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Math 240, FINAL EXAM
December 18, 2012
INSTRUCTIONS OFFICIAL USE
ONLY
INSTRUCTIONS:
- Please complete the information requested below. There are 9 multiple choice problems and 1 True/False problem. Partial credit will be given on the multiple choice ques- tions.
- Please show all your work on the exam itself. Correct answers with little or no supporting work will not be given credit.
- You are allowed to use one hand-written sheet of paper with formulas. No calculators, books or other aids are al- lowed. Please turn in your crib sheet together with your exam.
- Name (please print):
- Name of your Professor: © Ryan Blair © Vasile Brinzanescu © Tony Pantev
- Name of your TA: © Shiying Dong © Ryan Eberhart © Ryan Manion © Sebastian Moore
- Recitation day and time:
- My signature below certifies that I have complied with the University of Pennsylvania’s Code of Academic In- tegrity in completing this exam. Signature:
Problem Points
Total
(1) 10 points Find the outward flux of the vector field
F = x̂ı + ŷ + z k̂ through the boundary of the solid region R bounded by the paraboloid z = 4 − x^2 − y^2 and the plane z = 0.
(a) 1 (b) 6π (c) 24π (d) 2 (e) π
(f)
2 π 3
(3) 10 points Consider the homogeneous linear system Ax = 0, where
A =
Which of the following statements is correct.
(a) Ax = 0 has no solutions. (b) Ax = 0 has one free variable. (c) Ax = 0 has two free variables. (d) Ax = 0 has three free variables. (e) Ax = 0 has a unique solution. (f) x 1 and x 2 are pivot variables for Ax = 0.
(4) 10 points Which one of the following is not a basis for the vector space of all symmetric 2 × 2 matrices. Justify that the set you pick is not a basis.
(a)
[
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[
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[
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(b)
[
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[
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[
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(c)
[
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[
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[
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(d)
[
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[
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[
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(e)
[
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[
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(f)
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(6) 10 points Which of the following matrices have two linearly independent eigenvectors. Circle all that apply and justify your answers.
(a)
(b)
(c)
(d)
(e)
(f)
(7) 10 points Please circle “T” for true or “F” for false in the space provided to the left
of the following statements. You do not need to justify your answer for full credit.
T F The sum of two solutions to a linear, non-homogeneous differential equation is
always a solution.
T F Three column vectors in V 2 (I) = Fun(I, R^2 ) must be linearly dependent.
T F The fundamental solution set to a vector differential equation given by x′^ = Ax
forms a spanning set for the vector space of solutions to x′^ = Ax.
T F The zero function is a solution to every linear differential equation.
T F An initial value problem for an n-th order, homogeneous, constant coefficient,
linear differential equation always has n linearly independent solutions.
(9) 10 points Let y(x) be the general solution of the linear differential equation
y′′^ − 6 y′^ + 9y = ex^ + 1.
Find (^) x→−∞lim y(x).
(a) 4
(b)
(c)
(d) − 6 (e) − 9 (f) e^3 + 1
(10) 10 points Let x be the solution of the system of differential equations
x′^ =
(^) x
which satisfies the initial condition x(0) =
. What is x 3 (1)?
(a) −e (b) 2e (c) − 3 e^2 (d) e^2 (e) e − e^2 (f) 2e^2 − e
