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

68 Chapter 1 Functions and Models
CAS
CAS
58. (a) What are the values of e ln 300 and lnse 300 d?
(b) Use your calculator to evaluate e ln 300 and lnse 300 d. What
do you notice? Can you explain why the calculator has
trouble?
59. Graph the function f sxd − sx 3 1 x 2 1 x 1 1 and explain
why it is one-to-one. Then use a computer algebra system
to find an explicit expression for f 21 sxd. (Your CAS will
produce three possible expressions. Explain why two of
them are irrelevant in this context.)
60. (a) If tsxd − x 6 1 x 4 , x > 0, use a computer algebra system
to find an expression for t 21 sxd.
(b) Use the expression in part (a) to graph y − tsxd, y − x,
and y − t 21 sxd on the same screen.
61. If a bacteria population starts with 100 bacteria and doubles
every three hours, then the number of bacteria after t hours
is n − f std − 100 ∙ 2 ty3 .
(a) Find the inverse of this function and explain its meaning.
(b) When will the population reach 50,000?
62. When a camera flash goes off, the batteries immediately
begin to recharge the flash’s capacitor, which stores electric
charge given by
Qstd − Q 0s1 2 e 2tya d
(The maximum charge capacity is Q 0 and t is measured in
seconds.)
(a) Find the inverse of this function and explain its meaning.
(b) How long does it take to recharge the capacitor to 90%
of capacity if a − 2?
63–68 Find the exact value of each expression.
63. (a) cos 21 s21d (b) sin 21 s0.5d
64. (a) tan 21 s3
(b) arctans21d
;
;
65. (a) csc 21 s2 (b) arcsin 1
66. (a) sin 21 (21ys2 ) (b) cos 21 (s3 y2)
67. (a) cot 21 (2s3 ) (b) sec 21 2
68. (a) arcsinssins5y4dd (b) cos(2 sin 21 ( 13))
5
69. Prove that cosssin 21 xd − s1 2 x 2 .
70–72 Simplify the expression.
70. tanssin 21 xd 71. sinstan 21 xd 72. sins2 arccos xd
73-74 Graph the given functions on the same screen. How are
these graphs related?
73. y − sin x, 2y2 < x < y2; y − sin 21 x; y − x
74. y − tan x, 2y2 , x , y2; y − tan 21 x; y − x
75. Find the domain and range of the function
tsxd − sin 21 s3x 1 1d
76. (a) Graph the function f sxd − sinssin 21 xd and explain the
appearance of the graph.
(b) Graph the function tsxd − sin 21 ssin xd. How do you
explain the appearance of this graph?
77. (a) If we shift a curve to the left, what happens to its
reflection about the line y − x? In view of this geometric
principle, find an expression for the inverse of
tsxd − f sx 1 cd, where f is a one-to-one function.
(b) Find an expression for the inverse of hsxd − f scxd,
where c ± 0.
1 Review
CONCEPT CHECK
1. (a) What is a function? What are its domain and range?
(b) What is the graph of a function?
(c) How can you tell whether a given curve is the graph of
a function?
2. Discuss four ways of representing a function. Illustrate your
discussion with examples.
3. (a) What is an even function? How can you tell if a function
is even by looking at its graph? Give three examples of an
even function.
(b) What is an odd function? How can you tell if a function
is odd by looking at its graph? Give three examples of
an odd function.
Answers to the Concept Check can be found on the back endpapers.
4. What is an increasing function?
5. What is a mathematical model?
6. Give an example of each type of function.
(a) Linear function
(b) Power function
(c) Exponential function (d) Quadratic function
(e) Polynomial of degree 5 (f) Rational function
7. Sketch by hand, on the same axes, the graphs of the following
functions.
(a) f sxd − x (b) tsxd − x 2
(c) hsxd − x 3 (d) jsxd − x 4
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