Name
Student ID #
Math 3150 Summer 2011
PRACTICE EXAM I
Friday, May 27, 2011
Important Note:
This practice test is representative of the style of Exam I which will be given
on June 1. It is not reasonable for you to expect that all of the problems on
the actual exam will be similar to the practice problems presented here.
The following identities may or may not be useful for the test.
Z
sin((a − b)x) sin((a + b)x
−
+ C, a2 6= b2
(i)
sin ax sin bx dx =
2(a − b)
2(a + b)
Z
sin((a − b)x) sin((a + b)x
(ii)
cos ax cos bx dx =
+
+ C, a2 6= b2
2(a − b)
2(a + b)
Z
cos((a − b)x) cos((a + b)x
−
+ C, a2 6= b2
(iii)
cos ax sin bx dx =
2(a − b)
2(a + b)
Z
x
1
(iv)
sin2 ax dx = −
sin 2ax + C
2 4a
Z
x
1
(iv)
cos2 ax dx = +
sin 2ax + C
2 4a
Z
x
1
(vi)
x sin ax dx = 2 sin ax − cos ax + C
a
a
Z
1
x
(vii)
x cos ax dx = 2 cos ax + sin ax + C
a
a
(10 points) 1. State whether or not
f (x) =
√0, −1 ≤ x ≤ 0,
x, 0 < x ≤ 1.
is (a) continuous, (b) piecewise continuous, and (c) piecewise smooth.
Note there are three separate questions here, please answer and provide
justification for each one.
1
(20 points) 2. Solve the following initial value problem for u(x, t):
∂u ∂u
−
= 0,
x ∈ R, t > 0,
∂t
∂x
2
u(x, 0) = e−x , x ∈ R.
(15 points) 3. For the 2-periodic function

(x − 1)2 , −1 ≤ x ≤ − 21 ,



sin(πx), − 12 < x < 0,
f (x) =
1,
0 ≤ x < 31 ,



1
≤ x < 1,
ex ,
3
determine the value of the Fourier series representation at the points
x = − 12 , 0, and 3.
2
(15 points) 4. Find a Fourier series representation for the function
1
3
f (x) = cos x + 5 sin 2x − 2 sin x.
2
2
(20 points) 5. Given that the Fourier series representation for the 2-periodic function
1,
− 12 < x < 12 ,
f (x) =
0, −1 < x < − 12 , or 12 < x < 1,
is
∞
1 2X1
nπ
+
sin( ) cos(nπx)
2 π n=1 n
2
find the Fourier series representation for the 6-periodic function
5,
− 32 < x < 32 ,
g(x) =
0, −3 < x < − 32 , or 32 < x < 3,
3
(20 points) 6. Find the sine series expansion for the function f (x) = cos x on the
interval 0 < x < 1.
4