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Main points of this past exam are: Rigorous Argument, Determine, Improper Integrals, Converges, Diverges, Calculate, Value
Typology: Exercises
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Math 106-C (Salomone) February 27, 2009 Show all your work!
Name:
Score (25 points possible):
Problem 1. (10 points) Determine whether each of the following improper integrals converges or diverges. You do
not need to calculate their value, but you should provide a rigorous argument. (Write a sentence or two.)
(a) (4 points)
0
1 + x 2 1 − x^2
dx
(b) (3 points)
2
dx x 4 + x + 12
(c) (3 points)
1
5 − z + 3 z 2 9 − 9 z − 17 z 2
dz
Problem 2. (10 points) Evaluate the improper integral
∫ (^) ∞
0
x 3 e −x^ dx.
Problem 3. (5 points) Is there any real number n for which
0 x^
n (^) e −x (^) dx diverges? If so, give an example and explain
why it works. If not, explain why not.
