Math 126
Carter
Final Exam
Spring 2011
General Instructions: Write your name on only the outside of your blue book. Do all
of your work and write your answers inside your blue book. Put your test paper inside your
blue book as you leave. Solve each of the following problems; point values are indicated on
the problems. There are 150 total points available on the test.
A nice tomato and cucumber salad can be made with 1 medium union chopped, 1 large
cucumber sliced, 3 to 4 medium tomatoes chopped, some balsamic vinegar, olive oil, sugar,
the juice of one lemon squeezed, and possibly some added white vinegar. Serve within 3
hours of preparation.
Good Luck. I enjoyed teaching you this semester!
1. (5 points). Let y = f (x) denote a continuous function. State one version of the
Fundamental Theorem of Calculus.
2. (5 points each). Compute the following:
Z
(a)
Z
(b)
(c)
Z
(e)
(f)
Z
x
dx
+1
x2
x cos (3x) dx
sin3 (x) cos3 (x) dx
Z
(d)
√
√
x2
dx
9 − x2
1
dx
(x − 2)2 (x − 1)
Z ∞
1
dx
(x + 4)2
3. (10 points). Compute the volume that is obtained by rotating the region in the plane
that is bounded between the curves y = x2 − 1 and y = 2x − 1 about the line y = 4.
4. (10 points). Set up, but don’t solve, an integral that computes the arc length of the
curve y = x2/3 for x ∈ [1, 3].
5. (10 points). Compute the radius of convergence for the series:
∞
X
2n xn
n=1
1
n!
6. Consider the function f (x) = (1 − x)−1 .
(a) (10 points). Systematically compute the quantities in the table below:
f (x)
f 0 (x)
f 00 (x)
f (3) (x)
..
.
f (0)
f 0 (0)
f 00 (0)
f (3) (0)
..
.
f (n) (x) f (n) (0)
(b)
(c)
(d)
(e)
(5 points). Give the nth Taylor polynomial for f at a = 0.
1
(5 points) Express 1−x
as an infinite series.
(5 points). Compute the radius of convergence for the series.
(10 points). Use substitution and integration to compute the series expansion for
f (x) = arctan (x).
7. (5 points each). Compute the following limits (or indicate why they do not exist):
(a)
8n
lim
n→∞ n!
(b)
lim 1 +
n→∞
(c)
lim
n→∞
h√
5
n
n2 + n −
n
i
√
n2 − n
8. (10 points each). Use the root test, ratio test, comparison theorem, integral comparison, or limit comparison to determine if the following series converge:
(a)
∞
X
n2
3
n=1 n + 1
(b)
∞ n
X
2
n=1
3
(c)
∞
X
n4
n=1 n!
9. (10 points). Give a parametrization of the ellipse
(x − 2)2 (y + 3)2
+
=1
25
144
that starts at the point (7, −3) and travels once around the ellipse counterclockwise in
a time interval [0, 2π).
2