Calculus II, Quiz Questions
Name:
1. True or False: Every point (x, y) has precisely two representations in polar coordinates.
Solution:
False
2. Consider the sequence an = n sin( n1 ). Select the true statement:
A. The sequence {an } converges.
B.
∞
X
an converges.
n=1
C.
∞
X
an converges, AND the sequence {an } converges.
n=1
D. 0 ≤ an ≤ 1 for any n.
3. For what value of a does the curve x(t) = at2 , y(t) = t + 1 pass through (2, 0)?
Solution:
a=2
4. True or False: Two parametric curves can generate the same curve.
Solution:
True
5. Consider the function f (x) = 5(x + 1)17 + 20(x + 1)2 . If f (x) =
explicit formula for the cn ’s.
Solution:

 20
5
cn =

0
P
cn (x + 1)n , write an
:n=2
: n = 17
: n 6= 2, 17
6. True or False: There exists a power series
convergence.
P
Solution:
False
Page 0
cn (x − 1)n with (−5, 4) as its interval of
Calculus II, Quiz Questions
Name:
7. True or False: If lim an = 0, then
n→∞
∞
X
an converges.
n=1
Solution:
False
8. Let x(t) = 2 sin(t) and y(t) = cos(t). Eliminate the parameter t to give the cartesian
equation for the curve.
Solution:
x2
4
+ y2 = 1
9. True or False: For any infinitely differentiable function f , f (x) =
∞
X
f (n) (0)
n=0
Solution:
False
10. Let {an }∞
n=1 be given. Select the true statement:
A. If an is monotone, then it converges.
B. If an isn’t monotone, then it diverges.
C. If an is bounded, then it converges.
D. If an converges, then it is bounded.
11. True or False: if lim an = 0, then
n→∞
∞
X
an converges.
n=1
Solution:
False
12. The following identity is true for all x:
A. sin2 (x) cos2 (x) = 1.
B. sin2 (x) + cos2 (x) = 1.
C. sin(x) cos2 (x) = sin2 (x) cos(x).
D. sin(2x) = 2 sin(x).
13. The formula for integration by parts is given by:
n!
xn .
Calculus II, Quiz Questions
Name:
Solution:
R
R
u dv = uv − v du
14. For what values of p does
∞
X
1
np
converge?
n=1
Solution:
p>1
15. Consider the sequence an = en −
n2
2 .
Select the true statement:
A. lim an = 0
n→∞
B. The sequence {an } is increasing.
C. The sequence {an } is decreasing.
D. 0 ≤ an ≤ 1 for any n.
16. True or False: If {an } and {bn } are both divergent, then {an + bn } is also divergent.
Solution:
False
17. True or False: If
convergent.
∞
X
(an + bn ) and
n=1
∞
X
bn are both convergent, then
n=1
Solution:
True
18. True or False: if lim |an − L| = 0, then lim an = L
n→∞
n→∞
Solution:
True
19. If f (x) = ln(x), let an = f (n + 17). Then a3 is:
Page 2
∞
X
n=1
an is also
Calculus II, Quiz Questions
Name:
Solution:
ln(20)
20. Express the point (0, 1) in polar coordinates in two distinct ways, with θ ∈ [0, 2π].
Solution:
3π
(1, pi
2 ), (−1, 2 )
21. True or False: The curve (t3 , 2t − 1) connects the points (1, 1) and (3, 4).
Solution:
False
22. The graph of f (x) for x in [0, 1] is a curve, give a parameterized curve that generates
it.
Solution:
x(t) = t, y(t) = f (t) for t ∈ [0, 1]
23. One of the following integrals is improper–choose it from the rest.
Z 5
1
A.
dx
x
1
Z π
4
tan(x) dx
B.
0
Z
π
sec(x) dx
C.
0
Z
15
D.
0
x2
1
dx
+1
24. If x = 2 tan(θ), then csc(θ) =
Solution:
√
4+x2
x
25. The Fundamental theorem of calculus states that, for a real valued, differentiable function f :
Page 3
Calculus II, Quiz Questions
Name:
Solution:
Rb 0
a f (s) ds = f (b) − f (a)
26. The graph of r = cos(2θ) will have how many petals?
Solution:
4
27. One of the integrals below diverges–choose it from the rest.
Z ∞
1
√ dx
A.
x
1
Z ∞
1
dx
B.
x2
1
Z 5
1
C.
dx
1 x
Z ∞
2
D.
e−x dx
0
28. If {an } and {bn } are given, and 0 ≤ an ≤ bn , which of the following is a true statement?
A. If bn converges, so does an .
B. If an diverges, so does bn .
C. If limn→∞ bn = 0, then limn→∞ an = 0.
29. True or False: Every polynomial with real coefficients may be decomposed into a product of linear and quadratic terms.
Solution:
True
30. If
∞
X
cn xn =
n=−
Solution:
n
2!
cn =
0
∞
X
n!x2n is a power series, write a formula for cn .
n=0
: nis even
: nis odd
31. If f and g are continuous, and 0 ≤ f (x) ≤ g(x), which of the following is a true
statement?
Page 4
Calculus II, Quiz Questions
A. If
R∞
Name:
g(x) dx diverges, so too does
R∞
f (x) dx
R∞
B. If 0 f (x) dx converges, so too does 0 g(x) dx
R∞
R∞
C. If 0 f (x) dx diverges, so too does 0 f (x) dx
0
0
R∞
32. Consider a power series
∞
X
cn xn . Which of the following is not a possible set of con-
n=1
vergence for this power series?
A. {0}
B. [0, 1)
C. (−1, 1) ∪ (17, 23)
D. R
(−1)n
. Select the true statement:
n17
P
A. The ratio test shows that
an converges.
P
B. The ratio test shows that
an diverges.
P
C. The ratio test is inconclusive for
an .
33. Let an =
D. This sequence has negative terms, and thus the ratio test cannot be applied.
34. If x = 2 tan(θ), then csc(θ) =
Solution:
√
4+2
x
35. True or False:
∞
X
n
n!x 2 is a power series.
n=0
Solution:
False
36. Consider the sequence cn =
1
n.
The interval of convergence for the power series
∞
X
n=1
is?
Solution:
[−1, 1)
Page 5
cn xn
Calculus II, Quiz Questions
Name:
37. True or False: There exists a power series whose interval of convergence is [0, ∞).
Solution:
False
38.
R1√
0
1 − x2 dx =
Solution:
π
2
39. Suppose
P
an is conditionally convergent, |an | > 0. Select the true statement:
A. limn→∞ | an+1
an | < 1
P
B.
|an | converges.
C. Any reordering of the terms of
P
an will give the same value.
D. an must be negative for infinitely many n.
R∞
40. Compute 0 e−x dx.
Solution:
1
41. Give a polar expression for x2 − y 2 = 1.
Solution:
r2 cos(θ) − r2 sin(θ) = 1
42. Suppose
ex
=
∞
X
cn xn . What is C0 ?
n=0
Solution:
1
R∞
R∞
R∞
43. True or False: if 0 f (x) dx converges and 0 g(x) dx diverges, then 0 f (x) + g(x) dx
also diverges. (Hint: Think about limits! )
Page 6
Calculus II, Quiz Questions
Name:
Solution:
True
44.
R1
0
xex dx =
Solution:
2−e
45. Find a partial fractions decomposition for
x
:
(x − 2)3
Solution:
1
(x−2)2
+
2
(x−2)3
46. If β 2 −4γ < 0, write the general form for a partial fraction decomposition of
(use letters to denote constants).
1
(x−1)2 (x2 +βx+γ)2
Solution:
A
x−1
+
B
(x−1)2
+
Cx+D
x2 +βx+γ
+
Ex+F
(x2 +βx+γ)2
47. True or False: If {an } is a strictly positive sequence such that limn→∞
p
limn→∞ n |an | = 1. (Hint: Think about the series involved!)
Solution:
True
Page 7
|an+1 |
|an |
= 1, then