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Unformatted text preview: 18.02 Problem Set 4 Due Thursday 10/2/08, 12:45 pm in 2-106. Part A (15 points) Hand in the underlined problems only; the others are for more practice. Lecture 10. Thu Sept. 25 Maxima and minima. Least squares. Read: 13.5 pp. 878881, 884885; Notes LS Work: 2F/ 1a b, 2 ; 2G/ 1a b, 4 . Lecture 11. Fri Sept. 26 Second derivative test. Boundaries and infinity. Read: 13.10 through the top of p. 930; Notes SD. Work: 2H/ 1ac , 3 , 4 , 6 ; 13.10/ 32. Lecture 12. Tue Sept. 30 Differentials. Chain rule. Read: 13.6 pp. 889892 ; 13.7. Work *: 2C/ 1a bcd , 2 , 3 , 5ab ; 2E/ 1abc , 2bc , 5, 8a b. Warning: Dont mix differentials like df with differences 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 definition of gradient in 13.8 (top of p. 910) to know what it is before you do them. Part B (26 points) 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 collabo- rated 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 = [ x 1 x 2 . . . x n ], y = [ y 1 y 2 . . . y n ] and u = [1 1 . . . 1] ( n ones). Let y = ax + b be the best-fitting line for the n points ( x i , y i ). Translate the formula (4) in LS into a single 2 2 matrix equation A z = r , z = bracketleftbigg a b bracketrightbigg Write the entries of A and r in Matlab-ready form. Dont use summations, instead use, for example, x * u for...
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This note was uploaded on 04/28/2009 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
- Fall '08
- Least Squares