Math 51 Second Exam May 18, 2017

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1 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 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 9 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, June 1, 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, Spring 2017 Second Exam May 18, 2017 Page 1 of 9 1. (8 points) Let S be the surface in R 3 with equation z = x 3 + y 3. (a) Find the equation of the tangent plane to S at the point where (x, y) = (1, 2). Show all of your steps. (b) Use linear approximation to estimate whether is bigger or smaller than = 3. Show all the steps of your reasoning.

3 Math 51, Spring 2017 Second Exam May 18, 2017 Page 2 of 9 2. (12 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.

4 Math 51, Spring 2017 Second Exam May 18, 2017 Page 3 of 9 Please refer to the level-set diagram of the function f : R 2 R, as well as the stated assumptions on f, as given on the facing page. For the questions below, you do not need to justify your answers. (a) Sketch, on the plot, the direction of the gradient vector f at D. (b) Sketch, on the plot, the direction of the gradient vector f at G. (c) (Circle one) f x at B is: NEGATIVE ZERO POSITIVE (d) (Circle one) f y at A is: NEGATIVE ZERO POSITIVE [ ] (e) (Circle one) Let u = ; then D 1 u f at A is: NEGATIVE ZERO POSITIVE [ ] (f) (Circle one) Let v = ; then D 1 v f at H is: NEGATIVE ZERO POSITIVE (g) Which partial derivative is larger, in absolute value? Circle one: f x (A) f x (C) (h) Which partial derivative is larger, in absolute value? Circle one: f y (H) f y (J) (i) (Circle one) 2 f at B is: NEGATIVE POSITIVE x2 (j) (Circle one) 2 f at A is: NEGATIVE POSITIVE y2 (k) (Circle one) 2 f x y at H is: NEGATIVE POSITIVE [ ] (l) (Circle one) Let v = and M = Hf at C; then v 1 T Mv is: NEGATIVE POSITIVE (Here Hf is the Hessian matrix of f.)

5 Math 51, Spring 2017 Second Exam May 18, 2017 Page 4 of 9 3. (10 points) Let a be the point (7, 8, 9) in R 3. Suppose all we know about a certain differentiable function G : R 3 R 3 is: G(a) = (0, 0, 0), and 3/2 0 0 the derivative matrix of G at a is DG(a) = 0 1/ In addition, let F : R 3 R 3 be defined by: F (x) = x G(x). (a) Compute DF (a), the derivative matrix of F at a = (7, 8, 9). (b) Start with b = (7.1, 8.1, 9.1) and apply F to it 10 times: y = F (F (F (... (F (b))))). }{{} 10 times Estimate y using all information available; give all reasoning. (c) Is your estimate for the point y: (i) near a and b, but much closer to a than to b; (ii) near a and b, but much closer to b than to a; (iii) much closer to (0, 0, 0) than to either of a or b; or (iv) far away from all of these points? Explain your reasoning completely.

6 Math 51, Spring 2017 Second Exam May 18, 2017 Page 5 of 9 4. (12 points) Suppose the air temperature at any point in R 3, in degrees, is given by f(x, y, z) = x 2 y 2 + 4xz (a) A fly is at the point (1, 1, 1); in what unit-vector direction should it start flying in order to cool down as quickly as possible? (b) Instead, suppose the fly at (1, 1, 1) wishes to maintain a constant temperature. Give an example of a direction in which it could start flying. (c) Can the fly start flying away from (1, 1, 1) in such a way that it can maintain both a constant temperature and a constant distance from the origin? If so, give an example of a direction in which it can start flying to do this; if not, explain why not.

7 Math 51, Spring 2017 Second Exam May 18, 2017 Page 6 of (10 points) Suppose A is the 4 4 matrix for which A 1 = (a) Find all solutions x to the equation Ax = (b) It is a fact that det(a 1 ) = 11; you do not have to prove this. What is det(2a)? Explain. (c) Find det(rref(a)); give all reasoning.

8 Math 51, Spring 2017 Second Exam May 18, 2017 Page 7 of 9 6. (7 points) Let C be the curve in R 2 with equation (x 3) 2 + (y 7) 2 = 5. [ ] 3 2 (a) Let T : R 2 R 2 be the linear transformation whose matrix is given by A =. Determine 3 1 an equation satisfied by the points of T (C), the image of C under T ; show all reasoning. Your final answer does not have to be algebraically simplified (but it should not involve matrices). (b) The curve C is a circle of radius 5, and the curve T (C) is an ellipse (you don t have to prove these facts). What is the area of the region enclosed by the ellipse T (C)?

9 Math 51, Spring 2017 Second Exam May 18, 2017 Page 8 of 9 7. (12 points) Let S be the surface in R 3 with equation x 2 + y 2 + z 2 = 14. (a) Find an equation of the plane tangent to S at the point a = (1, 3, 2). (b) Let P be the plane in R 3 with equation x + y + z = 0. Determine, in parametric form, a line lying in P that also lies in the tangent plane you found in part (a). (c) The plane P intersects the surface S to make a curve C; note that this curve contains the point a. If an ant positioned on S at a starts crawling so that its y-coordinate changes by 0.01, what are the approximate changes it needs to make in each of its x- and z-coordinates so that it can remain on the curve C?

10 Math 51, Spring 2017 Second Exam May 18, 2017 Page 9 of 9 8. (10 points) Let f(x, y) = 2x 4 + 6xy 2y 4. (a) The equation f(x, y) = 2 defines a curve C in R 2. Find, showing all steps, an equation of the line tangent to C at (1, 0). (b) Find Hf(1, 0), the Hessian matrix of f at (1, 0). Show all steps. (c) Use quadratic (second-order Taylor) approximation to estimate the value of f(1.1, 0.1); simplify your answer as much as possible.

Math 51 First Exam October 19, 2017

Math 51 First Exam October 19, 2017 Math 5 First Exam October 9, 27 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