Math 121, section 1
Final Exam
1. [10 points] Evaluate each limit. Justify your answer.
x3
sin x − arctan x
(a) lim
x→0
(b) lim (1 + ln(2x))tan(πx)
x→ 12
2. [15 points] Evaluate each integral.
Z π/2
cos4 t dt
(a)
0
√
Z
2
1
dx
+ 2)5/2
0
Z
arctan x
(c)
dx
x2
(b)
(x2
3. [15 points] For each improper integral, evaluate it if possible, or determine that it diverges.
Z ∞
1
√
√ dx
(a)
x
x
+ x
1
Z ∞
x
(b)
dx
2
x + 3x + 2
0
Z ∞
1
dx
(c)
x ((ln x)2 + ln x)
e
4. [15 points] Evaluate each of the following sums (you may assume that each sum converges).
∞
X
x2n
(express your answer in terms of x).
(a)
n!
n=0
4
(b) 4 − 43 + 45 − 47 + 94 − 11
+ ···
1
1
1
1
(c) − 2
+ 3
− 4
+ ···
2 2 ·2 2 ·3 2 ·4
∞
X
1 + 2n
(d)
n!
(e)
n=1
∞
X
n=1
(−1)n−1 n
x , assuming that x > 0 (express your answer in terms of x).
(2n + 1)!
5. [20 points] For each series, determine whether it is absolutely convergent, conditionally
convergent, or divergent. Justify your answers.
√
∞
X
n−1 n + n
(a)
(−1)
n3 + 1
n=1
∞ √ 2
X
n +1
(b)
n
(c)
n=1
∞
X
(−1)n−1
n=1
n!
(2n)n Math 121, section 1
(d)
(e)
Final Exam
∞
X
n2 + 100n
n=0
∞
X
n=0
3n
arctan(n)
n2
6. [7 points] Find the interval of convergence for the following power series. Analyze carefully
with full justification.
∞
X
1
xn
2
n +n+1
n=0
7. [4 points] Suppose that a patient takes a single dose of a certain medication every day for a
long period of time. Each dose is 100mg of medication, which is slowly removed from the
patient’s bloodstream according to the following rule: t days after the patient takes the dose,
t
the number of milligrams left from that dose in the patient’s bloodstream is 100 · 45 .
If the patient continues to take this medication once a day for a very long period of time, how
many milligrams of medication will be in her bloodstream immediately after taking a dose?
8. [8 points] Consider the region bounded by the curves y = ex , y = e, and the y-axis. This
region is revolved around the line x = −1 to obtain a solid. Compute the volume of this solid.
9. [10 points] Consider the paramteric curve given by the following equations.
x(t) = 4 sinh(t)
y(t) = 8 cosh(t)
(a) Determine the tangent line to this curve when t = ln 2.
(b) Set up but do not evaluate an integral computing the arc length of this curve between
t = 0 and t = ln 2.
(c) Set up but do not evaluate an integral for the surface area obtained by revolving this
curve around the y-axis.
10. [12 points] For each region described, set up but do not evaluate an integral (or sum of
integrals) for its area.
(a) The area bounded inside the polar curve r = 2 − 2 cos θ.
(b) The area outside the polar curve r = 1 and inside the polar curve r = 1 − sin θ.
(c) The area inside both of the polar curves r = 1 and r = 2 − 2 cos θ.
√
(d) The image below depicts the polar curve r = 1 − 2 sin θ. Set up but do not evaluate
an integral computing the area of the shaded region (enclosed by the “inner loop” of the
curve). Math 121, section 1
Final Exam