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Makeup Final Exam, Math 240: Calculus III
September, 2011
No books, papers or may be used, other than a hand-written
note sheet at most 8. 5
′′
× 11
′′
in size. All electronic devices
must be switched off during the exam.
This examination consists of nine (9) long answer questions and one (1) true/false question. Partial
credits will be given only when a substantial part of a long answer question has been worked out.
Merely displaying some formulas is not sufficient ground for receiving partial credits.
- Your name, printed:
- Your Penn ID (last 4 of the middle 8 digits):
My signature below certifies that I have complied with the University of Pennsylvania’s
code of academic integrity in completing this examination.
Your signature
1 2 3 4 5 6 7 8 9 10 Total
- For what values of k is the following matrix singular?
2 k 0
−k − 6 4
Ans. k =.
- Find a solution of the differential equation
d
dt
~x(t) =
~x(t), ~x(t) =
x 1 (t)
x 2
(t)
which satisfies
lim
t→∞
~x(t) = 0 and x 2 (0) = 1.
Ans. ~x(t) =
- Let A be the 4 × 4 matrix.
A =
(a) (4 pts) Compute the matrix A
2 and find all eigenvalues of A
Ans. A
2
The eigenvalues of A are.
- Let D = {(x, y, z) ∈ R
3 | 4 ≤ x
2
2
2 ≤ 9 }, the solid region between the sphere of
radius 3 and the sphere of radius 2, both centered at the origin. Let S be the boundary of
D, consisting of the sphere S 3
of radius 3 and the sphere S 2
of radius 2, both centered at
the origin. Orient S by the unit normal vector field on S such that
N (x, y, z) =
1
3
(x~i + y ~j + z
k) if (x, y, z) ∈ S 3
1
2
(x~i + y ~j + z
k) if (x, y, z) ∈ S 2
Compute the surface integral ∫ ∫
S
x~i ·
N dA ,
i.e. the flux of the vector field
F (x, y, z) = x~i through the boundary S of the solid D.
Ans. The surface integral is.
- Let A and B be two 5 × 5 matrices such that AB = B
3 and 4 is an eigenvalue of B, find
one eigenvalue of A.
Ans. One eigenvalue of A is.
- Find the general solution to the following system of differential equations
d
dt
x 1 (t)
x 2
(t)
x 3
(t)
x 1 (t)
x 2
(t)
x 3
(t)
Ans.
x 1
(t)
x 2 (t)
x 3
(t)
- Let S = {(x, y, z) ∈ R
3 | x
2
2
2 = 1 be the unit sphere in R
3 centered about the
origin. Orient S by the unit normal vector field
N := x~i + y~j + z
k on S. Compute the
oriented surface integral
S
curl
x
3 ~ i + y
3 ~ j + z
k
x
2
2
2
N dA
Ans. This integral, also written as
S
curl
x
3 ~ i + y
3 ~ j + z
k
x
2
2
2
N dS, is.
