Thomas Calculus 13th [Solutions]

116 Chapter 3 Derivatives
9.
m
3 3
( 2 h) ( 2)
lim
h 0
h
2
lim (12 6 h h ) 12;
h
0
2 3
lim 8 12h 6h h 8
h 0
h
at ( 2, 8): y 8 12( x ( 2)) y 12x
16,
tangent line
10.
1 1
( 2 h) 3
( 2)
3
8 ( 2 h)
m lim
lim
h 0
h
h 0 8 h( 2 h)
2 3
(12h 6 h h )
2
lim
lim 12 6h h
3
3
h 0 8 h( 2 h)
h 0 8( 2 h)
12 3
8( 8) 16 ;
at 2, 1 : y 1 3
8 8 16 ( x ( 2))
y 3 x 1
16 2 , tangent line
3
3
11.
m
[(2 h) 1] 5
lim
h 0
h
2
2
lim (5 4 h h ) 5 lim h(4 h)
4;
h 0
h
h 0
h
at (2,5): y 5 4( x 2), tangent line
12.
m
[(1 h) 2(1 h) ] ( 1)
lim
h 0
h
2
(1 2 4 2 ) 1
lim h h h
h 0
h
2
h( 3 2 h)
lim 3;
h 0
h
at (1, 1) : y 1 3( x 1), tangent line
13.
m
lim
h 0
3 h
(3 h) 2 3
h
lim (3 h ) 3( h 1)
h 0
h( h 1)
lim 2h
0
h( h 1)
2; at (3,3): y 3 2( x 3), tangent line
h
14.
m
lim
h 0
8
(2 h )
2
h
2
8 2(2 h)
lim
2
h 0 h(2 h)
2
8 2(4 4 h h )
lim
2
h 0 h(2 h)
2
2 h(4 h) lim 8 2;
2
h 0 h(2 h)
4
at (2,2): y 2 2( x 2)
15.
m
3
(2 h) 8
lim
h 0
h
2 3
(8 12h 6 h h ) 8
lim
h 0
h
h(12 6 h h )
lim 12;
h 0
h
2
at (2,8): y 8 12( t 2), tangent line
16.
m
3
[(1 h) 3(1 h)] 4
lim
h 0
h
2 3
(1 3h 3h h 3 3 h) 4
lim
h 0
h
2
h(6 3 h h )
lim 6; at (1, 4): y 4 6( t 1), tangent line
h 0
h
17.
4 h 2 4 h 2 4 h 2
m lim lim
h 0
h
h 0
h 4 h 2
at (4,2): y 2 1 ( x 4), tangent line
4
(4 h) 4
lim
h 0 h 4 h 2
lim
h 0 h
h
4 h 2
1
4 2
1
4 ;
18.
(8 h) 1 3 9 h 3 9 h 3
m lim
lim
h 0
h
h 0
h 9 h 3
at (8,3): y 3 1 ( x 8), tangent line
6
(9 h) 9
lim
lim h
h 0 h 9 h 3 h 0 h 9 h 3
1 1 ;
9 3 6
19. At x 1, y 5 m
5( 1 h) 5
lim
h 0
h
2
5(1 2 h h ) 5
lim
h 0
h
2
5 h( 2 h)
lim
h 0
h
10, slope
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