Math 115 Exam #4 Practice Problems
1. Solve the initial-value problem
y + 8y + 16y = 0,
y (0) = 3,
y (0) = 6.
2. Solve the dierential equation
y + 2y + 37y = 0.
3. Solve the initial-value problem
y + 5y 24y = 0,
y (0) = 0,
y (0) = 3.
4. Solve the dierenti
Math 115 Exam #3 Practice Problems
1. Solve the initial-value problem
dx
dt
+ 2tx = x, x(0) = 5. Use your solution to compute x(3).
2. Solve the dierential equation 7yy = 5x.
3. Solve the initial-value problem y + y = 2, y (0) = 1.
4. Solve the initial-va
Math 115 Exam #2 Practice Problems
1. Find the Maclaurin series for tan1 (x) (feel free just to write out the rst few terms).
2. Use the rst two non-zero terms of an appropriate series to give an approximation of
1
sin(x2 )dx.
0
Give (with explanation) an
Math 115 Exam #1 Practice Problems
For each of the following, say whether it converges or diverges and explain why.
1.
n3
n=1 n5 +3
Answer: Notice that
n3
1
n3
< 5= 2
n5 + 3
n
n
1
for all n. Therefore, since
n2 converges (its a p-series with p = 2 > 1), t
Math 115 Exam #3 Solutions
1. Solve the initial-value problem
dy
dx
= xy + x, y (0) = 10.
Answer: Factoring out an x on the left hand side yields
dy
= x(y + 1),
dx
which is a separable equation. Separate variables and integrate:
dy
=
y+1
xdx.
ln |y + 1| =
Math 115 Exam #2
1. Find the limit
cos x 1
.
ex2 1
lim
x0
Answer: The Maclaurin series for cos x is
cos x = 1
x2
x4
+
.,
2!
4!
so the numerator can be written as
cos x 1 =
1
x2
x4
x2
x4
+
. 1 = +
.
2!
4!
2!
4!
On the other hand, using the Maclaurin se
Math 115 Exam #1 Solutions
1. Does
2n + 3n
4n
n=0
converge? If so, what is the sum?
Answer: We can re-write this as the sum of two geometric series:
2n
3n
2n + 3n
=
+
=
n
n
4
4
4n
n=0
n=0
n=0
n=0
1
2
n
+
n=0
3
4
n
.
Using what we know about the sums of ge