Math 121 Final Exam
May 9, 2017
• This is a closed-book examination. No books, notes, calculators, cell phones, communication
devices of any sort, or other aids are permitted.
• You
However, numerical answers such as
π need3 not simplify algebraically complicated answers.
√
ln
4
7
−
ln
5
3
ln
3
sin
, e
, arctan( 3), or cosh(ln 3) should be simplified.
, 4 2 , e , ln(e ), e
6
• Please show all of your work and justify all of your answers. (You may use the backs of pages for
additional work space.)
1. [12 Points] Evaluate the following limit. Please justify your answer. Be clear if the limit
equals a value, +∞ or −∞, or Does Not Exist. Simplify.
x
5xex − arctan(5x)
5
(a) lim
1 − arcsin
(b) Compute lim
x→∞
x→0 sinh x + ln(1 − x)
x
2.
[16 Points] Evaluate each of the following integrals.
Z
Z 0
cos x
(b)
(a)
x arcsin x dx
7 dx
−1
1 + sin2 x 2
3.
[40 Points] For each of the following improper integrals, determine whether it converges
or diverges. If it converges, find its value. Simplify.
Z
e3
(a)
0
Z
(c)
1
4.
2
(b)
0
Z
2
dx
2
x − 6x + 8
[20 Points]
ln x
√ dx
x
∞
(d)
5
x2
1
dx
− 6x + 13
Find the sum of each of the following series (which do converge). Simplify.
∞
X
(−1)n 32n−1
(a)
42n+1
n=1
(d) 1 −
e
Z
1
dx
x [9 + (ln x)2 ]
1 1 1 1
+ − + −...
3 5 7 9
∞
X
(−1)n (ln 8)n
(b)
3n+1 n!
(c)
n=0
(e) −
π2
2!
+
π4
4!
−
∞
X
n=0
π6
6!
+
π8
8!
−...
(−1)n π 2n
(36)n (2n + 1)!
(f) −1 +
1 1 1
− + −...
2 3 4
5.
[30 Points] In each case determine whether the given series is absolutely convergent,
conditionally convergent, or divergent. Justify your answers.
∞
X
(−1)n (n3 + 7)
(a)
n7 + 3
n=1
(c)
∞
X
(−1)n arctan(n2 )
n2 + 1
n=1
∞
X
n+1
n2
n=1
∞
X
n2
(d)
arctan
n2 + 1
(b)
(−1)n
n=1
1
(e)
∞
X
(−1)n (3n)! ln n
(n!)2 24n nn
n=1 6.
[12 Points] Find the Interval and Radius of Convergence for the following power series
∞
X
(−1)n (3x + 1)n
. Analyze carefully and with full justification.
(n + 7) · 7n
n=1
Z 1
1
7. [10 Points] (a) Use MacLaurin series to Estimate x sin x2 dx with error less than
.
1000
0
Please analyze with detail and justify carefully. Simplify.
1
1
1
. Justify in words that your error is indeed less than
.
(b) Estimate √ with error less than
100
100
e
8.
[8 Points] For each of the following functions, find the MacLaurin Series and, then State the
Radius of Convergence.
1
1
(a) f (x) = sinh x
(b) f (x) =
. Hint: Differentiate
(1 − x)2
1−x
9.
[18 Points]
π
, and x = 0. Rotate the region about
2
the vertical line x = 3 . Set-up, BUT DO NOT EVALUATE!!, the integral to compute the
volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one
of the approximating shells.
(a) Consider the region bounded by y = arcsin x, y =
(b) Consider the region bounded by y = arctan x, y = 0, x = 0 and x = 1. Rotate the
region about the vertical line y-axis . COMPUTE the volume of the resulting solid using the
Cylindrical Shells Method. Sketch the solid, along with one of the approximating shells.
10.
[14 Points]
√
x = ln t + ln(1 − t2 ) and y = 8 arcsin t.
1
1
≤ t ≤ . Show that the answer
COMPUTE the arclength of this parametric curve for
4
2
5
simplifies to ln
2
Consider the Parametric Curve represented by
11.
[18 Points]
For each of the following problems, do the following two things:
1. Sketch the Polar curves and shade the described bounded region.
2. Set-Up but DO NOT EVALUATE the Integral representing the area of the described bounded
region.
(a) The area bounded outside the polar curve r = 1+cos θ and inside the polar curve r = 3 cos θ.
(b) The area bounded outside the polar curve r = 2 and inside the polar curve
r = 4 sin θ.
(c) The area that lies inside both of the curves r = 1+sin θ and inside the polar curve r = 1−sin θ.
2