MATH 51 MIDTERM 2November 16, 2000Brumfiel Hutchings Levandosky Staffilani White11:00 01 05 09 13 171:15 03 07 11 15 19Name:Student ID:Signature:Instructions: Print your name and student ID number and write your signature to indicatethat you accept the honor code. Circle the number of the section for which you are registeredon Infopier. During the test, you may not use notes, books, or calculators. Read eachquestion carefully, and show all your work. Put a box around your final answer to eachquestion. You have 90 minutes to do all the problems.Question Score12345678910Total1. (a) Compute the inverse of the matrix1 3 −20 2 40 0 −11(b) For which value(s) of x is the matrix below not invertible? Explain your answer.1 1 10 1 25 x 62. (a) SupposeA =2/3 −1/3 2/32/3 2/3 −1/3−1/3 2/3 2/3is the matrix of a linear transformation which is geometrically a 60 degree rotationabout a line L in R3. Find the matrix of a 120 degree rotation about L. Hint:Think about composition.(b) LetB =2 2 3 54 3 2 1−1 2 −1 29 8 5 8v =64−210Compute B−1v. Hint: You do not need to compute B−1. Compare v with thecolumns of B.3. Let ∆1be the triangle with vertices (0, 0), (−1, 0) and (0, 2) and let ∆2be the trianglewith vertices (0 , 0), (2, 0) and (3, 3). Suppose T : R2→ R2is a linear transformationsuch that T (∆1) = ∆2.T(a) There are exactly two such linear transformations. Find the matrix for one ofthem.(b) Let E represent the region bounded by the ellipsex24+y225= 1The area of E is 10π. Find the area of T (E). Note: The answer is the same forboth linear transformations T which satisfy T (∆1) = ∆2.24. LetA =·5 11 3¸B =2 −1 02 1 15 −7 6(a) Find the eigenvalues of A.(b) λ = 3 is an eigenvalue of B. (You do not need to check this.) Find all eigenvectorsof B with eigenvalue 3.5. LetA =·0 −1−1 0¸B =·0 1−1 0¸C =·1 00 −1¸D =·−1 00 −1¸E =·0 −11 0¸Let S denote the set of points in the face shown below.Each figure below is the image of S under the linear transformation corresponding toone of the matrices above. Match each figure with the corresponding matrix.(a)(b)3(c)(d)6. (a) Compute the following limit. Explain your answer.lim(x,y,z)→(2,3,−1)xy2z − 2xyzx2y + xz + y2z2(b) Show that the following limit does not exist.lim(x,y)→(0,0)2x2+ y2x2+ 2y27. Let f(x, y) = xy + sin(2x − 4y).(a) Suppose an ant is crawling on a surface whose height in cm at the point (x, y)is given by f(x, y). If the ant is crawling in such a way that its x-coordinate isincreasing at 2cm/sec and its y-coordinate is increasing at 1cm/sec, at what rateis its height changing when the (x, y) coordinates of the ant are (2, 1)?(b) Find∂2f∂y∂x(x, y) and∂2f∂x2(x, y).8. Let f : D ⊂ R2→ R be defined by f(x, y) =pxy + y2.(a) Sketch the domain D of f. Hint: xy + y2= y(x + y).(b) Find Jf(3, 1).(c) Use the answer to part (b) to find an approximation of f(3.01, 1.02).49. Define f : R2→ R3and g : R3→ R2byf(x, y) = (xy, x2+ y2, 2x − 2y)g(x, y, z) = (x2+ y2+ z2, xyz)Find the following Jacobian matrices.(a) Jf(1, 1).(b) Jg(1, 2, 0).(c) J(g ◦ f )(1, 1).10. Each figure below represents the level curves of some function. (The graphs are shownin the usual orientation, with the x-axis horizontal and the y-axis vertical.)−1 0 1−1−0.500.51Figure 1−1 0 1−1−0.500.51Figure 2−1 0 1−1−0.500.51Figure 3−1 0 1−1−0.500.51Figure 4−1 0 1−1−0.500.51Figure 5−1 0 1−1−0.500.51Figure 6For each function below, indicate which figure represents its level curves.(a) x − 2y(b) xy(c) x2+
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