Quiz I18.085 (Dr. Christianson)2/March/09 PRINTED Name:• Do all your work on these pages. No calculators or computers may be used. Notes and thetext may be used. The point value (out of 100) of each problem is marked in the margin.1. (25 points) This problem concerns solving equations with point masses (delta functions).The next page is blank for extra work.a. (10 pts) Solve the following equation for u(x) with a point mass at x = a:(−d2dx2u = δ(x − a)u(0) = 1, u0(1) = −1.b. (5 pts)Use part (a) to solve the following equation for U(x)(−d2dx2U = x2U(0) = 1, U0(1) = −1.(Hint: If G(x, a) is the solution to part (a), thenU(x) =Z10G(x, a)a2da + null solution.Don’t forget to check the boundary conditions and add an appropriate nullspace solution.)c. (10 pts) Solve the following matrix equation for a = 1 and also for a = 2:K3u = δa,where as usualK3=2 −1 0−1 2 −10 −1 2,andδ1=100, δ2=010.2. (25 points) Consider the function f(x, y) = 4x2− 4xy + 2cy2.a. (15 pts) Where does f have critical points (the points where ∇f = 0)? For what valuesof c does f have a minimum? a maximum? a trough (positive or negative semi-definite)? asaddle point?b. (10 pts) For the value of c giving a trough (the semi-definite case), along which linedoes the bottom of the trough go (the line where f (x, y) = 0)?3. (25 points) This problem concerns SVD and singular values. The next page is blank forextra work.a. (10 pts) Compute the factors U, Σ, V in the SVD for the 1 × 2 matrix A = [1 −2].b. (10 pts) For the matrix M given byM = √32√30√5√3!,compute the condition number c(M ) = σmax/σmin.c. (5 pts) Find vectors u, f , ∆u, and ∆f for which the condition number gives a sharpbound:k∆ukkuk= c(M)k∆fkkfk.(Hint: think in terms of eigenvectors of MTM or MMT.)c3m1m2c1c2Figure 1: The spring/mass setup for Problem 4.4. (25 points) This problem concerns the spring/mass setup in Figure 1. The next page isblank for extra work.a. (15 pts) The masses are m1and m2, and the springs have spring constants c1, c2and c3respectively. There are no external forces (e.g. no gravity). Find the stiffness matrixK = ATCA in terms of the cjso that the displacements u = (u1, u2)Tof the masses satisfies anequation of the formMu00+ Ku = 0,where M is the matrix of masses as usual.b. (10 pts) For the particular case when m1= 1, m2= 1/4, c1= 2, and c2= c3= 1, findthe eigenvalues and eigenvectors of M−1K, and find the general solution to Newton’s equation(e.g. a combination of pure eigenvector solutions, the normal modes):Mu00+ Ku =
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