MATH 51 MIDTERM IINovember 17, 2005Name:Numeric Student ID:Instructor’s Name:I agree to abide by the terms of the honor code:Signature:Instructions: Print your name, student ID number and instructor’s name in the spaceprovided. During the test you may not use notes, books or calculators. Read eachquestion carefully and show all your work; full credit cannot be obtained withoutsufficient justification for your answer unless explicitly stated otherwise. Underline yourfinal answer to each question. There are 9 questions. You have 90 minutes to do all theproblems.Question Score Maximum1 102 63 84 65 106 107 158 109 10Total 85Question 5 of 9, Page 2 of 4 Solutions1. Consider the functionf(x, y) = x4y3and the point P = (1, 1, 1) on its graph.(a) Write down the equation of the tangent plane at the graph of the function atthe point P .(b) Using your answer from (a), write down an expression for the change, ∆z, inz = f(x, y) depending on ∆x and ∆y, the change in x and y, respectively,near the point P = (1, 1, 1). Is the function f(x, y) more sensitive to a changein x or to a change in y? Explain.(c) Using your answer to (b), find the approximate value of f(1.01, 1.02).2. (a) The steady state temperature function T (x, y) for a thin flat plate satisfiesthe equationTxx+ Tyy= 0.Does the functionT (x, y) = ln(x2+ y2)satisfy the equation above? Show your work.(b) Givenf(x, y, z) = z sin x +cos y ln(z + y)z,compute the partial derivative∂∂x∂f∂z.3. (a) Compute the determinant of the matrix5 11−2 3.(b) What are the values of the parameter a for which the matrixM =1 1 10 1 a + 21 4 − a 5is invertible?4. Find the inverse of the matrixA =1 −1 1−1 1 13 3 3.Question 8 of 9, Page 3 of 4 Solutions5. (a) The linear operator T : R2→ R2is defined byT (x1, x2) = (3x1+ 4x2, x1+ 3x2).Find a basis of R2such that the matrix of T with respect to that basis isdiagonal.(b) Are there real numbers a and b such that the matricesA =0 0 a1 0 b0 1 0and B =1 0 00 1 00 0 −2are similar (i.e. A = CBC−1)? If yes, find a and b. If no, explain why not.6. Determine the definiteness (p ositive/negative definite or semidefinite, indefinite)of the f ollowing quadratic forms. Explain your answers.(a) The quadratic form in two variables,Q(x.y) = x2+ 6xy + 2y2.(b) The quadratic form associated to the matrix1 0 20 1 02 0 1.7. Determine the matrix associated to the following linear transformations. Expressyour answers in terms of the standard basis.(a) The transformation T (x1, x2) = (7x1− x2, 8x1+ 3x2) where (x1, x2) ∈ R2.(b) Rotation in R2by an angle of π/3 radians.(c) Projection of vectors in R3onto the line L given by the points3t−2ttt ∈ R.(d) Reflection in R3in the plane given by3y − z = 0.8. Let V be the 2-dimensional subspace of R3with basisB =3−51,209.Question 9 of 9, Page 4 of 4 Solutions(a) Given the vector in B-coordinatesv =42B,express v in standard coordinates.(b) Express the vectorv =1−12written in s tandard coordinates in terms of the B-coordinates.9. Let A and B be two invertible n × n matrices. Show that AB and BA have thesame characteristic
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