MATH 141, 141E, 141H
FINAL EXAM
1. If f is invertible, f (3) = 5, f (4) = 3, f 0 (3) = 7 and f 0 (4) = 11, find
(f −1 )0 (3).
a) 1/7
b) 1/11
SAMPLE A
Z
1
dx.
x2 + 3x − 4
5. Evaluate the integral
a)
1 x + 4 ln +C
5
x − 1
b)
1
ln |(x − 1)(x + 4)4 | + C
5
c) 1/5
1
ln |(x − 1)4 (x + 4)| + C
5
1 x − 1 d)
ln +C
5
x + 4
c)
d) 1/3
e) Cannot be determined
e) ln |x2 + 3x − 4| + C
2
2. Find the limit lim e x−3 .
x→3−
π/2
Z
a) 0
6. Evaluate the integral
3 sin2 θ cos3 θdθ.
0
b) ∞
c) e2/3
a) 2/5
d) e
b) 2/15
e) e−2/3
c) −2/5
d) 0
x
3. Find the derivative of arcsin(e ).
a) √
e) The integral is divergent
1
7. Evaluate the integral
d) √
e) √
Z
4. Evaluate
x ln xdx.
1
ex
b)
1 − e2x
c) √
2
Z
1 − e2x
a) ln 2 − 1
1
b) ln 16 − 3
e2x − 1
c) ln 2
ex
1 − ex
d) ln 4 −
3
2
e) ln 4 −
3
4
ex
1 − e2x
1
√
dx.
x2 x2 − 9
8. For the series
3
+C
x
x
b) √
+C
2
x −9
+∞
X
n=1
a)
(−1)n
√
determine which of the following staten2 + 1
ments is/are true?
(i) The series diverges by the test for divergence.
+∞
X 1
(ii) The series diverges by comparison with
.
n
n=1
(iii) The series converges by the alternating series test.
x2 + 1
c) √
+C
x2 − 9
√
x2 − 9
d)
+C
9x
√
x2 − 9
e) −
+C
3x
a) Only (i) is true
b) Only (ii) is true
c) Only (iii) is true
d) Only (i) and (ii) are true
e) None of (i), (ii), or (iii) is true
1
MATH 141, 141E, 141H
FINAL EXAM
9. Find the Taylor series of f (x) = ex centered at a = 3.
a)
∞
X
en (x − 3)n
n!
n=0
b)
∞
X
e3 (x − 3)n
n!
n=0
13. Find dy/dx for the parametric curve y = tan−1 (t2 ), x = ln t.
∞
X
xn
c)
n!
n=0
d)
∞
X
(x − 3)n
n!
n=0
e)
∞
X
exn
n!
n=0
10. Express f (x) =
x
as a power series.
x+3
a)
∞
X
(−1)n xn+1
3n+1
n=0
b)
∞
X
(−1)n+1 xn+1
3n
n=0
c)
∞
X
(−1)n xn
3n
n=0
d)
∞
X
(−1)n+1 xn+1
3n−1
n=0
e)
∞
X
(−1)n xn+1
3n−1
n=0
(R = 3)
(R = 3)
(R = 3)
(R = 3)
(R = 3)
11. Write the Maclaurin series of cos 2x.
a) 1 −
x4
x6
x2
+
−
+ ...
2!
4!
6!
b) 2 −
2x2
2x4
2x6
+
−
+ ...
2!
4!
6!
c) 1 −
16x4
64x6
4x2
+
−
+ ...
2!
4!
6!
d) 2x −
32x5
128x7
8x3
+
−
+ ...
3!
5!
7!
e) 2x −
2x3
2x5
2x7
+
−
+ ...
3!
5!
7!
12. Find the polar equation for the parabola y =
1 2
x .
2
a) r = 2 tan θ
b) r = 2 tan θ sec θ
c) r =
1
cot θ csc θ
2
d) r =
1
tan θ cos θ
2
SAMPLE A
e) r = 2 sin θ
2
a)
t
1 + t2
b)
t
1 + t4
c)
2t
t + t5
d)
1
t + t3
e)
2t2
1 + t4
MATH 141, 141E, 141H
FINAL EXAM
SAMPLE A
14. Identify the curve described by the equation r = 1−cos θ, θ ∈ (0, 2π].
15. If lim an = 0, then
n→∞
+∞
X
an is convergent.
n=1
a)
a) True
b) False
16. If
+∞
X
an is a convergent series with positive terms, then
n=1
+∞
X
n=1
1
an
is divergent.
a) True
b)
b) False
17.
lim (sin x)1/x is an indeterminate form.
x→0+
a) True
b) False
c)
Z
18. The improper integral
2
∞
1
dx converges.
x(ln x)2
a) True
b) False
19. If the radius of convergence of the power series
+∞
X
cn (x − 2)n is 1,
n=0
d)
then the series converges for x = 0.
a) True
b) False
For Problems 20 – 24, determine whether each series is absolutely
convergent, conditionally convergent, or divergent. Code on your
scantron sheet:
e)
A - if the series is Absolutely convergent,
C - if the series is Conditionally Convergent,
D - if the series is Divergent.
+∞
X (−1)n
20.
√
1+ n
n=1
21.
+∞
X
(−1)n+1
n=1
For Problems 15–19, mark (a) or (b) on your scantron sheet.
22.
+∞
X
n=2
23.
+∞
X
3
(−1)n−1
(ln n)n
(−1)n+1
n=1
n
2 + 3n
3n n 6
n!
MATH 141, 141E, 141H
FINAL EXAM
Evaluate the following limits. Choose each answer from options a)
through e). Note that an answer choice can be used more than once.
Be sure to bubble your answer choice on the scantron. Each limit is
worth 3 points.
a) 1
b) −1
c) 0
d) +∞
e) −∞
24.
lim
x→∞
2
arctan x
π
SAMPLE A FINAL EXAM
1. B 2. A 3. E 4. D 5. D 6. A 7. E 8. C 9. B 10. A 11. C 12. B
13. E 14. D 15. B 16. A 17. B 18. A 19. B 20. C 21. D 22. A 23.
A 24. A 25. B 26. C 27. E
28. R = 2; (4, 8]
29. a)ex =
2
b)ex =
25.
sin x − 1
cos x
lim
x→0
SAMPLE A
Z
c)
∞
X
xn
x2
x3
=1+x+
+
+ · · · , (−∞, ∞)
n!
2!
3!
n=0
∞
X
x2n
x4
x6
= 1 + x2 +
+
+ · · · , (−∞, ∞)
n!
2!
3!
n=0
2
ex dx = C +
∞
X
n=0
x2n+1
x3
x5
x7
= C +x+
+
+
+· · · ,
(2n + 1)n!
3
5 · 2! 7 · 3!
(−∞, ∞)
26.
√
lim
t→0+
30. a) 2 circles, both through pole, 1 symmetric to y-axis, 1 sym√ π
metric to x-axis. Intersection points: ( 3, ) and pole.
6
t ln t
π
6
Z
b) A =
0
27.
lim
t→0+
ln t
√
t
28. For the power series
∞
X
(−1)n (x − 6)n
,
n2n
n=1
a) (2pts) find the radius of convergence.
b) (6pts) Find the interval of convergence.
29.
a) (2 pts) Write the Maclaurin series for f (x) = ex in both
summation notation and term by term (4 terms).
2
b) (2 pts) Write the Maclaurin series for g(x) = ex in both
summation notation and term by term (4 terms).
Z
2
c) (5 pts) Express
ex dx as a power series in both summation notation and term by term (4 terms).
30.
√
a) (4 pts) Sketch the curves r = 2 cos θ and r = 2 3 sin θ on
one pair of axes, labeling all points of intersection.
b) (6 pts) Set up - but do not evaluate - an integral expression for the area of the intersection of the regions contained
inside both curves.
4
1 √
(2 3 sin θ)2 dθ +
2
Z
π
2
π
6
1
(2 cos θ)2 dθ
2