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

168 Chapter 2 Limits and Derivatives
(b) What does the statement f 9s10d − 1200 mean?
(c) Is f 9srd always positive or does it change sign?
42–44 Trace or copy the graph of the function. Then sketch a
graph of its derivative directly beneath.
42.
y 43.
y
49. Sketch the graph of a function f that satisfies all of the
following conditions: The domain of f is all real numbers
except 0, lim f sxd − 1, lim fsxd − 0, f 9sxd . 0 for
x l 02 x l 01
all x in the domain of f , lim f9sxd − 0, lim f9sxd − 1.
xl 2` xl`
50. Let Pstd be the percentage of Americans under the age of 18 at
time t. The table gives values of this function in census years
from 1950 to 2010.
44.
0 x
y
0 x
t Pstd t Pstd
1950 31.1 1990 25.7
1960 35.7 2000 25.7
1970 34.0 2010 24.0
1980 28.0
;
;
0
x
45. (a) If f sxd − s3 2 5x , use the definition of a derivative
to find f 9sxd.
(b) Find the domains of f and f 9.
(c) Graph f and f 9 on a common screen. Compare the graphs
to see whether your answer to part (a) is reasonable.
46. (a) Find the asymptotes of the graph of f sxd − 4 2 x
3 1 x and
use them to sketch the graph.
(b) Use your graph from part (a) to sketch the graph of f 9.
(c) Use the definition of a derivative to find f 9sxd.
(d) Use a graphing device to graph f 9 and compare with
your sketch in part (b).
47. The graph of f is shown. State, with reasons, the numbers
at which f is not differentiable.
y
_1
0
2
4 6
48. The figure shows the graphs of f , f 9, and f 0. Identify each
curve, and explain your choices.
y
a
x
(a) What is the meaning of P9std? What are its units?
(b) Construct a table of estimated values for P9std.
(c) Graph P and P9.
(d) How would it be possible to get more accurate values
for P9std?
51. Let Bstd be the number of US $20 bills in circulation at time t.
The table gives values of this function from 1990 to 2010, as
of December 31, in billions. Interpret and estimate the value
of B9s2000d.
t 1990 1995 2000 2005 2010
Bstd 3.45 4.21 4.93 5.77 6.53
52. The total fertility rate at time t, denoted by Fstd, is an estimate
of the average number of children born to each woman
(assuming that current birth rates remain constant). The
graph of the total fertility rate in the United States shows
the fluctuations from 1940 to 2010.
(a) Estimate the values of F9s1950d, F9s1965d, and F9s1987d.
(b) What are the meanings of these derivatives?
(c) Can you suggest reasons for the values of these derivatives?
y
3.5
3.0
2.5
2.0
1.5
baby
boom
y=F(t)
baby
bust
baby
boomlet
0
b
c
x
1940 1950 1960 1970 1980 1990 2000 2010
53. Suppose that | f sxd |
Find lim x l a f sxd.
< tsxd for all x, where lim x l a tsxd − 0.
54. Let f sxd − v x b 1 v2x b.
(a) For what values of a does lim x l a f sxd exist?
(b) At what numbers is f discontinuous?
t
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.