James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

126 Chapter 2 Limits and Derivatives
;
53– 56 Use the Intermediate Value Theorem to show that
there is a root of the given equation in the specified interval.
53. x 4 1 x 2 3 − 0, s1, 2d
54. ln x − x 2 sx , s2, 3d
55. e x − 3 2 2x, s0, 1d
56. sin x − x 2 2 x, s1, 2d
57 – 58 (a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that
contains a root.
57. cos x − x 3 58. ln x − 3 2 2x
59– 60 (a) Prove that the equation has at least one real root.
(b) Use your graphing device to find the root correct to three
decimal places.
59. 100e 2xy100 − 0.01x 2
60. arctan x − 1 2 x
61– 62 Prove, without graphing, that the graph of the function
has at least two x-intercepts in the specified interval.
61. y − sin x 3 , s1, 2d
62. y − x 2 2 3 1 1yx, s0, 2d
63. Prove that f is continuous at a if and only if
lim f sa 1 hd − f sad
h l 0
64. To prove that sine is continuous, we need to show
that lim x l a sin x − sin a for every real number a.
By Exercise 63 an equivalent statement is that
lim sinsa 1 hd − sin a
h l 0
Use (6) to show that this is true.
65. Prove that cosine is a continuous function.
66. (a) Prove Theorem 4, part 3.
(b) Prove Theorem 4, part 5.
67. For what values of x is f continuous?
f sxd −H 0 1
if x is rational
if x is irrational
68. For what values of x is t continuous?
tsxd −H 0 x
if x is rational
if x is irrational
69. Is there a number that is exactly 1 more than its cube?
70. If a and b are positive numbers, prove that the equation
a
x 3 1 2x 2 2 1 1
b
x 3 1 x 2 2 − 0
has at least one solution in the interval s21, 1d.
71. Show that the function
f sxd −H x 4 sins1yxd
0
is continuous on s2`, `d.
if x ± 0
if x − 0
72. (a) Show that the absolute value function Fsxd − | x | is
continuous everywhere.
(b) Prove that if f is a continuous function on an interval,
then so is | f | .
(c) Is the converse of the statement in part (b) also true? In
other words, if | f | is continuous, does it follow that f is
continuous? If so, prove it. If not, find a counterexample.
73. A Tibetan monk leaves the monastery at 7:00 am and
takes his usual path to the top of the mountain, arriving at
7:00 pm. The following morning, he starts at 7:00 am at the
top and takes the same path back, arriving at the monastery
at 7:00 pm. Use the Intermediate Value Theorem to show
that there is a point on the path that the monk will cross at
exactly the same time of day on both days.
In Sections 2.2 and 2.4 we investigated infinite limits and vertical asymptotes. There we
let x approach a number and the result was that the values of y became arbitrarily large
(positive or negative). In this section we let x become arbitrarily large (positive or negative)
and see what happens to y.
Let’s begin by investigating the behavior of the function f defined by
f sxd − x 2 2 1
x 2 1 1
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