.890 ","End":"13:18.910","Text":"For example here, x is y minus 1."},{"Start":"13:18.910 ","End":"13:27.415","Text":"This gives us that x is equal to 4/3 minus 1 is just 1/3."},{"Start":"13:27.415 ","End":"13:30.370","Text":"If I substitute y in the other 1,"},{"Start":"13:30.370 ","End":"13:33.310","Text":"I\u0027ll get z. Now, minus 1/2y."},{"Start":"13:33.310 ","End":"13:39.190","Text":"1/2y is 2/3, so it\u0027s minus 2/3."},{"Start":"13:39.190 ","End":"13:44.050","Text":"Minus 2/3 plus 4 is 3 and 1/3,"},{"Start":"13:44.050 ","End":"13:47.800","Text":"3 and 1/3 is 10 over 3."},{"Start":"13:47.800 ","End":"13:55.330","Text":"Now I found my points x, y, z."},{"Start":"13:55.330 ","End":"13:59.020","Text":"Sometimes you would write this with an asterisk to indicate a minimum."},{"Start":"13:59.020 ","End":"14:00.880","Text":"You know what, I\u0027ll do it this time."},{"Start":"14:00.880 ","End":"14:04.420","Text":"Minimum x, y, z is equal"},{"Start":"14:04.420 ","End":"14:12.040","Text":"to 1/3, 4/3, 10/3."},{"Start":"14:12.040 ","End":"14:17.470","Text":"Now I didn\u0027t go into the rigorous methods of showing that it\u0027s a minimum."},{"Start":"14:17.470 ","End":"14:18.955","Text":"That\u0027s beyond the scope here."},{"Start":"14:18.955 ","End":"14:20.830","Text":"But we know there\u0027s a minimum because we know there\u0027s"},{"Start":"14:20.830 ","End":"14:24.535","Text":"a shortest distance from a point to a plane,"},{"Start":"14:24.535 ","End":"14:29.020","Text":"and so we\u0027ll just declare that this is the minimum."},{"Start":"14:29.020 ","End":"14:37.730","Text":"Now this actually only answers part B and we\u0027ve got part B answered first."},{"Start":"14:38.370 ","End":"14:42.985","Text":"In fact, I\u0027ll highlight it,"},{"Start":"14:42.985 ","End":"14:46.000","Text":"but this is the answer to part B."},{"Start":"14:46.000 ","End":"14:48.340","Text":"Now the shortest distance, well,"},{"Start":"14:48.340 ","End":"14:52.060","Text":"we can substitute in this target function f and then take"},{"Start":"14:52.060 ","End":"14:56.814","Text":"the square root or we can just substitute straight away in the distance formula."},{"Start":"14:56.814 ","End":"14:58.750","Text":"We get that the shortest distance,"},{"Start":"14:58.750 ","End":"15:00.655","Text":"I\u0027ll call it d with an asterisk,"},{"Start":"15:00.655 ","End":"15:05.245","Text":"would be the square root of, let\u0027s see."},{"Start":"15:05.245 ","End":"15:10.630","Text":"I have to plug in these 3 values into here. What do we get?"},{"Start":"15:10.630 ","End":"15:13.705","Text":"1/3 minus 1 squared,"},{"Start":"15:13.705 ","End":"15:19.210","Text":"and then y minus 2 squared is 4/3 minus 2 squared."},{"Start":"15:19.210 ","End":"15:27.075","Text":"Then this 10 over 3 minus 3 squared,"},{"Start":"15:27.075 ","End":"15:29.530","Text":"it\u0027s a little bit longer."},{"Start":"15:29.640 ","End":"15:33.550","Text":"Let\u0027s just compute this. What do we get?"},{"Start":"15:33.550 ","End":"15:36.205","Text":"1/3 minus 1 is minus 2/3,"},{"Start":"15:36.205 ","End":"15:38.395","Text":"but when you square it, it\u0027s 4/9."},{"Start":"15:38.395 ","End":"15:42.840","Text":"4/3 minus 2 is 1 and 1/3,"},{"Start":"15:42.840 ","End":"15:46.395","Text":"minus 2 is minus 2/3 squared."},{"Start":"15:46.395 ","End":"15:49.235","Text":"Again, we get 4/9."},{"Start":"15:49.235 ","End":"15:52.540","Text":"10/3 is 3 and 1/3 minus 3 is 1/3."},{"Start":"15:52.540 ","End":"15:54.685","Text":"When you square it, it\u0027s 1/9."},{"Start":"15:54.685 ","End":"15:57.355","Text":"We need the square root of all this."},{"Start":"15:57.355 ","End":"15:59.890","Text":"4 plus 4 plus 1 is 9,"},{"Start":"15:59.890 ","End":"16:04.405","Text":"so it\u0027s the square root of 9 over 9."},{"Start":"16:04.405 ","End":"16:08.845","Text":"This is 1, and the answer is 1."},{"Start":"16:08.845 ","End":"16:11.590","Text":"This is the answer to part A."},{"Start":"16:11.590 ","End":"16:18.100","Text":"We found the point that\u0027s closest and the distance from that point to the plane."},{"Start":"16:18.100 ","End":"16:23.320","Text":"Now we\u0027ll go on to part C. Just to show you what it was."},{"Start":"16:23.320 ","End":"16:25.975","Text":"It was to check our answer using the formula."},{"Start":"16:25.975 ","End":"16:27.400","Text":"I\u0027ll give you the formula."},{"Start":"16:27.400 ","End":"16:33.370","Text":"Suppose we have the point and let\u0027s call the point x_1,"},{"Start":"16:33.370 ","End":"16:39.835","Text":"y_1, z_1, and suppose we have a plane."},{"Start":"16:39.835 ","End":"16:46.450","Text":"The general form of a plane is Ax plus By plus"},{"Start":"16:46.450 ","End":"16:54.010","Text":"Cz plus D equals 0 and the distance formula from the point of the plane,"},{"Start":"16:54.010 ","End":"16:56.770","Text":"what we do is as follows."},{"Start":"16:56.770 ","End":"17:00.925","Text":"We first of all substitute the point into the plane."},{"Start":"17:00.925 ","End":"17:10.045","Text":"So we get Ax_1 plus BY_1 plus Cz_1 plus"},{"Start":"17:10.045 ","End":"17:13.810","Text":"D. Then we divide"},{"Start":"17:13.810 ","End":"17:21.145","Text":"by the square root of the first 3 coefficients squared,"},{"Start":"17:21.145 ","End":"17:26.935","Text":"except the D. A squared plus B squared plus C squared."},{"Start":"17:26.935 ","End":"17:28.870","Text":"There\u0027s one final thing."},{"Start":"17:28.870 ","End":"17:30.640","Text":"If this comes out negative,"},{"Start":"17:30.640 ","End":"17:31.750","Text":"we need to make it positive."},{"Start":"17:31.750 ","End":"17:34.330","Text":"In other words, we take the absolute value."},{"Start":"17:34.330 ","End":"17:38.110","Text":"Here, denominator is positive of course."},{"Start":"17:38.110 ","End":"17:41.485","Text":"Our point, it\u0027s just off the screen,"},{"Start":"17:41.485 ","End":"17:44.259","Text":"was 1, 2, 3,"},{"Start":"17:44.259 ","End":"17:45.775","Text":"and the plane which is here,"},{"Start":"17:45.775 ","End":"17:48.700","Text":"means that A is minus 2,"},{"Start":"17:48.700 ","End":"17:50.950","Text":"B is minus 2,"},{"Start":"17:50.950 ","End":"17:54.640","Text":"C is 1, and D is 0."},{"Start":"17:54.640 ","End":"18:02.515","Text":"Now we can apply the formula and get that D equals,"},{"Start":"18:02.515 ","End":"18:07.420","Text":"we have A times x1,"},{"Start":"18:07.420 ","End":"18:11.440","Text":"which is minus 2 times 1."},{"Start":"18:11.440 ","End":"18:15.430","Text":"Then By_1 minus 2 times"},{"Start":"18:15.430 ","End":"18:26.440","Text":"2Cz_1 is 1 times 3 and then plus 0."},{"Start":"18:26.440 ","End":"18:29.800","Text":"Just doing it to show that I haven\u0027t forgotten it."},{"Start":"18:29.800 ","End":"18:32.810","Text":"This is an absolute value,"},{"Start":"18:33.210 ","End":"18:40.330","Text":"then over the square root of A"},{"Start":"18:40.330 ","End":"18:48.130","Text":"squared plus B squared plus C squared."},{"Start":"18:48.130 ","End":"18:50.455","Text":"Let\u0027s see what that gives us."},{"Start":"18:50.455 ","End":"18:53.890","Text":"Now here we have minus 2,"},{"Start":"18:53.890 ","End":"18:57.340","Text":"minus 4, that\u0027s minus 6."},{"Start":"18:57.340 ","End":"19:01.674","Text":"Minus 6 plus 3 is minus 3,"},{"Start":"19:01.674 ","End":"19:05.470","Text":"so we have the absolute value of minus 3 over,"},{"Start":"19:05.470 ","End":"19:08.755","Text":"now this is 4 plus 4 plus 1,"},{"Start":"19:08.755 ","End":"19:11.200","Text":"so it\u0027s the square root of 9."},{"Start":"19:11.200 ","End":"19:13.645","Text":"This is 3 and this is 3,"},{"Start":"19:13.645 ","End":"19:20.470","Text":"so this equals 1 and so we have confirmed that these 2 are the same."},{"Start":"19:20.470 ","End":"19:24.950","Text":"The shortest distance really is 1, and we\u0027re done."}],"ID":9679},{"Watched":false,"Name":"Exercise 13","Duration":"16m 52s","ChapterTopicVideoID":9789,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.650 ","End":"00:04.665","Text":"In this exercise, we have to find the point or points"},{"Start":"00:04.665 ","End":"00:09.075","Text":"on the following surface closest to the origin."},{"Start":"00:09.075 ","End":"00:12.555","Text":"This is a surface in 3D."},{"Start":"00:12.555 ","End":"00:18.150","Text":"The reason, there might be more than 1 point is it could be a tie."},{"Start":"00:18.150 ","End":"00:23.845","Text":"It could be 2 points equally distant to the origin or more."},{"Start":"00:23.845 ","End":"00:28.160","Text":"I also noticed that the origin is not on the surface."},{"Start":"00:28.160 ","End":"00:32.570","Text":"If it was, I could just stop right away and say,"},{"Start":"00:32.570 ","End":"00:36.980","Text":"yeah, the closest is the origin itself and the distance is 0."},{"Start":"00:36.980 ","End":"00:38.970","Text":"But it\u0027s not that easy."},{"Start":"00:38.970 ","End":"00:43.205","Text":"This is clearly an optimization problem with constraint,"},{"Start":"00:43.205 ","End":"00:46.490","Text":"and we\u0027re going to use the method of Lagrange multipliers."},{"Start":"00:46.490 ","End":"00:50.230","Text":"We need 2 functions,"},{"Start":"00:50.230 ","End":"00:53.390","Text":"we need a constraint function."},{"Start":"00:53.390 ","End":"00:55.715","Text":"I usually start with the constraint."},{"Start":"00:55.715 ","End":"00:58.760","Text":"The constraint is just the equation of the surface,"},{"Start":"00:58.760 ","End":"01:03.155","Text":"except that we put everything to 1 side."},{"Start":"01:03.155 ","End":"01:10.334","Text":"Z squared minus xy plus 1 equals 0,"},{"Start":"01:10.334 ","End":"01:16.575","Text":"and this bit is the constraint function, it\u0027s minus 1."},{"Start":"01:16.575 ","End":"01:20.270","Text":"Then we have a target function."},{"Start":"01:20.270 ","End":"01:24.110","Text":"That\u0027s the thing we want to maximize or minimize."},{"Start":"01:24.110 ","End":"01:28.854","Text":"In this case, the word closest implies that it\u0027s a minimum problem,"},{"Start":"01:28.854 ","End":"01:33.500","Text":"and that\u0027s the distance of the point to the origin."},{"Start":"01:33.500 ","End":"01:37.790","Text":"I guess I should have just said that we\u0027re taking a general point P,"},{"Start":"01:37.790 ","End":"01:40.790","Text":"maybe which is x, y,"},{"Start":"01:40.790 ","End":"01:46.169","Text":"z, and that the origin O is the point 0,"},{"Start":"01:46.169 ","End":"01:49.580","Text":"0, 0, That\u0027s a letter O, and this is a 0."},{"Start":"01:49.580 ","End":"01:52.880","Text":"We want x, y, z closest to this."},{"Start":"01:52.880 ","End":"01:54.290","Text":"Under this constraint anyway,"},{"Start":"01:54.290 ","End":"01:57.380","Text":"the target would be the distance,"},{"Start":"01:57.380 ","End":"02:01.620","Text":"and that would be the square root"},{"Start":"02:03.470 ","End":"02:09.065","Text":"of x minus 0 squared would just be x squared."},{"Start":"02:09.065 ","End":"02:13.620","Text":"It\u0027s going to be x squared plus y squared plus z squared."},{"Start":"02:14.720 ","End":"02:21.260","Text":"We can state our problem formally as yes,"},{"Start":"02:21.260 ","End":"02:25.775","Text":"so we write minimum,"},{"Start":"02:25.775 ","End":"02:28.559","Text":"and then it\u0027s a curly brace,"},{"Start":"02:28.559 ","End":"02:31.480","Text":"and then you put the target function,"},{"Start":"02:31.480 ","End":"02:38.180","Text":"the square root of x squared plus y squared plus z squared,"},{"Start":"02:38.180 ","End":"02:40.050","Text":"and then we write s. t,"},{"Start":"02:40.050 ","End":"02:41.880","Text":"which means subject to,"},{"Start":"02:41.880 ","End":"02:44.770","Text":"and then we put the constraint."},{"Start":"02:44.770 ","End":"02:51.460","Text":"We can either write it in this form or you can also write it in the original form,"},{"Start":"02:51.460 ","End":"02:56.364","Text":"z squared equals xy plus 1."},{"Start":"02:56.364 ","End":"03:02.320","Text":"Sometimes we put a restriction on the domain x, y, z,"},{"Start":"03:02.320 ","End":"03:05.680","Text":"but there\u0027s no problem in substituting x,"},{"Start":"03:05.680 ","End":"03:08.810","Text":"y, z in any of these."},{"Start":"03:08.960 ","End":"03:12.920","Text":"What we\u0027re going to do is a standard trick."},{"Start":"03:12.920 ","End":"03:20.930","Text":"This function is actually the distance function from the point P to the point 0."},{"Start":"03:20.930 ","End":"03:24.575","Text":"That would be d of x, y,"},{"Start":"03:24.575 ","End":"03:30.935","Text":"z, which is the square root of x squared plus y squared plus z squared."},{"Start":"03:30.935 ","End":"03:34.685","Text":"But it\u0027s cumbersome to work with distances and square roots."},{"Start":"03:34.685 ","End":"03:40.520","Text":"The standard trick is to take the square of the distance,"},{"Start":"03:40.520 ","End":"03:42.455","Text":"a different target function,"},{"Start":"03:42.455 ","End":"03:47.329","Text":"which is just x squared plus y squared, plus z squared."},{"Start":"03:47.329 ","End":"03:50.240","Text":"Because the square roots a positive function,"},{"Start":"03:50.240 ","End":"03:58.325","Text":"the square distance, I\u0027ll just write that that\u0027s the square of the distance."},{"Start":"03:58.325 ","End":"04:01.970","Text":"It\u0027s least whenever the distance is least."},{"Start":"04:01.970 ","End":"04:06.845","Text":"This is the square of the distance."},{"Start":"04:06.845 ","End":"04:10.010","Text":"The thing though is that even though the distance"},{"Start":"04:10.010 ","End":"04:13.430","Text":"and the square of the distance occur at the same point,"},{"Start":"04:13.430 ","End":"04:18.590","Text":"the value is different when I find the actual minimum value of this,"},{"Start":"04:18.590 ","End":"04:22.640","Text":"I have to take d equals"},{"Start":"04:22.640 ","End":"04:28.045","Text":"the square root of f when I get to the answer because f is d squared."},{"Start":"04:28.045 ","End":"04:30.465","Text":"Just not to forget to do that."},{"Start":"04:30.465 ","End":"04:33.950","Text":"Here, they didn\u0027t ask for the shortest distance,"},{"Start":"04:33.950 ","End":"04:38.210","Text":"but we usually do that also and not just the points."},{"Start":"04:39.550 ","End":"04:44.280","Text":"Yeah, that\u0027s the target function."},{"Start":"04:44.280 ","End":"04:46.504","Text":"I didn\u0027t write the constraint function."},{"Start":"04:46.504 ","End":"04:49.685","Text":"The constraint function we usually call g,"},{"Start":"04:49.685 ","End":"04:53.525","Text":"target is f, constraint is g, just customary,"},{"Start":"04:53.525 ","End":"04:58.500","Text":"would be the left side of this,"},{"Start":"04:58.500 ","End":"05:05.180","Text":"the thing that\u0027s equal to 0 is c squared minus xy minus 1."},{"Start":"05:05.180 ","End":"05:07.765","Text":"These are our 2 functions."},{"Start":"05:07.765 ","End":"05:16.940","Text":"Constraint and target, and the Lagrange method introduces a new variable called Lambda,"},{"Start":"05:16.940 ","End":"05:20.405","Text":"and then we get 4 equations and 4 unknowns."},{"Start":"05:20.405 ","End":"05:26.230","Text":"The first 3 equations come from, I\u0027ll write it over here."},{"Start":"05:26.230 ","End":"05:36.785","Text":"The gradient of f is equal to Lambda times the gradient of g. These are vectors in 3D,"},{"Start":"05:36.785 ","End":"05:40.475","Text":"if you just write what this means component-wise,"},{"Start":"05:40.475 ","End":"05:43.800","Text":"then we get 3 equations."},{"Start":"05:43.800 ","End":"05:47.600","Text":"We get derivative of f with respect to"},{"Start":"05:47.600 ","End":"05:53.390","Text":"x equals Lambda times partial derivative of g with respect to x,"},{"Start":"05:53.390 ","End":"05:56.155","Text":"and the same thing for y,"},{"Start":"05:56.155 ","End":"05:59.480","Text":"just copying, with y instead of x,"},{"Start":"05:59.480 ","End":"06:02.480","Text":"and the same thing for z, the third variable,"},{"Start":"06:02.480 ","End":"06:09.830","Text":"Lambda g, z, and that gives us 3 equations,"},{"Start":"06:09.830 ","End":"06:14.980","Text":"and the 4th one is the constraint."},{"Start":"06:14.980 ","End":"06:21.060","Text":"Either this equals 0 or in the original form."},{"Start":"06:21.410 ","End":"06:25.305","Text":"Well, in general it\u0027s g equals 0."},{"Start":"06:25.305 ","End":"06:28.100","Text":"Then when we get to our specific case,"},{"Start":"06:28.100 ","End":"06:35.660","Text":"I can write that either this equals 0,"},{"Start":"06:35.660 ","End":"06:38.540","Text":"I usually like to go with the original form,"},{"Start":"06:38.540 ","End":"06:42.995","Text":"z squared equals x, y plus 1."},{"Start":"06:42.995 ","End":"06:45.905","Text":"Last equation is always the constraint."},{"Start":"06:45.905 ","End":"06:48.410","Text":"The first 3, let\u0027s just see what they are."},{"Start":"06:48.410 ","End":"06:55.455","Text":"We need to do some partial differentiation of f and g, so let\u0027s see."},{"Start":"06:55.455 ","End":"06:59.610","Text":"F with respect to x is 2x. You know what?"},{"Start":"06:59.610 ","End":"07:03.740","Text":"I\u0027m just going to go with the left-hand side first here we\u0027ll get 2y,"},{"Start":"07:03.740 ","End":"07:05.465","Text":"here we\u0027ll get 2z,"},{"Start":"07:05.465 ","End":"07:07.355","Text":"we\u0027re going to get Lambda something,"},{"Start":"07:07.355 ","End":"07:09.440","Text":"Lambda something, Lambda something."},{"Start":"07:09.440 ","End":"07:19.005","Text":"Now g with respect to x is minus y, with respect to y,"},{"Start":"07:19.005 ","End":"07:23.175","Text":"it\u0027s minus x, and with respect to z,"},{"Start":"07:23.175 ","End":"07:25.890","Text":"it\u0027s going to be 2z,"},{"Start":"07:25.890 ","End":"07:29.195","Text":"4 equations, 4 unknowns."},{"Start":"07:29.195 ","End":"07:32.390","Text":"The standard trick is to divide 1 equation"},{"Start":"07:32.390 ","End":"07:36.200","Text":"by another from the top 3 and get rid of Lambda."},{"Start":"07:36.200 ","End":"07:38.420","Text":"I tried this out earlier,"},{"Start":"07:38.420 ","End":"07:40.880","Text":"that actually is not the best approach here."},{"Start":"07:40.880 ","End":"07:45.530","Text":"It gets a bit messy with various cases of denominators being 0 or not."},{"Start":"07:45.530 ","End":"07:50.495","Text":"Actually, the best place to attack is this 3rd equation,"},{"Start":"07:50.495 ","End":"07:54.345","Text":"which I can rewrite if I divide by 2,"},{"Start":"07:54.345 ","End":"08:01.515","Text":"as z equals Lambda z."},{"Start":"08:01.515 ","End":"08:06.350","Text":"From here, I can split off into 2 possibilities."},{"Start":"08:06.350 ","End":"08:09.485","Text":"Either z is 0,"},{"Start":"08:09.485 ","End":"08:15.030","Text":"so I\u0027ll do that in 1 color and write z equals 0,"},{"Start":"08:15.030 ","End":"08:17.660","Text":"and that will be my Case 1."},{"Start":"08:17.660 ","End":"08:20.930","Text":"The other possibility if z is not 0,"},{"Start":"08:20.930 ","End":"08:25.405","Text":"then I can divide by it and I can get that Lambda equals 1,"},{"Start":"08:25.405 ","End":"08:28.350","Text":"and I\u0027ll write that in this color."},{"Start":"08:28.350 ","End":"08:31.640","Text":"Then we\u0027ll work on the 2 cases separately."},{"Start":"08:31.640 ","End":"08:35.255","Text":"Let\u0027s start out with the z equals 0 case."},{"Start":"08:35.255 ","End":"08:41.189","Text":"In this case, I can substitute it in the other 3 equations,"},{"Start":"08:41.189 ","End":"08:42.665","Text":"the 1st, 2nd, and 4th,"},{"Start":"08:42.665 ","End":"08:51.510","Text":"and we\u0027ll get that 2x equals minus Lambda y,"},{"Start":"08:51.510 ","End":"08:57.340","Text":"2y equals minus Lambda x."},{"Start":"08:58.790 ","End":"09:03.500","Text":"Bringing the 1 over to the other side, I get x,"},{"Start":"09:03.500 ","End":"09:08.095","Text":"y equals minus 1 because z is 0."},{"Start":"09:08.095 ","End":"09:10.470","Text":"3 equations and 3 unknowns."},{"Start":"09:10.470 ","End":"09:12.605","Text":"Let me just get some more space here."},{"Start":"09:12.605 ","End":"09:14.690","Text":"Now, in this case,"},{"Start":"09:14.690 ","End":"09:21.400","Text":"why don\u0027t I divide the first equation by the second,"},{"Start":"09:21.400 ","End":"09:23.685","Text":"and I can get rid of Lambda that way,"},{"Start":"09:23.685 ","End":"09:28.710","Text":"and what we\u0027ll get will be that 2x over"},{"Start":"09:28.710 ","End":"09:38.605","Text":"2y is equal to minus Lambda y over minus Lambda x."},{"Start":"09:38.605 ","End":"09:43.485","Text":"Now we have to make sure that we\u0027re not dividing by 0."},{"Start":"09:43.485 ","End":"09:45.610","Text":"There are 3 things to consider,"},{"Start":"09:45.610 ","End":"09:47.710","Text":"x, y, and Lambda."},{"Start":"09:47.710 ","End":"09:57.760","Text":"I think it\u0027s clear that x is not 0 and y is not 0 because their product is minus 1,"},{"Start":"09:57.760 ","End":"10:00.355","Text":"so neither of them could be 0."},{"Start":"10:00.355 ","End":"10:02.250","Text":"We\u0027re okay with y and x,"},{"Start":"10:02.250 ","End":"10:03.630","Text":"but what about Lambda?"},{"Start":"10:03.630 ","End":"10:06.375","Text":"Suppose that Lambda equals 0,"},{"Start":"10:06.375 ","End":"10:07.875","Text":"what would happen then?"},{"Start":"10:07.875 ","End":"10:10.350","Text":"Well, if you just put it in either one of these,"},{"Start":"10:10.350 ","End":"10:11.655","Text":"let\u0027s say in the first one,"},{"Start":"10:11.655 ","End":"10:18.410","Text":"we\u0027d get that 2x equals 0 and that would mean that x is 0."},{"Start":"10:18.960 ","End":"10:23.725","Text":"We already showed that x is not 0, so that\u0027s wrong."},{"Start":"10:23.725 ","End":"10:27.860","Text":"That means that Lambda is also not 0."},{"Start":"10:28.440 ","End":"10:30.850","Text":"We can continue here."},{"Start":"10:30.850 ","End":"10:35.335","Text":"A bit of tidying up now 2 cancels with 2."},{"Start":"10:35.335 ","End":"10:39.535","Text":"If we can divide by the whole minus Lambda and minus Lambda."},{"Start":"10:39.535 ","End":"10:45.010","Text":"We have x/y equals y/x."},{"Start":"10:45.010 ","End":"10:49.030","Text":"x squared equals y squared."},{"Start":"10:49.030 ","End":"10:51.025","Text":"That\u0027s the cross multiplication."},{"Start":"10:51.025 ","End":"10:58.210","Text":"Now I want to emphasize this equation and this equation,"},{"Start":"10:58.210 ","End":"11:01.960","Text":"2 equations and 2 unknowns, x and y."},{"Start":"11:01.960 ","End":"11:05.000","Text":"Let\u0027s continue."},{"Start":"11:05.340 ","End":"11:09.340","Text":"What I suggest is we take from this equation,"},{"Start":"11:09.340 ","End":"11:17.200","Text":"the fact that y equals minus 1/x and substitute it into this"},{"Start":"11:17.200 ","End":"11:25.060","Text":"1 and then we\u0027ll get that x squared equals minus 1/x squared."},{"Start":"11:25.060 ","End":"11:26.605","Text":"Need the brackets notice."},{"Start":"11:26.605 ","End":"11:29.860","Text":"That gives us this 1/x squared."},{"Start":"11:29.860 ","End":"11:32.739","Text":"We get x^4 equals 1,"},{"Start":"11:32.739 ","End":"11:37.150","Text":"and therefore x is plus or minus 1."},{"Start":"11:37.150 ","End":"11:40.825","Text":"Now, each of these gives a possibility."},{"Start":"11:40.825 ","End":"11:44.080","Text":"What we get, let\u0027s say that x equals 1."},{"Start":"11:44.080 ","End":"11:48.400","Text":"If x equals 1, y is minus 1/x,"},{"Start":"11:48.400 ","End":"11:51.445","Text":"is minus 1/1 is minus 1,"},{"Start":"11:51.445 ","End":"11:54.880","Text":"and z we already know is 0."},{"Start":"11:54.880 ","End":"12:00.295","Text":"The other possibility is if we take the minus 1 here,"},{"Start":"12:00.295 ","End":"12:09.265","Text":"we\u0027ve got minus 1. y is minus 1/x makes it plus 1 and still z is 0."},{"Start":"12:09.265 ","End":"12:15.415","Text":"But we\u0027ve got 2 critical points which are suspects for minima."},{"Start":"12:15.415 ","End":"12:19.870","Text":"That\u0027s meanwhile for this branch,"},{"Start":"12:19.870 ","End":"12:21.280","Text":"for the z equals 0."},{"Start":"12:21.280 ","End":"12:25.580","Text":"Let\u0027s go to the other possibility, Lambda equals 1."},{"Start":"12:25.950 ","End":"12:31.195","Text":"Just like in the case here,"},{"Start":"12:31.195 ","End":"12:35.200","Text":"we don\u0027t need this third equation because that\u0027s where we got this split."},{"Start":"12:35.200 ","End":"12:37.540","Text":"We need the first, second, and fourth."},{"Start":"12:37.540 ","End":"12:43.645","Text":"Lambda is 1, we have 2x equals minus"},{"Start":"12:43.645 ","End":"12:50.305","Text":"y from here 2y equals minus x."},{"Start":"12:50.305 ","End":"12:58.960","Text":"The same constraint, the last equation from here,"},{"Start":"12:58.960 ","End":"13:04.150","Text":"just copying it, z squared is xy plus 1."},{"Start":"13:04.150 ","End":"13:06.850","Text":"The top 2 are 2 equations in x and y."},{"Start":"13:06.850 ","End":"13:10.315","Text":"When I find those, I can substitute in the third, that\u0027s the strategy."},{"Start":"13:10.315 ","End":"13:14.575","Text":"What I suggest is that we take from this first equation,"},{"Start":"13:14.575 ","End":"13:19.795","Text":"just isolate y as minus 2x and substitute it"},{"Start":"13:19.795 ","End":"13:26.920","Text":"into this 1 and then we will get that twice,"},{"Start":"13:26.920 ","End":"13:34.135","Text":"y is minus 2x equals minus x."},{"Start":"13:34.135 ","End":"13:38.815","Text":"If I bring all the x\u0027s to one side,"},{"Start":"13:38.815 ","End":"13:41.170","Text":"let\u0027s say everything to the left-hand side,"},{"Start":"13:41.170 ","End":"13:47.560","Text":"I\u0027ve got minus 4x plus x. I\u0027ve got minus 3x equals 0,"},{"Start":"13:47.560 ","End":"13:52.220","Text":"divide by minus 3, x equals 0."},{"Start":"13:52.230 ","End":"13:55.480","Text":"Once I have x equals 0,"},{"Start":"13:55.480 ","End":"14:01.105","Text":"then we have x is 0."},{"Start":"14:01.105 ","End":"14:07.360","Text":"We have that y is minus 2x,"},{"Start":"14:07.360 ","End":"14:09.340","Text":"so y is also 0."},{"Start":"14:09.340 ","End":"14:14.485","Text":"But we can get z squared."},{"Start":"14:14.485 ","End":"14:17.140","Text":"I\u0027m going to wait on this."},{"Start":"14:17.140 ","End":"14:20.410","Text":"We would get that x is 0, y is 0,"},{"Start":"14:20.410 ","End":"14:24.985","Text":"so we get that z squared equals 1."},{"Start":"14:24.985 ","End":"14:31.210","Text":"So z is plus or minus 1."},{"Start":"14:31.210 ","End":"14:33.610","Text":"We have 2 possibilities, 0, 0,"},{"Start":"14:33.610 ","End":"14:38.560","Text":"1 and 0, 0 minus 1."},{"Start":"14:38.560 ","End":"14:42.610","Text":"Altogether, 4 possibilities."},{"Start":"14:42.610 ","End":"14:45.610","Text":"I\u0027d like to label these 4 points we found."},{"Start":"14:45.610 ","End":"14:48.550","Text":"Let\u0027s called this one A, this one B,"},{"Start":"14:48.550 ","End":"14:55.090","Text":"this one C, and this one D. Just off the board,"},{"Start":"14:55.090 ","End":"14:57.460","Text":"I want to rewrite our target function,"},{"Start":"14:57.460 ","End":"15:01.260","Text":"which was the distance squared f of x, y,"},{"Start":"15:01.260 ","End":"15:08.925","Text":"z was x squared plus y squared plus z squared."},{"Start":"15:08.925 ","End":"15:14.900","Text":"We want the least distance or the least distance squared."},{"Start":"15:14.900 ","End":"15:19.900","Text":"Let\u0027s try substituting all of the 4 points we found."},{"Start":"15:19.900 ","End":"15:23.920","Text":"Lets see, f of a is equal to,"},{"Start":"15:23.920 ","End":"15:25.390","Text":"we can do this mentally,"},{"Start":"15:25.390 ","End":"15:30.040","Text":"0 squared plus 0 squared plus 1 squared is 1."},{"Start":"15:30.040 ","End":"15:33.140","Text":"Same thing for B."},{"Start":"15:33.240 ","End":"15:37.270","Text":"Minus 1 won\u0027t change the result."},{"Start":"15:37.270 ","End":"15:44.065","Text":"For C we will get 1 squared plus minus 1 squared,"},{"Start":"15:44.065 ","End":"15:45.865","Text":"this will give us 2."},{"Start":"15:45.865 ","End":"15:51.415","Text":"The same here, f of D is 2."},{"Start":"15:51.415 ","End":"15:57.490","Text":"It looks like we have a tie for the minimum and if I want to highlight the answers,"},{"Start":"15:57.490 ","End":"16:00.130","Text":"we were asked for the points that give the least."},{"Start":"16:00.130 ","End":"16:02.125","Text":"This is one answer,"},{"Start":"16:02.125 ","End":"16:04.420","Text":"and this is the other answer."},{"Start":"16:04.420 ","End":"16:11.095","Text":"These 2 points are equally least distance from the origin that are on the surface."},{"Start":"16:11.095 ","End":"16:13.000","Text":"Now, although it didn\u0027t ask,"},{"Start":"16:13.000 ","End":"16:17.275","Text":"I\u0027d like to find out what this least distance is."},{"Start":"16:17.275 ","End":"16:19.510","Text":"The answer is not 1."},{"Start":"16:19.510 ","End":"16:23.650","Text":"Well it is, but not exactly."},{"Start":"16:23.650 ","End":"16:29.995","Text":"What I mean is that the distance is not f but D and that was the square root."},{"Start":"16:29.995 ","End":"16:35.215","Text":"The distance of A from the origin is the square root of 1."},{"Start":"16:35.215 ","End":"16:39.850","Text":"That is equal to 1 and the same thing for B."},{"Start":"16:39.850 ","End":"16:43.795","Text":"The least distance is 1,"},{"Start":"16:43.795 ","End":"16:45.310","Text":"and these are the 2 points."},{"Start":"16:45.310 ","End":"16:47.350","Text":"But if it happened to be one of these,"},{"Start":"16:47.350 ","End":"16:48.850","Text":"then it wouldn\u0027t be 2,"},{"Start":"16:48.850 ","End":"16:53.780","Text":"it would be the square root of 2 for the distance. We\u0027re done."}],"ID":9680},{"Watched":false,"Name":"Exercise 14","Duration":"19m 37s","ChapterTopicVideoID":9783,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.265","Text":"In this exercise, we\u0027re given an ellipsoid and a plane."},{"Start":"00:05.265 ","End":"00:08.490","Text":"This is not to scale or accurate."},{"Start":"00:08.490 ","End":"00:10.380","Text":"Just for illustration."},{"Start":"00:10.380 ","End":"00:14.010","Text":"We want to find the greatest and least distances"},{"Start":"00:14.010 ","End":"00:18.600","Text":"from the ellipsoid to the plane where the equations are given."},{"Start":"00:18.600 ","End":"00:21.985","Text":"I mean, there\u0027s going to be 1 point somewhere, I\u0027m not sure,"},{"Start":"00:21.985 ","End":"00:25.335","Text":"say here that is the closest to the plane."},{"Start":"00:25.335 ","End":"00:26.895","Text":"It\u0027s going to be another point,"},{"Start":"00:26.895 ","End":"00:30.900","Text":"maybe here that is furthest away from the plane."},{"Start":"00:30.900 ","End":"00:34.239","Text":"But let\u0027s take a general point."},{"Start":"00:34.239 ","End":"00:37.520","Text":"Let\u0027s say this is our general point."},{"Start":"00:37.520 ","End":"00:42.230","Text":"We\u0027ll call it x, y, z."},{"Start":"00:42.230 ","End":"00:49.290","Text":"It has a certain distance where it hits the plane at right angles."},{"Start":"00:50.420 ","End":"00:53.620","Text":"We\u0027ll call the distance d,"},{"Start":"00:53.620 ","End":"00:55.700","Text":"which of course depends on x, y,"},{"Start":"00:55.700 ","End":"01:00.320","Text":"and z. I\u0027ll describe the general strategy first."},{"Start":"01:00.320 ","End":"01:02.240","Text":"We\u0027re going to do it as an optimization on"},{"Start":"01:02.240 ","End":"01:04.895","Text":"the constraint or [inaudible] of the constraint,"},{"Start":"01:04.895 ","End":"01:08.739","Text":"we have both a maximum and a minimum."},{"Start":"01:08.739 ","End":"01:15.530","Text":"What we\u0027re going to do is the constraint will be for this point to be on the ellipsoid."},{"Start":"01:15.530 ","End":"01:20.959","Text":"The target function will be the distance from the point to the plane."},{"Start":"01:20.959 ","End":"01:23.305","Text":"There\u0027s a formula for that."},{"Start":"01:23.305 ","End":"01:27.710","Text":"We\u0027ll look for both the minimum and the maximum of the target function."},{"Start":"01:27.710 ","End":"01:30.210","Text":"That\u0027s the idea."},{"Start":"01:30.460 ","End":"01:33.695","Text":"Let\u0027s start with the constraint."},{"Start":"01:33.695 ","End":"01:37.260","Text":"The constraint on the point x, y,"},{"Start":"01:37.260 ","End":"01:40.699","Text":"z is simply that it beyond the ellipsoid."},{"Start":"01:40.699 ","End":"01:48.125","Text":"In other words, just the equation of the ellipsoid plus y squared plus z squared."},{"Start":"01:48.125 ","End":"01:52.890","Text":"But I like to write it as something equals 0."},{"Start":"01:53.110 ","End":"02:00.900","Text":"The target is this distance"},{"Start":"02:00.900 ","End":"02:05.495","Text":"and there was a formula given in the exercise book."},{"Start":"02:05.495 ","End":"02:08.555","Text":"But let me write the formula again."},{"Start":"02:08.555 ","End":"02:16.935","Text":"We let the point be in general, x_1, y_1, z_1."},{"Start":"02:16.935 ","End":"02:23.870","Text":"The plane in general will be something of"},{"Start":"02:23.870 ","End":"02:33.600","Text":"the form ax plus by plus cz plus d equals 0."},{"Start":"02:38.120 ","End":"02:41.795","Text":"The distance is equal to,"},{"Start":"02:41.795 ","End":"02:46.420","Text":"we substitute the point in the plane."},{"Start":"02:46.420 ","End":"02:53.650","Text":"That\u0027s a ax_1 plus by_1 plus"},{"Start":"02:53.650 ","End":"02:57.620","Text":"cz_1 plus d. Then we divide"},{"Start":"02:57.620 ","End":"03:02.600","Text":"that by the square root of just the first 3 coefficients squared."},{"Start":"03:02.600 ","End":"03:06.620","Text":"In other words, a squared plus b squared plus c squared."},{"Start":"03:06.620 ","End":"03:08.930","Text":"This could come out negative."},{"Start":"03:08.930 ","End":"03:13.030","Text":"We don\u0027t want that, so we put an absolute value here as well."},{"Start":"03:13.030 ","End":"03:16.080","Text":"That\u0027s the formula for point plane."},{"Start":"03:16.080 ","End":"03:19.500","Text":"In our case, instead of x_1,"},{"Start":"03:19.500 ","End":"03:22.905","Text":"y_1, z_1, we just have a general x, y, z."},{"Start":"03:22.905 ","End":"03:26.700","Text":"The a, b, c, and d are taken from here."},{"Start":"03:26.700 ","End":"03:32.309","Text":"It\u0027s slightly different. There could have written minus 288 equals 0."},{"Start":"03:32.540 ","End":"03:35.860","Text":"Maybe I should do that."},{"Start":"03:35.860 ","End":"03:39.725","Text":"There. I put the 288 on the other side."},{"Start":"03:39.725 ","End":"03:47.105","Text":"Now I can write the target function using this formula."},{"Start":"03:47.105 ","End":"03:50.090","Text":"I also noticed that instead of x_1,"},{"Start":"03:50.090 ","End":"03:52.940","Text":"y_1, z_1, we just have plane x, y, z."},{"Start":"03:52.940 ","End":"03:54.380","Text":"We just put x, y,"},{"Start":"03:54.380 ","End":"03:57.890","Text":"z in the plane equation here."},{"Start":"03:57.890 ","End":"04:03.545","Text":"We get 3x plus"},{"Start":"04:03.545 ","End":"04:09.680","Text":"4y plus 12z minus 288."},{"Start":"04:09.680 ","End":"04:15.375","Text":"Then we divide by the square root"},{"Start":"04:15.375 ","End":"04:22.560","Text":"of 3 squared plus 4 squared plus 12 squared."},{"Start":"04:22.560 ","End":"04:26.240","Text":"Then there\u0027s the matter of the absolute value."},{"Start":"04:26.240 ","End":"04:28.880","Text":"So I put an absolute value here,"},{"Start":"04:28.880 ","End":"04:31.975","Text":"and that\u0027s the target."},{"Start":"04:31.975 ","End":"04:35.440","Text":"We want both the maximum and the minimum."},{"Start":"04:35.440 ","End":"04:40.645","Text":"In general, this optimization"},{"Start":"04:40.645 ","End":"04:46.195","Text":"under constraints kind of problem is written as either min or max."},{"Start":"04:46.195 ","End":"04:47.800","Text":"In our case actually both,"},{"Start":"04:47.800 ","End":"04:51.645","Text":"I\u0027ll write min/max and understand it\u0027s 2 separate things."},{"Start":"04:51.645 ","End":"04:54.700","Text":"Then we write a curly brace and in general,"},{"Start":"04:54.700 ","End":"04:57.580","Text":"we write the target function."},{"Start":"04:57.580 ","End":"05:00.400","Text":"In general will be a function f of x, y,"},{"Start":"05:00.400 ","End":"05:03.640","Text":"z. I\u0027ll define that in a moment to be this."},{"Start":"05:03.640 ","End":"05:06.940","Text":"Then we write subject to,"},{"Start":"05:06.940 ","End":"05:11.220","Text":"and then we write the constraint function equals 0."},{"Start":"05:11.220 ","End":"05:12.430","Text":"Sum g of x,"},{"Start":"05:12.430 ","End":"05:15.425","Text":"y, z equals 0."},{"Start":"05:15.425 ","End":"05:17.255","Text":"That\u0027s the mathematical phrasing."},{"Start":"05:17.255 ","End":"05:19.775","Text":"I just need to tell you what f and g are,"},{"Start":"05:19.775 ","End":"05:21.845","Text":"the target and the constraint."},{"Start":"05:21.845 ","End":"05:25.435","Text":"I\u0027ll save space and I\u0027ll just write it here."},{"Start":"05:25.435 ","End":"05:27.270","Text":"This instead of the 0,"},{"Start":"05:27.270 ","End":"05:30.600","Text":"I\u0027ll write g of x, y,"},{"Start":"05:30.600 ","End":"05:36.870","Text":"and z. Yeah, it\u0027s equal to 0 for the constraint."},{"Start":"05:36.870 ","End":"05:42.990","Text":"In fact I\u0027ll also highlight this, there."},{"Start":"05:42.990 ","End":"05:47.660","Text":"That takes care of g. I don\u0027t want to copy this as is."},{"Start":"05:47.660 ","End":"05:49.775","Text":"I want to tidy it up a bit."},{"Start":"05:49.775 ","End":"05:53.135","Text":"My f of x, y, z."},{"Start":"05:53.135 ","End":"05:57.355","Text":"There is couple of things I want to do with this."},{"Start":"05:57.355 ","End":"06:00.875","Text":"1 thing is I want to evaluate this denominator,"},{"Start":"06:00.875 ","End":"06:08.000","Text":"3 squared plus 12 squared is 9 plus 16 plus a 144. This is a 169."},{"Start":"06:08.000 ","End":"06:12.800","Text":"The square root of 169 is 13 precisely."},{"Start":"06:12.800 ","End":"06:16.235","Text":"I want to put the 13 inside."},{"Start":"06:16.235 ","End":"06:24.750","Text":"I\u0027ve got the absolute value of 3 thirteenths x plus"},{"Start":"06:24.750 ","End":"06:29.250","Text":"4 thirteenths y plus"},{"Start":"06:29.250 ","End":"06:40.565","Text":"12 thirteenths z minus 288 over 13."},{"Start":"06:40.565 ","End":"06:45.470","Text":"But there\u0027s another minor nuisance and that\u0027s the absolute value."},{"Start":"06:45.470 ","End":"06:48.380","Text":"It\u0027s not good for differentiating."},{"Start":"06:48.380 ","End":"06:52.100","Text":"But absolute value just means that it\u0027s plus or minus."},{"Start":"06:52.100 ","End":"06:54.965","Text":"Turns out that on 1 side of the plane it\u0027s a plus,"},{"Start":"06:54.965 ","End":"06:57.365","Text":"on the other side of the plane it\u0027s a minus."},{"Start":"06:57.365 ","End":"06:59.695","Text":"But I don\u0027t know which."},{"Start":"06:59.695 ","End":"07:02.180","Text":"Instead of the absolute value,"},{"Start":"07:02.180 ","End":"07:04.760","Text":"I\u0027ll put plus or minus."},{"Start":"07:04.760 ","End":"07:10.789","Text":"In fact, let me say now that just for expedience instead of the plus or minus,"},{"Start":"07:10.789 ","End":"07:15.770","Text":"I\u0027m going to just take it as a plus and ignore the plus or minus."},{"Start":"07:15.770 ","End":"07:18.155","Text":"But I have to remember at the end that"},{"Start":"07:18.155 ","End":"07:24.870","Text":"these distances have to be made positive if they come out negative."},{"Start":"07:24.980 ","End":"07:31.849","Text":"My target function is just this without the absolute value or the plus or minus."},{"Start":"07:31.849 ","End":"07:36.459","Text":"But we have to remember to relate to it at the end if we get negative distances."},{"Start":"07:36.459 ","End":"07:44.150","Text":"Now we\u0027re going to start solving it using the technique of the Lagrange multipliers."},{"Start":"07:44.150 ","End":"07:49.360","Text":"Basically, we\u0027re going to get 4 equations and 4 unknowns,"},{"Start":"07:49.360 ","End":"07:53.120","Text":"xyz and a new auxiliary variable Lambda."},{"Start":"07:53.120 ","End":"07:57.995","Text":"The first thing we do is interpret the following."},{"Start":"07:57.995 ","End":"08:02.270","Text":"The gradient of f equals lambda times the"},{"Start":"08:02.270 ","End":"08:06.860","Text":"gradient of g. This is just a shorthand, if you like,"},{"Start":"08:06.860 ","End":"08:11.030","Text":"for 3 equations in 1 because what this means is that f with"},{"Start":"08:11.030 ","End":"08:16.220","Text":"respect to x equals lambda times g with respect to x."},{"Start":"08:16.220 ","End":"08:20.390","Text":"The same thing with y instead of x,"},{"Start":"08:20.390 ","End":"08:25.415","Text":"and the same thing with z instead of x."},{"Start":"08:25.415 ","End":"08:31.160","Text":"That\u0027s only 3 equations and the last equation is always the constraint,"},{"Start":"08:31.160 ","End":"08:33.050","Text":"which is that g equals 0."},{"Start":"08:33.050 ","End":"08:35.440","Text":"I\u0027m just writing all this in shorthand."},{"Start":"08:35.440 ","End":"08:40.550","Text":"That\u0027s in general. Let\u0027s see what this comes down to in our case."},{"Start":"08:40.550 ","End":"08:43.160","Text":"I\u0027ll start with the curly brace."},{"Start":"08:43.160 ","End":"08:46.490","Text":"I want to write 4 equations."},{"Start":"08:46.490 ","End":"08:49.860","Text":"Now let\u0027s see, f with respect to x. I\u0027ve got f"},{"Start":"08:49.860 ","End":"08:53.685","Text":"highlighted and I\u0027ve got g highlighted that makes it easier."},{"Start":"08:53.685 ","End":"09:01.175","Text":"F with respect to x is just 3 thirteenths because everything else is a constant."},{"Start":"09:01.175 ","End":"09:05.960","Text":"We have 3 over 13 equals lambda."},{"Start":"09:05.960 ","End":"09:08.255","Text":"Now g with respect to x,"},{"Start":"09:08.255 ","End":"09:15.020","Text":"this is g. G with respect to x is just 2x over 96,"},{"Start":"09:15.020 ","End":"09:22.170","Text":"which if I divide by the 2 comes out to be x over 48."},{"Start":"09:22.560 ","End":"09:25.495","Text":"F with respect to y,"},{"Start":"09:25.495 ","End":"09:30.235","Text":"that\u0027s just 4/13 equals Lambda."},{"Start":"09:30.235 ","End":"09:35.095","Text":"G with respect to y is just 2y."},{"Start":"09:35.095 ","End":"09:43.615","Text":"Here we get 12/13 equals Lambda times 2z."},{"Start":"09:43.615 ","End":"09:49.210","Text":"Finally, the constraint g equals 0,"},{"Start":"09:49.210 ","End":"09:51.880","Text":"which is x squared"},{"Start":"09:51.880 ","End":"10:00.970","Text":"over 96 plus y squared plus z squared minus 1 equals 0."},{"Start":"10:00.970 ","End":"10:05.330","Text":"You know what? I\u0027ll go back to the original form, say equals 1."},{"Start":"10:06.090 ","End":"10:13.930","Text":"Now comes our usual trick where we divide 1 of these first 3 equations by the other,"},{"Start":"10:13.930 ","End":"10:15.624","Text":"and actually we do that twice."},{"Start":"10:15.624 ","End":"10:19.990","Text":"But before I do that, I want to note that Lambda"},{"Start":"10:19.990 ","End":"10:25.045","Text":"can\u0027t be 0 otherwise I\u0027d get 3/13 is equals 0."},{"Start":"10:25.045 ","End":"10:28.210","Text":"Similarly, x can\u0027t be 0."},{"Start":"10:28.210 ","End":"10:31.030","Text":"Likewise here y can\u0027t be 0."},{"Start":"10:31.030 ","End":"10:35.050","Text":"Basically, all of these quantities Lambda x,"},{"Start":"10:35.050 ","End":"10:37.840","Text":"y, and z are all non 0."},{"Start":"10:37.840 ","End":"10:41.260","Text":"Otherwise, I get 0 equals something not 0."},{"Start":"10:41.260 ","End":"10:46.310","Text":"That means that I can do some divisions safely."},{"Start":"10:46.440 ","End":"10:50.095","Text":"If I divide this one by this one,"},{"Start":"10:50.095 ","End":"10:58.165","Text":"I\u0027ll get 3/13 over 4/13"},{"Start":"10:58.165 ","End":"11:03.650","Text":"equals Lambda times x/48"},{"Start":"11:03.930 ","End":"11:09.610","Text":"over Lambda times 2y,"},{"Start":"11:09.610 ","End":"11:13.105","Text":"and nothing in the denominator is 0."},{"Start":"11:13.105 ","End":"11:17.560","Text":"This could be simplified."},{"Start":"11:17.560 ","End":"11:22.615","Text":"The 13 and the denominator here can go with this 13."},{"Start":"11:22.615 ","End":"11:26.020","Text":"Lambda can go with Lambda."},{"Start":"11:26.020 ","End":"11:29.905","Text":"Then I can cross multiply,"},{"Start":"11:29.905 ","End":"11:36.190","Text":"and this will give me 3 times 2y is 6y."},{"Start":"11:36.190 ","End":"11:42.310","Text":"The other way I\u0027ll get 4x/48."},{"Start":"11:42.310 ","End":"11:47.005","Text":"4 goes into 48, 12 times,"},{"Start":"11:47.005 ","End":"11:56.860","Text":"so I end up with x equals 72y."},{"Start":"11:56.860 ","End":"11:58.600","Text":"Now let me take the other pair,"},{"Start":"11:58.600 ","End":"12:01.900","Text":"this with this, and also divide."},{"Start":"12:01.900 ","End":"12:07.520","Text":"We get 4/13 over"},{"Start":"12:07.710 ","End":"12:16.165","Text":"12/13 equals Lambda times 2y over Lambda times 2z."},{"Start":"12:16.165 ","End":"12:22.210","Text":"I can cancel the 13 with the 13 here,"},{"Start":"12:22.210 ","End":"12:26.950","Text":"Lambda and the 2 here and here."},{"Start":"12:26.950 ","End":"12:29.515","Text":"That\u0027s not all, 4/12,"},{"Start":"12:29.515 ","End":"12:36.820","Text":"I can make it as 1/3."},{"Start":"12:36.820 ","End":"12:44.620","Text":"All I\u0027m left with is 1 times z is equal to y times 3,"},{"Start":"12:44.620 ","End":"12:48.110","Text":"so z equals 3y."},{"Start":"12:48.690 ","End":"12:52.780","Text":"Now I\u0027d like to highlight this equation,"},{"Start":"12:52.780 ","End":"12:58.730","Text":"this equation, and this equation."},{"Start":"12:58.980 ","End":"13:06.835","Text":"What we have is 3 equations and 3 unknowns, x, y, and z."},{"Start":"13:06.835 ","End":"13:12.430","Text":"What I suggest is that since I have x and z in terms of y,"},{"Start":"13:12.430 ","End":"13:15.070","Text":"if I plug these 2 into here,"},{"Start":"13:15.070 ","End":"13:17.605","Text":"I can just get an equation in y."},{"Start":"13:17.605 ","End":"13:20.875","Text":"What I\u0027ll get is x squared,"},{"Start":"13:20.875 ","End":"13:28.135","Text":"which is 72 squared y squared"},{"Start":"13:28.135 ","End":"13:35.725","Text":"over 96 plus y squared as is"},{"Start":"13:35.725 ","End":"13:44.455","Text":"plus z squared is 9y squared is equal to 1."},{"Start":"13:44.455 ","End":"13:48.160","Text":"Now, if we look at the coefficient of y squared here,"},{"Start":"13:48.160 ","End":"13:52.105","Text":"it\u0027s 72 squared over 96."},{"Start":"13:52.105 ","End":"13:53.950","Text":"I did it on the calculator,"},{"Start":"13:53.950 ","End":"13:57.130","Text":"comes out evenly to be 54."},{"Start":"13:57.130 ","End":"14:03.350","Text":"Basically, I can replace this by 54."},{"Start":"14:04.440 ","End":"14:09.510","Text":"What we get, we can do it in our heads,"},{"Start":"14:09.510 ","End":"14:15.810","Text":"54 plus 1 plus 9 is like 54 plus 10, is 64."},{"Start":"14:15.810 ","End":"14:23.830","Text":"So I get 64y squared is equal to 1."},{"Start":"14:24.180 ","End":"14:33.130","Text":"That gives me that y squared is 1/64."},{"Start":"14:33.130 ","End":"14:42.320","Text":"That gives that y equals plus or minus 1/8."},{"Start":"14:42.470 ","End":"14:45.960","Text":"We now have 2 possibilities."},{"Start":"14:45.960 ","End":"14:48.299","Text":"We have the plus and we have the minus."},{"Start":"14:48.299 ","End":"14:50.565","Text":"If y is plus an 1/8,"},{"Start":"14:50.565 ","End":"14:53.390","Text":"then I can substitute here and here,"},{"Start":"14:53.390 ","End":"14:57.810","Text":"x will equal 72/8,"},{"Start":"14:57.810 ","End":"15:05.655","Text":"which is 9, and z is 3 times y,"},{"Start":"15:05.655 ","End":"15:10.780","Text":"so that is 3/8."},{"Start":"15:10.780 ","End":"15:12.190","Text":"If I take the minus,"},{"Start":"15:12.190 ","End":"15:13.690","Text":"it\u0027ll just be the same thing,"},{"Start":"15:13.690 ","End":"15:14.830","Text":"but with a minus,"},{"Start":"15:14.830 ","End":"15:18.925","Text":"minus 9 and minus 3/8."},{"Start":"15:18.925 ","End":"15:21.595","Text":"In other words, we get the 2 points."},{"Start":"15:21.595 ","End":"15:24.799","Text":"If I take this, it\u0027s going to be 9,"},{"Start":"15:25.530 ","End":"15:30.820","Text":"1/8, 3/8, that\u0027s 1 point."},{"Start":"15:30.820 ","End":"15:36.470","Text":"The other point will be minus 9,"},{"Start":"15:37.140 ","End":"15:44.150","Text":"minus 1/8, and minus 3/8."},{"Start":"15:44.550 ","End":"15:47.740","Text":"Let me give these points names."},{"Start":"15:47.740 ","End":"15:52.975","Text":"I\u0027ll call this point A and this point I\u0027ll call B."},{"Start":"15:52.975 ","End":"15:55.840","Text":"Now, we\u0027re going to expect that 1 of these will come out to"},{"Start":"15:55.840 ","End":"15:58.360","Text":"be the minimum or least distance and 1 of them will"},{"Start":"15:58.360 ","End":"16:03.460","Text":"be the maximum or greatest distance from the plane. Let\u0027s see."},{"Start":"16:03.460 ","End":"16:09.760","Text":"Let\u0027s substitute in the target function f. We\u0027ll see"},{"Start":"16:09.760 ","End":"16:16.449","Text":"what f of A equals and what f of B equals."},{"Start":"16:16.449 ","End":"16:19.150","Text":"Now, f of A is,"},{"Start":"16:19.150 ","End":"16:20.410","Text":"I\u0027ll just write what A is,"},{"Start":"16:20.410 ","End":"16:28.660","Text":"we\u0027ll copy it, 9, 1/8, 3/8."},{"Start":"16:28.660 ","End":"16:31.480","Text":"Let me just simplify this."},{"Start":"16:31.480 ","End":"16:36.805","Text":"For this computation, I\u0027d rather have a single 13 on the denominator."},{"Start":"16:36.805 ","End":"16:40.990","Text":"I\u0027m going to put a dividing line over 13,"},{"Start":"16:40.990 ","End":"16:44.650","Text":"and then it\u0027ll be just 3x plus"},{"Start":"16:44.650 ","End":"16:52.900","Text":"4y plus 12z minus 288."},{"Start":"16:52.900 ","End":"16:57.980","Text":"But there\u0027s still the plus-minus or absolute value."},{"Start":"16:58.200 ","End":"17:01.765","Text":"We get from the numerator,"},{"Start":"17:01.765 ","End":"17:08.260","Text":"3 times 9 plus 4 times"},{"Start":"17:08.260 ","End":"17:12.430","Text":"1/8 plus 12 times"},{"Start":"17:12.430 ","End":"17:20.950","Text":"z is 3/8 minus 288,"},{"Start":"17:20.950 ","End":"17:26.155","Text":"all this over 13."},{"Start":"17:26.155 ","End":"17:33.050","Text":"This computation comes out to be minus 256/13."},{"Start":"17:33.810 ","End":"17:36.640","Text":"Because of the plus or minus, you know what,"},{"Start":"17:36.640 ","End":"17:41.680","Text":"I\u0027ll rewrite this back as the absolute value,"},{"Start":"17:41.680 ","End":"17:45.145","Text":"and so what I have is the absolute value of this,"},{"Start":"17:45.145 ","End":"17:49.645","Text":"and that\u0027s just 256/13."},{"Start":"17:49.645 ","End":"17:56.410","Text":"That\u0027s the distance of the point A on the ellipsoid from the plane."},{"Start":"17:56.410 ","End":"17:59.020","Text":"Now, for the second 1,"},{"Start":"17:59.020 ","End":"18:02.200","Text":"this is f of the other point,"},{"Start":"18:02.200 ","End":"18:07.480","Text":"minus 9, minus an 1/8, minus 3/8."},{"Start":"18:07.480 ","End":"18:08.889","Text":"It\u0027s very similar."},{"Start":"18:08.889 ","End":"18:17.760","Text":"I get 3 times minus 9 plus 4 times minus an 1/8 plus"},{"Start":"18:17.760 ","End":"18:28.050","Text":"12 times minus 3/8 minus 288/13."},{"Start":"18:28.860 ","End":"18:33.130","Text":"I should put an absolute value here,"},{"Start":"18:33.130 ","End":"18:36.445","Text":"by the way, just to be strictly speaking."},{"Start":"18:36.445 ","End":"18:40.765","Text":"This comes out to be the absolute value of"},{"Start":"18:40.765 ","End":"18:50.570","Text":"minus 320/13, so it\u0027s 320/13."},{"Start":"18:50.570 ","End":"18:54.830","Text":"Now, we can see that this is bigger than this."},{"Start":"18:54.830 ","End":"19:00.880","Text":"This one is our minimum,"},{"Start":"19:00.880 ","End":"19:05.635","Text":"and this one is the maximum,"},{"Start":"19:05.635 ","End":"19:10.885","Text":"which means that this one"},{"Start":"19:10.885 ","End":"19:16.160","Text":"is the one with the least distance to the plane."},{"Start":"19:16.160 ","End":"19:17.734","Text":"I should write a full sentence."},{"Start":"19:17.734 ","End":"19:23.550","Text":"A is the point on the ellipsoid with the least distance to the plane, and the other one,"},{"Start":"19:23.550 ","End":"19:25.810","Text":"we\u0027re asked for the greatest distance,"},{"Start":"19:25.810 ","End":"19:30.350","Text":"and that is the point B."},{"Start":"19:30.350 ","End":"19:33.840","Text":"It has the greatest distance to the plane."},{"Start":"19:34.420 ","End":"19:37.920","Text":"I think that\u0027s it. We\u0027re done."}],"ID":9681},{"Watched":false,"Name":"Exercise 15","Duration":"15m 57s","ChapterTopicVideoID":9784,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"This exercise is a bit different from some of the previous."},{"Start":"00:03.450 ","End":"00:09.600","Text":"It\u0027s an extremum or optimization question but with multiple constraints. Let me read it."},{"Start":"00:09.600 ","End":"00:13.995","Text":"You want to find the minimum and maximum distances from the origin of the curve."},{"Start":"00:13.995 ","End":"00:15.540","Text":"Now, which curve is that?"},{"Start":"00:15.540 ","End":"00:19.125","Text":"It\u0027s the 1 obtained by intersecting 2 surfaces,"},{"Start":"00:19.125 ","End":"00:22.620","Text":"the cylinder, this 1 and the plane, this 1."},{"Start":"00:22.620 ","End":"00:27.420","Text":"Typically a cylinder and a plane intersect in some ellipse but that doesn\u0027t matter."},{"Start":"00:27.420 ","End":"00:31.605","Text":"But we get a curve and on that curve, on that ellipse,"},{"Start":"00:31.605 ","End":"00:33.480","Text":"there will be a point that\u0027s closest to"},{"Start":"00:33.480 ","End":"00:37.155","Text":"the origin and the 1 that\u0027s furthest to the origin."},{"Start":"00:37.155 ","End":"00:40.035","Text":"The origin is the point 0,"},{"Start":"00:40.035 ","End":"00:44.130","Text":"0, 0 in 3D."},{"Start":"00:44.130 ","End":"00:51.125","Text":"What we\u0027re going to do is define a target function and 2 constraint functions."},{"Start":"00:51.125 ","End":"00:55.140","Text":"Usually I thought with the constraints time and start with the target."},{"Start":"00:55.580 ","End":"01:00.785","Text":"Let\u0027s say that we\u0027re talking about a point x, y, z."},{"Start":"01:00.785 ","End":"01:03.305","Text":"Now, that\u0027s our general point."},{"Start":"01:03.305 ","End":"01:12.035","Text":"Now, the target is the distance of this from the origin."},{"Start":"01:12.035 ","End":"01:15.750","Text":"Now, the distance from the origin d,"},{"Start":"01:15.750 ","End":"01:17.420","Text":"it\u0027s a function of x, y,"},{"Start":"01:17.420 ","End":"01:22.625","Text":"and z is going to be equal the square root"},{"Start":"01:22.625 ","End":"01:28.715","Text":"of x minus 0 squared using the distance formula here,"},{"Start":"01:28.715 ","End":"01:31.369","Text":"plus y minus 0 squared,"},{"Start":"01:31.369 ","End":"01:35.725","Text":"plus z minus 0 squared."},{"Start":"01:35.725 ","End":"01:42.440","Text":"So that this is the distance of the point x,"},{"Start":"01:42.440 ","End":"01:44.239","Text":"y, z from the origin."},{"Start":"01:44.239 ","End":"01:49.160","Text":"Now, as usual and this is our standard trick when we have a square root,"},{"Start":"01:49.160 ","End":"01:51.740","Text":"instead of looking at the distance as the target,"},{"Start":"01:51.740 ","End":"01:53.750","Text":"we look at the distance squared."},{"Start":"01:53.750 ","End":"01:57.424","Text":"Let\u0027s define f of x, y,"},{"Start":"01:57.424 ","End":"02:02.225","Text":"and z to be the distance of this from the origin but squared,"},{"Start":"02:02.225 ","End":"02:06.455","Text":"so we don\u0027t need the square root and also this minus 0 is useless."},{"Start":"02:06.455 ","End":"02:13.580","Text":"Basically, our target becomes x squared plus y squared plus z squared."},{"Start":"02:13.580 ","End":"02:18.590","Text":"Since we\u0027re asked for the maximum and minimum distances not just the points,"},{"Start":"02:18.590 ","End":"02:21.860","Text":"when we found the points and then we found the distances,"},{"Start":"02:21.860 ","End":"02:26.390","Text":"we\u0027ll have to go back in the end and take d as the square root of"},{"Start":"02:26.390 ","End":"02:31.590","Text":"f. This is the distance squared so at the end,"},{"Start":"02:31.590 ","End":"02:35.284","Text":"we\u0027re going to have to remember that because f is d squared,"},{"Start":"02:35.284 ","End":"02:39.110","Text":"d is the square root of f so at the end,"},{"Start":"02:39.110 ","End":"02:40.940","Text":"sometimes we forget to take the square root."},{"Start":"02:40.940 ","End":"02:43.140","Text":"I\u0027m mentioning it now."},{"Start":"02:43.880 ","End":"02:47.970","Text":"This one here is our target."},{"Start":"02:47.970 ","End":"02:50.910","Text":"Now, I said there\u0027s going to be 2 constraints."},{"Start":"02:50.910 ","End":"02:52.860","Text":"If the point x, y,"},{"Start":"02:52.860 ","End":"02:56.480","Text":"z is going to be on the intersection of these 2 surfaces,"},{"Start":"02:56.480 ","End":"02:58.745","Text":"it has to satisfy both of them."},{"Start":"02:58.745 ","End":"03:03.045","Text":"Both of these equations become constraints."},{"Start":"03:03.045 ","End":"03:07.370","Text":"Only I prefer to write them as something equals 0 as usual method."},{"Start":"03:07.370 ","End":"03:09.920","Text":"I\u0027m going to define g of x, y,"},{"Start":"03:09.920 ","End":"03:15.860","Text":"z to be, but we customarily put everything on the left."},{"Start":"03:15.860 ","End":"03:20.930","Text":"That\u0027s going to be x squared plus y squared minus 1,"},{"Start":"03:20.930 ","End":"03:24.870","Text":"and next letter in the alphabet is h,"},{"Start":"03:24.870 ","End":"03:26.840","Text":"so we\u0027ll use h of x, y,"},{"Start":"03:26.840 ","End":"03:33.545","Text":"z to be the other one z minus x minus y."},{"Start":"03:33.545 ","End":"03:40.050","Text":"The constraints are that g equals 0 and that h equals 0."},{"Start":"03:40.900 ","End":"03:43.995","Text":"I\u0027m going to highlight these also."},{"Start":"03:43.995 ","End":"03:47.760","Text":"We have 3 functions, target, constraint, constraint."},{"Start":"03:47.760 ","End":"03:50.235","Text":"Now, if we wanted to phrase our problem,"},{"Start":"03:50.235 ","End":"03:52.470","Text":"it\u0027s usually min or max."},{"Start":"03:52.470 ","End":"03:53.880","Text":"In this case, it\u0027s both."},{"Start":"03:53.880 ","End":"04:00.125","Text":"We want min and we want max curly brace the target function,"},{"Start":"04:00.125 ","End":"04:04.610","Text":"x squared plus y squared plus z squared,"},{"Start":"04:04.610 ","End":"04:08.315","Text":"subject to a constraint,"},{"Start":"04:08.315 ","End":"04:12.890","Text":"x squared plus y squared minus 1 equals 0,"},{"Start":"04:12.890 ","End":"04:22.495","Text":"and the other constraint z minus x minus y equals 0."},{"Start":"04:22.495 ","End":"04:29.270","Text":"We need the technique for optimization with multiple constraints."},{"Start":"04:29.270 ","End":"04:34.460","Text":"With 1 constraint we had a Lambda here turns out we\u0027re going to have 2 Greek letters,"},{"Start":"04:34.460 ","End":"04:39.890","Text":"Lambda and Mu which is actually going to give us 5 variables,"},{"Start":"04:39.890 ","End":"04:41.570","Text":"x, y, z, Lambda, Mu."},{"Start":"04:41.570 ","End":"04:48.360","Text":"Now, the first 3 are obtained from the gradient and we want the gradient of f to be,"},{"Start":"04:48.360 ","End":"04:53.895","Text":"with the 1 case constraint it was Lambda times grad of g,"},{"Start":"04:53.895 ","End":"05:00.685","Text":"but here we get the extra bit Mu times grad of"},{"Start":"05:00.685 ","End":"05:03.990","Text":"h. This gives us 3 equations and then"},{"Start":"05:03.990 ","End":"05:08.510","Text":"the 2 constraints will give us 5 equations and 5 unknowns."},{"Start":"05:08.510 ","End":"05:13.240","Text":"Now in general, this gives us that f with respect to x"},{"Start":"05:13.240 ","End":"05:20.430","Text":"equals Lambda g with respect to x plus Mu h with respect to x,"},{"Start":"05:20.430 ","End":"05:22.580","Text":"and now this is tiresome,"},{"Start":"05:22.580 ","End":"05:28.600","Text":"just the same thing but with y in place of x,"},{"Start":"05:28.600 ","End":"05:39.779","Text":"and again with z Lambda g with respect to z plus Mu h with respect to z,"},{"Start":"05:39.779 ","End":"05:44.720","Text":"and then we also need the 2 constraints in"},{"Start":"05:44.720 ","End":"05:52.205","Text":"which case g equals 0 and h equals 0."},{"Start":"05:52.205 ","End":"05:58.940","Text":"This is a system of 5 equations and 5 unknowns."},{"Start":"05:58.940 ","End":"06:03.200","Text":"In general, let\u0027s see what it will be for our particular f,"},{"Start":"06:03.200 ","End":"06:07.610","Text":"g, and h. I\u0027ll start with the curly braces."},{"Start":"06:07.610 ","End":"06:10.520","Text":"That\u0027s straightforward enough and"},{"Start":"06:10.520 ","End":"06:13.880","Text":"now could have prepared all these partial derivatives in advance,"},{"Start":"06:13.880 ","End":"06:17.285","Text":"but I just do them as I need them."},{"Start":"06:17.285 ","End":"06:25.880","Text":"F with respect to x is 2x and that what\u0027s going to be easy for me to just go vertical,"},{"Start":"06:25.880 ","End":"06:30.200","Text":"f with respect to y is going to be 2y,"},{"Start":"06:30.200 ","End":"06:35.925","Text":"f with respect to z is going to be 2z that is going to be a Lambda,"},{"Start":"06:35.925 ","End":"06:45.000","Text":"Lambda and Lambda, then g with respect to x is 2x with respect to y,"},{"Start":"06:45.000 ","End":"06:51.425","Text":"it\u0027s 2y, and with respect to z it\u0027s 0."},{"Start":"06:51.425 ","End":"06:53.600","Text":"Then I\u0027m going to get plus Mu,"},{"Start":"06:53.600 ","End":"06:57.645","Text":"plus Mu, plus Mu and let\u0027s see."},{"Start":"06:57.645 ","End":"06:59.330","Text":"H with respect to x, y,"},{"Start":"06:59.330 ","End":"07:01.970","Text":"and z with respect to x,"},{"Start":"07:01.970 ","End":"07:05.105","Text":"it\u0027s minus 1 with respect to y,"},{"Start":"07:05.105 ","End":"07:09.865","Text":"it\u0027s minus 1, with respect to z, it\u0027s just 1."},{"Start":"07:09.865 ","End":"07:15.060","Text":"3 equations and then g equals 0 is this equation,"},{"Start":"07:15.060 ","End":"07:18.605","Text":"but I often prefer to just write it in the original form,"},{"Start":"07:18.605 ","End":"07:21.125","Text":"x squared plus y squared equals 1."},{"Start":"07:21.125 ","End":"07:24.275","Text":"Somehow more aesthetic to me unless writing."},{"Start":"07:24.275 ","End":"07:27.830","Text":"On this one, we can write as this equals"},{"Start":"07:27.830 ","End":"07:34.945","Text":"0 or z equals x plus y as it was originally, doesn\u0027t really matter."},{"Start":"07:34.945 ","End":"07:38.180","Text":"Wow, 5 equations and 5 unknowns."},{"Start":"07:38.180 ","End":"07:42.290","Text":"Usually you stare at this awhile and then you get some ideas what to do."},{"Start":"07:42.290 ","End":"07:45.485","Text":"If I subtract this one minus this one,"},{"Start":"07:45.485 ","End":"07:51.290","Text":"then I will get that from the left-hand sides,"},{"Start":"07:51.290 ","End":"08:00.540","Text":"2x minus 2y is equal to Lambda."},{"Start":"08:02.060 ","End":"08:07.800","Text":"I can write it as 2x minus 2y,"},{"Start":"08:07.800 ","End":"08:09.690","Text":"I can only take Lambda out the brackets."},{"Start":"08:09.690 ","End":"08:14.010","Text":"Then this minus this just cancels itself out."},{"Start":"08:14.690 ","End":"08:17.320","Text":"If we divide now by 2,"},{"Start":"08:17.320 ","End":"08:25.409","Text":"we get that x minus y equals Lambda times x minus y."},{"Start":"08:26.520 ","End":"08:31.404","Text":"I could try and divide both sides by x minus y,"},{"Start":"08:31.404 ","End":"08:36.865","Text":"but I don\u0027t know that x minus y is not 0, it could be."},{"Start":"08:36.865 ","End":"08:41.305","Text":"At this point, we split up into 2 cases."},{"Start":"08:41.305 ","End":"08:46.825","Text":"We either have that x minus y is 0,"},{"Start":"08:46.825 ","End":"08:50.830","Text":"which means that x equals y,"},{"Start":"08:50.830 ","End":"08:56.290","Text":"or x minus y is not 0 and then we can divide both sides by it,"},{"Start":"08:56.290 ","End":"08:59.380","Text":"so, we get Lambda equals 1."},{"Start":"08:59.380 ","End":"09:02.725","Text":"Let me do each of these separately."},{"Start":"09:02.725 ","End":"09:06.475","Text":"Let me deal first of all with this case and I\u0027ll start with a different color,"},{"Start":"09:06.475 ","End":"09:09.460","Text":"and here, Lambda equals 1."},{"Start":"09:09.460 ","End":"09:11.515","Text":"Then what can I see?"},{"Start":"09:11.515 ","End":"09:16.130","Text":"If I put Lambda equals 1 in the first equation,"},{"Start":"09:17.400 ","End":"09:25.885","Text":"then that will give me that 2x equals 2x minus Mu."},{"Start":"09:25.885 ","End":"09:29.875","Text":"I get that Mu is equal to 0,"},{"Start":"09:29.875 ","End":"09:36.190","Text":"and if we put Mu equals 0 in this third equation,"},{"Start":"09:36.190 ","End":"09:39.685","Text":"then everything, all the whole right-hand side is 0,"},{"Start":"09:39.685 ","End":"09:46.720","Text":"so that gives us now that z equals 0,"},{"Start":"09:46.720 ","End":"09:51.770","Text":"and if z is 0,"},{"Start":"09:52.320 ","End":"09:56.080","Text":"then in the last equation,"},{"Start":"09:56.080 ","End":"10:00.640","Text":"I will get x plus y equals 0."},{"Start":"10:00.640 ","End":"10:05.035","Text":"Now I have 2 equations and 2 unknowns with x and y,"},{"Start":"10:05.035 ","End":"10:06.940","Text":"because I can also take this 1,"},{"Start":"10:06.940 ","End":"10:11.860","Text":"x squared plus y squared equals 1."},{"Start":"10:11.860 ","End":"10:15.100","Text":"Now if we can take y out of here,"},{"Start":"10:15.100 ","End":"10:19.735","Text":"y equals minus x and substitute it in here,"},{"Start":"10:19.735 ","End":"10:27.385","Text":"we will get that x squared plus minus x squared equals 1."},{"Start":"10:27.385 ","End":"10:32.680","Text":"In other words, 2x squared equals 1, divide by 2,"},{"Start":"10:32.680 ","End":"10:34.090","Text":"take the square root,"},{"Start":"10:34.090 ","End":"10:39.025","Text":"x equals plus or minus the square root of 1/2."},{"Start":"10:39.025 ","End":"10:43.260","Text":"Actually, we can get everything from this because we get 2 points."},{"Start":"10:43.260 ","End":"10:47.705","Text":"If x is the square root of 1/2,"},{"Start":"10:47.705 ","End":"10:53.920","Text":"then y being minus x is minus the square root of 1/2,"},{"Start":"10:53.920 ","End":"10:58.000","Text":"and we also have that z equals 0."},{"Start":"10:58.000 ","End":"11:04.869","Text":"This is the point we get and the other point we get is"},{"Start":"11:04.869 ","End":"11:12.370","Text":"if we take that minus the square root of 1/2 for x,"},{"Start":"11:12.370 ","End":"11:13.870","Text":"then y is minus x,"},{"Start":"11:13.870 ","End":"11:16.720","Text":"so it\u0027s plus the square root of 1/2,"},{"Start":"11:16.720 ","End":"11:19.615","Text":"and z is still 0."},{"Start":"11:19.615 ","End":"11:22.390","Text":"I\u0027ll give these points names,"},{"Start":"11:22.390 ","End":"11:25.630","Text":"we\u0027ll call this 1 A and this 1 B."},{"Start":"11:25.630 ","End":"11:29.260","Text":"But that\u0027s not all because we still have the other path,"},{"Start":"11:29.260 ","End":"11:32.740","Text":"and in this path we take the other alternative and"},{"Start":"11:32.740 ","End":"11:37.400","Text":"I\u0027ll use a different color, x equals y."},{"Start":"11:37.710 ","End":"11:44.710","Text":"Now, this case is very similar to the other case because here we had y equals minus x,"},{"Start":"11:44.710 ","End":"11:48.130","Text":"but we also had this equation,"},{"Start":"11:48.130 ","End":"11:52.975","Text":"a copy from here, x squared plus y squared equals 1."},{"Start":"11:52.975 ","End":"11:56.215","Text":"If we replace y by x instead of minus x,"},{"Start":"11:56.215 ","End":"11:59.245","Text":"we\u0027ll still get this same equation here."},{"Start":"11:59.245 ","End":"12:04.240","Text":"We\u0027ll still get the 2x squared equals 1 and we\u0027ll still"},{"Start":"12:04.240 ","End":"12:10.990","Text":"get that x equals plus or minus the square root of 1/2."},{"Start":"12:10.990 ","End":"12:19.855","Text":"The difference is that if x is the square root of 1/2 before y was minus x,"},{"Start":"12:19.855 ","End":"12:22.510","Text":"here, y equals x, so,"},{"Start":"12:22.510 ","End":"12:28.270","Text":"y is also root of 1/2 and this time we need to compute z."},{"Start":"12:28.270 ","End":"12:33.430","Text":"There z was 0, but here z we can get from this equation, x plus y."},{"Start":"12:33.430 ","End":"12:39.925","Text":"This plus this is just twice this, twice root 1/2."},{"Start":"12:39.925 ","End":"12:48.115","Text":"The other point, we would get from the minus square root of 1/2,"},{"Start":"12:48.115 ","End":"12:50.050","Text":"and this time again,"},{"Start":"12:50.050 ","End":"12:55.070","Text":"y equals x, so it\u0027s also minus root of 1/2,"},{"Start":"12:55.070 ","End":"12:58.080","Text":"and z is x plus y, so,"},{"Start":"12:58.080 ","End":"13:03.090","Text":"it\u0027s minus twice root 1/2 and we\u0027ll give these points names."},{"Start":"13:03.090 ","End":"13:10.465","Text":"Also, this 1 will be C and this 1 will be D. Now we have the points,"},{"Start":"13:10.465 ","End":"13:17.870","Text":"but we wanted the distances and which is least and which is greatest."},{"Start":"13:18.660 ","End":"13:25.585","Text":"I\u0027ll remind you that the target function was f of xy and z,"},{"Start":"13:25.585 ","End":"13:31.525","Text":"and that was x squared plus y squared plus z squared,"},{"Start":"13:31.525 ","End":"13:33.550","Text":"but that wasn\u0027t the distance,"},{"Start":"13:33.550 ","End":"13:35.590","Text":"that was the square of the distance and at the end"},{"Start":"13:35.590 ","End":"13:37.750","Text":"we have to say that d is the square root of"},{"Start":"13:37.750 ","End":"13:43.795","Text":"f. Let\u0027s apply this to each of the 4 points."},{"Start":"13:43.795 ","End":"13:50.485","Text":"We\u0027ll need f of A, f of B."},{"Start":"13:50.485 ","End":"13:56.980","Text":"Over here, I\u0027ll do f of C and f of D,"},{"Start":"13:56.980 ","End":"13:58.900","Text":"and afterwards we\u0027ll take the square root."},{"Start":"13:58.900 ","End":"14:06.925","Text":"Now, f of A would be this thing squared is 1/2,"},{"Start":"14:06.925 ","End":"14:10.675","Text":"this thing squared is also 1/2,"},{"Start":"14:10.675 ","End":"14:14.335","Text":"and this thing squared is 0,"},{"Start":"14:14.335 ","End":"14:18.055","Text":"this equals 1 so"},{"Start":"14:18.055 ","End":"14:26.215","Text":"the distance from the origin of point a is the square root of 1, which is 1."},{"Start":"14:26.215 ","End":"14:28.720","Text":"As for B, well,"},{"Start":"14:28.720 ","End":"14:32.530","Text":"it\u0027s just the same thing because there are just minuses. It\u0027s the same thing."},{"Start":"14:32.530 ","End":"14:35.140","Text":"F of b is also 1,"},{"Start":"14:35.140 ","End":"14:40.165","Text":"and the distance of B from the origin is 1."},{"Start":"14:40.165 ","End":"14:46.030","Text":"Now for C, we get x squared plus y squared plus z squared from here."},{"Start":"14:46.030 ","End":"14:48.415","Text":"This will be 1/2 when it\u0027s squared,"},{"Start":"14:48.415 ","End":"14:50.755","Text":"this will be 1/2 when it\u0027s squared,"},{"Start":"14:50.755 ","End":"14:57.110","Text":"and this will be 4 times 1/2,"},{"Start":"14:57.810 ","End":"15:01.975","Text":"so, that\u0027s 2 plus 1/2 plus 1/2."},{"Start":"15:01.975 ","End":"15:06.055","Text":"That\u0027s equal to 3."},{"Start":"15:06.055 ","End":"15:12.085","Text":"The distance of the point C from the origin is the square root of 3."},{"Start":"15:12.085 ","End":"15:15.490","Text":"Now, same thing happens with D,"},{"Start":"15:15.490 ","End":"15:17.230","Text":"the minuses when squared,"},{"Start":"15:17.230 ","End":"15:19.825","Text":"this is also equal to 3,"},{"Start":"15:19.825 ","End":"15:26.425","Text":"and the distance of the point D from the origin is also root 3,"},{"Start":"15:26.425 ","End":"15:31.480","Text":"so, we have a tie in both cases for maximum and minimum."},{"Start":"15:31.480 ","End":"15:43.600","Text":"The minimum distance is 1,"},{"Start":"15:43.600 ","End":"15:46.090","Text":"and I got lazy and did a copy paste."},{"Start":"15:46.090 ","End":"15:48.160","Text":"If I erase that,"},{"Start":"15:48.160 ","End":"15:50.515","Text":"and I erase that,"},{"Start":"15:50.515 ","End":"15:54.025","Text":"the maximum distance is root 3."},{"Start":"15:54.025 ","End":"15:57.830","Text":"This answers the question. We\u0027re done."}],"ID":9682},{"Watched":false,"Name":"Exercise 16","Duration":"23m 15s","ChapterTopicVideoID":9785,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.090","Text":"In this exercise, we have to find the minimum and the maximum distances or"},{"Start":"00:06.090 ","End":"00:09.900","Text":"least and greatest distances from the origin"},{"Start":"00:09.900 ","End":"00:14.955","Text":"of the curve obtained by intersecting these 2 surfaces."},{"Start":"00:14.955 ","End":"00:18.585","Text":"One surface is an ellipsoid given by this equation,"},{"Start":"00:18.585 ","End":"00:21.914","Text":"and the other is the plane given by this equation."},{"Start":"00:21.914 ","End":"00:27.675","Text":"The origin, at least in 3D is of course 000."},{"Start":"00:27.675 ","End":"00:36.960","Text":"Now this is actually a problem of optimization or extramum under 2 constraints."},{"Start":"00:36.960 ","End":"00:44.245","Text":"Let me explain. In order to be on the curve of the intersection of 2 surfaces,"},{"Start":"00:44.245 ","End":"00:51.230","Text":"a typical point, call it xyz has to be on both of them."},{"Start":"00:51.230 ","End":"00:54.200","Text":"It has to be on the ellipsoid and on the plane,"},{"Start":"00:54.200 ","End":"00:56.510","Text":"and then it\u0027ll be on the intersection curve."},{"Start":"00:56.510 ","End":"00:58.540","Text":"These are 2 constraints."},{"Start":"00:58.540 ","End":"01:02.850","Text":"The target function, is the distance from the origin."},{"Start":"01:02.850 ","End":"01:13.310","Text":"The target function would be the distance of the point xyz from the origin."},{"Start":"01:13.310 ","End":"01:15.200","Text":"It\u0027s a function of 3 variables,"},{"Start":"01:15.200 ","End":"01:17.179","Text":"and using the distance formula,"},{"Start":"01:17.179 ","End":"01:22.860","Text":"it\u0027s the square root of x minus 0 squared plus"},{"Start":"01:22.860 ","End":"01:30.090","Text":"y minus 0 squared plus z minus 0 squared."},{"Start":"01:30.090 ","End":"01:38.480","Text":"If you want to phrase our question as optimization and the constraint, we want both."},{"Start":"01:38.480 ","End":"01:40.690","Text":"We want minimum and we want maximum,"},{"Start":"01:40.690 ","End":"01:43.460","Text":"so I\u0027ll write it like this of"},{"Start":"01:43.460 ","End":"01:51.680","Text":"the function square root of x squared plus y squared plus z squared."},{"Start":"01:51.680 ","End":"01:55.105","Text":"I eliminated these minus 0 here."},{"Start":"01:55.105 ","End":"02:00.290","Text":"Subject to the first constraint is this 1,"},{"Start":"02:00.290 ","End":"02:06.450","Text":"just copy it as is y squared of 5,"},{"Start":"02:06.450 ","End":"02:11.025","Text":"z squared over 25 equals 1."},{"Start":"02:11.025 ","End":"02:18.415","Text":"There\u0027s 2 constraints, and z equals x plus y."},{"Start":"02:18.415 ","End":"02:22.970","Text":"Now, let\u0027s define some functions."},{"Start":"02:22.970 ","End":"02:25.070","Text":"We need 3 functions,"},{"Start":"02:25.070 ","End":"02:28.310","Text":"we need a target and 2 constraints."},{"Start":"02:28.310 ","End":"02:30.650","Text":"But the target, we\u0027re going to use"},{"Start":"02:30.650 ","End":"02:34.375","Text":"our usual trick because we don\u0027t like to work with square roots."},{"Start":"02:34.375 ","End":"02:36.215","Text":"We\u0027ve done this before."},{"Start":"02:36.215 ","End":"02:40.980","Text":"We take instead the distance squared instead of the distance,"},{"Start":"02:40.980 ","End":"02:47.764","Text":"we\u0027ll take f of xyz to be the square of the distance,"},{"Start":"02:47.764 ","End":"02:50.770","Text":"which means without the square root,"},{"Start":"02:50.770 ","End":"02:55.185","Text":"just x squared plus y squared plus z squared,"},{"Start":"02:55.185 ","End":"02:59.565","Text":"and this is the distance squared."},{"Start":"02:59.565 ","End":"03:02.570","Text":"I\u0027m emphasizing this because we have to remember at"},{"Start":"03:02.570 ","End":"03:07.370","Text":"the end that if f is d squared at the end,"},{"Start":"03:07.370 ","End":"03:09.980","Text":"when we get the distances,"},{"Start":"03:09.980 ","End":"03:11.935","Text":"we have to get back to d,"},{"Start":"03:11.935 ","End":"03:17.630","Text":"which is the square root of f. The maximum and minimum occur at the same values of x,"},{"Start":"03:17.630 ","End":"03:19.940","Text":"y and z but the values themselves,"},{"Start":"03:19.940 ","End":"03:21.650","Text":"one is the square of the other."},{"Start":"03:21.650 ","End":"03:23.870","Text":"That\u0027s our target."},{"Start":"03:23.870 ","End":"03:26.955","Text":"Now we have 2 constraints."},{"Start":"03:26.955 ","End":"03:31.625","Text":"The constraints are just what are written here as equations except"},{"Start":"03:31.625 ","End":"03:36.885","Text":"that we take everything to 1 side and make it a function."},{"Start":"03:36.885 ","End":"03:42.990","Text":"G of xyz is this first constraint."},{"Start":"03:42.990 ","End":"03:46.610","Text":"But I\u0027ll bring everything, in other words,"},{"Start":"03:46.610 ","End":"03:53.700","Text":"I\u0027ll bring the 1 over to the other side just a moment, 25 minus 1."},{"Start":"03:53.700 ","End":"03:57.645","Text":"Then our constraint is that g equals 0."},{"Start":"03:57.645 ","End":"04:02.805","Text":"The other constraint function is from here."},{"Start":"04:02.805 ","End":"04:07.975","Text":"I\u0027ll take it as z minus x minus y."},{"Start":"04:07.975 ","End":"04:10.670","Text":"We can rephrase this."},{"Start":"04:10.670 ","End":"04:14.195","Text":"Once we\u0027ve replaced this g by f,"},{"Start":"04:14.195 ","End":"04:15.560","Text":"we can actually say,"},{"Start":"04:15.560 ","End":"04:26.015","Text":"we want the minimum and maximum of F of xy and z."},{"Start":"04:26.015 ","End":"04:32.825","Text":"Subject to G of xyz equals 0,"},{"Start":"04:32.825 ","End":"04:39.590","Text":"and h of xyz equals 0."},{"Start":"04:39.590 ","End":"04:43.790","Text":"I\u0027m just rephrasing this in terms of f, g, and h,"},{"Start":"04:43.790 ","End":"04:48.850","Text":"not something you would have to do either 1."},{"Start":"04:48.850 ","End":"04:53.330","Text":"But it\u0027s important that we\u0027ve defined these 3 functions because"},{"Start":"04:53.330 ","End":"04:57.800","Text":"now we\u0027re going to use the method of Lagrange multipliers."},{"Start":"04:57.800 ","End":"05:01.370","Text":"The method of Lagrange multipliers."},{"Start":"05:01.370 ","End":"05:04.925","Text":"It starts off with the following equation."},{"Start":"05:04.925 ","End":"05:11.210","Text":"That the gradient of f is lambda times the gradient of"},{"Start":"05:11.210 ","End":"05:19.040","Text":"g plus another letter Mu times the gradient of h. It introduces 2 new variables,"},{"Start":"05:19.040 ","End":"05:21.740","Text":"Lambda and Mu, together with x,"},{"Start":"05:21.740 ","End":"05:27.274","Text":"y and z will need 5 equations and 5 unknowns."},{"Start":"05:27.274 ","End":"05:30.245","Text":"This will give us 3 because it\u0027s a vector function in"},{"Start":"05:30.245 ","End":"05:34.175","Text":"3D and the 2 others will be the constraints."},{"Start":"05:34.175 ","End":"05:41.195","Text":"We\u0027re going to write down in general 5 equations."},{"Start":"05:41.195 ","End":"05:43.430","Text":"This 1, if I write it out,"},{"Start":"05:43.430 ","End":"05:47.030","Text":"says that f with respect to x is"},{"Start":"05:47.030 ","End":"05:55.020","Text":"Lambda g with respect to x plus Mu h with respect to x."},{"Start":"05:55.020 ","End":"05:56.705","Text":"Then the same thing,"},{"Start":"05:56.705 ","End":"06:04.504","Text":"I\u0027m just copying it but replacing x with y and again"},{"Start":"06:04.504 ","End":"06:14.400","Text":"replacing x with z. Lambda G with respect to z plus mu H with respect to z."},{"Start":"06:14.400 ","End":"06:23.480","Text":"Then we need the 2 constraints which that g equals 0 and h equals 0."},{"Start":"06:23.480 ","End":"06:26.140","Text":"Just what it says here in shorthand."},{"Start":"06:26.140 ","End":"06:30.409","Text":"I\u0027ll put a curly brace around"},{"Start":"06:30.409 ","End":"06:35.960","Text":"these functions and let\u0027s see what this translates to in our particular case."},{"Start":"06:35.960 ","End":"06:40.610","Text":"I\u0027d like to highlight these just to make it easier."},{"Start":"06:40.610 ","End":"06:48.360","Text":"This and this, target, constraint, constraint."},{"Start":"06:48.360 ","End":"06:55.190","Text":"F with respect to x would be 2x. You know what?"},{"Start":"06:55.190 ","End":"06:56.800","Text":"I prefer to work vertically,"},{"Start":"06:56.800 ","End":"07:00.600","Text":"f with respect to y is 2y,"},{"Start":"07:00.600 ","End":"07:04.875","Text":"f with respect to z is 2z."},{"Start":"07:04.875 ","End":"07:10.735","Text":"Then I\u0027ll do the derivatives of g I\u0027ll put a Lambda here, here and here."},{"Start":"07:10.735 ","End":"07:17.655","Text":"Now, G derivative with respect to x is 2x over 4,"},{"Start":"07:17.655 ","End":"07:20.445","Text":"which is x over 2,"},{"Start":"07:20.445 ","End":"07:26.370","Text":"and then 2y over 5,"},{"Start":"07:26.370 ","End":"07:33.690","Text":"and then 2z over 25 plus,"},{"Start":"07:33.690 ","End":"07:39.240","Text":"plus, plus Mu, Mu, Mu."},{"Start":"07:39.240 ","End":"07:44.530","Text":"Then I need the derivatives of H with respect to x."},{"Start":"07:44.530 ","End":"07:48.780","Text":"It\u0027s minus 1 with respect to y,"},{"Start":"07:48.780 ","End":"07:54.435","Text":"minus 1 with respect to z, it\u0027s just 1."},{"Start":"07:54.435 ","End":"07:57.275","Text":"Then we have the constraints,"},{"Start":"07:57.275 ","End":"08:06.295","Text":"which is that g equals 0 or I could write it in the original form."},{"Start":"08:06.295 ","End":"08:09.750","Text":"Think I\u0027ll copy paste from here."},{"Start":"08:09.750 ","End":"08:12.930","Text":"The other 1, H equals 0,"},{"Start":"08:12.930 ","End":"08:18.200","Text":"or I could take the original form and copy paste from here and here we are,"},{"Start":"08:18.200 ","End":"08:22.325","Text":"and I just forgot the curly braces."},{"Start":"08:22.325 ","End":"08:25.460","Text":"5 equations and 5 unknowns."},{"Start":"08:25.460 ","End":"08:29.705","Text":"I\u0027ll tell you what my strategy is after looking at it for a while."},{"Start":"08:29.705 ","End":"08:31.715","Text":"In the first equation,"},{"Start":"08:31.715 ","End":"08:35.060","Text":"I\u0027ll extract x in terms of lambda and mu."},{"Start":"08:35.060 ","End":"08:36.455","Text":"In the second equation,"},{"Start":"08:36.455 ","End":"08:40.095","Text":"I\u0027ll extract y, here I\u0027ll extract z."},{"Start":"08:40.095 ","End":"08:46.980","Text":"Then I\u0027ll plug in to these 2 equations and I\u0027ll have just 2 equations in Lambda and Mu."},{"Start":"08:46.980 ","End":"08:48.755","Text":"That\u0027s the strategy."},{"Start":"08:48.755 ","End":"08:50.015","Text":"Let\u0027s get started."},{"Start":"08:50.015 ","End":"08:57.100","Text":"The first 1. I\u0027ll bring this term over to the left and take x outside the brackets."},{"Start":"08:57.100 ","End":"09:02.140","Text":"I get that 2 minus"},{"Start":"09:02.140 ","End":"09:10.050","Text":"Lambda over 2x equals minus mu."},{"Start":"09:10.050 ","End":"09:18.500","Text":"Then I\u0027ll extract x by saying that x is this over this. you know what?"},{"Start":"09:18.500 ","End":"09:22.115","Text":"Instead of the minus, I can reverse the order of the subtraction."},{"Start":"09:22.115 ","End":"09:25.235","Text":"It\u0027s going to be Mu over the opposite order,"},{"Start":"09:25.235 ","End":"09:29.660","Text":"Lambda over 2 minus 2."},{"Start":"09:29.660 ","End":"09:33.710","Text":"A similar thing here."},{"Start":"09:33.710 ","End":"09:37.670","Text":"First of all, I bring this over to the other side and get"},{"Start":"09:37.670 ","End":"09:43.720","Text":"2 minus 2 Lambda"},{"Start":"09:43.720 ","End":"09:49.980","Text":"over 5 times y equals minus Mu."},{"Start":"09:49.980 ","End":"09:55.170","Text":"That will give me the y equals,"},{"Start":"09:55.170 ","End":"09:58.910","Text":"again I\u0027ll get rid of the minus on"},{"Start":"09:58.910 ","End":"10:03.410","Text":"the numerator and just reverse the order of the subtraction."},{"Start":"10:03.410 ","End":"10:10.390","Text":"2 Lambda over 5 minus 2."},{"Start":"10:10.390 ","End":"10:12.960","Text":"Now the third in this series,"},{"Start":"10:12.960 ","End":"10:17.080","Text":"I\u0027m bringing the z over and taking it out."},{"Start":"10:17.080 ","End":"10:24.945","Text":"I\u0027ve got 2 minus 2 over 25 lambda,"},{"Start":"10:24.945 ","End":"10:28.150","Text":"2 lambda over 25,"},{"Start":"10:28.150 ","End":"10:33.989","Text":"z equals Mu and"},{"Start":"10:33.989 ","End":"10:41.170","Text":"that gives me that z is equal to Mu,"},{"Start":"10:42.440 ","End":"10:54.250","Text":"same order 2 minus 2 Lambda over 25."},{"Start":"10:55.020 ","End":"10:57.620","Text":"I\u0027d like to emphasize these."},{"Start":"10:57.620 ","End":"11:00.460","Text":"I think I\u0027ll just box them."},{"Start":"11:01.400 ","End":"11:05.975","Text":"Now I\u0027ve got x, y, and z in terms of lambda and Mu."},{"Start":"11:05.975 ","End":"11:08.845","Text":"Now I\u0027ve got still 2 other equations."},{"Start":"11:08.845 ","End":"11:13.510","Text":"Please allow me to change my mind and I\u0027m going to put this in the form that"},{"Start":"11:13.510 ","End":"11:19.840","Text":"z minus x minus y equals 0 as it was from here."},{"Start":"11:19.840 ","End":"11:22.975","Text":"Slightly more convenient, now I\u0027m going to substitute"},{"Start":"11:22.975 ","End":"11:29.040","Text":"these 3 values into this last equation."},{"Start":"11:29.040 ","End":"11:31.020","Text":"Let\u0027s see what we get."},{"Start":"11:31.020 ","End":"11:36.720","Text":"z is Mu over 2"},{"Start":"11:36.720 ","End":"11:43.950","Text":"minus and I\u0027m going to work in decimals 225th is 0.08."},{"Start":"11:44.700 ","End":"11:51.775","Text":"Lambda minus x is Mu"},{"Start":"11:51.775 ","End":"12:00.880","Text":"over 0.5 lambda minus 2."},{"Start":"12:00.880 ","End":"12:03.970","Text":"Then we have minus,"},{"Start":"12:03.970 ","End":"12:10.440","Text":"then there\u0027s a y which is Mu over 2/5"},{"Start":"12:10.440 ","End":"12:17.900","Text":"is 0.4 lambda minus 2."},{"Start":"12:18.390 ","End":"12:21.565","Text":"This equals 0."},{"Start":"12:21.565 ","End":"12:28.570","Text":"What I\u0027d like to do next is take Mu outside the brackets and then I can divide by it."},{"Start":"12:28.570 ","End":"12:30.280","Text":"Before I divide by Mu,"},{"Start":"12:30.280 ","End":"12:34.315","Text":"let me note that Mu is not equal to 0."},{"Start":"12:34.315 ","End":"12:37.900","Text":"Because if Mu was 0, look if I plug it in here, here,"},{"Start":"12:37.900 ","End":"12:39.640","Text":"and here, I get that x, y,"},{"Start":"12:39.640 ","End":"12:41.665","Text":"and Z are all 0."},{"Start":"12:41.665 ","End":"12:46.330","Text":"If x, y, and Z are all 0 and I put them in here,"},{"Start":"12:46.330 ","End":"12:49.060","Text":"then I\u0027ll get that 0 equals 1."},{"Start":"12:49.060 ","End":"12:53.035","Text":"We\u0027re safe with Mu not equals 0 and now that I look at it."},{"Start":"12:53.035 ","End":"12:59.200","Text":"I would actually prefer to divide both sides by minus Mu."},{"Start":"12:59.200 ","End":"13:05.260","Text":"Then this 1 I can just reverse"},{"Start":"13:05.260 ","End":"13:07.840","Text":"the denominator order because I noticed here"},{"Start":"13:07.840 ","End":"13:10.795","Text":"it\u0027s the Lambda before the number and here not."},{"Start":"13:10.795 ","End":"13:12.610","Text":"If I divide by minus Mu,"},{"Start":"13:12.610 ","End":"13:17.365","Text":"I get 1 over and here I reverse the order,"},{"Start":"13:17.365 ","End":"13:21.625","Text":"0.08 lambda minus 2."},{"Start":"13:21.625 ","End":"13:29.500","Text":"These will all come out plus 1 over 0.5 lambda minus 2,"},{"Start":"13:29.500 ","End":"13:39.385","Text":"and this will be 1 over 0.4 lambda minus 2 equals 0."},{"Start":"13:39.385 ","End":"13:42.670","Text":"Next, and this is getting messy."},{"Start":"13:42.670 ","End":"13:44.515","Text":"You put a common denominator."},{"Start":"13:44.515 ","End":"13:47.785","Text":"We\u0027ll multiply everything by this times this, times this."},{"Start":"13:47.785 ","End":"13:51.100","Text":"Here I\u0027ll get these 2 factors remaining."},{"Start":"13:51.100 ","End":"13:55.625","Text":"I\u0027ll get 0.4 lambda minus 2,"},{"Start":"13:55.625 ","End":"13:59.160","Text":"0.5 Lambda minus 2."},{"Start":"13:59.160 ","End":"14:05.470","Text":"Then I\u0027ll get here the 2 missing factors are 0.08 lambda minus 2,"},{"Start":"14:05.470 ","End":"14:08.365","Text":"0.4 Lambda minus 2."},{"Start":"14:08.365 ","End":"14:10.780","Text":"From here I get this times this,"},{"Start":"14:10.780 ","End":"14:15.249","Text":"which is 0.08 lambda minus 2"},{"Start":"14:15.249 ","End":"14:21.610","Text":"and 0.5 lambda minus 2 equals 0."},{"Start":"14:21.610 ","End":"14:24.505","Text":"This is getting to be quite tedious,"},{"Start":"14:24.505 ","End":"14:27.025","Text":"I\u0027ll spare you all the details."},{"Start":"14:27.025 ","End":"14:28.540","Text":"Cut to the chase."},{"Start":"14:28.540 ","End":"14:32.305","Text":"We get the quadratic equation,"},{"Start":"14:32.305 ","End":"14:37.360","Text":"17 lambda squared minus"},{"Start":"14:37.360 ","End":"14:47.150","Text":"245 lambda plus 750 equals 0."},{"Start":"14:47.280 ","End":"14:50.440","Text":"I\u0027ll just give you straight away the solutions."},{"Start":"14:50.440 ","End":"14:51.790","Text":"We have 2 solutions,"},{"Start":"14:51.790 ","End":"14:57.099","Text":"we have lambda equals 10 and we have Lambda"},{"Start":"14:57.099 ","End":"15:04.670","Text":"equals 75 over 17."},{"Start":"15:05.490 ","End":"15:08.935","Text":"What I want to do is,"},{"Start":"15:08.935 ","End":"15:11.485","Text":"let me copy this,"},{"Start":"15:11.485 ","End":"15:17.665","Text":"it\u0027s going off screen I\u0027ll copy it down here and I\u0027ll scroll down a bit."},{"Start":"15:17.665 ","End":"15:25.210","Text":"But wait, I think I\u0027m going to need the ellipsoid later,"},{"Start":"15:25.210 ","End":"15:29.755","Text":"I\u0027ll stick it here meanwhile then later I\u0027ll move it again."},{"Start":"15:29.755 ","End":"15:33.970","Text":"Now what I\u0027m going to do is I\u0027m going to work in parallel"},{"Start":"15:33.970 ","End":"15:38.515","Text":"with the 2 different values of Lambda."},{"Start":"15:38.515 ","End":"15:43.510","Text":"I\u0027ll start with the lambda equals 10."},{"Start":"15:43.510 ","End":"15:46.510","Text":"I\u0027ll keep working in this color and later,"},{"Start":"15:46.510 ","End":"15:47.830","Text":"just to remind myself,"},{"Start":"15:47.830 ","End":"15:54.530","Text":"we\u0027ll do lambda equals 75 over 17 and I\u0027ll work in this color."},{"Start":"15:55.350 ","End":"15:58.330","Text":"If I plug lambda into these 3,"},{"Start":"15:58.330 ","End":"16:00.820","Text":"I can get what x, y, and z are."},{"Start":"16:00.820 ","End":"16:04.345","Text":"Let me just write x, y, z."},{"Start":"16:04.345 ","End":"16:06.790","Text":"If lambda is 10,10 over 2 is 5."},{"Start":"16:06.790 ","End":"16:11.470","Text":"5 minus 2 is 3 so it\u0027s 1/3 Mu."},{"Start":"16:11.470 ","End":"16:19.970","Text":"Let me just write down the others for you just to save time.1/2 Mu and 5/6 Mu."},{"Start":"16:20.520 ","End":"16:25.615","Text":"While I\u0027m at it, I\u0027ll just substitute the other value of lambda in here too."},{"Start":"16:25.615 ","End":"16:36.455","Text":"Won\u0027t bore you with the tedious calculations of x equals 34 over 7 times Mu,"},{"Start":"16:36.455 ","End":"16:45.355","Text":"and y comes out to be minus 17 over 4 Mu,"},{"Start":"16:45.355 ","End":"16:53.110","Text":"and z comes out to be 17 over 28 Mu."},{"Start":"16:53.110 ","End":"16:55.480","Text":"Now what are we going to do with these?"},{"Start":"16:55.480 ","End":"16:57.190","Text":"We\u0027re going to substitute x, y,"},{"Start":"16:57.190 ","End":"17:00.760","Text":"and z into the ellipsoid."},{"Start":"17:00.760 ","End":"17:04.345","Text":"Let me move it from here down to here."},{"Start":"17:04.345 ","End":"17:06.865","Text":"Why don\u0027t I put a box around it?"},{"Start":"17:06.865 ","End":"17:09.730","Text":"Now if we substitute these 3; x, y,"},{"Start":"17:09.730 ","End":"17:12.460","Text":"z into the ellipsoid, well,"},{"Start":"17:12.460 ","End":"17:15.445","Text":"everything\u0027s going to contain a Mu squared on the left."},{"Start":"17:15.445 ","End":"17:18.460","Text":"Let\u0027s just take Mu"},{"Start":"17:18.460 ","End":"17:23.035","Text":"squared straightaway outside the brackets and see what we\u0027re left with."},{"Start":"17:23.035 ","End":"17:26.785","Text":"X squared, without the Mu squared is"},{"Start":"17:26.785 ","End":"17:33.760","Text":"1/9 and it\u0027s going to be over 4 plus this squared,"},{"Start":"17:33.760 ","End":"17:39.025","Text":"1/2 squared is a quarter, over 5."},{"Start":"17:39.025 ","End":"17:46.180","Text":"Here, z squared will be 25 over 36"},{"Start":"17:46.180 ","End":"17:59.215","Text":"over 25 close brackets equals 1."},{"Start":"17:59.215 ","End":"18:07.190","Text":"If you calculate, it comes out to be 19 over a 180."},{"Start":"18:07.560 ","End":"18:12.190","Text":"Mu squared equals, bring it over to the other side,"},{"Start":"18:12.190 ","End":"18:18.490","Text":"180 over 19 which means that Mu"},{"Start":"18:18.490 ","End":"18:26.455","Text":"is plus or minus the square root of 180 over 19."},{"Start":"18:26.455 ","End":"18:29.425","Text":"I\u0027m going to simplify this slightly."},{"Start":"18:29.425 ","End":"18:33.535","Text":"Because 180 is 36 times 5."},{"Start":"18:33.535 ","End":"18:37.270","Text":"I\u0027m going to take the square root of 36 times square root of 5."},{"Start":"18:37.270 ","End":"18:44.320","Text":"In short, the 6 comes out and it\u0027s just root of 5 over 19."},{"Start":"18:44.320 ","End":"18:47.679","Text":"What I can now do is now that I have Mu,"},{"Start":"18:47.679 ","End":"18:55.825","Text":"I can substitute Mu into these 3 equations for x, y, and z."},{"Start":"18:55.825 ","End":"18:58.404","Text":"Let\u0027s take the plus first."},{"Start":"18:58.404 ","End":"19:06.205","Text":"What I\u0027ll get for x y z will be 1/3 of 6 is 2,"},{"Start":"19:06.205 ","End":"19:11.590","Text":"and then it\u0027s root 5 over 19."},{"Start":"19:11.590 ","End":"19:16.870","Text":"Then 1/2 of it will make it 3 root 5 over 19,"},{"Start":"19:16.870 ","End":"19:22.765","Text":"and the 5/6 will make it 5 root 5 over 19."},{"Start":"19:22.765 ","End":"19:26.155","Text":"I\u0027ll give the point and name A."},{"Start":"19:26.155 ","End":"19:30.280","Text":"Similarly, we get a point B,"},{"Start":"19:30.280 ","End":"19:32.470","Text":"which will be same as this,"},{"Start":"19:32.470 ","End":"19:34.375","Text":"but with the minuses."},{"Start":"19:34.375 ","End":"19:39.835","Text":"It\u0027ll be minus 2 root 5 over 19,"},{"Start":"19:39.835 ","End":"19:45.920","Text":"minus 3 root 5 over 19,"},{"Start":"19:47.610 ","End":"19:53.440","Text":"minus 5 root 5 over 19."},{"Start":"19:53.440 ","End":"20:00.055","Text":"This is just the 2 points from this case where lambda equals 10."},{"Start":"20:00.055 ","End":"20:03.129","Text":"Now let\u0027s go with the other case."},{"Start":"20:03.129 ","End":"20:06.265","Text":"I copied part of this over here."},{"Start":"20:06.265 ","End":"20:08.050","Text":"The part that\u0027s going to be the same."},{"Start":"20:08.050 ","End":"20:10.420","Text":"We need x squared, y squared,"},{"Start":"20:10.420 ","End":"20:14.185","Text":"and z squared, and it\u0027s going to be messy,"},{"Start":"20:14.185 ","End":"20:22.240","Text":"but it\u0027s 34 over 7 squared and here we\u0027re going to get 17 over 4 squared."},{"Start":"20:22.240 ","End":"20:27.580","Text":"We don\u0027t need the minus and 17 over 28 squared."},{"Start":"20:27.580 ","End":"20:29.350","Text":"I\u0027m just going to give you the answer."},{"Start":"20:29.350 ","End":"20:33.610","Text":"Otherwise, we\u0027re just wasting time with arithmetic and boring numbers."},{"Start":"20:33.610 ","End":"20:35.800","Text":"It comes out something quite awful."},{"Start":"20:35.800 ","End":"20:38.320","Text":"Mu comes out to be plus or"},{"Start":"20:38.320 ","End":"20:48.535","Text":"minus 140 over 17 root 646."},{"Start":"20:48.535 ","End":"20:50.635","Text":"You see what thing we\u0027re dealing with."},{"Start":"20:50.635 ","End":"20:58.930","Text":"Anyway, this time we substitute the Mu in these 3 and we get 2 pairs of x,"},{"Start":"20:58.930 ","End":"21:00.100","Text":"y, z, 1 with the plus,"},{"Start":"21:00.100 ","End":"21:07.490","Text":"and 1 with the minus and we\u0027ll get the point C with the plus comes out."},{"Start":"21:07.740 ","End":"21:12.220","Text":"40 over root 6,"},{"Start":"21:12.220 ","End":"21:20.320","Text":"46 minus 35 over root 646"},{"Start":"21:20.320 ","End":"21:29.470","Text":"and 5 over root 646 and D is just the same thing,"},{"Start":"21:29.470 ","End":"21:31.810","Text":"but with the opposite signs."},{"Start":"21:31.810 ","End":"21:34.480","Text":"We just fit it in."},{"Start":"21:34.480 ","End":"21:37.675","Text":"Yes, I know these are horrible numbers."},{"Start":"21:37.675 ","End":"21:39.745","Text":"I\u0027m sorry about that."},{"Start":"21:39.745 ","End":"21:41.560","Text":"Now that we\u0027ve got the 4 points,"},{"Start":"21:41.560 ","End":"21:44.920","Text":"what we have to do is substitute them in the target function."},{"Start":"21:44.920 ","End":"21:46.450","Text":"We had this function f,"},{"Start":"21:46.450 ","End":"21:48.355","Text":"which was the distance squared."},{"Start":"21:48.355 ","End":"21:50.995","Text":"We had f of x, y,"},{"Start":"21:50.995 ","End":"21:56.950","Text":"z was equal to x squared plus y squared plus z squared."},{"Start":"21:56.950 ","End":"22:00.400","Text":"As you recall, it wasn\u0027t the distance it was the square of the distance."},{"Start":"22:00.400 ","End":"22:06.640","Text":"If we substitute A and if we substitute B we\u0027ll get"},{"Start":"22:06.640 ","End":"22:09.520","Text":"the same thing because these are just negatives of each"},{"Start":"22:09.520 ","End":"22:13.860","Text":"other and it turns out that this is 10."},{"Start":"22:13.860 ","End":"22:15.360","Text":"If you do this squared, plus this squared,"},{"Start":"22:15.360 ","End":"22:17.460","Text":"plus this squared, you\u0027ll get 10."},{"Start":"22:17.460 ","End":"22:20.640","Text":"This is just minus of this it\u0027s the same thing."},{"Start":"22:20.640 ","End":"22:26.359","Text":"If you do it for C and for D,"},{"Start":"22:26.359 ","End":"22:33.925","Text":"you will get 75 over 17."},{"Start":"22:33.925 ","End":"22:38.050","Text":"Now clearly 10 is bigger than 75 over 17."},{"Start":"22:38.050 ","End":"22:40.880","Text":"We\u0027re going to get a max here."},{"Start":"22:41.160 ","End":"22:45.910","Text":"The maximum distance is not 10 though it\u0027s"},{"Start":"22:45.910 ","End":"22:50.800","Text":"the square root of 10 because f was little d squared,"},{"Start":"22:50.800 ","End":"22:56.005","Text":"remember, or D was square root of f and little d with the distance."},{"Start":"22:56.005 ","End":"22:59.020","Text":"The maximum distance is that and"},{"Start":"22:59.020 ","End":"23:08.185","Text":"the minimum distance is root 75 over 17."},{"Start":"23:08.185 ","End":"23:10.840","Text":"I\u0027ll just leave it briefly like that."},{"Start":"23:10.840 ","End":"23:15.020","Text":"Finally got rid of this exercise"}],"ID":9683},{"Watched":false,"Name":"Exercise 17","Duration":"16m 12s","ChapterTopicVideoID":9886,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this exercise, we have a word problem. Let\u0027s read it."},{"Start":"00:06.030 ","End":"00:11.310","Text":"Joe goes to the market and he buys a certain number of kilograms of cucumber,"},{"Start":"00:11.310 ","End":"00:15.465","Text":"and a certain number of kilograms of tomatoes."},{"Start":"00:15.465 ","End":"00:19.875","Text":"There\u0027s utility function, this is the concept in economics."},{"Start":"00:19.875 ","End":"00:24.585","Text":"It\u0027s how much he enjoys the product,"},{"Start":"00:24.585 ","End":"00:28.050","Text":"if he buys x kilo of cucumbers,"},{"Start":"00:28.050 ","End":"00:33.810","Text":"and y kilo of tomatoes then the utility is given by this formula."},{"Start":"00:33.810 ","End":"00:44.810","Text":"Next, we\u0027re given the cost of a kilo of tomatoes and cost of a kilo of cucumbers."},{"Start":"00:44.810 ","End":"00:47.940","Text":"Next, we have a goal."},{"Start":"00:48.010 ","End":"00:52.640","Text":"This actually will turn out to be our constraint."},{"Start":"00:52.640 ","End":"00:58.710","Text":"Anyway, its goal is to achieve a certain utility level of natural log 32,"},{"Start":"00:58.710 ","End":"01:02.480","Text":"and he wants to reach it with the least possible cost."},{"Start":"01:02.480 ","End":"01:05.825","Text":"We have to formulate this mathematically,"},{"Start":"01:05.825 ","End":"01:10.895","Text":"as 1 of those optimization problems under constraint,"},{"Start":"01:10.895 ","End":"01:14.495","Text":"optimization means the extremum, maximum or minimum."},{"Start":"01:14.495 ","End":"01:17.940","Text":"In general, we have 2 functions that we need to consider."},{"Start":"01:17.940 ","End":"01:19.510","Text":"1 is the target function,"},{"Start":"01:19.510 ","End":"01:20.960","Text":"1 is the constraint."},{"Start":"01:20.960 ","End":"01:24.830","Text":"Let\u0027s start then with the constraint."},{"Start":"01:24.830 ","End":"01:31.220","Text":"The constraint is that the utility level should be natural log of 32,"},{"Start":"01:31.220 ","End":"01:37.580","Text":"in other words, u of x and y equals natural log of 32."},{"Start":"01:37.580 ","End":"01:42.560","Text":"If I write down explicitly what u of x and y is,"},{"Start":"01:42.560 ","End":"01:46.520","Text":"then that gives me the natural log of x,"},{"Start":"01:46.520 ","End":"01:52.990","Text":"plus natural log of y equals natural log of 32."},{"Start":"01:52.990 ","End":"01:56.430","Text":"Next the constraint, I want the target,"},{"Start":"01:56.430 ","End":"01:59.935","Text":"and the target is the cost."},{"Start":"01:59.935 ","End":"02:04.100","Text":"I don\u0027t have to necessarily give it a name,"},{"Start":"02:04.100 ","End":"02:10.450","Text":"I could call it f, but I\u0027ll just call it cost in terms of x and y."},{"Start":"02:10.450 ","End":"02:16.610","Text":"What it is equal to is the total cost would be the number"},{"Start":"02:16.610 ","End":"02:26.140","Text":"of x kilograms at 1 dollar per kilo is just x dollar, x times 1."},{"Start":"02:26.140 ","End":"02:29.690","Text":"You can write x times 1 because that\u0027s the 1 from here,"},{"Start":"02:29.690 ","End":"02:33.530","Text":"but he buys y kilos of tomatoes,"},{"Start":"02:33.530 ","End":"02:35.770","Text":"and each of these is 2 dollars,"},{"Start":"02:35.770 ","End":"02:38.780","Text":"so it\u0027s y times 2."},{"Start":"02:38.950 ","End":"02:43.600","Text":"This is what we want to minimize,"},{"Start":"02:43.600 ","End":"02:47.260","Text":"least possible cost, that means it\u0027s a minimum."},{"Start":"02:47.260 ","End":"02:52.845","Text":"I can phrase this as a problem of extremum with constraints,"},{"Start":"02:52.845 ","End":"02:55.425","Text":"I want the minimum,"},{"Start":"02:55.425 ","End":"02:59.110","Text":"and we put the curly braces of the target function,"},{"Start":"02:59.110 ","End":"03:02.720","Text":"which I\u0027ll write as x plus 2y."},{"Start":"03:03.770 ","End":"03:08.425","Text":"Then subject to the constraint,"},{"Start":"03:08.425 ","End":"03:12.370","Text":"which is that natural log of x,"},{"Start":"03:12.370 ","End":"03:18.395","Text":"plus natural log of y equals natural log of 32."},{"Start":"03:18.395 ","End":"03:24.864","Text":"That\u0027s just 1 other thing I have to include in this problem."},{"Start":"03:24.864 ","End":"03:27.640","Text":"This is a real-life problem,"},{"Start":"03:27.640 ","End":"03:35.305","Text":"and the number of kilograms of either vegetable is going to be positive,"},{"Start":"03:35.305 ","End":"03:41.740","Text":"so I would also like to add that x and y need to be positive."},{"Start":"03:41.740 ","End":"03:47.950","Text":"That also makes sense because the natural log is only defined on positive numbers,"},{"Start":"03:47.950 ","End":"03:51.490","Text":"there\u0027s an extra limitation on the domain."},{"Start":"03:51.490 ","End":"03:57.340","Text":"There is a simplification that I\u0027d like to do on the constraint."},{"Start":"03:57.340 ","End":"04:00.900","Text":"Using the properties of the natural logarithm,"},{"Start":"04:00.900 ","End":"04:03.020","Text":"I have a sum of natural log,"},{"Start":"04:03.020 ","End":"04:06.215","Text":"I can take it as natural log of the product,"},{"Start":"04:06.215 ","End":"04:12.325","Text":"so this says natural log of x times y is natural log of 32,"},{"Start":"04:12.325 ","End":"04:16.100","Text":"and if the natural log of 2 quantities are equal,"},{"Start":"04:16.100 ","End":"04:21.355","Text":"then they are equal xy equals 32."},{"Start":"04:21.355 ","End":"04:25.290","Text":"Let me just rephrase this problem."},{"Start":"04:25.290 ","End":"04:32.290","Text":"We have again minimum of x plus 2y,"},{"Start":"04:32.290 ","End":"04:39.830","Text":"subject to the condition xy equals 32,"},{"Start":"04:39.830 ","End":"04:46.675","Text":"and a certain restriction on the domain that x and y have to both be positive."},{"Start":"04:46.675 ","End":"04:51.335","Text":"Now I\u0027m going to use the method of Lagrange multipliers,"},{"Start":"04:51.335 ","End":"04:55.165","Text":"the target function which is this,"},{"Start":"04:55.165 ","End":"05:01.500","Text":"we usually often call f. So I have f of xy,"},{"Start":"05:01.500 ","End":"05:05.950","Text":"is equal to x plus 2y,"},{"Start":"05:06.460 ","End":"05:13.745","Text":"and the constraint function is usually written as something equals 0."},{"Start":"05:13.745 ","End":"05:20.310","Text":"In other words, I take g of xy to be xy minus 32,"},{"Start":"05:20.310 ","End":"05:26.210","Text":"and then this constraint effectively says that g equals 0,"},{"Start":"05:26.210 ","End":"05:29.820","Text":"xy minus 32 equals 0."},{"Start":"05:30.640 ","End":"05:32.960","Text":"To proceed, we\u0027re going to need"},{"Start":"05:32.960 ","End":"05:38.245","Text":"the partial derivatives of f and of g up to the second-order,"},{"Start":"05:38.245 ","End":"05:41.685","Text":"and so let\u0027s do that technical part first."},{"Start":"05:41.685 ","End":"05:45.480","Text":"F with respect to x is just 1,"},{"Start":"05:45.480 ","End":"05:49.005","Text":"f with respect to y is 2,"},{"Start":"05:49.005 ","End":"05:53.869","Text":"and then the second-order partial derivatives will all be 0,"},{"Start":"05:53.869 ","End":"05:58.460","Text":"fx_x is 0, fx_y is 0,"},{"Start":"05:58.460 ","End":"06:02.270","Text":"and fy_y is 0."},{"Start":"06:02.270 ","End":"06:11.170","Text":"As for g, g with respect to x is equal to y,"},{"Start":"06:11.170 ","End":"06:17.090","Text":"g with respect to y is equal to x."},{"Start":"06:17.090 ","End":"06:24.835","Text":"Then gx_x is equal to,"},{"Start":"06:24.835 ","End":"06:27.375","Text":"this would be 0,"},{"Start":"06:27.375 ","End":"06:30.090","Text":"because y is a constant,"},{"Start":"06:30.090 ","End":"06:36.800","Text":"gx_y is equal to this with respect to y is 1,"},{"Start":"06:36.800 ","End":"06:39.990","Text":"and also gy_x should be the same."},{"Start":"06:39.990 ","End":"06:42.615","Text":"Let\u0027s just check gy_x here also 1,"},{"Start":"06:42.615 ","End":"06:51.245","Text":"and gy_y is equal to this with respect to y is 0."},{"Start":"06:51.245 ","End":"06:57.110","Text":"So there\u0027s not very much that\u0027s not 0 as far as the second-order,"},{"Start":"06:57.110 ","End":"07:00.360","Text":"this is the only non-zero 1."},{"Start":"07:00.760 ","End":"07:03.875","Text":"Just a quick summary before the next step,"},{"Start":"07:03.875 ","End":"07:10.325","Text":"we\u0027ve phrased the problem, formulated it mathematically."},{"Start":"07:10.325 ","End":"07:13.575","Text":"That would be this part here,"},{"Start":"07:13.575 ","End":"07:15.700","Text":"and I\u0027ll highlight it,"},{"Start":"07:15.700 ","End":"07:17.500","Text":"that\u0027s the mathematical formulation."},{"Start":"07:17.500 ","End":"07:24.100","Text":"We\u0027ve identified a target function and the constraint function."},{"Start":"07:24.100 ","End":"07:31.150","Text":"We\u0027ve done the technical work of the partial derivatives up to the 2nd order,"},{"Start":"07:31.150 ","End":"07:34.615","Text":"and now we\u0027re going to use the Lagrange multiplier method,"},{"Start":"07:34.615 ","End":"07:38.165","Text":"to write 3 equations in 3 unknowns."},{"Start":"07:38.165 ","End":"07:40.950","Text":"In general, I\u0027ll write them at the side,"},{"Start":"07:40.950 ","End":"07:43.990","Text":"the 3 equations that we generally write,"},{"Start":"07:43.990 ","End":"07:45.760","Text":"this is for critical points,"},{"Start":"07:45.760 ","End":"07:51.725","Text":"is we write f with respect to x is Lambda g with respect to x,"},{"Start":"07:51.725 ","End":"07:55.060","Text":"same thing with respect to y."},{"Start":"07:55.640 ","End":"08:02.540","Text":"The 3rd equation is the constraint equals 0."},{"Start":"08:02.540 ","End":"08:06.289","Text":"We can write that g equals 0."},{"Start":"08:06.289 ","End":"08:08.000","Text":"Now in our case,"},{"Start":"08:08.000 ","End":"08:11.645","Text":"the equations come out to be as follows."},{"Start":"08:11.645 ","End":"08:19.980","Text":"The first 1 fx is 1 equals Lambda times y,"},{"Start":"08:19.980 ","End":"08:26.805","Text":"the 2nd 1 is we have 2 equals lambda times x,"},{"Start":"08:26.805 ","End":"08:32.134","Text":"and the last 1 I either write xy minus 32 is 0,"},{"Start":"08:32.134 ","End":"08:35.990","Text":"or I can write it as xy equals 32,"},{"Start":"08:35.990 ","End":"08:38.400","Text":"it doesn\u0027t really matter."},{"Start":"08:38.950 ","End":"08:41.750","Text":"The technique is always the same,"},{"Start":"08:41.750 ","End":"08:45.740","Text":"we divide 1 of the first 2 equations by the other."},{"Start":"08:45.740 ","End":"08:48.685","Text":"Let\u0027s say we\u0027ll do the first divided by the second,"},{"Start":"08:48.685 ","End":"08:53.960","Text":"so we get from these 2 that 1 over"},{"Start":"08:53.960 ","End":"09:00.140","Text":"2 equals Lambda y over Lambda x."},{"Start":"09:00.140 ","End":"09:07.490","Text":"But we have to be careful not to divide by 0. Now,"},{"Start":"09:07.490 ","End":"09:10.345","Text":"lambda x cannot be 0,"},{"Start":"09:10.345 ","End":"09:12.595","Text":"I mean neither lambda nor x,"},{"Start":"09:12.595 ","End":"09:15.160","Text":"because lambda x is equal to 2,"},{"Start":"09:15.160 ","End":"09:17.035","Text":"so it wouldn\u0027t work."},{"Start":"09:17.035 ","End":"09:20.065","Text":"We\u0027re not dividing by 0."},{"Start":"09:20.065 ","End":"09:25.555","Text":"We can cancel the lambda and then cross multiply,"},{"Start":"09:25.555 ","End":"09:32.470","Text":"and then we get that x equals 2y."},{"Start":"09:32.470 ","End":"09:38.290","Text":"Now I have 2 equations and 2 unknowns, x and y."},{"Start":"09:38.290 ","End":"09:41.560","Text":"That would be this 1 and this 1."},{"Start":"09:41.560 ","End":"09:44.575","Text":"That\u0027s fairly straightforward to solve."},{"Start":"09:44.575 ","End":"09:55.810","Text":"What I\u0027ll do is I\u0027ll just put x into this equation and get that xy is 2yy."},{"Start":"09:55.810 ","End":"09:58.255","Text":"I\u0027ll continue over here."},{"Start":"09:58.255 ","End":"10:04.390","Text":"2yy equals the 32,"},{"Start":"10:04.390 ","End":"10:11.125","Text":"dividing by 2, I get that y squared equals 16."},{"Start":"10:11.125 ","End":"10:14.515","Text":"Because y is positive,"},{"Start":"10:14.515 ","End":"10:18.430","Text":"so I get that y equals 4,"},{"Start":"10:18.430 ","End":"10:20.290","Text":"can\u0027t have minus 4,"},{"Start":"10:20.290 ","End":"10:27.380","Text":"can\u0027t buy minus 4 kilo of whatever it was, tomatoes or cucumbers."},{"Start":"10:29.250 ","End":"10:37.150","Text":"Now I can find x because I can put that into this equation here,"},{"Start":"10:37.150 ","End":"10:40.210","Text":"x is 2y so that gives us"},{"Start":"10:40.210 ","End":"10:48.040","Text":"that x equals 8."},{"Start":"10:48.040 ","End":"10:51.880","Text":"Then we also need lambda for later on,"},{"Start":"10:51.880 ","End":"10:58.000","Text":"so I\u0027ve got that lambda y equals 1 for example,"},{"Start":"10:58.000 ","End":"11:05.170","Text":"so lambda times y is 4 equals 1."},{"Start":"11:05.170 ","End":"11:07.045","Text":"That\u0027s when I put y in here."},{"Start":"11:07.045 ","End":"11:15.220","Text":"So I\u0027ve got that lambda is equal to 1/4."},{"Start":"11:15.220 ","End":"11:18.340","Text":"The xy lambda for the critical point often"},{"Start":"11:18.340 ","End":"11:21.670","Text":"is indicated with an asterisk so this is x asterisk,"},{"Start":"11:21.670 ","End":"11:25.225","Text":"y asterisk, and lambda asterisk."},{"Start":"11:25.225 ","End":"11:28.850","Text":"It wouldn\u0027t hurt to highlight them all."},{"Start":"11:29.310 ","End":"11:32.560","Text":"The next thing we have to do is decide whether"},{"Start":"11:32.560 ","End":"11:35.485","Text":"this critical point is a maximum or minimum."},{"Start":"11:35.485 ","End":"11:39.979","Text":"There is this horrible expression for H,"},{"Start":"11:40.470 ","End":"11:46.480","Text":"it\u0027s equal to f_xx minus"},{"Start":"11:46.480 ","End":"11:54.610","Text":"lambda g_xx times g_y squared."},{"Start":"11:54.610 ","End":"12:00.415","Text":"Then plus the same thing with x and y reversed"},{"Start":"12:00.415 ","End":"12:09.025","Text":"f_yy minus lambda g_yy times g_x squared."},{"Start":"12:09.025 ","End":"12:12.595","Text":"Sorry about this, it\u0027s just 1 of those things,"},{"Start":"12:12.595 ","End":"12:17.155","Text":"minus twice, and then we take a hybrid of these 2."},{"Start":"12:17.155 ","End":"12:23.530","Text":"Here we have a mixture f_xy minus lambda."},{"Start":"12:23.530 ","End":"12:27.170","Text":"Then from mixture of these 2, g_xy."},{"Start":"12:27.780 ","End":"12:30.790","Text":"Instead of g_y squared or g_x squared,"},{"Start":"12:30.790 ","End":"12:33.080","Text":"we have 1 of each g_x, g_y."},{"Start":"12:34.590 ","End":"12:39.040","Text":"The idea is that when we evaluate it at"},{"Start":"12:39.040 ","End":"12:42.850","Text":"this particular point with this particular lambda,"},{"Start":"12:42.850 ","End":"12:45.910","Text":"if it\u0027s bigger than 0,"},{"Start":"12:45.910 ","End":"12:48.595","Text":"this H known as the Hessian,"},{"Start":"12:48.595 ","End":"12:50.740","Text":"if it\u0027s bigger than 0,"},{"Start":"12:50.740 ","End":"12:58.630","Text":"then it\u0027s a minimum and if it comes out less than 0, it\u0027s a maximum."},{"Start":"12:58.630 ","End":"13:01.580","Text":"Let\u0027s see what we have."},{"Start":"13:02.280 ","End":"13:09.775","Text":"Now, we were looking for h at our particular point,"},{"Start":"13:09.775 ","End":"13:17.930","Text":"which is when x is 8,"},{"Start":"13:19.650 ","End":"13:25.870","Text":"y is 4, and lambda is 1/4,"},{"Start":"13:25.870 ","End":"13:31.790","Text":"I\u0027ll just write that again, x, y lambda."},{"Start":"13:31.800 ","End":"13:34.420","Text":"Now a lot of these are already 0."},{"Start":"13:34.420 ","End":"13:36.055","Text":"If I look over here,"},{"Start":"13:36.055 ","End":"13:39.320","Text":"then I see that all the f_xx,"},{"Start":"13:39.320 ","End":"13:43.230","Text":"all the second-order derivatives of f are 0,"},{"Start":"13:43.230 ","End":"13:46.515","Text":"this is 0, this is 0, this is 0."},{"Start":"13:46.515 ","End":"13:49.740","Text":"Also g_xx and g_yy is 0,"},{"Start":"13:49.740 ","End":"13:53.380","Text":"so this is 0, this is 0."},{"Start":"13:53.380 ","End":"13:57.815","Text":"This 1 is not 0, this is 1."},{"Start":"13:57.815 ","End":"14:02.100","Text":"The first 2 terms here will be 0."},{"Start":"14:02.100 ","End":"14:05.040","Text":"I need to know what g_x and g_y are."},{"Start":"14:05.040 ","End":"14:10.330","Text":"Now in general, g_x is y and g_y is x."},{"Start":"14:10.330 ","End":"14:19.780","Text":"But I\u0027m substituting when x is 8 and y is 4 so g_x here is y,"},{"Start":"14:19.780 ","End":"14:27.250","Text":"which is 4 and g_y is equal to x,"},{"Start":"14:27.250 ","End":"14:31.045","Text":"which in our case is 8,"},{"Start":"14:31.045 ","End":"14:37.260","Text":"and the lambda is 1/4."},{"Start":"14:37.260 ","End":"14:43.405","Text":"What I end up getting there\u0027s a minus and a minus is a plus,"},{"Start":"14:43.405 ","End":"14:45.520","Text":"so I get a plus."},{"Start":"14:45.520 ","End":"14:55.340","Text":"Then I\u0027ve got 1/4 times 1 times 4 times 8."},{"Start":"14:55.770 ","End":"14:58.870","Text":"I don\u0027t really need the actual answer though it\u0027s"},{"Start":"14:58.870 ","End":"15:01.300","Text":"easy enough to compute that this is equal to 8."},{"Start":"15:01.300 ","End":"15:05.289","Text":"The point is, I can see it\u0027s positive, it\u0027s bigger than 0."},{"Start":"15:05.289 ","End":"15:11.479","Text":"Therefore, I do have a minimum point."},{"Start":"15:11.670 ","End":"15:16.915","Text":"What I\u0027d actually like to do is write it in human-friendly terms that the minimum,"},{"Start":"15:16.915 ","End":"15:20.800","Text":"the x, which is 8,"},{"Start":"15:20.800 ","End":"15:25.764","Text":"so it\u0027s 8 kilograms of"},{"Start":"15:25.764 ","End":"15:34.795","Text":"cucumbers and y is 4,"},{"Start":"15:34.795 ","End":"15:42.140","Text":"so it\u0027s 4 kilograms of tomatoes."},{"Start":"15:42.750 ","End":"15:47.380","Text":"We could end there but I would also like to know what is the cost,"},{"Start":"15:47.380 ","End":"15:49.000","Text":"what was the minimum cost?"},{"Start":"15:49.000 ","End":"15:54.610","Text":"We substitute in this function, x plus 2y."},{"Start":"15:54.610 ","End":"15:58.990","Text":"I could say the cost x plus"},{"Start":"15:58.990 ","End":"16:06.460","Text":"2y is 8 plus twice 4 and it\u0027s in dollars, is $16."},{"Start":"16:06.460 ","End":"16:09.265","Text":"That\u0027s the least cost."},{"Start":"16:09.265 ","End":"16:12.620","Text":"Okay, so that\u0027s it."}],"ID":9783},{"Watched":false,"Name":"Exercise 18","Duration":"4m 59s","ChapterTopicVideoID":9887,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.765","Text":"This is a question from economics,"},{"Start":"00:03.765 ","End":"00:06.615","Text":"where there\u0027s something called a transformation curve,"},{"Start":"00:06.615 ","End":"00:10.035","Text":"also known as a production possibility curve,"},{"Start":"00:10.035 ","End":"00:17.445","Text":"and it\u0027s an equation between 2 quantities, in this case,"},{"Start":"00:17.445 ","End":"00:19.890","Text":"x mangoes and y pineapples,"},{"Start":"00:19.890 ","End":"00:25.445","Text":"and it\u0027s given by x squared plus y squared is equal to 13."},{"Start":"00:25.445 ","End":"00:27.819","Text":"There\u0027s a utility function,"},{"Start":"00:27.819 ","End":"00:32.940","Text":"which means how much we enjoy the product."},{"Start":"00:34.900 ","End":"00:40.940","Text":"Jane is looking for a basket of x mangoes and y pineapples"},{"Start":"00:40.940 ","End":"00:45.100","Text":"that satisfies this equation and its on the transformation curve,"},{"Start":"00:45.100 ","End":"00:50.355","Text":"in such a way that she gets a maximum utility from the fruit."},{"Start":"00:50.355 ","End":"00:53.465","Text":"We have to just put this in mathematical terms,"},{"Start":"00:53.465 ","End":"00:56.400","Text":"this problem, and to solve it."},{"Start":"00:56.530 ","End":"00:59.840","Text":"We\u0027re going to do it as an optimization,"},{"Start":"00:59.840 ","End":"01:07.265","Text":"meaning maximum or minimum of some target function under a constraint,"},{"Start":"01:07.265 ","End":"01:10.620","Text":"and let\u0027s do the constraint first."},{"Start":"01:10.690 ","End":"01:17.435","Text":"The constraint is just the transformation curve."},{"Start":"01:17.435 ","End":"01:24.860","Text":"It gives a relation between the number of pineapples and the number of mangoes."},{"Start":"01:24.860 ","End":"01:31.115","Text":"That is just x squared plus y squared equals 13."},{"Start":"01:31.115 ","End":"01:41.090","Text":"The target function and the constraint function is just the utility,"},{"Start":"01:41.090 ","End":"01:47.160","Text":"which is 4x plus 6y."},{"Start":"01:48.560 ","End":"01:51.560","Text":"The question states maximize."},{"Start":"01:51.560 ","End":"01:55.220","Text":"Usually utility is something that you want to maximize."},{"Start":"01:55.220 ","End":"01:59.630","Text":"We write, it\u0027s always max or min,"},{"Start":"01:59.630 ","End":"02:02.300","Text":"and in this case max for maximum,"},{"Start":"02:02.300 ","End":"02:07.280","Text":"use curly braces and then the target function."},{"Start":"02:07.280 ","End":"02:12.100","Text":"We want the maximum of 4x plus 6y,"},{"Start":"02:12.100 ","End":"02:15.575","Text":"and then we write st, subject to,"},{"Start":"02:15.575 ","End":"02:21.770","Text":"and then the constraint x squared plus y squared equals 13,"},{"Start":"02:21.770 ","End":"02:27.860","Text":"and then sometimes we write some limitations on the domain"},{"Start":"02:27.860 ","End":"02:32.750","Text":"in this case because it\u0027s a real life problem with fruits,"},{"Start":"02:32.750 ","End":"02:39.005","Text":"we need to have x and y both positive."},{"Start":"02:39.005 ","End":"02:41.660","Text":"Suppose you could say bigger or equal to 0."},{"Start":"02:41.660 ","End":"02:48.560","Text":"Theoretically, you could have all pineapples and no mangoes or vice versa,"},{"Start":"02:48.560 ","End":"02:51.190","Text":"so write it like that."},{"Start":"02:51.190 ","End":"02:58.640","Text":"I\u0027ll highlight this, and that answers the formulate part."},{"Start":"02:58.640 ","End":"03:02.340","Text":"Now we have to do the solve part."},{"Start":"03:02.360 ","End":"03:10.760","Text":"Now, it just so happens that this is the same problem"},{"Start":"03:10.760 ","End":"03:15.965","Text":"as in exercise 3 of this chapter"},{"Start":"03:15.965 ","End":"03:23.525","Text":"on Lagrange multipliers and optimization and the constraint."},{"Start":"03:23.525 ","End":"03:26.980","Text":"Unless someone\u0027s changed the numbering since I did this,"},{"Start":"03:26.980 ","End":"03:29.975","Text":"and so let\u0027s just take the solution from there."},{"Start":"03:29.975 ","End":"03:33.365","Text":"There was a difference though because in this exercise,"},{"Start":"03:33.365 ","End":"03:38.655","Text":"there was not this extra restriction on the domain,"},{"Start":"03:38.655 ","End":"03:51.210","Text":"and in fact, there was a maximum at the values x and y,"},{"Start":"03:51.210 ","End":"03:55.694","Text":"2 and 3 actually, I\u0027ll use an asterisk,"},{"Start":"03:55.694 ","End":"03:58.295","Text":"it is often used to indicate these."},{"Start":"03:58.295 ","End":"03:59.930","Text":"There was also a minimum."},{"Start":"03:59.930 ","End":"04:02.490","Text":"I\u0027m just mentioning this."},{"Start":"04:04.670 ","End":"04:14.794","Text":"Turns out to be minus 2, minus 3, and we rule this one out for 2 reasons."},{"Start":"04:14.794 ","End":"04:18.560","Text":"First of all, it\u0027s a minimum, not a maximum, we want the maximum."},{"Start":"04:18.560 ","End":"04:20.989","Text":"Secondly, even if it was maximum,"},{"Start":"04:20.989 ","End":"04:28.569","Text":"it would be ruled out because it doesn\u0027t belong to the domain of non-negative numbers,"},{"Start":"04:28.569 ","End":"04:35.930","Text":"and the maximum value was 26 there."},{"Start":"04:35.930 ","End":"04:44.580","Text":"I would just say something like 2 mangoes and 3 pineapples,"},{"Start":"04:44.580 ","End":"04:48.815","Text":"and that would be a little bit friendlier than just saying this."},{"Start":"04:48.815 ","End":"04:51.860","Text":"The utility, I might add in brackets,"},{"Start":"04:51.860 ","End":"04:57.050","Text":"should get to utility of 26 in this case."},{"Start":"04:57.050 ","End":"04:59.970","Text":"That\u0027s it."}],"ID":9784},{"Watched":false,"Name":"Exercise 19","Duration":"5m 20s","ChapterTopicVideoID":9888,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.450","Text":"We have here a word problem in the world of economics. Let\u0027s read it."},{"Start":"00:06.450 ","End":"00:12.870","Text":"In the market, Danny buys x kilograms of cucumbers and y kilograms of tomatoes."},{"Start":"00:12.870 ","End":"00:16.544","Text":"The utility function is given as follows."},{"Start":"00:16.544 ","End":"00:23.400","Text":"This is like a measure of the pleasure or enjoyment he gets out of his purchase."},{"Start":"00:23.400 ","End":"00:26.325","Text":"You want that to be as great as possible."},{"Start":"00:26.325 ","End":"00:30.210","Text":"We\u0027re given the price of cucumbers, $1 a kilo."},{"Start":"00:30.210 ","End":"00:34.065","Text":"Tomatoes are a bit more expensive, $3 a kilo."},{"Start":"00:34.065 ","End":"00:38.055","Text":"But Danny has a fixed budget, he has $12."},{"Start":"00:38.055 ","End":"00:44.245","Text":"We have to formulate mathematically and to solve this problem."},{"Start":"00:44.245 ","End":"00:48.630","Text":"Now, obviously, we want to do it in terms"},{"Start":"00:48.630 ","End":"00:55.249","Text":"of an optimization or extremum problem of a function under a constraint."},{"Start":"00:55.249 ","End":"00:56.870","Text":"That\u0027s the chapter we\u0027re in."},{"Start":"00:56.870 ","End":"01:00.230","Text":"The constraint is his fixed budget,"},{"Start":"01:00.230 ","End":"01:05.435","Text":"$12, which means that the cost of what he buys has got to be $12."},{"Start":"01:05.435 ","End":"01:07.175","Text":"I\u0027m assuming he spends it all."},{"Start":"01:07.175 ","End":"01:10.055","Text":"Let\u0027s figure it out. You have x kilos,"},{"Start":"01:10.055 ","End":"01:13.805","Text":"$1 per kilo, that\u0027s 1 times x."},{"Start":"01:13.805 ","End":"01:17.040","Text":"Let me just write the word constraint."},{"Start":"01:17.740 ","End":"01:23.150","Text":"The constraint is 1 times x and $3"},{"Start":"01:23.150 ","End":"01:29.614","Text":"times the number of kilos of tomatoes is 3y is going to equal 12."},{"Start":"01:29.614 ","End":"01:31.910","Text":"Then besides the constraint,"},{"Start":"01:31.910 ","End":"01:33.230","Text":"there\u0027s a target function."},{"Start":"01:33.230 ","End":"01:38.900","Text":"The thing that you want to either maximize or minimize,"},{"Start":"01:38.900 ","End":"01:41.405","Text":"optimize will be a general word."},{"Start":"01:41.405 ","End":"01:44.824","Text":"But the target is just the utility function."},{"Start":"01:44.824 ","End":"01:51.645","Text":"It\u0027s xy and we want to maximize it."},{"Start":"01:51.645 ","End":"01:56.210","Text":"We can now phrase this in mathematical terms"},{"Start":"01:56.210 ","End":"02:02.130","Text":"as maximum of the target,"},{"Start":"02:02.130 ","End":"02:08.775","Text":"which is xy and the constraint we write as s.t, means subject to."},{"Start":"02:08.775 ","End":"02:15.684","Text":"Then we write the constraint x plus 3y equals 12."},{"Start":"02:15.684 ","End":"02:17.900","Text":"But there\u0027s also an additional thing."},{"Start":"02:17.900 ","End":"02:24.210","Text":"Sometimes you want to restrict the domain because this is a real-life problem."},{"Start":"02:24.210 ","End":"02:30.690","Text":"x and y have to be positive or at least non-negative."},{"Start":"02:30.690 ","End":"02:37.770","Text":"We would like to say that x and y are bigger or equal to 0,"},{"Start":"02:37.770 ","End":"02:40.010","Text":"though, it really is bigger than 0."},{"Start":"02:40.010 ","End":"02:41.750","Text":"Because if x or y are 0,"},{"Start":"02:41.750 ","End":"02:43.550","Text":"the utility would be 0."},{"Start":"02:43.550 ","End":"02:46.210","Text":"I\u0027m going to highlight this."},{"Start":"02:46.210 ","End":"02:50.855","Text":"This is the formulate part of the problem."},{"Start":"02:50.855 ","End":"02:55.249","Text":"Now, we come to the solve part of the problem."},{"Start":"02:55.249 ","End":"02:59.930","Text":"Unless someone\u0027s changed the numbering since I did this,"},{"Start":"02:59.930 ","End":"03:10.175","Text":"if you look at exercise number 5 in this same chapter on Lagrange multipliers and so on,"},{"Start":"03:10.175 ","End":"03:17.000","Text":"then you\u0027ll see that this is exactly the problem that is phrased there."},{"Start":"03:17.000 ","End":"03:20.225","Text":"I\u0027m just going to quote the result we found there."},{"Start":"03:20.225 ","End":"03:23.570","Text":"We found that the x,"},{"Start":"03:23.570 ","End":"03:25.790","Text":"y for the maximum,"},{"Start":"03:25.790 ","End":"03:27.640","Text":"we often indicate this with an asterisk,"},{"Start":"03:27.640 ","End":"03:29.480","Text":"so this is the special x, y,"},{"Start":"03:29.480 ","End":"03:36.030","Text":"turned out to be 6, 2."},{"Start":"03:38.260 ","End":"03:47.115","Text":"The maximum value was equal to 12."},{"Start":"03:47.115 ","End":"03:51.200","Text":"Well, you could see this, its utility is x,"},{"Start":"03:51.200 ","End":"03:55.160","Text":"y, and the target 6 times 2 is 12."},{"Start":"03:55.160 ","End":"04:01.385","Text":"But I wouldn\u0027t like to leave it like this because the question was in words."},{"Start":"04:01.385 ","End":"04:11.505","Text":"I would say that the solution is that Danny bought 6 kilograms of"},{"Start":"04:11.505 ","End":"04:15.870","Text":"cucumbers and he bought"},{"Start":"04:15.870 ","End":"04:24.285","Text":"2 kilograms of tomatoes or if you\u0027re in America, that\u0027s tomatoes."},{"Start":"04:24.285 ","End":"04:35.940","Text":"The utility, the pleasure he got out of this purchase was equal to 12."},{"Start":"04:36.380 ","End":"04:40.610","Text":"That\u0027s just a coincidence that it\u0027s $12."},{"Start":"04:40.610 ","End":"04:41.870","Text":"This comes from another place."},{"Start":"04:41.870 ","End":"04:43.640","Text":"In fact, let\u0027s do a verification."},{"Start":"04:43.640 ","End":"04:51.630","Text":"6 kilograms times $1 per kilo plus"},{"Start":"04:51.630 ","End":"04:57.005","Text":"2 kilograms times $3"},{"Start":"04:57.005 ","End":"05:03.350","Text":"a kilo comes out to be $6 for the cucumbers,"},{"Start":"05:03.350 ","End":"05:08.235","Text":"2 times 3, $6 on tomatoes."},{"Start":"05:08.235 ","End":"05:10.700","Text":"It does come out to be $12."},{"Start":"05:10.700 ","End":"05:13.530","Text":"This was just a check."},{"Start":"05:14.080 ","End":"05:21.540","Text":"That\u0027s the answer and we used an exercise that was previously solved."}],"ID":9785},{"Watched":false,"Name":"Exercise 20","Duration":"4m 17s","ChapterTopicVideoID":9885,"CourseChapterTopicPlaylistID":114745,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"This exercise, we have a problem in economics or production."},{"Start":"00:05.460 ","End":"00:07.110","Text":"Let\u0027s read it."},{"Start":"00:07.110 ","End":"00:12.340","Text":"We have a manufacturer and there\u0027s a production function,"},{"Start":"00:14.390 ","End":"00:21.280","Text":"this formula here, where K is capital and L is labor,"},{"Start":"00:22.190 ","End":"00:24.450","Text":"capital is a bit abstract."},{"Start":"00:24.450 ","End":"00:31.900","Text":"Let\u0027s just say this is like number of machines and labor\u0027s like number of workers."},{"Start":"00:33.350 ","End":"00:37.140","Text":"They each have a price."},{"Start":"00:37.140 ","End":"00:41.670","Text":"Usually these are per unit time,"},{"Start":"00:41.670 ","End":"00:42.905","Text":"let\u0027s say per hour."},{"Start":"00:42.905 ","End":"00:44.900","Text":"It might be in dollars,"},{"Start":"00:44.900 ","End":"00:52.010","Text":"$2 for the machinery and $1 for the worker."},{"Start":"00:52.010 ","End":"00:54.695","Text":"Maybe it\u0027s not per hour or maybe it\u0027s per minute,"},{"Start":"00:54.695 ","End":"00:57.260","Text":"doesn\u0027t really matter per some unit time,"},{"Start":"00:57.260 ","End":"01:01.450","Text":"and Q is the output of say the manufacturing chairs."},{"Start":"01:01.450 ","End":"01:04.340","Text":"This would be the number of chairs per unit time,"},{"Start":"01:04.340 ","End":"01:06.600","Text":"minute or hour or whatever,"},{"Start":"01:06.600 ","End":"01:12.585","Text":"and he\u0027s operating in an output level of like a 100 chairs a minute."},{"Start":"01:12.585 ","End":"01:15.400","Text":"Now, he wants a combination,"},{"Start":"01:15.400 ","End":"01:18.470","Text":"he wants to choose how many machines and"},{"Start":"01:18.470 ","End":"01:24.815","Text":"how many workers will minimize his cost per unit time."},{"Start":"01:24.815 ","End":"01:29.150","Text":"We\u0027re just asked to formulate the problem of the manufacturer,"},{"Start":"01:29.150 ","End":"01:30.770","Text":"but not to actually solve it."},{"Start":"01:30.770 ","End":"01:33.500","Text":"It probably comes out to be messy."},{"Start":"01:33.500 ","End":"01:39.230","Text":"Let\u0027s just formulate it in terms of a maximum or minimum"},{"Start":"01:39.230 ","End":"01:46.110","Text":"of some target function with some constraint."},{"Start":"01:46.240 ","End":"01:52.535","Text":"The constraint is just the production function,"},{"Start":"01:52.535 ","End":"01:57.900","Text":"and it says that Q equals"},{"Start":"01:57.900 ","End":"02:03.920","Text":"root K plus root L. K and L are the variables."},{"Start":"02:03.920 ","End":"02:07.810","Text":"Q is some constant or parameter."},{"Start":"02:07.810 ","End":"02:15.135","Text":"The only variables here are K and L. Then we also have a target."},{"Start":"02:15.135 ","End":"02:25.170","Text":"In this case, the target is the cost per unit time."},{"Start":"02:26.630 ","End":"02:30.645","Text":"Well, it\u0027s equal to, we can compute,"},{"Start":"02:30.645 ","End":"02:37.070","Text":"K machines and the machines price 2,"},{"Start":"02:37.070 ","End":"02:42.745","Text":"so we have 2 times K and"},{"Start":"02:42.745 ","End":"02:50.899","Text":"$1 times the number"},{"Start":"02:50.899 ","End":"02:56.625","Text":"of workers, L, the target."},{"Start":"02:56.625 ","End":"02:59.540","Text":"Now we just have to phrase it."},{"Start":"02:59.540 ","End":"03:03.950","Text":"We know it\u0027s a minimize problem, not max."},{"Start":"03:03.950 ","End":"03:09.450","Text":"We write min of and in curly braces,"},{"Start":"03:09.450 ","End":"03:15.040","Text":"we put the target 2K plus L,"},{"Start":"03:15.470 ","End":"03:20.845","Text":"and then the constraint to write st subject to"},{"Start":"03:20.845 ","End":"03:27.150","Text":"Q equals root K plus root L. As I said,"},{"Start":"03:27.150 ","End":"03:28.430","Text":"K and L are the variables,"},{"Start":"03:28.430 ","End":"03:31.500","Text":"Q is some given number we just don\u0027t know it."},{"Start":"03:31.500 ","End":"03:40.370","Text":"We also have a real life limitation that these numbers are going to be positive."},{"Start":"03:40.370 ","End":"03:42.710","Text":"You\u0027ve got to have some capital,"},{"Start":"03:42.710 ","End":"03:46.160","Text":"or in this case machines and some workers,"},{"Start":"03:46.160 ","End":"03:49.955","Text":"so K and L are positive,"},{"Start":"03:49.955 ","End":"03:54.170","Text":"and that\u0027s all there is to it."},{"Start":"03:54.170 ","End":"03:59.915","Text":"I\u0027ll just highlight this and this is the formulation of the problem."},{"Start":"03:59.915 ","End":"04:02.480","Text":"Like it says, we don\u0027t have to solve it."},{"Start":"04:02.480 ","End":"04:06.140","Text":"The asterisk here is just the notation that when we actually"},{"Start":"04:06.140 ","End":"04:09.335","Text":"find the solution, the critical point,"},{"Start":"04:09.335 ","End":"04:12.260","Text":"the minimum, then that particular K and L,"},{"Start":"04:12.260 ","End":"04:18.480","Text":"we call K and L with an asterisk. That\u0027s it."}],"ID":9786}],"Thumbnail":null,"ID":114745},{"Name":"Solving the Problem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"How to solve a problem","Duration":"24m 37s","ChapterTopicVideoID":9770,"CourseChapterTopicPlaylistID":114746,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:02.475","Text":"In a previous clip,"},{"Start":"00:02.475 ","End":"00:07.770","Text":"we talked about 1 side of the constraint extremum problem."},{"Start":"00:07.770 ","End":"00:12.945","Text":"We took a word problem and converted it to the abstract form,"},{"Start":"00:12.945 ","End":"00:17.310","Text":"the minimum of some objective function subject to a constraint."},{"Start":"00:17.310 ","End":"00:19.590","Text":"But this is as far as we got,"},{"Start":"00:19.590 ","End":"00:22.170","Text":"and we didn\u0027t learn how to actually solve it."},{"Start":"00:22.170 ","End":"00:24.600","Text":"Perhaps we went 1 step further and we talked"},{"Start":"00:24.600 ","End":"00:27.060","Text":"about an objective function and the constraint function,"},{"Start":"00:27.060 ","End":"00:28.995","Text":"but that\u0027s about it."},{"Start":"00:28.995 ","End":"00:33.720","Text":"Here we\u0027re going to start from this abstract problem and then I can solve it."},{"Start":"00:33.720 ","End":"00:35.870","Text":"It may have come from a word problem."},{"Start":"00:35.870 ","End":"00:38.840","Text":"It might have been something like a pound of apples"},{"Start":"00:38.840 ","End":"00:42.890","Text":"cost x dollars and the pound of grapes costs 2 dollars."},{"Start":"00:42.890 ","End":"00:49.160","Text":"You want to spend the least subject to your objective function,"},{"Start":"00:49.160 ","End":"00:53.555","Text":"which is how much you enjoy it,"},{"Start":"00:53.555 ","End":"00:57.550","Text":"your basket being 16 or something like that."},{"Start":"00:57.550 ","End":"01:00.120","Text":"Then we learned how to write it in an abstract form."},{"Start":"01:00.120 ","End":"01:01.895","Text":"Now we\u0027re going to do the second half."},{"Start":"01:01.895 ","End":"01:04.540","Text":"This is our starting point."},{"Start":"01:04.540 ","End":"01:07.060","Text":"We\u0027re taking a specific example,"},{"Start":"01:07.060 ","End":"01:08.695","Text":"that\u0027s the best way to teach it."},{"Start":"01:08.695 ","End":"01:10.070","Text":"This is our example."},{"Start":"01:10.070 ","End":"01:19.510","Text":"We want the minimum value of x plus 2y subject to or such that x squared y equals 16."},{"Start":"01:19.510 ","End":"01:27.180","Text":"What we did there already was define 2 functions."},{"Start":"01:27.870 ","End":"01:33.190","Text":"The constraint function is simply based on the constraint,"},{"Start":"01:33.190 ","End":"01:37.390","Text":"except that we put it as something equals 0 and it\u0027s a function of x and y,"},{"Start":"01:37.390 ","End":"01:41.175","Text":"lets call it the f of x and y."},{"Start":"01:41.175 ","End":"01:48.030","Text":"In our case it will be x squared y minus 16."},{"Start":"01:48.030 ","End":"01:53.905","Text":"The objective is similar to the concept we call the utility."},{"Start":"01:53.905 ","End":"01:56.915","Text":"Let\u0027s call it g of x, y."},{"Start":"01:56.915 ","End":"02:01.550","Text":"This is the thing that we\u0027re trying to extremize,"},{"Start":"02:01.550 ","End":"02:03.065","Text":"in this case minimize,"},{"Start":"02:03.065 ","End":"02:07.350","Text":"and that will be x plus 2y."},{"Start":"02:07.580 ","End":"02:19.294","Text":"All our problems in constrained extremum begin with 2 functions of x and y,"},{"Start":"02:19.294 ","End":"02:23.165","Text":"the constraint function and an objective function."},{"Start":"02:23.165 ","End":"02:27.200","Text":"From this point, the recipe has 3 steps."},{"Start":"02:27.200 ","End":"02:30.335","Text":"The recipe I just mean algorithm or method of solving it."},{"Start":"02:30.335 ","End":"02:36.980","Text":"Step 1 is to compute the partial derivatives of both of these functions,"},{"Start":"02:36.980 ","End":"02:38.975","Text":"the constraint and the objective,"},{"Start":"02:38.975 ","End":"02:43.385","Text":"and the derivatives of first and second order."},{"Start":"02:43.385 ","End":"02:46.250","Text":"They\u0027ll be 10 things to compute in total."},{"Start":"02:46.250 ","End":"02:49.790","Text":"Let\u0027s begin. It\u0027s a bit tedious but not too bad."},{"Start":"02:49.790 ","End":"02:56.760","Text":"Let\u0027s see, the derivative of f with respect to x is equal to,"},{"Start":"02:56.760 ","End":"02:59.095","Text":"y is a constant."},{"Start":"02:59.095 ","End":"03:01.070","Text":"This will just be"},{"Start":"03:01.070 ","End":"03:10.100","Text":"2xy because derivative of x squared is 2x times and the constant just stays there."},{"Start":"03:10.100 ","End":"03:13.355","Text":"It\u0027s 2xy and the 16 goes to nothing."},{"Start":"03:13.355 ","End":"03:17.575","Text":"Then we need f with respect to y."},{"Start":"03:17.575 ","End":"03:20.160","Text":"This time x is a constant,"},{"Start":"03:20.160 ","End":"03:23.115","Text":"so we just get x squared."},{"Start":"03:23.115 ","End":"03:27.400","Text":"Then we need f_xx,"},{"Start":"03:27.400 ","End":"03:32.040","Text":"second derivative with respect to x each time."},{"Start":"03:32.050 ","End":"03:37.055","Text":"We do that by taking the derivative of this 1 with respect to x,"},{"Start":"03:37.055 ","End":"03:38.710","Text":"y is a constant,"},{"Start":"03:38.710 ","End":"03:42.100","Text":"so it\u0027s just 2y."},{"Start":"03:43.360 ","End":"03:50.205","Text":"Then we need, with respect to second with respect to y both times."},{"Start":"03:50.205 ","End":"03:55.570","Text":"We\u0027ll take this one and differentiate it with respect to y,"},{"Start":"03:55.570 ","End":"03:57.110","Text":"but x is a constant,"},{"Start":"03:57.110 ","End":"03:59.375","Text":"so that would be 0."},{"Start":"03:59.375 ","End":"04:04.430","Text":"Next we need the mixed second derivative."},{"Start":"04:04.430 ","End":"04:11.765","Text":"It could be f_xy or f_yx as a theorem that in general they come out the same."},{"Start":"04:11.765 ","End":"04:16.490","Text":"I can either do this with respect to x and then I\u0027ll get 2x,"},{"Start":"04:16.490 ","End":"04:20.900","Text":"or I can do this with respect to y and I\u0027ll still get to 2x."},{"Start":"04:20.900 ","End":"04:22.885","Text":"Luckily, they both come out the same,"},{"Start":"04:22.885 ","End":"04:24.660","Text":"that\u0027s what they should do."},{"Start":"04:24.660 ","End":"04:28.865","Text":"Then we also have to do the same thing for g. Let\u0027s start,"},{"Start":"04:28.865 ","End":"04:32.185","Text":"g with respect to x is,"},{"Start":"04:32.185 ","End":"04:34.575","Text":"this time that y is the constant,"},{"Start":"04:34.575 ","End":"04:37.425","Text":"so it\u0027s just 1,"},{"Start":"04:37.425 ","End":"04:40.710","Text":"g with respect to y."},{"Start":"04:40.710 ","End":"04:43.260","Text":"Now x is the constant,"},{"Start":"04:43.260 ","End":"04:48.070","Text":"so that\u0027s just equal to 2."},{"Start":"04:52.220 ","End":"05:00.750","Text":"G_xx is, take this with respect to x, that\u0027s 0, g_yy."},{"Start":"05:02.140 ","End":"05:06.530","Text":"This with respect to y, 0."},{"Start":"05:06.530 ","End":"05:10.760","Text":"Then the mixed derivative of g, g_xy."},{"Start":"05:10.760 ","End":"05:13.770","Text":"Take your pick either this with respect to x,"},{"Start":"05:13.770 ","End":"05:15.140","Text":"so this with respect to y,"},{"Start":"05:15.140 ","End":"05:17.660","Text":"either way it comes out 0."},{"Start":"05:17.660 ","End":"05:20.200","Text":"That\u0027s the first step."},{"Start":"05:20.200 ","End":"05:23.160","Text":"Before I go to step 2,"},{"Start":"05:23.160 ","End":"05:28.085","Text":"there\u0027s something I\u0027d like to modify."},{"Start":"05:28.085 ","End":"05:33.340","Text":"I\u0027m much more used to the constraint function being g and"},{"Start":"05:33.340 ","End":"05:38.065","Text":"the objective function being f. In most books, it\u0027s that way."},{"Start":"05:38.065 ","End":"05:44.680","Text":"Let me just make that change of f to g and g to f. I just made the switch."},{"Start":"05:44.680 ","End":"05:47.890","Text":"It\u0027s better to get used to doing things one way."},{"Start":"05:47.890 ","End":"05:52.045","Text":"F is our objective for utility,"},{"Start":"05:52.045 ","End":"05:54.735","Text":"and g is the constraint."},{"Start":"05:54.735 ","End":"06:03.070","Text":"Step 2 is to write down 3 equations and 3 equations in 3 unknowns."},{"Start":"06:03.070 ","End":"06:07.670","Text":"Let me write them and then I\u0027ll do some explaining."},{"Start":"06:07.980 ","End":"06:11.515","Text":"We\u0027re going to have 3 equations."},{"Start":"06:11.515 ","End":"06:17.409","Text":"The first equation is f with respect to x,"},{"Start":"06:17.409 ","End":"06:26.795","Text":"is equal to Lambda times g with respect to x. I\u0027ll get to the Lambda in a minute."},{"Start":"06:26.795 ","End":"06:31.540","Text":"F with respect to y is equal to"},{"Start":"06:31.540 ","End":"06:39.540","Text":"Lambda times g with respect to y, it\u0027s number 2."},{"Start":"06:39.540 ","End":"06:46.230","Text":"Number 3 is just the original constraint,"},{"Start":"06:46.230 ","End":"06:48.240","Text":"set equals to 0."},{"Start":"06:48.240 ","End":"06:58.720","Text":"It\u0027s just g equals 0."},{"Start":"07:01.700 ","End":"07:06.225","Text":"This looks very curious, I\u0027ll soon."},{"Start":"07:06.225 ","End":"07:11.880","Text":"Give an example, but some general things I want to state."},{"Start":"07:11.880 ","End":"07:16.140","Text":"First of all, it\u0027s 3 equations and it\u0027s in 3 unknowns,"},{"Start":"07:16.140 ","End":"07:18.240","Text":"because I just abbreviate,"},{"Start":"07:18.240 ","End":"07:19.800","Text":"this is a function of x,"},{"Start":"07:19.800 ","End":"07:22.500","Text":"y, and this is a function of x, y,"},{"Start":"07:22.500 ","End":"07:26.790","Text":"so in general, the 3 variables are going to be x,"},{"Start":"07:26.790 ","End":"07:29.100","Text":"y, and there\u0027s a new variable,"},{"Start":"07:29.100 ","End":"07:31.530","Text":"and the Greek letter Lambda,"},{"Start":"07:31.530 ","End":"07:33.780","Text":"just a variable like anything else,"},{"Start":"07:33.780 ","End":"07:37.679","Text":"and we\u0027ll solve for x, y, and lambda,"},{"Start":"07:37.679 ","End":"07:39.630","Text":"and the points x,"},{"Start":"07:39.630 ","End":"07:41.010","Text":"y, which we find,"},{"Start":"07:41.010 ","End":"07:45.960","Text":"if any, are going to be the suspects for the extremum."},{"Start":"07:45.960 ","End":"07:53.460","Text":"This is very similar to the concept in Calculus 1"},{"Start":"07:53.460 ","End":"08:01.665","Text":"of finding first derivatives and setting it to 0 as suspects for maximum or minimum."},{"Start":"08:01.665 ","End":"08:05.910","Text":"We\u0027re going to use lambda to help us to decide afterwards if"},{"Start":"08:05.910 ","End":"08:10.005","Text":"this is a maximum or a minimum or something else,"},{"Start":"08:10.005 ","End":"08:12.360","Text":"so that\u0027s 1 remark."},{"Start":"08:12.360 ","End":"08:15.390","Text":"The other remark is it that some places, some books,"},{"Start":"08:15.390 ","End":"08:18.720","Text":"some professors don\u0027t write these equations right away,"},{"Start":"08:18.720 ","End":"08:24.270","Text":"but there are some preliminary preparatory steps before this, and that\u0027s okay too."},{"Start":"08:24.270 ","End":"08:27.430","Text":"I mean, everyone does it a bit differently."},{"Start":"08:27.640 ","End":"08:32.584","Text":"Let\u0027s see how this interprets in our case,"},{"Start":"08:32.584 ","End":"08:37.415","Text":"to make it look less abstract and more friendly we\u0027ll see."},{"Start":"08:37.415 ","End":"08:39.145","Text":"Now in our case,"},{"Start":"08:39.145 ","End":"08:45.330","Text":"f with respect to x is 1 and equals,"},{"Start":"08:45.330 ","End":"08:52.750","Text":"we just write here lambda and g with respect to x is 2xy lambda times 2xy."},{"Start":"08:52.940 ","End":"08:58.920","Text":"Second equation, f with respect to y is 2."},{"Start":"08:58.920 ","End":"09:05.640","Text":"This is going to equal Lambda g with respect to y is x squared,"},{"Start":"09:05.640 ","End":"09:07.335","Text":"and the last equation,"},{"Start":"09:07.335 ","End":"09:09.330","Text":"which is our constraint,"},{"Start":"09:09.330 ","End":"09:11.714","Text":"saying that g of x, y is 0,"},{"Start":"09:11.714 ","End":"09:14.790","Text":"is exactly the same as the original constraint."},{"Start":"09:14.790 ","End":"09:23.174","Text":"I can write it as is x squared y minus 16 equals 0,"},{"Start":"09:23.174 ","End":"09:28.330","Text":"but I would prefer to just write it as this equals 16,"},{"Start":"09:29.750 ","End":"09:33.720","Text":"so here we have 3 equations and 3 unknowns,"},{"Start":"09:33.720 ","End":"09:35.820","Text":"x, y, and lambda."},{"Start":"09:35.820 ","End":"09:40.605","Text":"Next, what we have to do is solve these equations."},{"Start":"09:40.605 ","End":"09:45.525","Text":"The way to solve these 3 equations is always the same."},{"Start":"09:45.525 ","End":"09:51.180","Text":"We divide 1 of these first 2 equations by the other."},{"Start":"09:51.180 ","End":"09:53.550","Text":"If you look at it in general, if I divide,"},{"Start":"09:53.550 ","End":"09:55.800","Text":"let\u0027s say this equation by this equation,"},{"Start":"09:55.800 ","End":"09:57.735","Text":"we get rid of Lambda,"},{"Start":"09:57.735 ","End":"10:00.135","Text":"and if we get rid of Lambda,"},{"Start":"10:00.135 ","End":"10:02.640","Text":"then we\u0027ll only have x and y,"},{"Start":"10:02.640 ","End":"10:05.025","Text":"and we\u0027ll have 2 equations in x and y."},{"Start":"10:05.025 ","End":"10:09.220","Text":"Let\u0027s see how this works out in our case."},{"Start":"10:09.650 ","End":"10:15.375","Text":"I\u0027ll divide, let\u0027s say the second 1 over the first 1,"},{"Start":"10:15.375 ","End":"10:18.700","Text":"and then I\u0027ll get 2/1."},{"Start":"10:21.890 ","End":"10:27.150","Text":"Then this over this will equal lambda times x"},{"Start":"10:27.150 ","End":"10:38.115","Text":"squared over Lambda times"},{"Start":"10:38.115 ","End":"10:45.180","Text":"2xy and now in this fraction the Lambda cancels."},{"Start":"10:45.180 ","End":"10:47.625","Text":"We\u0027re assuming Lambda\u0027s not 0."},{"Start":"10:47.625 ","End":"10:50.250","Text":"It\u0027s okay to assume that."},{"Start":"10:50.250 ","End":"10:57.540","Text":"In fact, we\u0027re also assuming that x and y are not 0 because of the denominator."},{"Start":"10:57.540 ","End":"10:59.340","Text":"If these things happen, we\u0027ll deal with them."},{"Start":"10:59.340 ","End":"11:00.960","Text":"It doesn\u0027t usually happen."},{"Start":"11:00.960 ","End":"11:03.690","Text":"Now that we have 2 fractions are equal,"},{"Start":"11:03.690 ","End":"11:08.190","Text":"then we cross-multiply this diagonal is equal to this diagonal,"},{"Start":"11:08.190 ","End":"11:10.810","Text":"so we get 2 times 2xy,"},{"Start":"11:12.890 ","End":"11:22.290","Text":"we get 4xy from this diagonal and 1 times x squared from this diagonal,"},{"Start":"11:22.290 ","End":"11:25.740","Text":"and this is an equation in x and y,"},{"Start":"11:25.740 ","End":"11:27.975","Text":"and I\u0027ll put it in a little box."},{"Start":"11:27.975 ","End":"11:34.695","Text":"That\u0027s 1 equation and the other equation is just our constraint equation."},{"Start":"11:34.695 ","End":"11:42.010","Text":"I can just repeat it over here x squared y equals 16,"},{"Start":"11:42.010 ","End":"11:48.270","Text":"and now this is 2 equations in 2 unknowns x and y."},{"Start":"11:48.750 ","End":"11:52.315","Text":"Why don\u0027t I highlight them?"},{"Start":"11:52.315 ","End":"11:56.120","Text":"1 equation, 2 equations."},{"Start":"11:56.120 ","End":"11:59.730","Text":"Let\u0027s go about solving these equations."},{"Start":"11:59.730 ","End":"12:06.180","Text":"The first 1, we can divide by x so from this 1 we\u0027ll get that"},{"Start":"12:06.180 ","End":"12:16.185","Text":"4y equals x and then I can substitute x equals 4y in here,"},{"Start":"12:16.185 ","End":"12:17.775","Text":"so from here I\u0027ll get"},{"Start":"12:17.775 ","End":"12:29.040","Text":"4y squared times y is equal to 16."},{"Start":"12:29.040 ","End":"12:37.860","Text":"That gives me 4 squared is 16y cubed is equal to 16,"},{"Start":"12:37.860 ","End":"12:41.235","Text":"so y cubed is equal to 1,"},{"Start":"12:41.235 ","End":"12:49.890","Text":"so y equals 1 there\u0027s only 1 cube root of 1 and that\u0027s 1 and if y equals 1,"},{"Start":"12:49.890 ","End":"12:58.260","Text":"then I can put it in here x is 4y so that gives us x is equal to 4,"},{"Start":"12:58.260 ","End":"13:04.330","Text":"and we would also like the value of Lambda."},{"Start":"13:04.670 ","End":"13:07.170","Text":"If we want Lambda,"},{"Start":"13:07.170 ","End":"13:11.175","Text":"we\u0027ll just go to 1 of these equations."},{"Start":"13:11.175 ","End":"13:17.460","Text":"This 1 looks good if I put Lambda in here and x equals 4,"},{"Start":"13:17.460 ","End":"13:24.495","Text":"I will get that 2 equals Lambda times 4 squared,"},{"Start":"13:24.495 ","End":"13:34.770","Text":"and that will give me that Lambda equals 2/4 squared 2/16 is 1/8, Lambda equals 1/8,"},{"Start":"13:34.770 ","End":"13:41.940","Text":"and the x, y Lambda that are solutions to this equation,"},{"Start":"13:41.940 ","End":"13:44.340","Text":"we put them with an asterisk,"},{"Start":"13:44.340 ","End":"13:46.185","Text":"let me just write this,"},{"Start":"13:46.185 ","End":"13:48.210","Text":"so we have that,"},{"Start":"13:48.210 ","End":"13:53.970","Text":"we\u0027ll call it x star, y star,"},{"Start":"13:53.970 ","End":"14:04.410","Text":"which is 4,1 is a suspect and a suspect for what?"},{"Start":"14:04.410 ","End":"14:10.120","Text":"For constrained extremum."},{"Start":"14:14.940 ","End":"14:17.665","Text":"I think we we\u0027re given,"},{"Start":"14:17.665 ","End":"14:21.320","Text":"whether it was minimum or maximum,"},{"Start":"14:21.570 ","End":"14:25.120","Text":"I think it was, let them just see."},{"Start":"14:25.120 ","End":"14:27.880","Text":"We wanted minimum."},{"Start":"14:27.880 ","End":"14:37.210","Text":"It was for a constrained minimum but just a suspect,"},{"Start":"14:37.210 ","End":"14:39.260","Text":"we don\u0027t know it yet."},{"Start":"14:40.230 ","End":"14:49.620","Text":"Lambda\u0027s going to be important and with Lambda equaling 1/8."},{"Start":"14:49.620 ","End":"14:52.290","Text":"Let\u0027s put a dividing line here."},{"Start":"14:52.290 ","End":"14:56.325","Text":"That\u0027s the end of Step 2."},{"Start":"14:56.325 ","End":"15:01.745","Text":"Now, some teachers, some books in some places just"},{"Start":"15:01.745 ","End":"15:09.565","Text":"stop here and the suspect is automatically guilty and we say that this must be the point."},{"Start":"15:09.565 ","End":"15:12.670","Text":"If in your institution this is how it\u0027s done,"},{"Start":"15:12.670 ","End":"15:15.985","Text":"then you\u0027ve got it made and you can stop here."},{"Start":"15:15.985 ","End":"15:22.410","Text":"But here we have to go ahead and approve or verify that 4,"},{"Start":"15:22.410 ","End":"15:25.890","Text":"1 is indeed a minimum and not a maximum or"},{"Start":"15:25.890 ","End":"15:30.135","Text":"something else and that\u0027s what Step 3 is going to be."},{"Start":"15:30.135 ","End":"15:34.250","Text":"If you\u0027re not required to demonstrate that it is indeed a minimum,"},{"Start":"15:34.250 ","End":"15:38.575","Text":"this is the end, but we\u0027re going to continue with Step 3."},{"Start":"15:38.575 ","End":"15:41.695","Text":"I\u0027ll just clean up a bit first."},{"Start":"15:41.695 ","End":"15:47.890","Text":"Next, what we have to do is to substitute the suspect,"},{"Start":"15:47.890 ","End":"15:50.410","Text":"the x asterisk, y asterisk,"},{"Start":"15:50.410 ","End":"15:59.020","Text":"and Lambda asterisk in a very strange expression called H. As a reason for the H,"},{"Start":"15:59.020 ","End":"16:03.850","Text":"this is something called a bordered Hessian matrix, doesn\u0027t matter."},{"Start":"16:03.850 ","End":"16:11.635","Text":"It\u0027s called H and it\u0027s very strange and let me write what it is. Here it is."},{"Start":"16:11.635 ","End":"16:21.310","Text":"It\u0027s f_xx second derivative minus Lambda times second derivative"},{"Start":"16:21.310 ","End":"16:26.200","Text":"of g with respect to x both times"},{"Start":"16:26.200 ","End":"16:32.950","Text":"times the first derivative of g with respect to y squared,"},{"Start":"16:32.950 ","End":"16:36.145","Text":"that\u0027s the first term and there\u0027s going to be 3 terms."},{"Start":"16:36.145 ","End":"16:38.754","Text":"Next term is similar,"},{"Start":"16:38.754 ","End":"16:40.600","Text":"just x and y reversed,"},{"Start":"16:40.600 ","End":"16:45.880","Text":"so we have f_yy minus Lambda times"},{"Start":"16:45.880 ","End":"16:54.710","Text":"g_yy times g_x squared."},{"Start":"16:54.920 ","End":"17:02.040","Text":"The last term is with a minus and it\u0027s a mixture of x and y"},{"Start":"17:02.040 ","End":"17:10.000","Text":"and it\u0027s minus twice f_xy,"},{"Start":"17:10.000 ","End":"17:11.770","Text":"we take mixed terms,"},{"Start":"17:11.770 ","End":"17:16.570","Text":"f_xy minus Lambda times g_xy,"},{"Start":"17:16.570 ","End":"17:20.020","Text":"the mixed second-order partial derivatives,"},{"Start":"17:20.020 ","End":"17:22.720","Text":"times not g_y squared,"},{"Start":"17:22.720 ","End":"17:27.190","Text":"not g_x squared, but g_x, g_y."},{"Start":"17:27.190 ","End":"17:32.785","Text":"All this expression is called H and we\u0027re going to substitute"},{"Start":"17:32.785 ","End":"17:41.725","Text":"the values of x asterisk and y asterisk and Lambda asterisk into this."},{"Start":"17:41.725 ","End":"17:45.265","Text":"Each of these we have because we did these computations here."},{"Start":"17:45.265 ","End":"17:47.660","Text":"We have all these,"},{"Start":"17:48.150 ","End":"17:51.625","Text":"perhaps a bit higher up maybe."},{"Start":"17:51.625 ","End":"17:56.830","Text":"They\u0027re all here and I\u0027m going to substitute our values of"},{"Start":"17:56.830 ","End":"18:04.045","Text":"xy Lambda into these and then depending on what sign I get, I\u0027ll draw conclusions."},{"Start":"18:04.045 ","End":"18:05.530","Text":"Here\u0027s what\u0027s going to happen."},{"Start":"18:05.530 ","End":"18:16.240","Text":"If H is bigger than 0,"},{"Start":"18:16.240 ","End":"18:24.590","Text":"then our point xy with asterisks are a minimum."},{"Start":"18:25.020 ","End":"18:30.070","Text":"It\u0027s a little bit like the second derivative test in"},{"Start":"18:30.070 ","End":"18:35.500","Text":"1 variable in Calculus 1 when we had second derivative was positive,"},{"Start":"18:35.500 ","End":"18:40.810","Text":"it was a minimum, and if the second derivative was negative,"},{"Start":"18:40.810 ","End":"18:42.235","Text":"it was a maximum."},{"Start":"18:42.235 ","End":"18:45.549","Text":"Only here we don\u0027t have second derivative,"},{"Start":"18:45.549 ","End":"18:49.510","Text":"we have this complicated expression H, it\u0027s intuitive."},{"Start":"18:49.510 ","End":"18:52.270","Text":"You\u0027ll have to remember it or hopefully it\u0027ll be on"},{"Start":"18:52.270 ","End":"18:58.930","Text":"the formula sheet and we\u0027ll evaluate it for our point in a moment."},{"Start":"18:58.930 ","End":"19:02.830","Text":"I just want to say that not every institution,"},{"Start":"19:02.830 ","End":"19:06.790","Text":"not every book or teacher uses this technique with the age,"},{"Start":"19:06.790 ","End":"19:13.550","Text":"sometimes it\u0027s just skipped and we just take it as if there\u0027s only 1 of them,"},{"Start":"19:13.550 ","End":"19:18.630","Text":"we take it as a minimum if we expected a minimum and if we want a minimum or a maximum,"},{"Start":"19:18.630 ","End":"19:21.420","Text":"then we substitute the value and then we say the bigger ones,"},{"Start":"19:21.420 ","End":"19:24.090","Text":"the maximum and the minimum."},{"Start":"19:24.090 ","End":"19:27.870","Text":"But that\u0027s very imprecise and I think it really should be"},{"Start":"19:27.870 ","End":"19:32.625","Text":"using this H test to see whether indeed it\u0027s a minimum or a maximum."},{"Start":"19:32.625 ","End":"19:34.410","Text":"Of course, if H comes out 0,"},{"Start":"19:34.410 ","End":"19:37.065","Text":"the test is inconclusive, so we don\u0027t know."},{"Start":"19:37.065 ","End":"19:39.670","Text":"We have to use other techniques."},{"Start":"19:39.750 ","End":"19:49.760","Text":"Having said that, let\u0027s get back to our case where we have to substitute,"},{"Start":"19:50.340 ","End":"19:53.529","Text":"but I can\u0027t see everything on the screen,"},{"Start":"19:53.529 ","End":"19:55.660","Text":"let me see what I can do."},{"Start":"19:55.660 ","End":"20:03.590","Text":"I need to see these values and I\u0027ll just copy them down here."},{"Start":"20:07.440 ","End":"20:14.965","Text":"Let me emphasize that our x and y are 4 and 1 and we\u0027ll be using these,"},{"Start":"20:14.965 ","End":"20:22.090","Text":"that\u0027s the suspect point and will also be needing the Lambda at that point which is 1/8."},{"Start":"20:22.090 ","End":"20:27.265","Text":"What I\u0027ll do first of all is put these values in here,"},{"Start":"20:27.265 ","End":"20:30.850","Text":"so x for y is 1."},{"Start":"20:30.850 ","End":"20:32.050","Text":"We\u0027ll compute each of these,"},{"Start":"20:32.050 ","End":"20:33.715","Text":"let\u0027s say I\u0027ll take a different color,"},{"Start":"20:33.715 ","End":"20:36.295","Text":"perhaps I\u0027ll use red."},{"Start":"20:36.295 ","End":"20:41.695","Text":"So twice 4 times 1 is 8,"},{"Start":"20:41.695 ","End":"20:44.215","Text":"so this is going to be 8."},{"Start":"20:44.215 ","End":"20:48.385","Text":"4 squared is 16,"},{"Start":"20:48.385 ","End":"20:52.585","Text":"twice 1 is 2,"},{"Start":"20:52.585 ","End":"20:57.865","Text":"0 here, and twice 4 is 8."},{"Start":"20:57.865 ","End":"21:01.225","Text":"Remember that we\u0027re saying x is 4, y is 1."},{"Start":"21:01.225 ","End":"21:03.700","Text":"Now here 1 is 1."},{"Start":"21:03.700 ","End":"21:05.995","Text":"Actually, just copy those."},{"Start":"21:05.995 ","End":"21:09.445","Text":"It\u0027s consistently red, 1 is just 1,"},{"Start":"21:09.445 ","End":"21:12.055","Text":"2 is just 2."},{"Start":"21:12.055 ","End":"21:15.130","Text":"Now I guess this whole line has no variables,"},{"Start":"21:15.130 ","End":"21:18.565","Text":"never mind 0, 0, and 0."},{"Start":"21:18.565 ","End":"21:29.695","Text":"Now I\u0027ll take these red terms and put them in here f_xx is this one is 0,"},{"Start":"21:29.695 ","End":"21:33.175","Text":"g_xx is over here."},{"Start":"21:33.175 ","End":"21:36.415","Text":"Now it\u0027s here, it\u0027s 2."},{"Start":"21:36.415 ","End":"21:43.210","Text":"G_y is here, it\u0027s 16. That\u0027s just the g_y."},{"Start":"21:43.210 ","End":"21:49.045","Text":"F_yy is 0, from here."},{"Start":"21:49.045 ","End":"21:55.460","Text":"G_y is here, 0."},{"Start":"21:57.090 ","End":"22:03.280","Text":"F_xy, the mixed second-order is 0 and for g,"},{"Start":"22:03.280 ","End":"22:06.235","Text":"the mixed second-order is 8."},{"Start":"22:06.235 ","End":"22:15.610","Text":"G_x is 8 and g_y is 16, 8 and 16."},{"Start":"22:15.610 ","End":"22:19.075","Text":"Now let\u0027s do the computation."},{"Start":"22:19.075 ","End":"22:22.160","Text":"Well, let\u0027s see."},{"Start":"22:24.150 ","End":"22:29.785","Text":"First term, Lambda is 1/8,"},{"Start":"22:29.785 ","End":"22:31.540","Text":"this term gives nothing,"},{"Start":"22:31.540 ","End":"22:36.265","Text":"so here we have 1/8."},{"Start":"22:36.265 ","End":"22:40.225","Text":"Everywhere I see Lambda, I\u0027ll write 1/8,"},{"Start":"22:40.225 ","End":"22:45.610","Text":"and here I have 1/8."},{"Start":"22:45.610 ","End":"22:47.365","Text":"I forgot to put in the g_x,"},{"Start":"22:47.365 ","End":"22:52.030","Text":"which was 8 here."},{"Start":"22:52.030 ","End":"22:54.820","Text":"Let\u0027s take each of the 3 terms separately."},{"Start":"22:54.820 ","End":"22:59.770","Text":"The first term, we get 0 minus 2"},{"Start":"22:59.770 ","End":"23:05.265","Text":"over 8 times 16 squared."},{"Start":"23:05.265 ","End":"23:09.705","Text":"16 squared is 256 and 2/8 is a quarter."},{"Start":"23:09.705 ","End":"23:16.115","Text":"256 over 4 is 64, I make it."},{"Start":"23:16.115 ","End":"23:19.240","Text":"Let me just quickly think about that again."},{"Start":"23:19.240 ","End":"23:22.385","Text":"That\u0027s 1/4 times 256,"},{"Start":"23:22.385 ","End":"23:27.000","Text":"64 and then we have a plus,"},{"Start":"23:27.000 ","End":"23:32.650","Text":"and then the next term we get is minus,"},{"Start":"23:33.600 ","End":"23:35.920","Text":"but this one is a minus,"},{"Start":"23:35.920 ","End":"23:38.665","Text":"I\u0027m sorry because we\u0027ve got a minus here."},{"Start":"23:38.665 ","End":"23:41.210","Text":"Cool, that in time."},{"Start":"23:41.210 ","End":"23:51.360","Text":"Total it\u0027s also going to be minus 0 because this bit is 0 and this bit is 0,"},{"Start":"23:51.360 ","End":"23:53.550","Text":"so it doesn\u0027t matter what this is."},{"Start":"23:53.550 ","End":"23:56.710","Text":"Then the last term,"},{"Start":"23:56.990 ","End":"24:03.130","Text":"this part is 0, we have a minus times something,"},{"Start":"24:03.130 ","End":"24:10.320","Text":"but it will cancel with this minus so it will basically be 2 times 1/8 times 8,"},{"Start":"24:10.320 ","End":"24:14.160","Text":"that still leaves us with 2 because these 2 together are just 1."},{"Start":"24:14.160 ","End":"24:21.340","Text":"2 times 8 times 16 is 16 times 16 is 256."},{"Start":"24:22.320 ","End":"24:26.590","Text":"Altogether, we can see that this thing is going to be"},{"Start":"24:26.590 ","End":"24:38.180","Text":"positive so it is indeed a minimum."}],"ID":9684},{"Watched":false,"Name":"An Important Remark on Notation","Duration":"5m 10s","ChapterTopicVideoID":9771,"CourseChapterTopicPlaylistID":114746,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.775","Text":"In this clip, I have an important remark to make."},{"Start":"00:02.775 ","End":"00:06.135","Text":"This is about extrema under constraint,"},{"Start":"00:06.135 ","End":"00:07.425","Text":"that type of problem."},{"Start":"00:07.425 ","End":"00:13.230","Text":"The typical problem is that we want the minimum or maximum,"},{"Start":"00:13.230 ","End":"00:14.955","Text":"one or the other,"},{"Start":"00:14.955 ","End":"00:19.604","Text":"of some function, which is the objective function,"},{"Start":"00:19.604 ","End":"00:21.595","Text":"f of x, y."},{"Start":"00:21.595 ","End":"00:24.435","Text":"It\u0027s always subject to a constraint,"},{"Start":"00:24.435 ","End":"00:29.760","Text":"which is some function g of x, y equaling 0."},{"Start":"00:29.760 ","End":"00:36.105","Text":"What we\u0027ve always been doing at some point in the process was to write down 3 equations,"},{"Start":"00:36.105 ","End":"00:46.025","Text":"and we would write them as f with respect to x equals lambda g with respect to x,"},{"Start":"00:46.025 ","End":"00:49.510","Text":"f with respect to y equals lambda g with respect to y,"},{"Start":"00:49.510 ","End":"00:53.990","Text":"and we\u0027d write the original constraint, g of x, y equals 0."},{"Start":"00:53.990 ","End":"01:01.160","Text":"But most books instructors start a bit before this and eventually get to this."},{"Start":"01:01.160 ","End":"01:05.075","Text":"There are 2 general approaches of how to start,"},{"Start":"01:05.075 ","End":"01:07.920","Text":"I\u0027ll call them 1 and 2."},{"Start":"01:07.960 ","End":"01:17.210","Text":"The first common approach is to use the gradient operator, this upside down Delta."},{"Start":"01:17.210 ","End":"01:22.865","Text":"What we do is we write 2 equations that"},{"Start":"01:22.865 ","End":"01:30.865","Text":"the gradient of f is equal to the gradient of g times lambda"},{"Start":"01:30.865 ","End":"01:35.465","Text":"and the original constraint g equals 0."},{"Start":"01:35.465 ","End":"01:41.600","Text":"Now the gradient is simply the derivative of f with respect to x,"},{"Start":"01:41.600 ","End":"01:43.190","Text":"this is as a vector,"},{"Start":"01:43.190 ","End":"01:45.860","Text":"derivative of f with respect to y."},{"Start":"01:45.860 ","End":"01:50.910","Text":"This is equal to lambda times the vector g with respect to x,"},{"Start":"01:50.910 ","End":"01:53.730","Text":"g with respect to y."},{"Start":"01:54.130 ","End":"01:59.755","Text":"Since this is a vector, I can take it coordinate-wise."},{"Start":"01:59.755 ","End":"02:00.949","Text":"In the first coordinate,"},{"Start":"02:00.949 ","End":"02:05.990","Text":"I get that f with respect to x equals lambda times g with respect to x,"},{"Start":"02:05.990 ","End":"02:08.090","Text":"this with this, and this with this,"},{"Start":"02:08.090 ","End":"02:14.075","Text":"I equate, I get f with respect to y equals lambda g with respect to y."},{"Start":"02:14.075 ","End":"02:16.940","Text":"This I copy as is, g equals 0,"},{"Start":"02:16.940 ","End":"02:20.615","Text":"so this gives us the 3 equations that we have here."},{"Start":"02:20.615 ","End":"02:28.055","Text":"The second approach is to define a function L for Lagrange."},{"Start":"02:28.055 ","End":"02:33.674","Text":"A function of 3 variables, x, y, and lambda,"},{"Start":"02:33.674 ","End":"02:45.785","Text":"to be equal to this f of x, y minus lambda times, the other function, g of x, y."},{"Start":"02:45.785 ","End":"02:48.565","Text":"This is a function of 3 variables."},{"Start":"02:48.565 ","End":"02:53.675","Text":"Then to take the 3 equations as follows:"},{"Start":"02:53.675 ","End":"02:57.230","Text":"we say L with respect to x equals 0,"},{"Start":"02:57.230 ","End":"03:00.170","Text":"L with respect to y equals 0,"},{"Start":"03:00.170 ","End":"03:03.620","Text":"and L with respect to lambda equals 0."},{"Start":"03:03.620 ","End":"03:08.695","Text":"All the partial derivatives of L with respect to its 3 variables are all 0."},{"Start":"03:08.695 ","End":"03:11.330","Text":"Let\u0027s see what each of these gives us."},{"Start":"03:11.330 ","End":"03:13.700","Text":"The derivative of L with respect to x,"},{"Start":"03:13.700 ","End":"03:15.245","Text":"let\u0027s see what that is."},{"Start":"03:15.245 ","End":"03:19.550","Text":"I have 2 bits. The first bit I differentiate with respect to x,"},{"Start":"03:19.550 ","End":"03:22.830","Text":"so that\u0027s just the partial derivative of x."},{"Start":"03:23.060 ","End":"03:27.995","Text":"Then lambda is a constant times a function,"},{"Start":"03:27.995 ","End":"03:30.410","Text":"so I can just leave that constant there,"},{"Start":"03:30.410 ","End":"03:34.750","Text":"minus lambda, and just differentiate g with respect to x"},{"Start":"03:34.750 ","End":"03:37.490","Text":"and say that this is 0."},{"Start":"03:37.490 ","End":"03:42.650","Text":"Similarly here, if I differentiate L with respect to y,"},{"Start":"03:42.650 ","End":"03:48.980","Text":"it\u0027s like taking f with respect to y minus lambda g with respect to y,"},{"Start":"03:48.980 ","End":"03:51.305","Text":"and that will also equal 0."},{"Start":"03:51.305 ","End":"03:54.580","Text":"L with respect to lambda,"},{"Start":"03:54.580 ","End":"03:56.805","Text":"x and y are constants,"},{"Start":"03:56.805 ","End":"03:59.505","Text":"so this thing is 0,"},{"Start":"03:59.505 ","End":"04:05.060","Text":"and what I get is that the derivative is just"},{"Start":"04:05.060 ","End":"04:12.815","Text":"minus g of x and y equals 0"},{"Start":"04:12.815 ","End":"04:16.670","Text":"because minus g of x, y is the derivative with respect to lambda."},{"Start":"04:16.670 ","End":"04:19.160","Text":"Now these 3 equations,"},{"Start":"04:19.160 ","End":"04:22.085","Text":"I can just rewrite them slightly"},{"Start":"04:22.085 ","End":"04:23.730","Text":"and I will get,"},{"Start":"04:23.730 ","End":"04:31.975","Text":"the first one will tell me that f with respect to x equals lambda g_x, that\u0027s this one."},{"Start":"04:31.975 ","End":"04:39.940","Text":"The second one says that f with respect to y equals lambda times g with respect to y."},{"Start":"04:39.940 ","End":"04:46.995","Text":"The last one, just throw out the minus and get g is equal to 0."},{"Start":"04:46.995 ","End":"04:51.740","Text":"That\u0027s the same 3, so if we look at it all,"},{"Start":"04:51.740 ","End":"04:54.989","Text":"these 3 and these 3 are the same"},{"Start":"04:54.989 ","End":"04:59.150","Text":"and they\u0027re also the same as the original 3 that we\u0027ve been using,"},{"Start":"04:59.150 ","End":"05:01.460","Text":"so it all works out."},{"Start":"05:01.460 ","End":"05:04.162","Text":"That was just to note because you may encounter"},{"Start":"05:04.162 ","End":"05:08.918","Text":"these other approaches in books, others teachers, and so on."},{"Start":"05:08.918 ","End":"05:11.130","Text":"We\u0027re done."}],"ID":9685}],"Thumbnail":null,"ID":114746},{"Name":"Formulating the Problem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction and worked example 1","Duration":"11m ","ChapterTopicVideoID":8776,"CourseChapterTopicPlaylistID":114747,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.775","Text":"In this clip, we\u0027ll learn about"},{"Start":"00:02.775 ","End":"00:08.295","Text":"extremum problems for functions of 2 variables under a constraint."},{"Start":"00:08.295 ","End":"00:12.840","Text":"The best way to explain this is through examples."},{"Start":"00:12.840 ","End":"00:16.590","Text":"We\u0027ll start off with some examples from the field of economics,"},{"Start":"00:16.590 ","End":"00:19.530","Text":"and then move on to other examples."},{"Start":"00:19.530 ","End":"00:22.620","Text":"Let\u0027s start with the first example."},{"Start":"00:22.620 ","End":"00:28.390","Text":"In this problem, Danny goes to the market, or supermarket,"},{"Start":"00:28.390 ","End":"00:36.160","Text":"to buy a basket of x pounds of tomatoes and y pounds of cucumbers."},{"Start":"00:36.160 ","End":"00:41.040","Text":"Now his utility from the basket x,"},{"Start":"00:41.040 ","End":"00:44.784","Text":"y, is x squared y."},{"Start":"00:44.784 ","End":"00:46.970","Text":"But those of you not from economics,"},{"Start":"00:46.970 ","End":"00:49.970","Text":"I need to explain what utility is."},{"Start":"00:49.970 ","End":"00:54.450","Text":"It\u0027s a quantitative measure of how"},{"Start":"00:54.450 ","End":"00:59.150","Text":"happy he is with his purchase, utilities like usefulness."},{"Start":"00:59.150 ","End":"01:05.030","Text":"It\u0027s a measure of his satisfaction or how much fun he gets out of it."},{"Start":"01:05.030 ","End":"01:08.030","Text":"It\u0027s a quantitative measure of how happy he is"},{"Start":"01:08.030 ","End":"01:11.410","Text":"with x pounds of tomatoes and y pounds of cucumbers,"},{"Start":"01:11.410 ","End":"01:15.270","Text":"and it\u0027s given by the formula x squared y."},{"Start":"01:15.710 ","End":"01:23.570","Text":"For example, if he buys 2 pounds of tomatoes and 1 pound of cucumbers,"},{"Start":"01:23.570 ","End":"01:26.290","Text":"then 2 squared times 1 is 4,"},{"Start":"01:26.290 ","End":"01:28.340","Text":"and his utility is 4."},{"Start":"01:28.340 ","End":"01:30.695","Text":"If he buys 10 pounds of each,"},{"Start":"01:30.695 ","End":"01:33.780","Text":"then 10 squared times 10 is 1,000,"},{"Start":"01:33.780 ","End":"01:38.220","Text":"and his utility is 1,000, and so on."},{"Start":"01:38.220 ","End":"01:39.950","Text":"Now let me continue,"},{"Start":"01:39.950 ","End":"01:43.190","Text":"and I\u0027ll give you the prices per pound of each of these."},{"Start":"01:43.190 ","End":"01:44.630","Text":"In this particular market,"},{"Start":"01:44.630 ","End":"01:47.495","Text":"a pound of tomatoes costs 40 cents,"},{"Start":"01:47.495 ","End":"01:51.985","Text":"and a pound of cucumbers costs only 20 cents."},{"Start":"01:51.985 ","End":"01:56.360","Text":"Now, Danny wants to choose a basket, in other words,"},{"Start":"01:56.360 ","End":"02:00.920","Text":"a certain number of pounds of tomatoes and a certain number of pounds of cucumbers,"},{"Start":"02:00.920 ","End":"02:04.060","Text":"that will give him a utility of 16,"},{"Start":"02:04.060 ","End":"02:07.745","Text":"in other words, x and y such that this comes out 16."},{"Start":"02:07.745 ","End":"02:13.330","Text":"We have to formulate Danny\u0027s problem as an extremum and the constraint,"},{"Start":"02:13.330 ","End":"02:15.555","Text":"and all will be clear soon."},{"Start":"02:15.555 ","End":"02:18.165","Text":"Let me first of all make a little table,"},{"Start":"02:18.165 ","End":"02:19.850","Text":"one column is x,"},{"Start":"02:19.850 ","End":"02:25.055","Text":"the other column is y, and there\u0027s another column you\u0027ll see in a moment."},{"Start":"02:25.055 ","End":"02:32.540","Text":"What I want to say is that there are several ways that Danny could get a utility of 16."},{"Start":"02:32.540 ","End":"02:37.625","Text":"For example, he could take 1 pound of tomatoes,"},{"Start":"02:37.625 ","End":"02:46.650","Text":"and he could take 16 pounds of cucumbers,"},{"Start":"02:46.650 ","End":"02:52.610","Text":"and then 1 squared times 16 from this formula would give me 16,"},{"Start":"02:52.610 ","End":"02:54.995","Text":"and that would be just fine."},{"Start":"02:54.995 ","End":"02:58.205","Text":"But he could also do it another way."},{"Start":"02:58.205 ","End":"03:02.210","Text":"He could take 2 pounds of tomatoes."},{"Start":"03:02.210 ","End":"03:04.325","Text":"Let\u0027s see, 2 squared is 4,"},{"Start":"03:04.325 ","End":"03:07.580","Text":"so I need another 4 pounds here,"},{"Start":"03:07.580 ","End":"03:10.975","Text":"and that would also give me a utility of 16."},{"Start":"03:10.975 ","End":"03:14.100","Text":"I could take x equals 4,"},{"Start":"03:14.100 ","End":"03:15.570","Text":"4 pounds of tomatoes,"},{"Start":"03:15.570 ","End":"03:18.495","Text":"and 4 squared is 16 times 1,"},{"Start":"03:18.495 ","End":"03:20.090","Text":"and 1 pound of cucumbers."},{"Start":"03:20.090 ","End":"03:22.564","Text":"That would also give me utility of 16."},{"Start":"03:22.564 ","End":"03:25.280","Text":"But of course I can\u0027t just take any x and y."},{"Start":"03:25.280 ","End":"03:26.765","Text":"If I took say x,"},{"Start":"03:26.765 ","End":"03:27.890","Text":"1 pound of tomatoes,"},{"Start":"03:27.890 ","End":"03:30.680","Text":"1 pound of cucumbers, that would not do,"},{"Start":"03:30.680 ","End":"03:33.079","Text":"because 1 squared times 1 is not 16,"},{"Start":"03:33.079 ","End":"03:41.080","Text":"and so you get the idea that we have here a constraint."},{"Start":"03:41.300 ","End":"03:50.780","Text":"In fact, this line is the constraint that the basket has to have a utility of 16."},{"Start":"03:50.780 ","End":"03:56.290","Text":"This is the constraint that I\u0027m talking about here."},{"Start":"03:56.600 ","End":"04:00.759","Text":"As we see, not every pair of values,"},{"Start":"04:00.759 ","End":"04:02.950","Text":"not every basket will do."},{"Start":"04:02.950 ","End":"04:05.320","Text":"I\u0027ll give 1 other example of what\u0027s possible."},{"Start":"04:05.320 ","End":"04:08.635","Text":"Suppose I took x equals 8,"},{"Start":"04:08.635 ","End":"04:12.160","Text":"8 pounds of tomatoes, 8 squared is 64,"},{"Start":"04:12.160 ","End":"04:16.380","Text":"times 1/4, would give me 16,"},{"Start":"04:16.380 ","End":"04:18.870","Text":"so I could take 8 pounds of tomatoes,"},{"Start":"04:18.870 ","End":"04:21.130","Text":"1 quarter of a pound of cucumbers."},{"Start":"04:21.130 ","End":"04:24.065","Text":"Not very many cucumbers, but there it is."},{"Start":"04:24.065 ","End":"04:26.985","Text":"Now we want to phrase this mathematically,"},{"Start":"04:26.985 ","End":"04:33.410","Text":"and we also want to know what is this extremum that is mentioned here."},{"Start":"04:33.410 ","End":"04:35.450","Text":"Get to the extremum in a moment,"},{"Start":"04:35.450 ","End":"04:39.020","Text":"I just want to write the constraint mathematically,"},{"Start":"04:39.020 ","End":"04:41.435","Text":"instead of this thing in words,"},{"Start":"04:41.435 ","End":"04:49.775","Text":"I can write that x squared times y equals 16."},{"Start":"04:49.775 ","End":"04:53.840","Text":"This is the constraint in mathematical terms."},{"Start":"04:53.840 ","End":"04:56.645","Text":"Now, back to the extremum."},{"Start":"04:56.645 ","End":"05:03.570","Text":"As you see, I have many possibilities of maintaining the constraint,"},{"Start":"05:03.570 ","End":"05:06.470","Text":"keeping it by having say x equals 1,"},{"Start":"05:06.470 ","End":"05:08.030","Text":"y equals 16, x equals 2,"},{"Start":"05:08.030 ","End":"05:09.875","Text":"y equals 4, and so on."},{"Start":"05:09.875 ","End":"05:17.855","Text":"The question is, is there any particular 1 of these baskets that I would prefer?"},{"Start":"05:17.855 ","End":"05:20.404","Text":"Now this is economics, remember,"},{"Start":"05:20.404 ","End":"05:23.330","Text":"and there are prices and costs."},{"Start":"05:23.330 ","End":"05:26.825","Text":"The natural thing in economics is to say, well,"},{"Start":"05:26.825 ","End":"05:29.165","Text":"if I have a choice of all of these,"},{"Start":"05:29.165 ","End":"05:32.405","Text":"wouldn\u0027t it makes sense to go for the 1 that is cheapest,"},{"Start":"05:32.405 ","End":"05:34.570","Text":"that costs the least?"},{"Start":"05:34.570 ","End":"05:38.340","Text":"Now each of these baskets has a cost,"},{"Start":"05:38.340 ","End":"05:40.425","Text":"and that\u0027s going to be our third column."},{"Start":"05:40.425 ","End":"05:44.010","Text":"This column is for the cost, but the cost,"},{"Start":"05:44.010 ","End":"05:46.375","Text":"I can express in terms of x and y,"},{"Start":"05:46.375 ","End":"05:49.025","Text":"because I have the prices per pound."},{"Start":"05:49.025 ","End":"05:52.025","Text":"If I take x pounds of tomatoes,"},{"Start":"05:52.025 ","End":"05:55.220","Text":"that will be 40 cents."},{"Start":"05:55.220 ","End":"05:56.975","Text":"Let\u0027s do it in dollars."},{"Start":"05:56.975 ","End":"06:02.580","Text":"In dollars this is going to be 0.4 dollars,"},{"Start":"06:02.580 ","End":"06:05.970","Text":"and this is going to be 0.2 dollars."},{"Start":"06:05.970 ","End":"06:09.349","Text":"The cost of this basket is going to be,"},{"Start":"06:09.349 ","End":"06:14.690","Text":"x pounds of tomatoes is 0.4 times"},{"Start":"06:14.690 ","End":"06:23.980","Text":"x plus 0.2 times y, for the cucumbers."},{"Start":"06:24.470 ","End":"06:30.945","Text":"In actual fact, for each basket I can compute this."},{"Start":"06:30.945 ","End":"06:38.625","Text":"For here, I\u0027d get 0.4 plus 16 times 0.2,"},{"Start":"06:38.625 ","End":"06:47.200","Text":"is 3.2, so it\u0027s $3.20 plus 40 cents, it\u0027s $3.60."},{"Start":"06:49.070 ","End":"06:52.425","Text":"This basket would be,"},{"Start":"06:52.425 ","End":"06:56.520","Text":"if I put x equals 2y equals 4 here,"},{"Start":"06:56.520 ","End":"06:58.785","Text":"0.4 times 2 is 0.8,"},{"Start":"06:58.785 ","End":"07:01.020","Text":"plus 0.8 is 1.6,"},{"Start":"07:01.020 ","End":"07:05.700","Text":"so it\u0027s $1.60, and so on."},{"Start":"07:05.700 ","End":"07:08.375","Text":"We can get a price here and the price here."},{"Start":"07:08.375 ","End":"07:11.840","Text":"What we want is the least price."},{"Start":"07:11.840 ","End":"07:17.150","Text":"This function becomes what we call the objective function."},{"Start":"07:17.150 ","End":"07:21.830","Text":"This is the function that we need to minimize or maximize in general,"},{"Start":"07:21.830 ","End":"07:23.270","Text":"but in this case minimize,"},{"Start":"07:23.270 ","End":"07:25.805","Text":"I want to pay the least, not the most."},{"Start":"07:25.805 ","End":"07:28.850","Text":"I\u0027m going to write the objective."},{"Start":"07:28.850 ","End":"07:32.455","Text":"Oh, I meant to write the word constraint."},{"Start":"07:32.455 ","End":"07:34.680","Text":"This was the constraint,"},{"Start":"07:34.680 ","End":"07:38.660","Text":"and now I have the other concept called the objective."},{"Start":"07:38.660 ","End":"07:43.070","Text":"The objective is to find the least cost,"},{"Start":"07:43.070 ","End":"07:52.765","Text":"which is the minimum of 0.4x plus 0.2y, constraint and objective."},{"Start":"07:52.765 ","End":"07:56.375","Text":"What we actually do is define 2 functions."},{"Start":"07:56.375 ","End":"07:59.240","Text":"Let\u0027s call them f and g. In general,"},{"Start":"07:59.240 ","End":"08:01.745","Text":"f will be the constraint function,"},{"Start":"08:01.745 ","End":"08:03.580","Text":"and we define f of x,"},{"Start":"08:03.580 ","End":"08:06.200","Text":"y is equal to, well,"},{"Start":"08:06.200 ","End":"08:07.960","Text":"the constraint has stuff on both sides,"},{"Start":"08:07.960 ","End":"08:11.980","Text":"so what we do is we put the 16 on the other side."},{"Start":"08:11.980 ","End":"08:15.290","Text":"The constraint is expressed as something equals 0."},{"Start":"08:15.290 ","End":"08:21.180","Text":"The constraint function is going to be x squared y minus 16,"},{"Start":"08:21.180 ","End":"08:23.210","Text":"and the objective function,"},{"Start":"08:23.210 ","End":"08:25.400","Text":"let\u0027s call it g of x, y,"},{"Start":"08:25.400 ","End":"08:30.140","Text":"is the thing that we\u0027re trying to minimize or maximize in other cases,"},{"Start":"08:30.140 ","End":"08:37.110","Text":"and that\u0027s going to be 0.4x plus 0.2y."},{"Start":"08:37.210 ","End":"08:41.270","Text":"Now that we have a constraint function and an objective function,"},{"Start":"08:41.270 ","End":"08:45.565","Text":"what we do is we formalize this word problem as follows."},{"Start":"08:45.565 ","End":"08:48.970","Text":"Let me have some more room here."},{"Start":"08:48.970 ","End":"08:52.775","Text":"We phrase the original problem as,"},{"Start":"08:52.775 ","End":"08:56.675","Text":"we want the minimum"},{"Start":"08:56.675 ","End":"09:04.710","Text":"of 0.4x plus 0.2y,"},{"Start":"09:04.710 ","End":"09:08.100","Text":"subject to, we write s.t,"},{"Start":"09:08.100 ","End":"09:11.250","Text":"it\u0027s standard abbreviation for subject to,"},{"Start":"09:11.250 ","End":"09:21.125","Text":"the constraint that x squared y minus 16 equals 0."},{"Start":"09:21.125 ","End":"09:26.300","Text":"Same thing as x squared y equals 16,"},{"Start":"09:26.300 ","End":"09:30.530","Text":"but usually we just make it equal to 0."},{"Start":"09:30.530 ","End":"09:33.530","Text":"This is specific for this case."},{"Start":"09:33.530 ","End":"09:37.940","Text":"In general we want a minimum or"},{"Start":"09:37.940 ","End":"09:45.220","Text":"maximum of the objective function g of x,"},{"Start":"09:45.220 ","End":"09:51.520","Text":"y, subject to the constraint function being 0."},{"Start":"09:51.520 ","End":"09:52.990","Text":"We do this in general,"},{"Start":"09:52.990 ","End":"09:58.200","Text":"which is why we like to write the constraint function as something which is equal to 0,"},{"Start":"09:58.200 ","End":"10:02.020","Text":"and we need to have the 16 in there somewhere."},{"Start":"10:02.020 ","End":"10:04.535","Text":"This is the word problem,"},{"Start":"10:04.535 ","End":"10:06.500","Text":"and this is its formalization;"},{"Start":"10:06.500 ","End":"10:13.640","Text":"minimum or maximum of an objective function subject to a constraint function equals 0."},{"Start":"10:13.640 ","End":"10:18.140","Text":"In general, there are 2 parts to this kind of problem."},{"Start":"10:18.140 ","End":"10:24.140","Text":"The first part is taking the word problem and formalizing it"},{"Start":"10:24.140 ","End":"10:31.390","Text":"as the minimum or maximum of a objective function subject to a constraint function,"},{"Start":"10:31.390 ","End":"10:37.715","Text":"and the next part is solving this using a recipe or algorithm."},{"Start":"10:37.715 ","End":"10:40.900","Text":"Now we\u0027re not going to do the solving part in this clip,"},{"Start":"10:40.900 ","End":"10:42.470","Text":"that will be for the next clip."},{"Start":"10:42.470 ","End":"10:48.530","Text":"I\u0027m going to do some more examples now of taking the word problem and expressing it or"},{"Start":"10:48.530 ","End":"10:55.090","Text":"formalizing it as a problem of extrema and the constraint,"},{"Start":"10:55.090 ","End":"10:58.370","Text":"minimum or maximum subject to a constraint."},{"Start":"10:58.370 ","End":"11:00.930","Text":"Next problem."}],"ID":8872},{"Watched":false,"Name":"worked example 2","Duration":"9m 16s","ChapterTopicVideoID":8777,"CourseChapterTopicPlaylistID":114747,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.530 ","End":"00:05.670","Text":"Here\u0027s problem number 2, which begins very,"},{"Start":"00:05.670 ","End":"00:09.675","Text":"very similarly to problem 1,"},{"Start":"00:09.675 ","End":"00:12.315","Text":"but you\u0027ll see that it\u0027s actually quite different."},{"Start":"00:12.315 ","End":"00:13.710","Text":"Again, at the market,"},{"Start":"00:13.710 ","End":"00:16.870","Text":"Danny buys x of"},{"Start":"00:19.970 ","End":"00:26.430","Text":"tomatoes and y of cucumbers and his utility from the basket,"},{"Start":"00:26.430 ","End":"00:28.650","Text":"x, y is x squared, y."},{"Start":"00:28.650 ","End":"00:31.695","Text":"Same as before, the same cost as before."},{"Start":"00:31.695 ","End":"00:35.955","Text":"A pound of tomato is 40 cents and a pound of cucumber is 20 cents."},{"Start":"00:35.955 ","End":"00:39.615","Text":"If we want that in dollars it\u0027s 0.4, 0.2."},{"Start":"00:39.615 ","End":"00:42.400","Text":"Now here comes the difference."},{"Start":"00:42.400 ","End":"00:49.910","Text":"This time we\u0027re given that Danny has a vegetable budget of $2.40."},{"Start":"00:49.910 ","End":"00:55.880","Text":"In other words, he can only buy tomatoes and cucumbers up to $2.40."},{"Start":"00:55.880 ","End":"01:00.740","Text":"This is going to be our constraint in this problem."},{"Start":"01:00.740 ","End":"01:02.990","Text":"The constraint is always the most important."},{"Start":"01:02.990 ","End":"01:06.485","Text":"At least that\u0027s where we start from, is the constraint."},{"Start":"01:06.485 ","End":"01:10.235","Text":"The rest of the question is also like before."},{"Start":"01:10.235 ","End":"01:15.800","Text":"In fact, I just copy pasted this now from problem 1."},{"Start":"01:15.800 ","End":"01:22.710","Text":"We have to formulate Danny\u0027s problem as one of an extremum under constraint."},{"Start":"01:22.880 ","End":"01:30.230","Text":"Just as before, we had a constraint and an objective. Let\u0027s see."},{"Start":"01:30.230 ","End":"01:33.260","Text":"Well, before I do that, perhaps I\u0027ll draw a table like"},{"Start":"01:33.260 ","End":"01:37.375","Text":"before so you can get the idea of what\u0027s going on."},{"Start":"01:37.375 ","End":"01:42.290","Text":"I write the constraint first because we already computed the cost function before."},{"Start":"01:42.290 ","End":"01:50.650","Text":"The cost function we computed in dollars as 0.4x plus 0.2y,"},{"Start":"01:50.650 ","End":"01:56.800","Text":"but this time instead of minimizing it as we did in the previous problem,"},{"Start":"01:56.800 ","End":"02:05.220","Text":"it\u0027s given, and it\u0027s given to be $2.40."},{"Start":"02:05.220 ","End":"02:09.765","Text":"So that\u0027s 2.4, and that\u0027s our constraint."},{"Start":"02:09.765 ","End":"02:12.345","Text":"In a moment we\u0027ll get to the objective."},{"Start":"02:12.345 ","End":"02:17.130","Text":"We actually have several possibilities, I\u0027ll show you."},{"Start":"02:17.130 ","End":"02:22.920","Text":"For example, if we let x be 4 pounds and this y is also 4 pounds,"},{"Start":"02:22.920 ","End":"02:33.015","Text":"4 pounds would be $1.60 and 4 pounds would be $0.8, so altogether $2.4."},{"Start":"02:33.015 ","End":"02:37.950","Text":"That\u0027s fine. There are other possibilities."},{"Start":"02:39.320 ","End":"02:43.550","Text":"6 pounds of tomatoes would already cost 2.40,"},{"Start":"02:43.550 ","End":"02:49.520","Text":"so no cucumbers and that would be within the budget, exactly budget."},{"Start":"02:49.520 ","End":"02:54.215","Text":"We could even take 10 pound,"},{"Start":"02:54.215 ","End":"02:56.695","Text":"no, that\u0027s too much."},{"Start":"02:56.695 ","End":"03:01.520","Text":"We could take 10 pounds of cucumbers and that would be $2"},{"Start":"03:01.520 ","End":"03:06.215","Text":"and we would still have $0.40 left over for 1 pound of tomatoes."},{"Start":"03:06.215 ","End":"03:09.710","Text":"In brief, there are several possibilities for x,"},{"Start":"03:09.710 ","End":"03:13.740","Text":"y, and they all satisfy the constraint."},{"Start":"03:13.740 ","End":"03:16.835","Text":"How does the objective come into this?"},{"Start":"03:16.835 ","End":"03:23.675","Text":"Well, previously, we wanted to minimize cost because we were given the utility."},{"Start":"03:23.675 ","End":"03:30.814","Text":"This time, it seems to me to make sense that if we have a fixed budget,"},{"Start":"03:30.814 ","End":"03:33.260","Text":"we want to maximize the utility."},{"Start":"03:33.260 ","End":"03:40.140","Text":"The utility is like the happiness or satisfaction that we get from the particular basket."},{"Start":"03:40.150 ","End":"03:44.540","Text":"What we want to do now is maximize the utility."},{"Start":"03:44.540 ","End":"03:48.155","Text":"We\u0027re given the utility which is x squared, y."},{"Start":"03:48.155 ","End":"03:54.890","Text":"The objective is to"},{"Start":"03:54.890 ","End":"04:02.520","Text":"maximize the utility function,"},{"Start":"04:02.560 ","End":"04:11.385","Text":"which is x squared times y. Constraint,"},{"Start":"04:11.385 ","End":"04:17.310","Text":"objective, and now we define the functions f of x,"},{"Start":"04:17.310 ","End":"04:18.560","Text":"y, and g of x, y,"},{"Start":"04:18.560 ","End":"04:21.785","Text":"the constraint function and the objective functions."},{"Start":"04:21.785 ","End":"04:28.395","Text":"What we have is that f of x, y."},{"Start":"04:28.395 ","End":"04:31.580","Text":"The constraint function is going to be,"},{"Start":"04:31.580 ","End":"04:35.855","Text":"and remember we just put everything on one side and leave 0 on the other,"},{"Start":"04:35.855 ","End":"04:37.790","Text":"and that\u0027s the function,"},{"Start":"04:37.790 ","End":"04:41.460","Text":"so it\u0027s equal to 0.4x"},{"Start":"04:41.740 ","End":"04:48.715","Text":"plus 0.2y minus 2.4."},{"Start":"04:48.715 ","End":"04:54.350","Text":"The objective function is just what we want to find the extremum of."},{"Start":"04:54.350 ","End":"04:57.155","Text":"Previously, it was a minimum here it\u0027s a maximum,"},{"Start":"04:57.155 ","End":"04:59.605","Text":"but in either case,"},{"Start":"04:59.605 ","End":"05:01.740","Text":"that\u0027s our g of x, y,"},{"Start":"05:01.740 ","End":"05:04.905","Text":"and that\u0027s equal to x squared, y."},{"Start":"05:04.905 ","End":"05:07.080","Text":"Once we have f and g,"},{"Start":"05:07.080 ","End":"05:09.705","Text":"we then formulate the problem."},{"Start":"05:09.705 ","End":"05:15.220","Text":"It\u0027s always the same formulation."},{"Start":"05:16.430 ","End":"05:25.580","Text":"Once we know the constraint and the objective or rather the functions,"},{"Start":"05:25.580 ","End":"05:27.470","Text":"then there\u0027s only 2 possibilities,"},{"Start":"05:27.470 ","End":"05:29.105","Text":"either a max or min."},{"Start":"05:29.105 ","End":"05:38.760","Text":"In our case, we want the maximum of the objective function,"},{"Start":"05:38.760 ","End":"05:41.160","Text":"which is x squared, y,"},{"Start":"05:41.160 ","End":"05:49.410","Text":"subject to the constraint function being 0."},{"Start":"05:49.410 ","End":"05:52.470","Text":"This just comes out to be the same."},{"Start":"05:52.470 ","End":"05:54.110","Text":"To say that this is 0,"},{"Start":"05:54.110 ","End":"05:56.660","Text":"is to say that this is 2.4."},{"Start":"05:56.660 ","End":"06:00.420","Text":"It\u0027s just a matter of 2.4 being on one side or the other."},{"Start":"06:07.940 ","End":"06:17.220","Text":"Once we have the constraint function and the objective function,"},{"Start":"06:18.170 ","End":"06:20.550","Text":"I see they\u0027re written here,"},{"Start":"06:20.550 ","End":"06:25.010","Text":"then the problem is always the same except for the variation of maximum or minimum."},{"Start":"06:25.010 ","End":"06:29.000","Text":"In our case, it\u0027s a maximum problem so it\u0027s always we"},{"Start":"06:29.000 ","End":"06:33.650","Text":"want the maximum of the objective of g of x, y,"},{"Start":"06:33.650 ","End":"06:36.810","Text":"which we usually write in curly brackets,"},{"Start":"06:36.810 ","End":"06:38.745","Text":"and then we write, s.t,"},{"Start":"06:38.745 ","End":"06:41.820","Text":"subject to, and then the constraint,"},{"Start":"06:41.820 ","End":"06:44.450","Text":"and the constraint is that f of x,"},{"Start":"06:44.450 ","End":"06:47.160","Text":"y is equal to 0."},{"Start":"06:47.680 ","End":"06:51.485","Text":"It\u0027s always this way or with a minimum here."},{"Start":"06:51.485 ","End":"06:53.120","Text":"In our particular case,"},{"Start":"06:53.120 ","End":"06:56.120","Text":"we just replace g and f with what they are."},{"Start":"06:56.120 ","End":"07:03.420","Text":"We want to find the maximum of the function x squared,"},{"Start":"07:03.420 ","End":"07:12.180","Text":"y subject to the constraint that f of x, y is 0,"},{"Start":"07:12.180 ","End":"07:17.820","Text":"means that this is 0.4x plus"},{"Start":"07:17.820 ","End":"07:26.525","Text":"0.2y minus 2.4 is equal to 0."},{"Start":"07:26.525 ","End":"07:32.255","Text":"Or I could just write as this equals 2.4 or minus 2.4 equals 0."},{"Start":"07:32.255 ","End":"07:33.635","Text":"It doesn\u0027t really matter."},{"Start":"07:33.635 ","End":"07:38.855","Text":"This is the formulation of the word problem as a mathematical problem."},{"Start":"07:38.855 ","End":"07:43.940","Text":"As before, we\u0027re going to stop here because in the following clip,"},{"Start":"07:43.940 ","End":"07:45.395","Text":"we\u0027ll learn how to solve."},{"Start":"07:45.395 ","End":"07:49.505","Text":"Our goal now is to do several examples of a word problem,"},{"Start":"07:49.505 ","End":"07:53.510","Text":"which we then formulate as a mathematical problem."},{"Start":"07:53.510 ","End":"07:59.435","Text":"The maximum of an objective function subject to a constraint."},{"Start":"07:59.435 ","End":"08:01.640","Text":"Let\u0027s get onto the next,"},{"Start":"08:01.640 ","End":"08:03.475","Text":"that would be example 3."},{"Start":"08:03.475 ","End":"08:06.480","Text":"There something I forgot,"},{"Start":"08:06.480 ","End":"08:09.060","Text":"sorry, with the last column in the table."},{"Start":"08:09.060 ","End":"08:10.670","Text":"Let me just complete that."},{"Start":"08:10.670 ","End":"08:15.830","Text":"The last column was supposed to be what we want to find the extremum of."},{"Start":"08:15.830 ","End":"08:19.760","Text":"In this case, it was the maximum of x squared,"},{"Start":"08:19.760 ","End":"08:24.060","Text":"y, which was our utility function."},{"Start":"08:27.470 ","End":"08:32.699","Text":"We had various combinations like, 4, 4,"},{"Start":"08:32.699 ","End":"08:37.860","Text":"which will give us a utility of 4 squared times 4 is 64, 6,"},{"Start":"08:37.860 ","End":"08:41.850","Text":"0 will give us a utility of x squared times 0,"},{"Start":"08:41.850 ","End":"08:43.930","Text":"well, this will be 0."},{"Start":"08:43.930 ","End":"08:48.205","Text":"The last example, 1 squared times 10 is 10."},{"Start":"08:48.205 ","End":"08:51.379","Text":"Although each of these has the same cost,"},{"Start":"08:51.379 ","End":"08:54.150","Text":"they all cost $2.40,"},{"Start":"08:54.150 ","End":"09:00.850","Text":"they give us different levels of utility, happiness, usefulness, satisfaction."},{"Start":"09:00.850 ","End":"09:04.790","Text":"From these 3, this of course is the highest and we would take this one,"},{"Start":"09:04.790 ","End":"09:07.450","Text":"it gives us most utility."},{"Start":"09:07.450 ","End":"09:12.065","Text":"This was just the table for example and I forgot to complete it and I apologize."},{"Start":"09:12.065 ","End":"09:15.750","Text":"Now we can get on to problem 3."}],"ID":8873},{"Watched":false,"Name":"worked example 3","Duration":"7m 12s","ChapterTopicVideoID":8778,"CourseChapterTopicPlaylistID":114747,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.445","Text":"In this problem number 3,"},{"Start":"00:02.445 ","End":"00:04.635","Text":"we have a chair factory,"},{"Start":"00:04.635 ","End":"00:06.330","Text":"factory that makes chairs,"},{"Start":"00:06.330 ","End":"00:11.220","Text":"and it has a production function given by f of K and L equals K"},{"Start":"00:11.220 ","End":"00:16.319","Text":"over L. I\u0027m not from economics myself or production,"},{"Start":"00:16.319 ","End":"00:25.335","Text":"but I know that K and L stand for capital and labor."},{"Start":"00:25.335 ","End":"00:28.395","Text":"Capital in German at any rate."},{"Start":"00:28.395 ","End":"00:33.090","Text":"For example, K could be the number of machines"},{"Start":"00:33.090 ","End":"00:41.110","Text":"and L could be how many workers there are."},{"Start":"00:41.810 ","End":"00:47.595","Text":"What matters is that there are 2 factors for production, K and L,"},{"Start":"00:47.595 ","End":"00:51.890","Text":"and the number of chairs produced per hour as a function of"},{"Start":"00:51.890 ","End":"00:56.540","Text":"K and L is given by K over L. Now,"},{"Start":"00:56.540 ","End":"00:58.355","Text":"the production factor k,"},{"Start":"00:58.355 ","End":"01:02.660","Text":"say the number of machines is $6 an hour,"},{"Start":"01:02.660 ","End":"01:05.675","Text":"I think the P stands for price, but I\u0027m not sure."},{"Start":"01:05.675 ","End":"01:09.559","Text":"The production factor L for labor,"},{"Start":"01:09.559 ","End":"01:11.810","Text":"say number of employees,"},{"Start":"01:11.810 ","End":"01:15.005","Text":"each employee might be $8 an hour."},{"Start":"01:15.005 ","End":"01:23.660","Text":"The factory wants to produce 100 shares an hour."},{"Start":"01:23.660 ","End":"01:29.285","Text":"We have to formulate the factory\u0027s problem mathematically."},{"Start":"01:29.285 ","End":"01:36.500","Text":"That is in terms of a constraint function and an optimization function."},{"Start":"01:36.500 ","End":"01:40.985","Text":"Like find the maximum of some function"},{"Start":"01:40.985 ","End":"01:45.695","Text":"under the constraint or subject to the following like we did before."},{"Start":"01:45.695 ","End":"01:48.815","Text":"Let\u0027s examine this and see what we have here."},{"Start":"01:48.815 ","End":"01:55.745","Text":"Just to be definite, let\u0027s say that K is the number of"},{"Start":"01:55.745 ","End":"02:05.500","Text":"machines and that L is the number of workers."},{"Start":"02:06.110 ","End":"02:11.700","Text":"It looked it up that P is price or cost."},{"Start":"02:11.700 ","End":"02:14.210","Text":"A machine is $6 an hour,"},{"Start":"02:14.210 ","End":"02:17.345","Text":"a worker is $8 an hour."},{"Start":"02:17.345 ","End":"02:21.830","Text":"This function tells us how many chairs we produce per"},{"Start":"02:21.830 ","End":"02:25.985","Text":"hour given the number of machines and the number of workers."},{"Start":"02:25.985 ","End":"02:28.490","Text":"Seems a strange function to me because"},{"Start":"02:28.490 ","End":"02:31.700","Text":"the more workers it seems that we produce less chairs,"},{"Start":"02:31.700 ","End":"02:36.030","Text":"but it\u0027s just an abstract problem."},{"Start":"02:39.530 ","End":"02:43.535","Text":"First thing we want to do is write solution."},{"Start":"02:43.535 ","End":"02:45.560","Text":"But after we\u0027ve written that,"},{"Start":"02:45.560 ","End":"02:48.635","Text":"what we want to do is find the constraint."},{"Start":"02:48.635 ","End":"02:50.630","Text":"Usually, we start with the constraint."},{"Start":"02:50.630 ","End":"02:54.200","Text":"The constraint is, what\u0027s written here"},{"Start":"02:54.200 ","End":"02:58.280","Text":"is that the factory wants to produce 100 chairs an hour,"},{"Start":"02:58.280 ","End":"03:00.815","Text":"and the production function is f,"},{"Start":"03:00.815 ","End":"03:08.000","Text":"which means that the constraint is given"},{"Start":"03:08.000 ","End":"03:16.980","Text":"by K over L is equal to 100."},{"Start":"03:17.860 ","End":"03:27.120","Text":"For example, K could be 100 and L could be 100 machines,"},{"Start":"03:27.120 ","End":"03:33.840","Text":"1 worker, or we could take 1,000 machines and 10 workers."},{"Start":"03:33.840 ","End":"03:42.635","Text":"There\u0027s all sorts of possibilities which will give us that K over L is 100."},{"Start":"03:42.635 ","End":"03:48.785","Text":"Now, what would we like to maximize or minimize?"},{"Start":"03:48.785 ","End":"03:52.325","Text":"Or the talks here about money and cost per hour."},{"Start":"03:52.325 ","End":"04:00.125","Text":"I think a sane factory would want to minimize the cost per hour."},{"Start":"04:00.125 ","End":"04:03.200","Text":"Let\u0027s assume that\u0027s what we\u0027re going to do."},{"Start":"04:03.200 ","End":"04:07.380","Text":"Our objective"},{"Start":"04:10.060 ","End":"04:17.820","Text":"is the cost per hour."},{"Start":"04:18.640 ","End":"04:24.740","Text":"I\u0027ll just write down, we want to do cost per hour."},{"Start":"04:24.740 ","End":"04:29.530","Text":"That\u0027s equal to K machines is $6 an hour,"},{"Start":"04:29.530 ","End":"04:35.365","Text":"so it\u0027s 6 times K. Let\u0027s work in dollars per hour."},{"Start":"04:35.365 ","End":"04:39.610","Text":"Cost in dollars per hour, so we have 6K,"},{"Start":"04:39.610 ","End":"04:46.840","Text":"L workers at $8 an hour is 8L,"},{"Start":"04:46.840 ","End":"04:47.860","Text":"costs to the machines,"},{"Start":"04:47.860 ","End":"04:49.150","Text":"costs to the workers,"},{"Start":"04:49.150 ","End":"04:52.420","Text":"and that\u0027s the total cost per hour."},{"Start":"04:52.420 ","End":"04:57.745","Text":"What we want to do is minimize this."},{"Start":"04:57.745 ","End":"05:03.430","Text":"Previously, we called the constraint function f and the objective g,"},{"Start":"05:03.430 ","End":"05:06.500","Text":"but f is taken already."},{"Start":"05:06.500 ","End":"05:09.800","Text":"Let\u0027s use a different letter."},{"Start":"05:09.800 ","End":"05:15.065","Text":"Let\u0027s say the constraint function h of"},{"Start":"05:15.065 ","End":"05:21.140","Text":"K and L is K over L minus 100."},{"Start":"05:21.140 ","End":"05:26.555","Text":"Remember, we just put everything on 1 side and call that the constraint function."},{"Start":"05:26.555 ","End":"05:30.035","Text":"The objective function g is still 3."},{"Start":"05:30.035 ","End":"05:32.115","Text":"G of K and L,"},{"Start":"05:32.115 ","End":"05:35.120","Text":"this is the thing we want to maximize or minimize, and in this case,"},{"Start":"05:35.120 ","End":"05:42.660","Text":"minimize, is 6K plus 8L."},{"Start":"05:43.750 ","End":"05:51.185","Text":"Now, we just follow the standard formulation."},{"Start":"05:51.185 ","End":"05:54.290","Text":"We always say that the problem is 2."},{"Start":"05:54.290 ","End":"05:56.630","Text":"The only variation is maximize or minimize."},{"Start":"05:56.630 ","End":"06:06.770","Text":"In this case, we want to minimize the 6K plus 8L subject to,"},{"Start":"06:06.770 ","End":"06:08.645","Text":"and this is where we put the constraint,"},{"Start":"06:08.645 ","End":"06:11.105","Text":"the constraint is equal to 0,"},{"Start":"06:11.105 ","End":"06:18.320","Text":"subject to K over L minus 100 equals 0,"},{"Start":"06:18.320 ","End":"06:21.740","Text":"which I prefer to write as this."},{"Start":"06:21.740 ","End":"06:24.485","Text":"If we wanted to write it more abstractly,"},{"Start":"06:24.485 ","End":"06:28.365","Text":"we would say in general,"},{"Start":"06:28.365 ","End":"06:35.510","Text":"minimize g of K L subject"},{"Start":"06:35.510 ","End":"06:43.305","Text":"to h of K L equals 0."},{"Start":"06:43.305 ","End":"06:45.980","Text":"That\u0027s the formulation. As I said,"},{"Start":"06:45.980 ","End":"06:49.430","Text":"the way of solving it is fairly routine,"},{"Start":"06:49.430 ","End":"06:51.805","Text":"but that will be in the following clip."},{"Start":"06:51.805 ","End":"06:54.940","Text":"We just leave it as this."},{"Start":"06:55.180 ","End":"07:00.140","Text":"More specifically, I\u0027d say the top row is"},{"Start":"07:00.140 ","End":"07:04.955","Text":"the mathematical formulation for our particular problem."},{"Start":"07:04.955 ","End":"07:11.850","Text":"Okay. Let\u0027s do problem 4 now."}],"ID":8874},{"Watched":false,"Name":"worked example 4","Duration":"3m 57s","ChapterTopicVideoID":9791,"CourseChapterTopicPlaylistID":114747,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"Problem 4 is not from economics or production and stuff like that."},{"Start":"00:05.550 ","End":"00:08.804","Text":"It\u0027s just an abstract mathematical problem."},{"Start":"00:08.804 ","End":"00:11.985","Text":"We\u0027re given that 2 positive numbers,"},{"Start":"00:11.985 ","End":"00:14.910","Text":"have a product of 36."},{"Start":"00:14.910 ","End":"00:17.325","Text":"Product is the multiplication here."},{"Start":"00:17.325 ","End":"00:21.150","Text":"We have to find the minimum possible sum."},{"Start":"00:21.150 ","End":"00:27.375","Text":"For example, 2 times 18 is 36,"},{"Start":"00:27.375 ","End":"00:30.405","Text":"but their sum is 20."},{"Start":"00:30.405 ","End":"00:35.760","Text":"I could say 4 times 9 is 36 and their sum is 13."},{"Start":"00:35.760 ","End":"00:39.765","Text":"I could try 12 times 3,"},{"Start":"00:39.765 ","End":"00:44.250","Text":"is 36, their sum is 15."},{"Start":"00:44.250 ","End":"00:47.270","Text":"From all these possible pairs of numbers,"},{"Start":"00:47.270 ","End":"00:52.880","Text":"call them x and y, where x times y is 36,"},{"Start":"00:52.880 ","End":"00:56.790","Text":"we want to find the minimum x plus y."},{"Start":"00:58.040 ","End":"01:01.955","Text":"This is actually our constraint."},{"Start":"01:01.955 ","End":"01:04.055","Text":"The product of the numbers is 36."},{"Start":"01:04.055 ","End":"01:08.765","Text":"If we agree to call them x and y and remember that there are positive,"},{"Start":"01:08.765 ","End":"01:16.115","Text":"then we have that x times y is equal to 36."},{"Start":"01:16.115 ","End":"01:19.310","Text":"That\u0027s the constraint."},{"Start":"01:19.310 ","End":"01:25.624","Text":"What we want to find is in this case minimum,"},{"Start":"01:25.624 ","End":"01:35.250","Text":"and we want the minimum of x plus y."},{"Start":"01:35.740 ","End":"01:39.005","Text":"This is not the way we write it."},{"Start":"01:39.005 ","End":"01:44.310","Text":"We usually write, let me just reformulate this."},{"Start":"01:47.570 ","End":"01:51.160","Text":"We just change the order when we formulate it,"},{"Start":"01:51.160 ","End":"01:57.230","Text":"we say find the minimum of the objective,"},{"Start":"01:57.230 ","End":"01:59.420","Text":"which is the sum,"},{"Start":"01:59.420 ","End":"02:03.450","Text":"the minimum of x plus y."},{"Start":"02:03.450 ","End":"02:06.785","Text":"Then we put the letters st subject to,"},{"Start":"02:06.785 ","End":"02:14.970","Text":"and then we put the constraint subject to x times y equals 36."},{"Start":"02:14.970 ","End":"02:20.015","Text":"Together with this, we also define 2 functions,"},{"Start":"02:20.015 ","End":"02:24.140","Text":"one for the constraint called the constraint function,"},{"Start":"02:24.140 ","End":"02:28.895","Text":"and 1 for what we\u0027re trying to maximize or minimize called the objective function."},{"Start":"02:28.895 ","End":"02:35.150","Text":"Here we\u0027ll have the objective function,"},{"Start":"02:35.150 ","End":"02:39.380","Text":"and I\u0027ll call that f of x,"},{"Start":"02:39.380 ","End":"02:45.935","Text":"y, which is the x plus y."},{"Start":"02:45.935 ","End":"02:52.430","Text":"In this case, I want the minimum and the constraint function,"},{"Start":"02:52.430 ","End":"02:56.870","Text":"objective, I should write a function and just put the word f here."},{"Start":"02:56.870 ","End":"03:05.930","Text":"Objective function and constraint function is g of x,"},{"Start":"03:05.930 ","End":"03:12.690","Text":"y, which is x, y minus 36."},{"Start":"03:12.690 ","End":"03:14.060","Text":"Remember, when we write the function,"},{"Start":"03:14.060 ","End":"03:15.580","Text":"we don\u0027t want to an equation,"},{"Start":"03:15.580 ","End":"03:19.320","Text":"we want everything on 1 side to be 0."},{"Start":"03:20.740 ","End":"03:23.360","Text":"I shouldn\u0027t have been lazy."},{"Start":"03:23.360 ","End":"03:26.250","Text":"I should have written the word function. Hang on."},{"Start":"03:27.130 ","End":"03:31.835","Text":"This is the formulation of the problem in mathematical terms,"},{"Start":"03:31.835 ","End":"03:34.070","Text":"and we haven\u0027t yet learned how to solve it."},{"Start":"03:34.070 ","End":"03:36.095","Text":"That will be like a recipe."},{"Start":"03:36.095 ","End":"03:39.380","Text":"These 2 functions will play a big part in the recipe,"},{"Start":"03:39.380 ","End":"03:42.475","Text":"the objective function, and the constraint function."},{"Start":"03:42.475 ","End":"03:46.010","Text":"We\u0027ll rely on the fact that we have an f and"},{"Start":"03:46.010 ","End":"03:49.370","Text":"a g or whatever letters and that will help us."},{"Start":"03:49.370 ","End":"03:51.875","Text":"But that, as I say will be in the following clip."},{"Start":"03:51.875 ","End":"03:57.120","Text":"Now let\u0027s go on to problem Number 5. What else?"}],"ID":9687},{"Watched":false,"Name":"worked example 5","Duration":"3m 38s","ChapterTopicVideoID":9790,"CourseChapterTopicPlaylistID":114747,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.500","Text":"We\u0027re asked to find the maximum area of a rectangle with perimeter 24."},{"Start":"00:07.500 ","End":"00:11.220","Text":"I hope you remember what perimeter is."},{"Start":"00:11.220 ","End":"00:14.640","Text":"If not, no worry, I\u0027ll show you."},{"Start":"00:14.640 ","End":"00:17.500","Text":"We have a rectangle."},{"Start":"00:19.430 ","End":"00:23.225","Text":"Let\u0027s say the sides are x and y."},{"Start":"00:23.225 ","End":"00:26.975","Text":"Of course this is also x and this is also y."},{"Start":"00:26.975 ","End":"00:31.790","Text":"The perimeter is the length of the board,"},{"Start":"00:31.790 ","End":"00:34.085","Text":"or if I walk around it,"},{"Start":"00:34.085 ","End":"00:36.950","Text":"how much do I travel?"},{"Start":"00:36.950 ","End":"00:39.020","Text":"Or let\u0027s just put it abstractly."},{"Start":"00:39.020 ","End":"00:41.465","Text":"That\u0027s x and y and x and y."},{"Start":"00:41.465 ","End":"00:46.800","Text":"In other words, the perimeter is 2x plus 2y."},{"Start":"00:47.060 ","End":"00:49.830","Text":"We\u0027re talking also about area."},{"Start":"00:49.830 ","End":"00:52.565","Text":"Let\u0027s also remember the formula for the area."},{"Start":"00:52.565 ","End":"00:56.315","Text":"The area is x times y."},{"Start":"00:56.315 ","End":"01:01.310","Text":"Now, the constraint which is most important to what we do first,"},{"Start":"01:01.310 ","End":"01:05.105","Text":"the constraint is that the perimeter is equal to 24."},{"Start":"01:05.105 ","End":"01:08.410","Text":"2x plus 2y is 24."},{"Start":"01:08.410 ","End":"01:11.120","Text":"What we find, the maximum marble,"},{"Start":"01:11.120 ","End":"01:15.940","Text":"the objective is x times y."},{"Start":"01:15.940 ","End":"01:21.230","Text":"We\u0027ve already had enough practice to formulate this problem as we want to"},{"Start":"01:21.230 ","End":"01:26.720","Text":"find the maximum of"},{"Start":"01:26.720 ","End":"01:31.820","Text":"x times y subject"},{"Start":"01:31.820 ","End":"01:39.655","Text":"to 2x plus 2y"},{"Start":"01:39.655 ","End":"01:44.620","Text":"equals 24, that\u0027s the constraint."},{"Start":"01:44.620 ","End":"01:49.900","Text":"I think we have to say somewhere that x and y are positive."},{"Start":"01:49.900 ","End":"01:53.650","Text":"X and y are both positive if we come to"},{"Start":"01:53.650 ","End":"01:57.985","Text":"that because length in geometry are usually positive."},{"Start":"01:57.985 ","End":"02:00.770","Text":"I\u0027ll just write that."},{"Start":"02:01.430 ","End":"02:06.685","Text":"That\u0027s basically it. Of course,"},{"Start":"02:06.685 ","End":"02:08.560","Text":"we haven\u0027t done the step of solving it,"},{"Start":"02:08.560 ","End":"02:11.960","Text":"but like I said, that will be in the following clip."},{"Start":"02:12.090 ","End":"02:17.590","Text":"It\u0027s customary to also write the functions,"},{"Start":"02:17.590 ","End":"02:21.295","Text":"the constraint function and the objective function."},{"Start":"02:21.295 ","End":"02:31.040","Text":"The objective function and we use these 2 functions in the solution,"},{"Start":"02:31.040 ","End":"02:33.334","Text":"which I haven\u0027t told you yet in the recipe."},{"Start":"02:33.334 ","End":"02:38.300","Text":"Objective function is the 1 with the maximum or the minimum."},{"Start":"02:38.300 ","End":"02:42.035","Text":"In this case it\u0027s x times y,"},{"Start":"02:42.035 ","End":"02:48.750","Text":"or it\u0027s called given the letter f of x, y, let\u0027s call it,"},{"Start":"02:48.750 ","End":"02:56.120","Text":"and the constraint function is the 1 we get from our constraint."},{"Start":"02:56.120 ","End":"02:59.030","Text":"But instead of an equation,"},{"Start":"02:59.030 ","End":"03:01.280","Text":"we want something equals 0."},{"Start":"03:01.280 ","End":"03:08.675","Text":"In this case, we just want the 2x plus 2y minus 24,"},{"Start":"03:08.675 ","End":"03:11.500","Text":"let\u0027s call that g of x, y."},{"Start":"03:11.500 ","End":"03:16.220","Text":"Again, 2x plus 2y minus 24,"},{"Start":"03:16.220 ","End":"03:20.585","Text":"that\u0027s the bit that we\u0027re going to force to equal 0."},{"Start":"03:20.585 ","End":"03:27.530","Text":"In general, we want maximum of the objective function, or minimum."},{"Start":"03:27.530 ","End":"03:32.645","Text":"Another problem subject to the constraint function equaling 0."},{"Start":"03:32.645 ","End":"03:34.415","Text":"I think we\u0027ll do 1 more."},{"Start":"03:34.415 ","End":"03:38.340","Text":"Let\u0027s just do Problem 7 and call it a day."}],"ID":9686},{"Watched":false,"Name":"worked example 6","Duration":"5m 50s","ChapterTopicVideoID":8779,"CourseChapterTopicPlaylistID":114747,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"The last 1. Among all points on the circle,"},{"Start":"00:04.590 ","End":"00:07.890","Text":"x squared plus y squared equals 32."},{"Start":"00:07.890 ","End":"00:13.455","Text":"We have to find the ones whose coordinates have the maximum sum,"},{"Start":"00:13.455 ","End":"00:17.460","Text":"and then the second problem is the same thing with minimum sum."},{"Start":"00:17.460 ","End":"00:20.010","Text":"That\u0027s what respectively means 1 time maximum,"},{"Start":"00:20.010 ","End":"00:21.210","Text":"1 time with minimum."},{"Start":"00:21.210 ","End":"00:23.650","Text":"This 2 in 1."},{"Start":"00:24.050 ","End":"00:28.695","Text":"Let me draw a sketch of this, makes it easier."},{"Start":"00:28.695 ","End":"00:33.555","Text":"Here is a rough sketch of this circle."},{"Start":"00:33.555 ","End":"00:40.065","Text":"The radius of the circle would be square root of 32."},{"Start":"00:40.065 ","End":"00:44.045","Text":"Because in general, when we have a circle,"},{"Start":"00:44.045 ","End":"00:49.300","Text":"its equation is x squared plus y squared is r squared."},{"Start":"00:49.300 ","End":"00:52.565","Text":"That\u0027s assuming it\u0027s centered at the origin, of course."},{"Start":"00:52.565 ","End":"00:55.115","Text":"In this case r squared is 32."},{"Start":"00:55.115 ","End":"00:58.430","Text":"The radius is square root of 32,"},{"Start":"00:58.430 ","End":"01:00.260","Text":"which means, for example,"},{"Start":"01:00.260 ","End":"01:07.325","Text":"that this point here is the point square root of 32,"},{"Start":"01:07.325 ","End":"01:11.130","Text":"0, and this point would be 0,"},{"Start":"01:11.130 ","End":"01:13.580","Text":"square root of 32."},{"Start":"01:13.580 ","End":"01:15.650","Text":"Other points on the circle,"},{"Start":"01:15.650 ","End":"01:18.800","Text":"because 32 is 16 plus 16 then, 4,"},{"Start":"01:18.800 ","End":"01:25.050","Text":"4 and would be on the circle,"},{"Start":"01:25.050 ","End":"01:27.285","Text":"and I guess so with all the others,"},{"Start":"01:27.285 ","End":"01:34.725","Text":"the 4 minus 4 and minus 4,4 and minus 4."},{"Start":"01:34.725 ","End":"01:37.545","Text":"Minus 4 is another example."},{"Start":"01:37.545 ","End":"01:41.650","Text":"There\u0027s an infinite number of points on this circle."},{"Start":"01:41.650 ","End":"01:45.875","Text":"In fact, this equation is the constraint of the problem."},{"Start":"01:45.875 ","End":"01:48.440","Text":"Usually, when you get some explicit formula,"},{"Start":"01:48.440 ","End":"01:51.860","Text":"it\u0027s the constraint, and the constraint is what we\u0027re really looking for."},{"Start":"01:51.860 ","End":"01:54.905","Text":"Now, for each of these points,"},{"Start":"01:54.905 ","End":"02:01.335","Text":"we could compute some of coordinates."},{"Start":"02:01.335 ","End":"02:03.060","Text":"For example, for this 1,"},{"Start":"02:03.060 ","End":"02:06.570","Text":"4 plus 4 would be 8."},{"Start":"02:06.570 ","End":"02:08.660","Text":"For this 1, we\u0027d get minus 4,"},{"Start":"02:08.660 ","End":"02:12.095","Text":"minus 4, which is minus 8."},{"Start":"02:12.095 ","End":"02:19.535","Text":"At this point, we\u0027d get 0 plus square root of 32,"},{"Start":"02:19.535 ","End":"02:22.500","Text":"which is, I don\u0027t know,"},{"Start":"02:22.500 ","End":"02:28.455","Text":"but it\u0027s between 5 and 6 because this is between 25 and 36."},{"Start":"02:28.455 ","End":"02:32.570","Text":"It\u0027s I don\u0027t know exactly what it is,"},{"Start":"02:32.570 ","End":"02:36.690","Text":"but let\u0027s say it\u0027s,"},{"Start":"02:36.690 ","End":"02:46.510","Text":"I don\u0027t know, 5.6 or something, and so on."},{"Start":"02:46.510 ","End":"02:48.055","Text":"I guess at this point,"},{"Start":"02:48.055 ","End":"02:52.420","Text":"minus 4 for the sum would be 0, and so on."},{"Start":"02:52.420 ","End":"02:53.920","Text":"For each point on the circle,"},{"Start":"02:53.920 ","End":"02:56.330","Text":"we can compute the sum of the coordinates,"},{"Start":"02:56.330 ","End":"02:58.510","Text":"and as it looks now,"},{"Start":"02:58.510 ","End":"03:00.700","Text":"this is the biggest we\u0027ve found so far."},{"Start":"03:00.700 ","End":"03:04.450","Text":"This looks like it might be the maximum and this might be the minimum."},{"Start":"03:04.450 ","End":"03:11.475","Text":"But will this is what we have to solve the problem to do."},{"Start":"03:11.475 ","End":"03:14.890","Text":"Anyway, I\u0027m just illustrating the concept."},{"Start":"03:16.520 ","End":"03:20.230","Text":"As always, the main things we need for"},{"Start":"03:20.230 ","End":"03:23.245","Text":"this first part of the problem for the formulation,"},{"Start":"03:23.245 ","End":"03:30.585","Text":"is the constraint and the objective function."},{"Start":"03:30.585 ","End":"03:37.550","Text":"It\u0027s quite easy to see that what we want to do is the maximum"},{"Start":"03:37.550 ","End":"03:47.580","Text":"of the function x plus y because x and y are the coordinates,"},{"Start":"03:47.580 ","End":"03:57.825","Text":"subject to the constraint that x squared plus y squared equals 32."},{"Start":"03:57.825 ","End":"04:02.255","Text":"Then, the same thing for minimum."},{"Start":"04:02.255 ","End":"04:04.430","Text":"I\u0027m not going to copy the whole thing again,"},{"Start":"04:04.430 ","End":"04:06.560","Text":"there\u0027s 2 problems in 1,"},{"Start":"04:06.560 ","End":"04:10.790","Text":"but it\u0027s the same objective function."},{"Start":"04:10.790 ","End":"04:12.410","Text":"It\u0027s just that once we want the maximum,"},{"Start":"04:12.410 ","End":"04:13.835","Text":"once you want the minimum."},{"Start":"04:13.835 ","End":"04:19.095","Text":"If I write the 2 functions as f of x and g of x,"},{"Start":"04:19.095 ","End":"04:22.800","Text":"f of x will be the constraint function,"},{"Start":"04:22.800 ","End":"04:29.795","Text":"will be x squared plus y squared minus 32."},{"Start":"04:29.795 ","End":"04:31.340","Text":"Remember for the constraint function,"},{"Start":"04:31.340 ","End":"04:34.120","Text":"you want to set something equal 0,"},{"Start":"04:34.120 ","End":"04:38.160","Text":"and the objective function,"},{"Start":"04:38.160 ","End":"04:40.829","Text":"the 1 we want to maximize,"},{"Start":"04:40.829 ","End":"04:44.070","Text":"add, or minimize is the other 1,"},{"Start":"04:44.070 ","End":"04:49.305","Text":"is just x plus y,"},{"Start":"04:49.305 ","End":"04:53.510","Text":"and I should have written the function of 2 variables."},{"Start":"04:53.510 ","End":"04:58.355","Text":"Hang on. Just go over the main things again."},{"Start":"04:58.355 ","End":"05:04.265","Text":"This bit is called the objective."},{"Start":"05:04.265 ","End":"05:07.670","Text":"This is the constraint."},{"Start":"05:07.670 ","End":"05:13.190","Text":"Altogether, this line is the mathematical formulation of the word problem."},{"Start":"05:13.190 ","End":"05:19.850","Text":"This is the constraint function comes from the constraint."},{"Start":"05:19.850 ","End":"05:23.510","Text":"This is the objective function comes from the objective,"},{"Start":"05:23.510 ","End":"05:27.200","Text":"and as I keep saying in the next clip,"},{"Start":"05:27.200 ","End":"05:32.210","Text":"we\u0027ll learn how to solve such a problem."},{"Start":"05:32.210 ","End":"05:36.680","Text":"Once we have the functions f and g, the constraint,"},{"Start":"05:36.680 ","End":"05:39.440","Text":"and objective functions, then we can"},{"Start":"05:39.440 ","End":"05:44.030","Text":"solve the problems using a standard recipe if you like."},{"Start":"05:44.030 ","End":"05:49.650","Text":"I\u0027m done here, for this clip."}],"ID":8875}],"Thumbnail":null,"ID":114747}]

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1.1

1