Thomas Calculus 13th [Solutions]

Chapter 4 Additional and Advanced Exercises 337
11. From (ii), f ( 1) 1 a
b c 2
0 a 1; from (iii), either 1 lim f ( x ) or 1 lim f ( x ). In either case,
x
x
1
1
lim ( ) lim x 1
1
1
x
x
f x lim 1 b 0 and c 1. For if b 1, then lim 0 and if
2 2
x x bx cx 2 x bx c
x c
2
x
c 0, then
lim
x
1
bx
1
x
2
x
1
x
2
x
1
lim .
x
x
Thus a 1, b 0, and c 1.
x
12.
dy
dx
2
3x
2kx
3
0
x
2
2k
4k
36
6
2 2
x has only one value when 4k 36 0 k 9 or k 3.
2
13. The area of the ABC is A( x) 1 (2) 1 x
2
2 1/2
(1 x ) , where 0 x 1. Thus A ( x) x
2
1 x
0 and 1 are critical points. Also A ( 1) 0 so
A (0) 1 is the maximum. When x 0 the ABC is
isosceles since AC BC 2.
14.
lim f ( c h) f ( c)
f ( c ) for 1
h 0
h
2 | f ( c ) | 0 there exists a 0 such that 0 | h |
f ( c h) f ( c) f ( c) 1 | f ( c) | . Then 1
f ( c h) f ( c) 0 | f ( c) |
f ( c) 1 | f ( c) |
h
2
2
h
2
1
f ( c h) f ( c) | f ( c) | f ( c) 1 | f ( c) | . If f ( c ) 0, then | f ( c) | f ( c)
2
h
2
3 f ( c h)
f ( c) 1 f ( c) 0; likewise if f ( c ) 0, then 1 f ( c h) 0 f ( c) 3 f ( c).
2 h 2
2 h 2
(a) If f ( c ) 0, then h 0 f ( c h ) 0 and 0 h f ( c h ) 0. Therefore, f ( c ) is a local
maximum.
(b) If f ( c ) 0, then h 0 f ( c h ) 0 and 0 h f ( c h ) 0. Therefore, f ( c ) is a local
minimum.
2 y
15. The time it would take the water to hit the ground from height y is
g , where g is the acceleration of gravity.
The product of time and exit velocity (rate) yields the distance the water travels:
D ( )
2 y
2 2 1/2
64( ) 8 ( ) ,
g
g
2 2 1/2
0 y h D ( y) 4 ( hy y ) ( h 2 y) 0, and h
g
2
are critical points. Now D(0) 0, D 2
1/2
h 8 2 h h 4 2
2 g
2 2
g
and D ( h ) 0 the best place to
drill the hole is at y h
2 .
16. From the figure in the text, tan( ) b a
tan tan
; tan( ) ; and tan a . These equations give
n
1 tan tan
h
tan
a
b a
h h tan a
. Solving for tan gives tan bh
h a
1 tan h a tan
or 2
( h a( b a)) tan bh.
2
h a( b a)
h
2
2 d
Differentiating both sides with respect to h gives 2h tan ( h a( b a))
sec b . Then
dh
d
0 2h tan b 2 bh
2 2
2
h
b 2 bh bh ab( b a)
h a( b a ) h a( a b).
dh
2
h a( b a)
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