MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Problem Set 4 Due Thursday 10 4 07 12 45 pm 15 points Part A Hand in the underlined problems only the others are for more practice Lecture 10 Thu Sept 27 Maxima and minima Least squares Read 13 5 pp 878 881 884 885 Notes LS Work 2F 1ab 2 2G 1ab 4 Lecture 11 Fri Sept 28 Second derivative test Boundaries and in nity Read 13 10 through the top of p 930 Notes SD Work 2H 1ac 3 4 6 13 10 32 Lecture 12 Tue Oct 2 Di erentials Chain rule Read 13 6 pp 889 892 13 7 Work 2C 1abcd 2 3 5ab 2E 1abc 2bc 5 8ab Warning Don t mix di erentials like df with di erences like x and y For instance equations 5 7 9 do not make sense Instead use 6 8 10 Some of the problems are written so as to depend on the notation for gradient Look ahead at the de nition of gradient in 13 8 top of p 910 to know what it is before you do them 25 points Part B Directions Attempt to solve each part of each problem yourself If you collaborate solutions must be written up independently It is illegal to consult materials from previous semesters With each problem is the day it can be done Write the names of all the people you consulted or with whom you collaborated and the resources you used Problem 1 Thursday 11 points 2 0 3 2 3 1 Least squares and data analysis Parts b f of this problem involve the use of Matlab You may optionally use any other software with similar features or even a calculator In that case indicate what you used and describe how you proceeded You must carry out the actual calculations rather than rely on the statistical functions that may be built into the software you are using a Before going to the terminal read Notes LS and do the following Consider the row vectors x x1 x2 xn y y1 y2 yn and u 1 1 1 n ones Let y ax b be the best tting line for the n points xi yi Translate the formula 4 in LS into a single 2 2 matrix equation a Az r z b Write the entries of A and r in Matlab ready form Don t use summations instead use for example x u for xi You will be able to con rm that your formulas are correct by testing them on a concrete example using Matlab in part c b The worldwide sales of iPods in million units for each year from 2001 to 2005 are given below sources Wikipedia Apple quarterly reports 1 Years xi Sales yi 01 0 125 02 0 470 03 1 451 04 8 263 05 31 960 To make the numerical answers easier to read we take the variable x to be the year minus 2000 so xi ranges from 1 to 5 for the given data points Look at the Matlab directions for plotting at the end of this problem set and make a scatter plot of these data points marked with s Nothing to hand in you will do this over again with part d c Use Matlab the data from b and the formulas you found in part a to nd the best line y ax b tted to the points Compute the di erence between the actual value of the data y and the predicted value y ax b Report a b and the worst case largest error Optional but recommended check that your answer for a and b agrees with the Matlab operation poly t x y 1 which computes the coe cients of the best polynomial of degree 1 tting the data x and y If c poly t x y 1 then c a b is the transpose of the column vector z in part a In this way you can con rm that you did part a correctly d When a new product is launched in the initial period the sales tend to grow expo nentially rather than linearly Use Matlab to nd the best t of the form ln y a1 x b1 Note Matlab uses the notation log for natural log and log10 for log in base 10 So in Matlab notation you will be using log y Report your values of a1 b1 If you exponentiate this equation you get y eb1 ea1 x Compute the di erence between y and the predicted value according to this formula and report the worst case largest error e Hand in a printout that shows on the same plot the scatter plot of x y labelled with s the straight line t x ax b as a dashed line and the curve x eb1 ea1 x connected by an ordinary line f According to the exponential best t how many iPods were sold in 2006 how many will be sold in 2015 for comparison the total world population is about 6 6 billions In fact the growth is no longer exponential only 46 4 million iPods were sold in 2006 Problem 2 Friday 8 points 2 2 2 2 Consider a triangle inscribed in the unit circle in the plane with one vertex at 1 0 and the two other vertices given by polar angles 1 and 2 in that order counterclockwise a Express the area A of the triangle in terms of 1 and 2 What is the set of possible values for 1 and 2 b Find the critical points of the function A in this region c By computing the values of A at the critical points and its behavior on the boundary of the region where it is de ned nd the maximum and the minimum of A justify your answer Describe the shapes of the triangles corresponding to these two situations d Use the second derivative test to con rm the nature of the critical points you found in b 2 Problem 3 Tuesday 6 points 1 2 2 1 see also 2E 5 a Let w f x y and suppose we change from rectangular to polar coordinates x r cos y r sin Using the chain rule derive the change of variables formula in matrix form wr wx A w wy writing the entries of the 2 2 matrix A as functions of r and x2 y 2 tan 1 y x to similarly derive the converse b Use the formulas r formula wx wr B wy w writing the entries of the 2 2 matrix B as functions of x and y c Check that B A 1 by computing the product A B and changing variables d If wr 2 and w 10 at the point of polar coordinates r 5 2 compute wx and wy at the same point 3 Matlab Directions You can reach Matlab in MIT Server by clicking …
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