Math 126-104 CRN20678
Final Exam
Spring 2016
Carter
General Instructions. Write your name on only the outside of the blue book. Do not write on
this test sheet. Write neat, complete, articulate solutions to each of the problems below. To make
my job easier, please also state clearly which problem you are working. There are 165 points.
If you are walking in Japan, England, or Australia, for example, look right then left before
crossing a street.
1. Compute the following integrals (5 points each):
(a)
Z
2
1
(b)
1
dx
x2
π
Z
(1 + cos (x)) dx
0
(c)
Z
sin (x) cos3 (x) dx
(d)
Z
x ln (x) dx
(e)
Z p
36 − x2 dx
(f)
Z
1
dx
(x + 4)(x − 7)
(g)
Z
1
∞
1
x5/4
dx
2. (10 points) Find the area of the region enclosed by y = x2 − x and y = x.
3. (10 points) Compute the volume obtained by rotating the region bounded by the curves
y = ex , y = 0, x = 0, x = 1 about the x-axis.
√
4. (10 points) Find the surface area that is obtained by rotating y = 64 − x2 about the x-axis
Z b
p
for x ∈ [0, 8]. Recall here that SA =
2πf (x) 1 + (f 0 (x))2 dx.
a
5. (10 points) The half-life of the plutonium isotope is 24,360 years. If 20 grams of plutonium
is released into the atmosphere by a nuclear accident, how many years will it take for 90% of
the isotope to decay?
6. (10 points) A leaky bucket is being lifted 100 meters. It originally holds 50 kilograms of
water, and at the end of its journey it holds 25 kilos. How much work does it take to lift the
bucket?
1
7. Compute the limit of the following sequences, or state that the sequence is divergent (5 points
each).
1 n
5
3n2 +n+3
n2 +2n+1
(a) an = 1 +
(b) an =
(c) an = ln(n) − ln(n + 1)
√
(d) an = n n
8. Determine a formula for the nth partial sum and sum the series (5 points):
1+
1 1
1
+ + ··· + n + ···
3 9
3
9. Use any test that is applicable to determine which of the following series converge (5 points
each).
(a)
∞
X
1
√
n
n=2
(b)
∞
X
n=1
1
n11/10
(c)
∞
X
n=2
n3
n
−1
10. Compute the interval of convergence for the series (5 points each):
(a)
∞
X
(x − 3)n
n=0
4n
(b)
∞
X
(−1)n xn
n=1
n
(c)
∞
X
xn
n=1
2
n!
11. (5 points) Use substitution and power series operations to find the Taylor series about x=0
of the function f (x) = xex .
12. (10 points) Give a parametrization of the ellipse
clockwise in an interval t ∈ [0, 2π].
x2
16
13. (10 points) Compute the areas of the polar graphs
r = 1 + cos (θ).
Z
θ1
Recall, A =
θ0
r2
dθ.
2
3
+
y2
36
= 1 that travels once counter-