Math 51 - Autumn 2010 - Fi nal Ex amName:Student ID:Select your section:Brandon Levin Amy Pang Yuncheng Lin Rebecca Bellovin05 (1:15-2:05 ) 14 (10:00-10:50 ) 06 (1:15-2:05 ) 09 (11:00-11:50 )15 (11:00- 11:50 ) 17 (1:15-2:05 ) 21 (11:00-11:50 AM) 23 (1:15-2:05 )Xin Zhou Simon Rubinstein-Salzedo Frederick Fong Jeff Danciger02 (11:00-11:50 ) 18 (2:15-3:05 ) 20 (10:00-10:50 ) ACE (1:15-3:05 )08 (10:00- 10:50 ) 24 (1:15-2:05 ) 03 (11:00-11:50 )Signature:Instructions:• Print your na me and student ID numb er, select your section number and TA’sname, and sign above t o indicate that you accept the Honor Code.• There are 11 problems on the pages numbered from 1 to 12, and each problem isworth 10 points. Please check that the version of the exam you have is completeand correctly stapled.• Read each question carefully. In order to receive full credit, please showall of your work and justify your answers unless specifically directedotherwise. If you use a result proved in class or in the text, you mustclearly state the result before applying it to your problem.• Unless otherwise specified, you may assume all vectors are written in standardcoordinates.• You have 3 hours. This is a closed-book, closed-notes exam. No calculators orother electronic aids will be permitted. If you finish early, you must hand yourexam paper to a member of the teaching staff.• If you need extra room, use the back sides of each page. If you must use extrapaper, make sure to write your name on it and a t t ach it to this exam. Do notunstaple or detach pa ges fr om this exam.Problem 1. Let V = Span1110, and let S be the set of all the vectors in R4whichare or t hogonal to V .a) Show that S is a subspace of R4.b) Find a matrix A with C(A) = V .1Problem 2. Suppose that {u, v} is a linearly independent set of vectors in Rn. For whatvalues o f t ∈ R is the set {v + tu, u − v} linearly independent?2Problem 3.a) Complete the following definition:A function f : X → Y is one-to-one ifb) Let L be a line through the origin in R2, and suppose that T : R2→ R2is projectionto L. Show that T is not one-to-one.3Problem 4. LetA =1 1 1 01 1 0 11 0 1 10 1 1 1.a) Is A invertible?b) Find the eigenvalues of A and compute the dimension of each eigenspace.4Problem 5. Let D = {(x, y) ∈ R2y ≥ 0}, and let f : D → R be defined byf(x, y) = ex√y.a) Find the linearization of f at (0, 1).b) Find the second order Taylor polynomial of f at (0, 1).c) Use the Taylor polynomial from the previous part to approximate e√2.5Problem 6.DefineA =0 1 21 0 12 1 0,and let Q : R3→ R be the quadratic form associated to A.a) Classify Q as positive definite, positive semidefinite, indefinite, negative semidefinite,or negative definite.b) Compute ∇Q(2, 1, 0).6Problem 7.Suppose t hat z(x, y) = x2+ y2, x(u, v) = uv, and y(u, v) = u2+ v.a) Compute∂z∂u(1, 0).b) Now suppose that u and v are functions of r, s, and t, withu(1, 2, 3) = 1, v(1, 2, 3) = 0,∂u∂r(1, 2, 3) = 2, and∂v∂r(1, 2, 3) = −1.Compute∂z∂r(1, 2, 3).7Problem 8. For each limit b elow, evaluate the limit or show it does not exist.a)lim(x,y)→(0,0)x3y2b)lim(x,y,z)→(0,0,0)yz2x2+ y2+ z28Problem 9. Let S be the surface defined byS = {(x, y, z)x2+ y2= 4z2+ 16.}.(This problem continues on the next page.)a) Define a function g : R3→ R with the property that S is a level set of g.b) Find the tangent plane to S at the point (4, 2, 1).9c) Let r(t) = (√20 cos t3,√20 sin t3, 1 ), and let t0∈ R satisfy r(t0) = (4, 2, 1). With g asin Part (a), find the directional derivative of g at (4, 2, 1) in the direction r′(t0).10Problem 10. Let g : R2→ R be defined by g(x, y) = ax2− 2ax −y2+ by2.a) Show that g has a critical po int at (1, 0).b) Under what conditions on the constants a and b does the Second Derivative Testguarantee that g has a local minimum at (1, 0)?11Problem 11. Let D be the discD = {(x, y)x2+ y2≤ 18 }and let f : D → R be defined by f (x, y) = x2+ y2+ 4x + 4y + 7.a) Explain why f must attain an absolute maximum on D.b) Find the point on D where f attains its absolute maximum.12The f ollowing boxes are strictly for grading purposes. Please do not mark.Question Score Maximum1 102 103 104 105 106 107 108 109 1010 1011 10Total
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