Thomas Calculus 13th [Solutions]

Chapter 7 Additional and Advanced Exercises 541
6. (a)
g g(0) g(0) 2 3 3
(0 0) 1 (0) (0) 2 (0) (0) (0) 2 (0) (0) (0) 0
1 g(0) g(0)
g g g g g g g g
2
g(0) g (0) 1 0 g(0) 0
(b)
2
g( x) g ( h)
1 g( x) g ( h)
( )
2
g( x h) g( x) g x
g( x) g( h) g( x) g ( x) g( h)
g ( x) lim lim lim
h 0
h
h 0
h
h 0 h 1 g( x) g( h)
g( h) 1 g ( x) 2 2
2
lim 1 1 g ( x) 1 g ( x) 1 g( x)
h 0
h 1 g( x) g( h)
dy 2 dy
1 1 1
(c) 1 y dx tan y x C tan g( x) x C; g(0) 0 tan 0 0 C
dx
2
1 y
1
C 0 tan g( x) x g( x) tan x
1 1
1
7. M 2 dx 2 tan x and
0
2
1 x
0 2
M 1 1
2x
2
M
ln 1 ln 2
y ln 2 ln 4
y
;
0 dx x x y
2
1 x
0
M
0 by
symmetry
2
8. (a)
(b)
V 4 2
1
4
1 4 1
4
1/4 dx ln | | ln 4 ln ln16 ln 2 ln 2
2
4 1/4 x
dx 4 x
x
1/4 4 4 4 4
4 4 1/2 3/2
4
M 1 1 1 8 1 64 1 63
y x dx x dx x
;
1/4 2 x 2 1/4
3 1/4 3 24 24 24
M 4
1 1 1 1
4
1 1
4
1 1
x
1/4 2 dx ln | | ln16 ln 2;
2 2 8 1/4 x
dx x x
8 x 1/4 8 2
M 4 4 1/2 1/2
4
1 1 1 3
1/4 dx 2 1/4 2 x dx x 2 ;
x
1/4 2 2
therefore, M
x
y 63 2 21 7 and
M 24 3 12 4
M
y x 1 ln 2 2 ln 2
M 2 3 3
rt
rt rt
9. A( t) A0
e ; A( t) 2A ln 2 .7 70 70
0 2A0 A0e e 2 rt ln 2 t t
r r 100 r ( r%)
10. In order to maximize the amount of sunlight, we need to maximize the angle formed by extending the two red
line segments to their vertex. The angle between the two lines is given by 1 ( 2) . From trig we
1
have tan 350 350
1 450 1 tan
x 450 x and 200 1
tan
200
2 2 tan
x
x
1 350 1
( 200
1 ( 2)) tan tan
450 x
x
d 1 350 1 200 350 200
dx 350
2 2
200
2 2 2 2
1 (450 x) 1
x (450 x) 122500 x 40000
450 x
x
d
2 2
0 350 200 0 200 (450 x) 122500 350( x 40000)
dx 2 2
(450 x) 122500 x 40000
2
3x 3600x 1020000 0 x 600 100 70. Since x 0, consider only x 600 100 70. Using the
first derivative test, d
9 0 and d
9 0 local max when
dx x 100 3500 dx x 400 5000
x 600 100 70 236.67 ft.
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