Calculus III – MTH 253
Calculus I & II review
Due Wednesday, June 24
Name:
Show all necessary work to justify your answer.
(1) Evaluate the following integral where a > 0. (Answer depends on a.)
Z∞
0
e−ax dx
(2) Evaluate the following integral in which bis arbitrary. (Answer depends on b.)
Zπ/2
0
√b+ sin xcos x dx
(3) Compute the following limits
(a) lim
x→∞
x3
ex2
(b) lim
x→∞
√x
ln x
(4) Consider the following functions: f(x) = 3x2−2x+ 2 and g(x) = x2+x+ 1. Prove
that f(x)≥g(x) for all x≥1 as follows.
(a) Show that f(1) −g(1) = 0.
(b) Show that f0(x)−g0(x)>0 for x≥1.
(c) Explain why items (a) and (b) above lead to the desired conclusion.
G. Lafferriere, June 22, 2009