Name ________________________
Math 241 Makeup Exam
Fall 2007
Write your name in the space provided. Be prepared to show your Penn ID if asked.
Indicate which professors class you are enrolled in.
You will have 2 hours to complete the exam.
No calculators or books are permitted, however you may use an 8 in. 11 in. sheet of notes.
Mark your answer on this sheet and on the problem itself. Do not detach this sheet.
Although this is a multiple choice exam, you must provide work to support your answer.
Use the space provided beneath each problem to show your work.
A correct answer with no work will be graded as an incorrect answer.
Each problem is equally weighted with no partial credit given for skipped or incorrect solutions.
1. Find the radius of convergence for the Taylor series of ( )
8
1
1
f z
z
=
about the point 2 2 2 2 z i = + .
(A) 1 (C) 2 (E) 3 (G) 4
(B)
3
2
(D)
5
2
(F)
7
2
(H)
2. Consider the Laurent series for the function
( )
2
2 3
2
z z
f z
z
+
=
in the region 1 1. z > What is the coefficient of the ( )
2
1 z
term?
(A) 6 (C) 3 (E) 1 (G) 3
(B) 4 (D) 0 (F) 2 (H) 6
3. Find the constant k such that the function ( )
2 3
, 3 1 v x y x y ky x = + + is a harmonic
conjugate of the function ( )
3 2
, 3 . u x y x xy y = +
(A) 3 (C) 1 (E) 1 (G) 3
(B) 2 (D) 0 (F) 2 (H) 4
4. Evaluate
2
0
i
iz
e dz
.
(A)
( )
2
1 i e
(C)
2
1 ie
(E)
2
i e
(G)
2
1 e
(B)
( )
2
1 i e
+ (D)
2
1 ie
+ (F)
2
i e
+ (H)
2
1 e
+
5. Evaluate
(A)
10
(C)
6
(E)
3
(G) 1
(B)
8
(D)
4
(F)
2
(H)
6. Evaluate
2
2 cos
0
d
.
(A) (C)
2
3
(E)
3
(G)
4
(B)
2
3
(D)
2
(F)
3
(H)
6
7. Evaluate
2
6 13
dx
x x
.
(A) (C)
24
(E)
24
(G)
2
(B)
12
(D) 0 (F)
12
(H)
8. In the Fourier series expansion of ( )
2
2 1 f x x = on (-1,1) find the coefficient on the ( ) cos 4 x
term.
(A) 0 (C)
1
2
(E)
1
2
(G) 1
(B)
2
1
2
(D)
2
2
(F)
2
(H) 2
9. Consider the Sturm-Liouville problem defined on 0
2
x
:
( ) 0 0 0, 0
2
y y y y
| |
+ = = =
|
\
.
Find all eigenvalues , 0,1, 2,
n
n = .
(A)
2
n
n = (C)
4
n
n
= (E)
( ) 2 1
4
n
n
= (G)
( )
2
2 1
2
n
n
=
(B)
2
4
n
n
= (D)
( ) 2 1
2
n
n
= (F) ( )
2
2 1
n
n = (H)
( )
2
2 1
4
n
n
=
10. The solution ( ) , u x t defined for 0 2, 0 x t to the wave equation
( ) ( ) with boundary conditions 0, 2, 0 is
tt xx x x
u u u t u t = = =
( ) ( ) ( ) ( )
2 2 2
0
, cos sin cos
n n n
n n
n
u x t A t B t x
=
( = +
.
Find
1 1
,
3 2
u
| |
|
\
with initial conditions ( ) ( ) ( ) ( ) , 0 3cos and , 0 2cos 3 .
t
u x x u x x = =
(A)
1
(C)
3
(E)
2
3
(G)
1
2
(B)
2
(D)
1
3
(F)
1
3
(H) 2
11. Let ( ) , u x t be a function defined for 0 , 0 x t such that
2
t xx x
u u u = +
with boundary conditions ( ) ( ) 0, , 0 for all 0 u t u t t = =
and initial condition ( ) ( ) , 0 sin 2
x
u x e x
= .
Use separation of variables to find ( )
4
,1 u
.
(With separation constant , you will find non-trivial solutions when 1 > , say
2
1 = + )
(A)
2
1
e
(C)
2
5
e
(E)
4
1
e
(G)
4
5
e
(B)
2
2
e
(D)
2
10
e
(F)
4
2
e
(H)
4
10
e
12. Consider a circular plate of radius 1 whose circular edge is maintained at the temperature
( ) 1, . u = The steady-state temperature ( ) ( ) ( )
0
1
, cos sin
n
n n
n
u r A r A n B n
=
( = + +
satisfies
2
1 1
0.
rr r r
r
u u u
+ + = Find the coefficient of the ( ) sin 3 term.
(A) 0 (C)
2
3
(E)
3
(G)
2
3
(B)
1
3
(D)
2
3
(F)
3
(H)
2
3
