Thomas Calculus 13th [Solutions]

Section 3.11 Linearization and Differentials 193
17. (a)
(b)
50
50
(1.0002) (1 0.0002) 1 50(0.0002) 1 .01 1.01
3 1/3
1.009 (1 0.009) 1 1 (0.009) 1 0.003 1.003
3
1/2
1/2
18. f ( x) x 1 sin x ( x 1) sin x f ( x)
1 ( x 1) cos x L ( )
2 f x f (0)( x 0) f (0)
3
2 ( x 0) 1 L f ( x ) 3
1/2
x 1, the linearization of f ( x); g( x)
x 1 ( x 1) g ( x)
1
2
2 ( x 1)
Lg ( x ) g (0)( x 0) g(0)
1 ( x 0) 1 L ( ) 1 1,
2 g x x the linearization of g( x ); h( x)
sin x
2
h ( x) cos x Lh
( x ) h (0)( x 0) h (0) (1)( x 0) 0 Lh
( x) x , the linearization of h( x).
L f ( x) Lg ( x) Lh
( x ) implies that the linearization of a sum is equal to the sum of the linearizations.
1/2
19.
3
y x 3 x
3 1/2
x 3x dy
2 3 1/2
3x x dx dy
2
2
3x
3
2 x
dx
20.
2 2 1/2
2 1/2 2 1/2
y x 1 x x (1 x ) dy (1) (1 x ) ( x) 1 (1 x ) ( 2 x)
dx
2
2
2
1/2
2 2
1 2x
1 x (1 x ) x dx dx
2
1 x
21. y 2x
2
1 x
dy
2 2
(2)(1 x ) (2 x)(2 x) dx 2 2x
(1 x ) (1 x )
2 2 2 2
dx
22.
y
1/2
2 x 2x
3 1 x 3 1 x
1/2
dy
1/2 1/2 1/2 3 1/2
3 1 2
2
1/2
2
x x x x
9 1
x
dx
1 2
3x
3 3 dx dy 1
1/2 2
2
9(1 x )
3 x(1 x)
dx
23.
3/2
2y xy x
1/2
0 3y dy y dx x dy dx 0
1/2
(3 y x)
dy (1 y)
dx
dy
1 y
dx
3 y x
24.
2 3/2
xy 4x y
2 1/2
0 y dx 2xy dy 6x dx dy 0 (2xy
1) dy
1/2 2
(6 x y ) dx dy
2
6 x y
dx
2xy
1
25. y sin (5 x)
1/2
sin (5 x )
dy
1/2
(cos (5 x ))
5 1/2
2 x dx dy 5cos 5
2 x
x
dx
26.
2
y cos ( x ) dy
2
[ sin ( x )](2 x)
dx
2
2xsin ( x ) dx
27.
y
3
4 tan x
3
dy
3
2 2
4 sec x ( x ) dx
3
2 2
dy 4x sec x dx
3
3
28.
2
y sec x 1 dy
2 2
[sec ( x 1) tan ( x 1)](2 x)
dx
2 2
2 x[sec( x 1) tan ( x 1)] dx
1/2
29. y 3csc(1 2 x)
3csc (1 2 x ) dy
dy 3 csc (1 2 x) cot (1 2 x)
dx
x
1/2 1/2 1/2
3( csc (1 2 x ))cot (1 2 x ) ( x ) dx
30. y 2cot 1
x
2cot x
1/2
dy
2 1/2 1 3/2
2csc ( x ) ( x ) dx dy
2
1 2
csc 1
3
x x
dx
31.
x
y e dy e dx
2 x
x
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