MATH 141
FINAL EXAM
1. Find (f −1 )0 (0) for f (x) = x2 − 4x − 5, where f (x) is defined for
x > 2.
SAMPLE C
π/2
Z
6. Evaluate
3 cos3 x dx.
0
a) 1/2
a) 1/6
b) −2
b) 1/5
c) 1/3
c) 6
d) 2
d) -1/4
e) 3
e) 5
2. Differentiate y = (cos x)x .
a) −(cos x)x tan x
b) (cos x)x [ln(cos x) − x tan x]
c) (cos x)x [ln(cos x) + x sec x]
d) ln(cos x) + x sec x
e) −(cos x)x sec x
3. Find f −1 (x) if f (x) =
ex + 1
.
ex − 1
Z
a) ln
7. After the appropriate substitution the integral
ex + 1 comes
ex − 1
b) − ln(x − 1)
x + 1
c) ln
x−1
d)
Z
a)
sec θ dθ.
Z
b)
x+1
x−1
cos θ dθ.
Z
cos3 θ
dθ.
sin θ
Z
csc2 θ dθ.
Z
cos2 θ
dθ.
sin θ
c)
x
e) x
e +1
d)
4. Find lim
x→∞
ln(3 +
4x
e2x )
.
e)
a) 0
b) 1/4
8. Find lim
x→0
sin x
.
x3
c) 1/2
a) ∞
d) 3/4
b) 0
e) ∞
c) -1/6
Z
5. Evaluate
d) 1/2
ln x
dx.
x2
e) 1
a) 1 + ln x + C
b) 1 − ln x + C
c)
1 + ln x
+C
x
d)
1 − ln x
+C
x
e)
−(1 + ln x)
+C
x
1
dx
√
bex2 x2 − 1
MATH 141
FINAL EXAM
∞
Z
9. Evaluate the improper integral
2
xe−x dx.
13. The series
1
∞
X
n=1
1
a)
2e
b)
SAMPLE C
1
n2 − ln n
a) converges by the direct comparison test with
n=1
1
+1
e
b) converges by the limit comparison test with
2
c)
e
d) −
∞
X
∞
X
n=1
1
.
n2
1
.
n2
c) converges by the ratio test.
2
e
d) diverges by the limit comparison test with
e) The integral diverges.
∞
X
n=1
1
.
n2
e) diverges by the test for divergence.
2n−1 + 1
converges or di10. Determine whether the sequence an = n−1
3
+1
14. Which of the following series converge?
verges. If it converges, find the limit.
∞
X
1
I.
π
n
n=14
a) converges to 1
b) converges to
2
3
II.
c) converges to 0
d) converges to
III.
1
3
e) The sequence diverges.
11. Find the sum of the series 5 −
∞
X
3n
2n
2
n=0
∞
X
(−1)n
√
n
n=1
a) I and II
b) I and III
20
40
10
+
−
+ ···.
3
9
27
c) I only
a) 3/5
d) III only
b) -2
e)
all of the above
c) 15
15. Find the Radius of Convergence, R, of the series
d) 3
e) 5
a) R = 1/3
12. Find the values of p for which the series
∞
X
p
n(1 + n2 ) is conver-
b) R = 3/2
c) R = 1
n=0
gent.
d) R = 2
e) R = 3
a) p = 0
b) p > 1
c) p < 1
d) p > −1
e)
p < −1
2
∞
X
(−1)n (3x − 2)n
.
n3n
n=1
MATH 141
FINAL EXAM
Z
16. Evaluate the indefinite integral
a)
b)
t
dt as a power series.
1 − t2
Arrows indicate the direction in which the curve is traced.
∞
X
t2n+1
+C
2n + 1
n=0
∞
X
t2n + C
n=1
c)
d)
∞
X
t2n+2
+C
2n + 2
n=0
∞
X
n=1
e)
tn
+C
n+1
∞ 2n
X
t
+C
2n
n=0
17. Find the Maclaurin series for f (x) = tan−1 (x2 ).
a)
∞
X
(−1)n x4n
(2n)
n=0
b)
∞
X
(−1)n x2n+1
(2n + 1)!
n=0
c)
∞
X
x4n
n=0
d)
∞
X
(−1)n x4n+2
(2n + 1)
n=0
e)
∞
X
(−1)n x4n+2
(4n + 2)
n=0
SAMPLE C
18. Find the curve described by the parametric equations
x = cos t, y = − sin t, 0 ≤ t ≤ π.
3
MATH 141
FINAL EXAM
19. Find the slope of the tangent line to the curve x = ln t, y = t + t2
at the point (0, 2).
24.
SAMPLE C
√
∞
X
(−1)n n
n
3
n=1
a) 3
b) 1
c) 1/3
d) 2/3
e) e
25.
20. Find a Cartesian equation for the curve described by the polar equation r = 6 sin θ.
a) x2 + y 2 = 32
b) (x − 3)2 + y 2 = 32
c) (x + 3)2 + y 2 = 32
d) x2 + (y − 3)2 = 32
e) x2 + (y + 3)2 = 32
For each of the series 21−25 below, determine whether it is absolutely convergent, conditionally convergent, or divergent. Code on
your scantron
A if the series is Absolutely convergent,
C if the series is Conditionally convergent,
D if the series is Divergent.
21.
∞
X
(−1)n 2n
3n + 1
n=1
22.
∞
X
cos(nπ)
√
n
n=1
23.
∞
X
(−1)n
ln(n3 )
n=2
4
∞
X
(−1)n (
n=1
3n + 1 n
)
2n + 5
MATH 141
FINAL EXAM
Z
26. (10 points) Evaluate
4
dx. Show all work for full credit.
x(x2 + 4)
27. (10 points)
a) Write the third degree Taylor polynomial T3 (x) for f (x) = ln(1 +
x) centered at 1.
Show all work for full credit.
b) Use Taylor’s Inequality to estimate the accuracy of the ap1
3
proximation f (x) ≈ T3 (x) when x lies in the interval ≤ x ≤ .
2
2
28. (15 points) Consider the region that lies inside the circle r = 3 sin θ
and outside the cardioid r = 1 + sin θ.
a) Find point(s) of intersection of the two curves.
b) Sketch the curves. Label any intersection point(s) and
shade the region inside the circle and outside the cardioid.
c) Set up but DO NOT EVALUATE the integral to find
the area of the region that lies inside the circle and outside
the cardioid.
KEY: 1. A 2. B 3. C 4. C 5. E 6. D 7. B 8. A 9. A 10. C 11. D
12. E 13. B 14. E 15. C 16. C 17. D 18. B 19. A 20. D 21. A 22.
C 23. C 24. A 25. D
26. ln √
|x|
+C
x2 + 4
27a) T3 (x) = ln 2 +
27b)
1
1
1
(x − 1) − (x − 1)2 +
(x − 1)3
2
8
24
1
324
3 π
3 5π
28a) ( , ), ( ,
) and pole
2 6
2 6
28b) picture of a circle intersecting with a cardiod at the above
intersection points.
π
2
Z
28c) A = 2
π
6
Z
OR A =
5π
6
π
6
1
(3 sin θ)2 − (1 + sin θ)2 dθ
2
1
(3 sin θ)2 − (1 + sin θ)2 dθ
2
5
SAMPLE C