Math 126-103 (CRN 12418)
Final
Fall 2014
Carter
General Instructions. Write your name on only the outside of your blue book. Do not write upon
this test sheet. Write neat complete solutions to each of the problems below. As a courtesy to
me, please write the problem down before you attempt a solution. Label your solutions carefully.
There are 160 points.
A basic tuna salad consists of a drained can of tunafish, a tablespoon of mayonnaise, and sweet
pickle relish (to taste). One can also add finely chopped celery and/or carrots. Serve with lettuce
on a hardy whole grain bread.
1. (5 points) Complete the statement of the Fundamental Theorem of Calculus:
Let y = f (t) denote a continuous function that is defined on a closed interval [a, b]. Define
Z x
A(x) =
f (t) dt.
a
Then
.
2. Compute the following definite, indefinite, and improper integrals (5 points each).
(a)
π
Z
(1 + cos (x)) dx
0
(b)
Z
x2 sin (x3 ) dx
(c)
1
Z
xe2x dx
0
(d)
Z
sin (x) cos3 (x) dx
(e)
Z
√
x dx
x2 + 49
(f)
Z
1
dx
(x + 3)(x − 2)
(g)
Z
0
∞
dx
x ln (x)
3. (10 points) Compute the area that lies between the curve
y = x3 − x
and the x-axis, for −1 ≤ x ≤ 1.
1
4. (10 points) Find the volume of the tetrahedron, by considering the triangular cross sections
perpendicular to the x-axis.
4
12
y
x
7
z
5. (10 points) Compute the volume obtained by rotating the region bounded by the curves
√
y = x, x = 4, x = 9 and y = 0 about the x-axis.
√
6. (10 points) Find the surface area that is obtained by rotating y = 64 − x2 about the x-axis
for x ∈ [2, 4]. Recall that the surface area is given by the formula
b
Z
A=
2πf (x)
p
1 + (f 0 (x))2 dx.
a
7. (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 15% of
the isotope to remain?
8. (10 points) Sum the geometric series
1+
1
1
1
+
+ ··· + n + ··· .
8 64
8
9. (10 points) Compute the interval of convergence for the series:
∞
X
xn
n=1
n
10. (10 points) Use the MacLaurin series for y = sin (x), to determine the series for
y = x sin (4x).
2
11. Use any test that is appropriate to determine if the series is convergent or divergent (5 points
each)
(a)
∞
X
1
n ln (n)
n=2
(b)
∞
X
n=1
1
n11/10
(c)
∞
X
n=1
n2
n
+1
(d)
∞
X
(−3)n
n=0
12. (5 points) Give a parametrization of the ellipse
in an interval t ∈ [0, 2π].
x2
9
n!
2
+ y25 = 1 that travels once counter-clockwise
13. (5 points) Give the equation of the tangent line to the curve at the given point
x(t) = 4 cos (t)
14. (10 points) Use the formula A =
1
2
R θ1
θ2
y(t) = 4 sin (t)
at t = 2π/3.
r2 (θ) dθ to compute the area enclosed by
r = sin (θ)
for θ ∈ [0, π].
3