Practice Final
Instructions
• Write your name here:
• Read each Exercise prompt fully, and be sure to address every part of it!
• Box every part of your final answer.
• Show your work.
GOOD LUCK!
1
Exercise 1
Use polynomial division to compute the antiderivative:
ż
x3
dx.
x2 ` 1
2
Exercise 2
Use integration by parts to compute the antiderivative:
ż
x2 arctanpxq dx.
3
Exercise 3
Compute the trigonometric integral:
ż
sec3 θ tan θ dθ.
4
Exercise 4
Compute the following antiderivative using trigonometric substitution:
ż
a
px ´ 3q x2 ´ 6x ` 18 dx.
5
Exercise 5
Compute the following antiderivative using partial fraction decomposition:
ż
8x2 ´ 3x ` 18
dx.
4x3 ` 9x
6
Exercise 6
Evaluate the following improper integral:
ż5
1
?
dx.
x´3
3
You must employ limit(s) for full credit.
7
Exercise 7
Does the series
∞
ÿ
p´1qn
n“1
4n
3n
converge or diverge? Explain why.
8
Exercise 8
Use the Limit Comparison Test to determine whether the series converges or diverges:
∞
ÿ
n“1
n3
n
.
´ n sinpnq
9
Exercise 9
Find the radius and interval of convergence for the following power series:
ˆ
˙n
∞
ÿ
p3 ¨ n!q2 1
x`1 .
p2nq!
2
n“0
10
Exercise 10
Find the Taylor series for
f pxq “
?
x
center at x “ 4. Calculate at least 4 terms explicitly. Then write the whole series in
sigma notation.
11
Exercise 11
Consider the vectors ~u “
1
2 , 1,
?
?
5 and ~v “ ´12, 1, 5 .
a) Find the unit vector in the direction of ~u.
b) Are ~u and ~v orthogonal?
c) Find proj~u~v .
d) Find the area of the parallelogram spanned by ~u and ~v . Be sure to simplify
expressions before taking a norm.
12
Exercise 12
A plane contains the points P “ p1, 1, 1q, Q “ p2, 5, 3q and R “ p0, 2, 3q. Is it parallel
to the plane whose equation is 2x ´ z “ 10?
13
Exercise 13
Consider a curve described by the vector-valued function
?
2 4
3
~rptq “ t ,
t , t .
3
a) What is this function’s domain, i.e. values of t for which ~rptq has a real output?
b) What arc length does the curve trace from t “ 1 to t “ 3?
c) Find the unit tangent vector at time t “ 2.
d) Find the curvature of this curve at time t “ 4. You can use either formula.
14
Exercise 14
a) Write your favorite vector-valued function describing a helix. Any one will do.
~ ptq, and Bptq
~
b) Find the unit tangent, normal, and binormal vectors T~ ptq, N
for
this helix.
15