PRACTICE MIDTERM 2
Math 1220 sec 006
Name: |
{z
by writing my name I swear by the honor code
}
Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take off points if I cannot see how you arrived at your answer (even if your final answer is
correct).
• No calculators or books are allowed for this test.
• You are allowed one 4”x6” notecard for the test. You must turn in your notecard with
your exam.
• Turn off cell phones, ipods, etc.
• No headphones allowed, turn hats backward, keep your eyes on your own paper.
• Please have your ID out and available.
• Please keep your written answers brief; be clear and to the point. I will take points off for
rambling and for incorrect or irrelevant statements.
• You have 100 minutes to work on this exam.
• Good luck!
Problem (value)
1 (20 points)
2 (10 points)
3 (10 points)
4 (10 points)
5 (15 points)
6 (10 points)
Total (75 points)
Score
1.
(20 points) Short Answer. For True/False questions you must write ’True’ or ’False’ and
justify your answer.
a. (4 pts)
four.
∞
We discussed seven indeterminate forms. Three are 00 , ∞
∞ , 1 . Name the other
R∞
b. (4 pts)
Complete the statement. The improper integral
c. (4 pts)
Find an explicit formula for the sequence 1−2 , 2−3/2 , 3−4/3 , 4−5/4 , ...
a
f (x)dx is said to converge
if ...
d. (4 pts)
Fill in the blank. A series of nonnegative terms converges if and only if its
sequence of partial sums are
.
e. (4 pts)
If f (x) =
∞
P
n=1
an xn write down a series that represents
df
dx .
2.
(10 points)
R∞
a. (5 pts)
For what values of p does the integral 1
does it diverge? Show your work.
b. (5 pts)
Does
R 10
3
√ 1 dx
x−3
1
xp dx
converge and if so, to what?
converge and for what values
3.
(10 points)
a. (5 pts)
Prove the following weaker form of l’Hopital’s rule: Let f and g be functions
df
dg
of x and let dx
= f 0 and dx
= g 0 . Suppose f (a) = g(a) = 0, that f 0 (a) and g 0 (a) exist, and that
0 (a)
(x)
g 0 (a) 6= 0. Then lim fg(x)
= fg0 (a)
.
x→a
b. (5 pts)
2
Find lim (3x)x .
x→0+
4.
(10 points)
a. (5 pts)
What is value of
∞
P
k=1
b. (5 pts)
1
(k+2)(k+3) ?
What is the convergence set for the power series
∞
P
n=0
xn e n
n+1 ?
5.
(20 points) Determine if the following series converge or diverge and give a reason for your
conclusion.
∞ 2
P
n
a. (5 pts)
n!
b. (5 pts)
n=1
∞
P
c. (5 pts)
k=1
∞
P
d. (5 pts)
n=1
∞ √
P
n=1
1
7k−1
cos((n + 1)π) 31n
n
n−1
6.
(10 points)
a. (5 pts)
about x = 0.
Calculate the first 3 nonzero terms of the Taylor series for cos kx centered
b. (5 pts)
Find an upper bound for the error in using the sum of the first 20 terms to
∞
P
1
approximate the sum of the convergent series
.
k3/2
k=1