Thomas Calculus 13th [Solutions]

Section 3.1 Tangents and the Derivative at a Point 117
20. At x 2, y 3 m
[1 (2 h) ] ( 3)
lim
h 0
h
2
(1 4 4 h h ) 3
lim
h 0
h
2
lim
h 0
h(4 h)
h
4, slope
21. At x 3, y 1 m
2
lim
h 0
1 1
(3 h ) 1 2
h
2 (2 h)
lim
h 0
2 h(2 h)
lim h 1 , slope
h 0
2 h(2 h) 4
22. At x 0, y 1 m
lim
h 0
h
h
1
1 ( 1)
h
( 1) ( 1)
lim h h
h 0
h( h 1)
lim 2h
0
h( h 1)
2
h
23. (a) It is the rate of change of the number of cells when t 5. The units are the number of cells per hour.
(b) P (3) because the slope of the curve is greater there.
2 2 2
6.10(5 h) 9.28(5 h) 16.43 [6.10(5) 9.28(5) 16.43] 61.0h 6.10h 9.28h
(c) P (5) lim lim
h 0 h
h 0 h
lim 51.72 6.10h
51.72 52 cells/hr.
h
0
24. (a) From t 0 to t 3, the derivative is positive.
(b) At t 3, the derivative appears to be 0. From t 2 to t 3, the derivative is positive but decreasing.
25. At a horizontal tangent the slope m
2 2 2 2
2 2
[( ) 4( ) 1] ( 4 1)
0 0 m lim x h x h x x
h 0
h
lim ( x 2xh h 4x 4h 1) ( x 4x 1) lim (2xh h 4 h)
lim (2 4) 2 4;
h 0
h
h 0
h
h 0
x 2. Then ( 2) 4 8 1 5 ( 2, 5)
x h x 2x
4 0
f is the point on the graph where there is
a horizontal tangent.
26.
27.
3 3
[( ) 3( )] ( 3 )
h 0
h
2 2
0 m lim x h x h x x
lim (3x 3xh h 3)
h
0
3 2 2 3 3
( 3 3 3 3 ) ( 3 )
lim x x h xh h x h x x
h 0
h
2 2 3
lim
3 x h 3 xh h 3 h
h 0
h
2 2
3x 3; 3x 3 0 x 1 or x 1. Then f ( 1) 2 and f (1) 2 ( 1, 2)
and (1, 2) are the points on the graph where a horizontal tangent exists.
1 1
( ) 1 1
0
h
1 m lim x h x
h
x( x 2) 0 0
( 1) ( x h 1)
0
h( x 1)( x h 1)
lim x
h
lim
h
h
1
0
h( x 1)( x h 1) ( x
2
1)
2 2
( x 1) 1 x 2x
0
x or x 2. If x 0, then y 1 and m 1 y 1 ( x 0) ( x 1). If x 2,
then y 1 and m 1 y 1 ( x 2) ( x 3).
28.
1
4
m
Thus, 1 1
4 2 x
lim x h x
h 0
h
lim x h x x h x
h 0
h x h x
( )
lim x h x
h 0 h x h x
h
h 0 h x h x
x x
4
4
1
2 x
lim .
x 2 x 4 y 2. The tangent line is y 2
1
( 4) 1.
29.
(2 ) (2)
lim f h f
h 0
h
lim
h
0
2 2
(100 4.9(2 ) ) (100 4.9(2) )
h
h
2
4.9(4 4 h h ) 4.9(4)
lim
h 0
h
h 0
The minus sign indicates the object is falling downward at a speed of 19.6 m/sec.
lim ( 19.6 4.9 h) 19.6.
30.
(10 ) (10)
lim f h f
h 0
h
lim
h
0
2 2
3(10 ) 3(10)
h
h
lim
h
0
2
3(20 )
h h
h
60 ft/sec.
31.
(3 ) (3)
lim f h f
h 0
h
lim
h
0
2 2
(3 ) (3) )
h
h
2
[9 6h h 9]
lim
h 0
h
h 0
lim (6 h)
6
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