Math 51 - Winter 2011 - Midterm Exam IIName:Student ID:Circle your section meting time:Nick Haber James Zhao Henry Adams11:00 AM 10:00 AM 11:00 AM1:15 PM 1:15 PM 1:15 PMRalph Furmaniak Jeremy Miller Ha Pham11:00 AM 11:00 AM 11:00 AM1:15 PM 2:15 PM 1:15 PMSukhada Fadnavis Max Murphy Jesse Gell-Redman10:00 AM 11:00 AM 1:15 PM1:15 PM 1:15 PMSignature:Instructions: Print your name and student ID number, select thetime at which your section meets, and write your signature to in-dicate that you accept the Honor Code. There are 10 problemson the pages numbered from 1 to 13. Each problem is worth 10 points.In problems with multiple parts, the parts are worth an equal numberof points unless otherwise noted. Please check that the version of theexam you have is complete, and correctly stapled. In order to receivefull credit, please show all of your work and justify your answers. Youmay use any result from class, but if you cite a theorem be sure toverify the hypotheses are satisfied. You have 1 hour and 30 min-utes. This is a closed-book, closed-notes exam. No calculators or otherelectronic aids will be permitted. GOOD LUCK!Question 1 2 3 4 5 6 7 8 9 10 TotalScore1(a). Find the inverse of the matrix A =1 2 22 2 22 2 1.11(b). Find all x for which the matrixx − 2 5 1−1 0 x−2 1 2is not invertible.22. Let T : R2→ R2be the linear transformation defined by:Txy=x + yy.(a). Find the matrix A that represents the linear transformation Twith respect to the standard basis S = {e1, e2}.(b). Consider the basis B = {v1, v2} given by: v1=31and v2=21.Find the change of basis matrix C for the basis B. That is, find thematrix C such that v = C[v]Bfor all vectors v.(c). Find the matrix B that represents the linear transformation Twith respect to the basis B.33(a). Find all eigenvalues of the matrix A =−1 2 22 2 −12 −1 2.43(b). Consider the matrix B =1 1 00 1 20 1 1.Find an eigenvector of B with eigenvalue λ = 1.54(a). Find the eigenvalues of the matrix A =4 2 02 1 00 0 2.4(b). Consider the quadratic form (Ax) · x (or in the other notationxTAx), where A is the matrix in part (a).Determine whether the quadratic form is positive definite, indefinite, ornegative definite. If it is none of those, determine whether the quadraticform is positive semidefinite or negative semidefinite.65.The position of a particle at time t is u(t) = (sin t, t2, cos t).(a). Find the velocity of the particle at time t.(b). Find the acceleration of the particle at time t.(c). Find the speed of the particle at time t.(d). Find the tangent line to the path of the particle at the point(0, 0, 1).76.Let T : R2→ R2be the reflection across the line y = −x.(a). Find the matrix for T (with respect to the standard basis of R2.)(b). Let R : R2→ R2be the rotation with angle π, and T the same asin 6(a). Find the matrix for T ◦ R (with respect to the standard basisof R2.)87. The temperature at a point x at time t on a heated wire is given byf(x, t) = sin((tx)2− 34)7(a). Compute both of the partial derivatives of f .7(b). Is the temperature at the point x = 2 decreasing or increasingat time t = 2?98. Suppose F : R3→ R2is defined byF (x, y, z) =sin(x cos y)x + 2y + sin x.Find the Jacobian matrix (i.e, the matrix for the total derivative)DF(0, 0, 1).109. Letf(x, y) =x2y + xy2+ y3x2+ y2.9(a). Findlim(x,y)→(0,0)f(x, y)if the limit exists.119(b). Compute∂f∂x, and use this to determinelim(x,y)→(0,0)∂f∂xif the limit exists.1210. Let g : R2−→R2be the functiong(x, y) = (sin(x + 3y), xy2+ y)and suppose that f is a function defined on a neighborhood of (0, 0),such that the composition f ◦ g is the identity function. Find Df(0,
View Full Document
Unlocking...