Math 51 Second Exam February 23, 2017

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1 Math 51 Second Exam February 23, 2017 Name: SUNet ID: ID #: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You may use any result proved in class or the text, but be sure to clearly state the result before using it, and to verify that all hypotheses are satisfied. Please check that your copy of this exam contains 12 numbered pages and is correctly stapled. This is a closed-book, closed-notes exam. No electronic devices, including cellphones, headphones, or calculation aids, will be permitted for any reason. You have 2 hours. Your organizer will signal the times between which you are permitted to be writing, including anything on this cover sheet, and to have the exam booklet open. During these times, the exam and all papers must remain in the testing room. When you are finished, you must hand your exam paper to a member of teaching staff. Paper not provided by teaching staff is prohibited. If you need extra room for your answers, use the extra space provided at the front of this packet, and clearly indicate that your answer continues there. Do not unstaple or detach pages from this exam. It is your responsibility to look over your graded exam in a timely manner. You have only until Thursday, March 9, to resubmit your exam for any regrade considerations; consult your section leader about the exact details of the submission process. Please sign the following: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. Signature:

2 Math 51, Winter 2017 Second Exam February 23, 2017 Page 1 of (12 points) (a) Let B be a 2 2 matrix such that Bv is given by rotating v R 2 by 45 degrees clockwise. Determine, with reasoning, the matrix B 4 = BBBB; simplify your answer as much as possible. (b) Let P be a 3 3 matrix for which: P v = v whenever v R 3 belongs to the xy-plane; and det(p ) = 0. Determine, with reasoning, as much of the matrix of P as you can. (The information you have been given is not enough to determine all the matrix entries; you can denote any matrix entry that you can t figure out by a? )

3 Math 51, Winter 2017 Second Exam February 23, 2017 Page 2 of 12 (Problem continuation: this page does not depend on the previous one.) (c) For this and part (d), let T : R 4 R 4 be the function which shifts each entry of a vector to the succeeding one, and puts the fourth component into the first component. For example, T(1, 2, 3, 4) = (4, 1, 2, 3) and T(5, 3, 1, 2) = (2, 5, 3, 1). It is a fact, which you do not have to prove, that T is a linear transformation. Determine, showing your steps, the matrix A of T. (d) Is the matrix A of part (c) invertible? Justify your answer.

4 Math 51, Winter 2017 Second Exam February 23, 2017 Page 3 of (10 points) Match the following functions f (x, y) to their level curves by writing the plot number in the blank to the right of the function. (In each diagram, lighter shades correspond to larger values of the function.) Every function has a matching picture, but two pictures will be unmatched; no justification is needed. (a) 2x + y 2x (e) sin(x) sin(y) (i) x + y2 + 1 (j) (b) y (x + 4)(x 4)x 30 (f) (c) sin( x + y ) (g) 1 x + y + 1 (d) sin(x + cos(y)) (h) x + cos(y) x p x + y x2 xy + y2 + 1

5 Math 51, Winter 2017 Second Exam February 23, 2017 Page 4 of (8 points) Below is a collection of level sets of a function f : R 2 R. You may assume that f and its first and second derivatives are continuous, and the length scales in the x- and y-directions are equal. For the questions below, you do not need to justify your answers; each has a unique best answer. (a) Draw, on the picture, a vector representing the direction of the gradient at point B. (b) (Circle one) f y at A is: NEGATIVE ZERO POSITIVE (c) (Circle one) f y at D is: NEGATIVE ZERO POSITIVE [ ] (d) (Circle one) Let v = ; then D 1 v f at A is: NEGATIVE ZERO POSITIVE [ ] (e) (Circle one) Let v = ; then D 1 v f at D is: NEGATIVE ZERO POSITIVE (f) (Circle one) 2 f at D is: NEGATIVE POSITIVE x2 (g) (Circle one) 2 f at C is: NEGATIVE POSITIVE x2 (h) (Circle one) 2 f x y at C is: NEGATIVE POSITIVE

6 Math 51, Winter 2017 Second Exam February 23, 2017 Page 5 of (12 points) Goldilocks has two dials which jointly control both the temperature t of her porridge (in F) and the width w of her chair (in inches). Specifically, if the first dial is set to a value x and the second dial is set to a value y, then Initially, both dials are set to 1. (a) If F : R 2 R 2 is the function t(x, y) = 100ye x cos(y 1) w(x, y) = 6x 2 + 6x + y. F (x, y) = [ ] t(x, y) w(x, y) calculate DF (1, 1), the derivative matrix of F evaluated at the point (x, y) = (1, 1). (b) Calculate F (1, 1).

7 Math 51, Winter 2017 Second Exam February 23, 2017 Page 6 of 12 (Problem continuation: see the facing page for formulas for F, your responses, etc.) (c) If Goldilocks moves the dials to and 0.999, use linear approximation to estimate the new values of t and w. (d) Goldilocks determines that the values t = and w = 13.1 are just right. Estimate the values x and y she should set on the dials to achieve this.

8 Math 51, Winter 2017 Second Exam February 23, 2017 Page 7 of (10 points) A certain city has two basketball teams, the Frogs and the Toads. Each citizen of the city is either a Frog supporter, a Toad supporter, or a non-watcher (doesn t watch basketball at all). Suppose the following: Every year, ten percent of Frog supporters give up on their team and become non-watchers; Every year, ten percent of Toad supporters give up on their team and become non-watchers; Yet hope springs eternal, for every year five percent of non-watchers become Frog supporters; another ten percent of non-watchers become Toad supporters. You may also ignore births, deaths, and migration; so there is no other mechanism for the number of each type of citizen to change. Let F, T and N respectively be the number of Frog supporters, the number of Toad supporters, and the number of Non-watchers. (a) Suppose that you know F, T and N for the year Write down a formula (involving matrices, vectors, and their operations) that tells you F, T and N for the year You do not need to work out a numerical answer. (b) Again suppose that you know F, T and N for Is it possible to determine, from this information, F, T and N for 2010? You do not need to work out a numerical formula if so; just explain completely whether it s possible or not. (c) Suppose additionally that the numbers F, T and N are actually stable from one year to the next (so there are the same number of Frog supporters in each year; same for Toads and non-watchers). [ FT ] Write down a nonzero 3 3 matrix B whose nullspace contains. You do not need to actually N compute the nullspace of B, but B should be given as a 3 3 matrix of numbers.

9 Math 51, Winter 2017 Second Exam February 23, 2017 Page 8 of 12 [ 3 6. (10 points) Let f : R 2 R 2 5 be the function f(x, y) = x y ] 4 5 x y. (a) Compute Df(x, y), the derivative matrix of f, at any point (x, y). [ t (b) Suppose g : R R 2 is the function g(t) = 3 + t 2 ]. Calculate the derivative matrix D(f g), 5 sin(t) evaluated at t = 1. [Hint: your answer should be a matrix with one column and two rows.] (c) Suppose h: R 2 R is some function such that h(2, 1) = vector (h f), evaluated at the point (x, y) = (2, 1). [ ] 7/5. Compute the gradient 1/5

10 Math 51, Winter 2017 Second Exam February 23, 2017 Page 9 of (10 points) A hill has elevation given by the function h(x, y) = 100 5x 2 2xy y 2. Here (x, y) measures position relative to the hill s summit; thus (0, 0) is the summit, x measures the distance east of the summit, and y measures distance north of the summit. x, y, h are all measured in meters. (a) Compute Dh(x, y), the derivative matrix of h at any point (x, y). (b) Gary the bear, being unimaginative, [ ] walks in a straight line, with his position at any time t R (in t 5 minutes) given by g(t) =. Gary s elevation at any given time is denoted f(t) = h(g(t)). 2t + 5 Compute df dt, the rate of change of Gary s elevation, as a function of t.

11 Math 51, Winter 2017 Second Exam February 23, 2017 Page 10 of 12 (Problem continuation: see the facing page for formulas, your responses, etc.) (c) Find a time t = t 0 at which df dt = 0; show your steps. (d) For the time t 0 of part (c), label Gary s position at time t 0 on the picture here (which depicts Gary s path and some level sets of h):

12 Math 51, Winter 2017 Second Exam February 23, 2017 Page 11 of (10 points) Shown below is a collection of level sets of the function f : R 2 R given by f(x, y) = xy 5x 2 5y 2. In addition, the point (0, 2) is marked. y x

13 Math 51, Winter 2017 Second Exam February 23, 2017 Page 12 of 12 For easy reference, f(x, y) = xy 5x 2 5y 2 ; please refer to the diagram of level sets for f on the facing page. (a) Compute the unit vector v that makes the directional derivative D v f(0, 2) as large as possible. Draw this vector on the diagram on the facing page, with its tail at the marked point (0, 2). [In your drawing, you need not worry about accurately depicting the length of the vector you ve computed.] (b) Calculate the equation of the tangent line to the curve f(x, y) = 0 at (0, 2). (c) An ant starts at the point (0, 2) in the plane. Wherever the ant is say, at position x it always walks in the direction of steepest ascent; i.e., in the unit direction w making the directional derivative D w f(x) as large as possible. If D w f(x) = 0 for every unit vector w, the ant stops. On the diagram on the facing page, draw the approximate path taken by the ant. [You do not need to explain your reasoning, or comment on whether this behavior is typical of ants.]

Math 51 Second Exam May 18, 2017

Math 51 Second Exam May 18, 2017 Name: SUNet ID: ID #: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify