Math 126-104
Final
Summer 2013
Carter
General Instructions: Write your name on only the outside of your blue book. Do all your work
inside your blue book. Write neat, complete, solutions to the problems below. Please label the
problem upon which you are working, and write a statement or paraphrase of the problem. Indicate
your solution clearly. There are 160 points. To make a nice Thai curry use about a tablespoon of
curry paste, some hot oil, various vegetables, tofu, and meat if desired. Fry the vegetables in oil
and curry paste, add meat, then add at least one can of coconut milk. Serve over rice. Add chili
peppers and fish sauce to taste.
1. Compute (5 points each):
(a)
2
Z
(x3 − 4x2 + 3) dx
0
(b)
Z
x sin (x) dx
(c)
π/6
Z
3 cos (3x)esin (3x) dx
0
(d)
Z
√
dx
9 − x2
(e)
Z
dx
(x + 1)(x + 3)
(f)
Z
∞
e−x dx
0
2. Compute the area that lies between the curves (10 points):
y = x2
and y = x
for x ∈ [0, 2].
3. (10 points) Compute the volume obtained by rotating the region bounded by the curves
√
y = x2 , y = x between x = 0 and x = 1 about the x-axis.
3
4. (10 points) Find the length
of the curve y = x 2 from x = 0 to x = 4. Recall that the formula
Rbq
for arc-length is L = a 1 + (f 0 (x))2 dx.
5. (10 points) A force of 25-Newtons will stretch a spring 5 meters beyond its natural length.
Assuming that Hooke’s Law applies, how much work does it take to stretch the spring 10
meters beyond its natural length? Recall: Hooke’s law states that the force required to stretch
a spring is proportional to the length that it is stretched.
1
6. (10 points) The half-life of the plutonium isotope is 24,360 years. If 10 grams of plotonium
is released into the atmosphere by a nuclear accident, how many years will it take for 90% of
the isotope to decay? You don’t need to simplify your answer.
7. Find the limit if it exists (5 point):
(a)
n2 − 3
n→∞ 2n2 − 2n + 1
lim
(b)
r
lim
n→∞
n
n+1
(c)
lim
n→∞
1
(−1) +
n
n
(d)
(−6)n
n→∞
n!
lim
8. Find the sum (5 points):
1+
2
4
8
2n
+
+
+ ··· + n + ···
5 25 125
5
9. Use any test that you like to determine if the given series converges (5 points each).
(a)
∞
X
1
n1.1
n=1
(b)
∞
X
n=1
n2
n
+1
6n
1
+1
(c)
∞
X
n=1
(d)
X n!
(2n)!
n=1
2
10. Compute the interval of convergence for the series (10 points):
∞
X
(x − 1)n
n=0
2n
11. (10 points) Use substitution to find the Taylor series about x=0 (MacLaurin series) of the
function f (x) = e−2x .
2
12. ( 5 points) Give a parametrization of the ellipse x16 +
once counter-clockwise in the interval t ∈ [0, 2π].
y2
9
= 1 that starts at (4, 0) and travels
13. (10 points) Compute the area enclosed by the polar graph: r = cos (θ) for θ ∈ [−π/2, π/2].
3