Exercises 1—14, to establish a big- O relationship, find witnesses C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k . Show that x 3 is O ( x 4 ) but that x 4 is not O ( x 3 ) .
Exercises 1—14, to establish a big- O relationship, find witnesses C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k . Show that x 3 is O ( x 4 ) but that x 4 is not O ( x 3 ) .
Solution Summary: The author explains that the given function x3 is Oleft, but it's not true.
Each of Exercises 15–30 gives a function f(x) and numbers L, c, and
ɛ > 0. In each case, find an open interval about c on which the inequal-
ity |f(x) – L| 0 such that for all
x satisfying 0 0, L= 2m, c = 2,
— тх,
ɛ = 0.03
28. f(x) = mx,
ɛ = c > 0
L = 3m,
c = 3,
m > 0,
L = (m/2) + b,
29. f(x)
c = 1/2,
m> 0,
ɛ = c > 0
= mx + b,
30. f(x) 3D тх + b, m> 0, L%3Dm+ b, с %3D 1, &%3D 0.05
Use the Extreme Value Theorem to show that each function f
in Exercises 49–54 has both a maximum and a minimum value
on [a, b]. Then use a graphing utility to approximate values M
and m in [a, b] at which f has a maximum and a minimum,
respectively. You may assume that these functions are contin-
uous everywhere.
49. f(x) = xª – 3x² – 2, [a, b] = [–2, 2]
50. f) — х1 — 3х? — 2, [а, b] — [0, 2]
51. f) %3D х* - Зx? - 2, [а, b] %—D [-1, 1]
52. f(x) = 3 – 2r² +x³, [a, b] = [-1, 2]
53. f(x) = 3 – 2x2 +x³, [a, b] = [0, 2]
54. f(x) = 3 – 2x?+x³, [a, b] = [–1, 1]
%3D
Chapter 3 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
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