Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 581 — #589
14.7. Asymptotic Notation 581
Using the Integral Method of Section 14.3, we can find integers a, b, c, d and a
real number e such that
Z b
a
x e dx S
What are appropriate values for a; : : : ; e?
Class Problems
Z d
c
x e dx
Problem 14.8.
Let f W R ! R be a continuous, weakly increasing function. Say that f grows
slowly when
Z n
f .n/ D o f .x/ dx :
(a) Prove that the function f a .n/ WWD n a grows slowly for any a > 0.
(b) Prove that the function e n does not grow slowly.
(c) Prove that if f grows slowly, then
1
Z n
1
f .x/ dx
nX
f .i/ :
iD1
Exam Problems
Problem 14.9.
Assume n is an integer larger than 1. Circle all the correct inequalities below.
Explanations are not required, but partial credit for wrong answers will not be
given without them. Hint: You may find the graphs in Figure 14.9 helpful.
nX
ln.i C 1/ ln 2 C
iD1
nX
ln.i C 1/
iD1
nX
iD1
Z n
0
Z
1 n
i 1
0 x C 1 dx
Z n
1
ln.x C 2/dx
ln.x C 1/dx