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

Section 4.3 How Derivatives Affect the Shape of a Graph 303
(b) When is this rate highest?
(c) On what intervals is the population function concave
upward or downward?
(d) Estimate the coordinates of the inflection point.
66. In an episode of The Simpsons television show, Homer
reads from a newspaper and announces “Here’s good news!
According to this eye-catching article, SAT scores are
declining at a slower rate.” Interpret Homer’s statement in
terms of a function and its first and second derivatives.
67. The president announces that the national deficit is increasing,
but at a decreasing rate. Interpret this statement in terms
of a function and its first and second derivatives.
68. Let f std be the temperature at time t where you live and suppose
that at time t − 3 you feel uncomfortably hot. How do
you feel about the given data in each case?
(a) f 9s3d − 2, f 0s3d − 4
(b) f 9s3d − 2, f 0s3d − 24
(c) f 9s3d − 22, f 0s3d − 4
(d) f 9s3d − 22, f 0s3d − 24
69. Let Kstd be a measure of the knowledge you gain by
studying for a test for t hours. Which do you think is larger,
Ks8d 2 Ks7d or Ks3d 2 Ks2d? Is the graph of K concave
upward or concave downward? Why?
70. Coffee is being poured into the mug shown in the figure at a
constant rate (measured in volume per unit time). Sketch a
rough graph of the depth of the coffee in the mug as a function
of time. Account for the shape of the graph in terms of
concavity. What is the significance of the inflection point?
71. A drug response curve describes the level of medication
in the bloodstream after a drug is administered. A surge
function Sstd − At p e 2kt is often used to model the response
curve, reflecting an initial surge in the drug level and then a
more gradual decline. If, for a particular drug, A − 0.01,
p − 4, k − 0.07, and t is measured in minutes, estimate the
times corresponding to the inflection points and explain their
significance. If you have a graphing device, use it to graph
the drug response curve.
72. The family of bell-shaped curves
y − 1
s2 e2sx2d 2 ys2 2 d
occurs in probability and statistics, where it is called the normal
density function. The constant is called the mean and
;
the positive constant is called the standard deviation. For
simplicity, let’s scale the function so as to remove the factor
1yss2 d and let’s analyze the special case where − 0.
So we study the function
f sxd − e 2x 2 ys2 2 d
(a) Find the asymptote, maximum value, and inflection
points of f .
(b) What role does play in the shape of the curve?
(c) Illustrate by graphing four members of this family on the
same screen.
73. Find a cubic function f sxd − ax 3 1 bx 2 1 cx 1 d that has a
local maximum value of 3 at x − 22 and a local minimum
value of 0 at x − 1.
74. For what values of the numbers a and b does the function
f sxd − axe bx2
have the maximum value f s2d − 1?
75. (a) If the function f sxd − x 3 1 ax 2 1 bx has the local minimum
value 2 2 9 s3 at x − 1ys3 , what are the values of a
and b?
(b) Which of the tangent lines to the curve in part (a) has the
smallest slope?
76. For what values of a and b is s2, 2.5d an inflection point of the
curve x 2 y 1 ax 1 by − 0? What additional inflection points
does the curve have?
77. Show that the curve y − s1 1 xdys1 1 x 2 d has three points
of inflection and they all lie on one straight line.
78. Show that the curves y − e 2x and y − 2e 2x touch the curve
y − e 2x sin x at its inflection points.
79. Show that the inflection points of the curve y − x sin x lie on
the curve y 2 sx 2 1 4d − 4x 2 .
80–82 Assume that all of the functions are twice differentiable
and the second derivatives are never 0.
80. (a) If f and t are concave upward on I, show that f 1 t is
concave upward on I.
(b) If f is positive and concave upward on I, show that the
function tsxd − f f sxdg 2 is concave upward on I.
81. (a) If f and t are positive, increasing, concave upward functions
on I, show that the product function ft is concave
upward on I.
(b) Show that part (a) remains true if f and t are both
decreasing.
(c) Suppose f is increasing and t is decreasing. Show, by
giving three examples, that ft may be concave upward,
concave downward, or linear. Why doesn’t the argument
in parts (a) and (b) work in this case?
82. Suppose f and t are both concave upward on s2`, `d.
Under what condition on f will the composite function
hsxd − f stsxdd be concave upward?
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