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

424 Chapter 5 Integrals
CAS
CAS
;
64. The Fresnel function Ssxd − y x 0
sin( 1 2 t2 ) dt was introduced
in Section 5.3. Fresnel also used the function
Csxd − y x
cos( 1 2 ) t2 dt
0
in his theory of the diffraction of light waves.
(a) On what intervals is C increasing?
(b) On what intervals is C concave upward?
(c) Use a graph to solve the following equation correct to
two decimal places:
y x
cos( 1 2 ) t2 dt − 0.7
0
(d) Plot the graphs of C and S on the same screen. How are
these graphs related?
65. Estimate the value of the number c such that the area
under the curve y − sinh cx between x − 0 and x − 1 is
equal to 1.
66. Suppose that the temperature in a long, thin rod placed
along the x-axis is initially Cys2ad if | x | < a and 0 if
| x | . a. It can be shown that if the heat diffusivity of the
rod is k, then the temperature of the rod at the point x at
time t is
C
Tsx, td −
as4kt ya e 2sx2ud 2 ys4ktd
du
0
To find the temperature distribution that results from an initial
hot spot concentrated at the origin, we need to compute
lim a l 0 Tsx, td. Use l’Hospital’s Rule to find this limit.
67. If f is a continuous function such that
y x
1
f std dt − sx 2 1de 2x 1 y x
e 2t f std dt
for all x, find an explicit formula for f sxd.
68. Suppose h is a function such that hs1d − 22, h9s1d − 2,
h0s1d − 3, hs2d − 6, h9s2d − 5, h0s2d − 13, and h0 is
continuous everywhere. Evaluate y 2 1 h0sud du.
69. If f 9 is continuous on fa, bg, show that
70. Find
2 y b
f sxd f 9sxd dx − f f sbdg 2 2 f f sadg 2
a
1
lim
h l 0
h y21h 2
1
s1 1 t 3 dt
71. If f is continuous on f0, 1g, prove that
72. Evaluate
y 1
0
1
lim FS 1 9
n l ` n nD
f sxd dx − y 1
f s1 2 xd dx
0
1S 1S 2
nD9 3 1 ∙ ∙ ∙ 1S
nD9
nD9G
n
73. Suppose f is continuous, f s0d − 0, f s1d − 1, f 9sxd . 0,
and y 1 0
f sxd dx − 1 3 . Find the value of the integral
y 1 f 21 0
syd dy.
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.