Practice Final Exam - Loyola University Chicago

Loyola University Chicago - Department of Mathematics and Statistics
MATH 162-005 Calculus II - Spring 2012 - Instructor: Marian Bocea
Practice Final Exam
NAME (please print clearly):
Question 1 (15 points):
Question 2 (15 points):
Question 3 (15 points):
Question 4 (15 points):
Question 5 (15 points):
Question 6 (15 points):
Question 7 (15 points):
Question 8 (15 points):
TOTAL SCORE:
Notes:
1. You have 120 minutes to complete the exam.
2. For full credit you must show your work completely. Simply writing down an
answer without justifying it will receive very little partial credit.
3. NO TEXTBOOKS, NOTES or CALCULATORS are allowed while you take this
exam.
1

1. (15 points) Find the arc length function for the curve y = x3 + 3 x2 + x + 1
What is the length of the curve from (0, 1/4) to (1, 59/24) ?
4x+4 .
2. (15 points) Let P (x) = a n x n +a n−1 x n−1 +· · · a 1 x+a 0 be any polynomial function
with a i ∈ R (i = 1, · · · , n), and let E(x) = e x . Show that P = o(E) as x → ∞.
2

3. (15 points) Use integration by parts to evaluate the integral
∫
1/ √ 2
2x sin −1 (x 2 )dx.
0
4. (15 points) Use a comparison test for improper integrals to determine whether
the integral
∫ ∞
1
1 − e −x
x
converges or diverges.
3

5. (15 points) For what values of the parameter a, if any, does the series ∞ ∑
converge?
n=1
( a
− )
1
n+2 n+4
6. (15 points) Show that the series
converges.
∞∑
n=1
1 · 3 · 5 · · · · (2n − 1)
[2 · 4 · 6 · · · · (2n)](3 n + 1)
4

7. (15 points) (a) Find the radius of convergence and the interval of convergence of
∑
the power series ∞ √ x n
n
; For what values of x ∈ R does the series converge
2 +3
n=1
(b) absolutely; (c) conditionally?
8. (15 points) (a) Use series to evaluate the limit lim x 2 (e −1/x2 − 1); (b) Evaluate
x→∞
the nonelementary integral
∫1/64
0
tan −1 x
√ x
using series.
5