*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **Math 51 Final Exam — June 6, 2008
Name : Section Leader: (Circle one) Fai Chandee Joseph Cheng 10:00 David Fernandez-Duque 11:00 Anca Vacarescu 1:15 Bezirgen Veliyev 2:15 Section Time: (Circle one) • Complete the following problems. You may use any result from class you like, but if you cite a theorem be sure to verify the hypotheses are satisﬁed. • 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 have 3 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you ﬁnish early, you must hand your exam paper to a member of teaching staﬀ. • If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. • 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: The following boxes are strictly for grading purposes. Please do not mark. 1 2 3 4 5 6 7 8 16 12 16 12 12 12 13 8 9 10 11 12 13 14 15 Total 12 15 12 15 15 15 15 200 Math 51, Spring 2008 Final Exam — June 6, 2008 Page 2 of 16 1 1. (16 points) Let f (x, y ) = x2 y − 4xy + y 2 + 1. 2 (a) Calculate formulas for the gradient of f and the Hessian matrix of f at the point (x, y ). (b) Find all critical points of f . (c) For each critical point, determine if it corresponds to a local maximum, local minimum or saddle point of f . Show your reasoning. (d) Write the quadratic approximation (that is, the degree-2 Taylor polynomial) for f at the point (x, y ) = (0, 0). Math 51, Spring 2008 Final Exam — June 6, 2008 Page 3 of 16 2. (12 points) Suppose S is the surface in R3 given by the equation xz 3 + yz 2 + x2 y = 18. (a) Find an equation of the plane tangent to S at (1, 2, 2). (b) Using linear approximations, estimate the z -coordinate of the point on the surface S that has x = 1.1 and y = 1.96. Math 51, Spring 2008 Final Exam — June 6, 2008 Page 4 of 16 3. (16 points) Let f (x, y ) be a function on R2 . Suppose that f (1, 1) = 6, and that we know the following information about the gradient of f : • • f (1, 1) · (1, 2) = 14, and f (1, 1) · (3, −1) = 0. Use this information to complete the following questions, showing all of your reasoning. (a) Find f (1, 1). (b) Estimate the value of f (1.02, 1.04). (c) If u can be any unit vector in R2 , what is the largest possible value for Du f (1, 1), the directional derivative of f at (1, 1) in the direction of u? (d) Find an equation of the line tangent to the level curve f (x, y ) = 6 at (1, 1). Math 51, Spring 2008 Final Exam — June 6, 2008 Page 5 of 16 4. (12 points) A circus performer is blowing up a sausage-shaped balloon for twisting into various animal shapes. At any point, the inﬂated portion consists of a cylinder of length h and radius r, and two hemispheres at either end; thus, its volume is the function 4 V (r, h) = πr2 h + πr3 . 3 (a) As the balloon is being inﬂated, we may view h and r as given by functions of time t; thus, volume is also a function of t. Suppose that at the time when the balloon’s length is 6 and the 1 radius is 1, the rates dr = 2 and dh = 3. Use the Chain Rule to ﬁnd the rate at which the dt dt volume of the balloon is increasing. (b) When the balloon is sealed shut, it has a length of 10 and a radius of 2. The performer begins to squeeze the balloon, slowly reducing its length by a rate dh = −1. Assuming the balloon’s dt volume remains constant and is described by the above formula, ﬁnd the rate at which the radius is increasing as the performer begins to squeeze. Math 51, Spring 2008 Final Exam — June 6, 2008 Page 6 of 16 5. (12 points) Suppose functions f : R3 → R3 and g : R3 → R are deﬁned by f (x, y, z ) = (x + y 2 , y + z 2 , z + x2 ) and g (x, y, z ) = ex+y+z (a) Find Df (1, 1, 1), the matrix of partial derivatives of f at the point (1, 1, 1). (b) Suppose h = g ◦ f . Find Dh(1, 1, 1). Math 51, Spring 2008 Final Exam — June 6, 2008 Page 7 of 16 6. (12 points) Find the point(s) in R2 lying on the line 7x + 12y = 120 where f (x, y ) = x7 y 3 is at a maximum. Show all reasoning. (You can take it as given that such a global maximum does exist.) Math 51, Spring 2008 Final Exam — June 6, 2008 Page 8 of 16 7. (13 points) Find the highest and lowest points (that is, the points with the largest and smallest z -coordinates) lying on the intersection of the two surfaces z = x2 + y 2 and 2x − y + z = 10 in R3 . You can take it as a fact that such points exist, but be sure to explain all reasoning. Math 51, Spring 2008 Final Exam — June 6, 2008 Page 9 of 16 Problem Problem 3)points) Problem 3) points) points) 3) (10 (10 (10 Problem 3) 8. (8points) Match each 3) (10 points) with its graph; keep in mind that one of the surfaces is not (10 points) Problem equation below
represented by an equation. No justiﬁcation is needed. (The coordinate axes are not shown so that Match the equation equation withbut the origindescribe x-y trace x-y trace intersection of surface the Match the equation with see, their graphs and the the x-y trace (the (the intersection ofsurface Match are easier to their graphs describe describe about the center of each graph, the the surfaces the with their graphs and and is located at the (the intersection of the theand surface Match the withwith withxy -plane) the equation withthewords in each each case. the of the surface intersection of the surfac equation xy -plane)points at and mostmost their graphscase. intersection x-y trace (the with their graphs at describe Match upward.) x-y each and the the the withwith with at threethree trace (the describe positive z -axis xy -plane)most three words in words in case. with the xy -plane) with atwith the xy -plane) in each case. three words in each case. most three words with at most II I II II II III III III I I II II III III I IV IV IV IV IV V V II V V V VI VI VI VI III VI Enter I,II,III,IV,V,VI herehere here Equation Describe Describetracex-y words in words Enter I,II,III,IV,V,VI Equation Enter I,II,III,IV,V,VI Equation Describe x-y x-y tracetrace the the the in in words Enter I,II,III,IV,V,VI here Enter I,II,III,IV,V,VI here Equation trace in words Equation Describe the x-y Describe the x-y trace in words
2 2 x2 −xy 2− xz 2− y 2 − z 2 = 1 2 I, II, III, −y2 − z1 = 1 2 =2 2 Equation − y − z = 1 x2 − y 2 − z 2 = 1 x IV V IV, or V x +y −z =1
22 x2 +x2y+ x2 z 2= y 222=2 z 2 2 =y 2 2 z 2+ x2 + 2y 2 = z 2 x 2x2 + y 2 = z+ 2y = z 2 2 2 −x2 + y 2 − z = 1 2x2 +xy 2+ x2z+ =z1 = z 2 = 1 2 2 +y2 + y 2 2 2 1 2 2222 + 2x2 + y 2 + 2z 2 =21+ 2y 2 + 22x2= 1y 2 + 2z 2 = 1 x z+
2 2 x2 −xy 2− x2 − y 2 = 5 2 =y5 = 5 2 x2 − y 2 = 5 x −y =5 2 2 x2 −xy 2− xz − y 2=2 1 =21 −y2 = z − z −1 x2 − y 2 − z = 1 x −y −z =1 2 2 x2 +xy 2+ xz + y 2=2 1 =21 −y2 = z − z −1 x2 + y 2 − z = 1 x +y −z =1 Math 51, Spring 2008 Final Exam — June 6, 2008 Page 10 of 16 9. (12 points) Complete the following sentences of deﬁnitions: (a) A real number λ is an eigenvalue for an n × n matrix A if (b) A linear transformation T : Rn → Rm is one-to-one if Math 51, Spring 2008 Final Exam — June 6, 2008 Page 11 of 16 x1 1 −1 1 2 x2 10. (15 points) Let A = 1 −1 −1 0. Also, let x = represent a vector of unknowns in R4 . x3 1 −1 0 1 x4 b1 b2 so that the system Ax = b has at least one solution. (a) Give conditions on the entries of b = b3 Express your answer as one or more linear equations involving only the entries of b. (b) Give a basis for the null space of A. 1 (c) Find all solutions to the linear system Ax = −1. 0 Math 51, Spring 2008 Final Exam — June 6, 2008 Page 12 of 16 11. (12 points) Let u, v, and w be three linearly independent vectors in Rn . (a) Is the set {u − v, v − w, u − w} linearly independent or linearly dependent, or is there not enough information to tell? Explain your answer. (b) Find all real numbers a such that the set u + v + w, u + 2v + aw, u + 4v + a2 w is linearly dependent. Math 51, Spring 2008 Final Exam — June 6, 2008 Page 13 of 16 12. (15 points) A matrix A is unknown, but its reduced row echelon form is given below: 1 −1 0 −1 01 2 , rref(A) = 0 . ?? A= 0 00 0 0 00 0 (a) Using this information, ﬁnd the following (no justiﬁcation necessary): A basis for the null space of A: dim(N (A)) : dim(C (A)) : (b) Now suppose you also know that 1 0 • is an eigenvector of A with eigenvalue 1, and 1 1 0 1 • is an eigenvector of A with eigenvalue 2. 1 1 With this additional information, ﬁnd the matrix A. You may leave your answer expressed as a product of matrices and inverses of matrices. (Hint: what are the eigenvectors of A with eigenvalue 0?) Math 51, Spring 2008 Final Exam — June 6, 2008 Page 14 of 16 13. (15 points) Suppose the linear transformation T : R2 → R2 is reﬂection across the line y = 2x. (a) Find the matrix of T with respect to the basis B = v1 = 1 2 , v2 = −2 1 . (b) Find the matrix of T with respect to the standard basis. (c) Now suppose the linear transformation S : R2 → R2 is counterclockwise rotation by π/2 radians. Is it true that T ◦ S = S ◦ T ? Justify your answer completely. Math 51, Spring 2008 Final Exam — June 6, 2008 x+y+z =0 . Page 15 of 16 x 3 given by P = y ∈ R3 14. (15 points) Let P be the plane in R z (a) Show that P is a subspace of R3 . (b) Consider the following four sets of vectors: 1 1 √ √ 1 1 2 16 √√ 1 −1 , 1 − 2 , 6 2 0 −2 0 −√ 6 1 0 √2 √ 1 − 2 , 0 1 0 1 √ 0 2 1 √√ 1 − 2 , 2 1 0 − √2 Which of the above sets form(s) an orthonormal basis for P ? Circle all that apply; you do not need to justify your answer. 1 (c) Let v = 1. Showing all steps, calculate ProjP (v), the orthogonal projection of v onto P . 2 (You may use one of your choices from part (b).) Math 51, Spring 2008 15. (15 points) Let A be the matrix Final Exam — June 6, 2008 Page 16 of 16 1 −1 2 1 2 . A = −1 −2 −2 6 It is a fact that two of the eigenvalues of A are 2 and 4. (You do not need to check this!) (a) Compute the characteristic polynomial of A, and simplify your answer. (b) Is A diagonalizable? Justify your answer completely. ...

View
Full
Document