Calculus II (MATH 1062)
Test #3 Practice Problems (Covering 11.8 −11.11, 10.1 – 10.4, 12.1 – 12.3) … 12.3 – 𝒑𝒓𝒐𝒋𝒗 𝒖 will not be
covered in Test 3. It could be tested in the Final Exam.
Part I. Free Response Questions

1. Find the radius and interval of convergence of

n0
(1) n 4n
(2 x  1) n .
n 1
2. (a) Find a power series representation for the function
1
10
(b) Use part (a) to express

0
1
and determine the radius of convergence.
1  x5
1
dx as the sum of a numerical series.
1  x5
1
10
(c) Use part (b) to approximate

0
1
dx correct to within 10−5.
5
1 x
3. Use a known power series to evaluate the sum of each series.
(a) 1  ln 2 

(b)

n 0
(ln 2) 2 (ln 2)3 (ln 2) 4


 ...
2!
3!
4!
(1) n  2 n
4 2 n ( 2n) !
ex  1  x2
4. Use power series to compute lim 4
.
x  0 x   x7
2
5. Find the 2rd degree Taylor polynomial for √𝑥 centered at 𝑎 = 4.
6. For the following, eliminate the parameter to find a Cartesian equation for each curve. Sketch the curve on
the xy-plane. Indicate with an arrow or arrows the direction the curve traces as the parameter increases.
(a) x  t  1 ,
y  t2  t ,
(b) 𝑥 = 2 cos 𝑡,
𝑦 = 2 sin 𝑡 ,
 2  t 1
0≤𝑡≤
𝜋
2
1
7. Find the exact length of the curve:
𝑥 = sin(𝑡) − 𝑡cos (𝑡),
𝑦 = cos(𝑡) + 𝑡sin (𝑡),
0≤𝑡≤2
x  1  t
8. (a) Find the slope of the tangent to the parametric curve 
at the point where 𝑡 = 1.
t2
y e
𝜋
(b) Find an equation for the line tangent to the polar curve r  2 cos at the point where 𝜃 = 3 .
9. The shaded region in the figure below is bounded by the polar curve r  1 cos .
Compute the area of the shaded region.
10. Set up the definite integral(s) for computing the area of each shaded region. Do not compute the integrals.
(a)
(b)
11. Let P = (1, 1, 1), Q = (2, 3, 2) and R = (0, 2, 3). Find
(a) The inner product of ⃗⃗⃗⃗⃗
𝑃𝑄 and ⃗⃗⃗⃗⃗
𝑃𝑅 .
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ .
(b) The angle between 𝑃𝑄 and 𝑃𝑅
⃗⃗⃗⃗⃗ direction.
(c) A unit vector in the 𝑄𝑅
2
12. Vectors a and b lie in the xy-plane as shown below.
(a) In the above figure, sketch the vector −2 𝒂 + 𝒃.
(b) Suppose |𝒂| = 2 and |𝒃| = 3, and that the angle between a and b is
𝜋
3
. Compute 𝒂 ∙ 𝒃 .
Part II. Multiple Choice Questions
13. If f ( x) 
(1) n
x 2 n  1 , then its derivative is
(2n  1)(2n)!


n  0
B. f ' ( x)  cos x
A. f ' ( x)  xe x
C. f ' ( x)  sin x

14. Find the exact value of the series  (1) n
n0
A.
1
B.
–𝜋
C. e 
D. 0
 2n  1
(2n  1)!

n 1
C. (3,  1)
x0
1
2
B. 1
C. −∞
D. 
(1) n ( x  2) n
is
n2
D. [3,  1)
16. Use a known Maclaurin expansion to find lim
A. 
f ' ( x)  x cos x
.

B. (3,  1]
E.
E. The series diverges.
15. The interval of convergence for the power series
A. (1, 3)
D. f ' ( x)  x sin x
2
3
E. [3,  1]
cos( x 3 )  1
.
x 6  3x 7
E.
0
3
17. Compute the 102nd derivative of f ( x)  e x at 𝑥 = 0.
2
B. 
A. 0
102!
51!
1
51!
C. 

a x
18. Assume a power series
n
n0
n
D. 
103!
52!
E. 
1
52!
converges for 𝑥 = 5 and diverges for 𝑥 = −10. Which of the following
series also converge(s)?

A.
 a (7
n0
n

n
)
 a (4)
B.

n
n0
 a (11)
C.
n
n0

n
n
D.
 a (5)
n0

n
E.
n
 a (8)
n0
n
n
 x  et sin t
19. The arc length of 
for 0 ≤ t ≤ 3 is
t
 y  e cos t
A. √2
B. 2(𝑒 6 − 1)
C. 𝑒 3/2 − 1
20. The slope of the line tangent to the curve {
A. −5
B. 2


n0
(1) n 2 n 1
x
3

B.
E. 
1
11
x
is
3  4x2
21. The Maclaurin series for
A.
E. √2 𝑒 3
𝑥 = ln 𝑡 + 𝑡 2
at the point where 𝑡 = 4 is
𝑦 = √𝑡 − 𝑡
D.  11
C. 11
D. √2 (𝑒 3 − 1)

n0
(1) n 2 n 1
x
3n

C.

n0
(1) n 4n 2 n
x
3n
(1) n 4n 2 n 1
 3n 1 x
n0

D.
22. The third degree Taylor polynomial 𝑇3 (𝑥) of cos 𝑥 centered at 𝑎 =

1

A. ( x  ) 2  ( x  ) 4
2
6
2
 1

D. ( x  )  ( x  )3
2 6
2
 1

B.  ( x  )  ( x  )3
2 6
2


𝜋
2
B. 3e
2

n0
(1) n 4n 2 n 1
x
3n 1
is

1

C. 1  ( x  )2  ( x  )4
2
6
2
E.  ( x  )  ( x  )3
2
2
23. The slope of the tangent to the polar curve r  e at the point where 𝜃 =
A. 3

E.
C. −2
D. 0
𝜋
4
is
E. The tangent is vertical.
4
24. The polar curve r  2 sin(3 ) is pictured below. Compute the area enclosed by one loop of the curve.

3
A.
B.

2
C.

6
E. 
D. 3
25. Which of the following values of a and b make the angle between the vectors 𝑎𝒊 + 2𝒋 and
3𝒊 − 4𝑏𝒋 obtuse?
A. 𝑎 = 1, 𝑏 = 0
B. 𝑎 = −1, 𝑏 = −1
C. 𝑎 = 0, 𝑏 = −1
D. 𝑎 = 1, 𝑏 = −1 E. 𝑎 = −1, 𝑏 = 1
26. The angle between the vector i  k and the vector 2 j  2k is
A.

6
B.

3
C.
5
6
D.
2
3
27. The Cartesian equation for the polar equation r 
A. x 2  y 2  4
B. x 2  y 2 
28. Use the power series expansion
1
A.
C.
1
 1 x
2
dx =

0
n0
1


0
1
E.


0
1
dx =
1  x2
1
dx =
1  x2
1
B.
n
2
dx =
0
D.

0
n
4 4

x y
D. 1 
4 4

x y
E. x  y  4
1
1
 1 x
2
dx .
0

1
 1 x
1
n0
 (1)
4
is
cos   sin 
C. x 2  y 2 
n


2

1

(1) n x 2 n to find a series representation of

2
1 x
n0
(1)
2n  1
 (1)
4
x y
E.
1
dx =
1  x2
1
1

2
1 x
n  0 2n  1

 (2n  1)
n0
(2n  1)
n0
5