Thomas Calculus 13th [Solutions]

Chapter 3 Practice Exercises 203
58.
1 2 1 2 1/2
y z cos z 1 z z cos z (1 z )
dy cos 1 2 1/2 1 1
z z 1
2 2
(1 z ) ( 2 z ) cos z z z cos z
dz
2 2
1 z 1 z 1 z
1
59. 1 dy 1 1
y t tan t ln t tan t t 1 1 1 tan t t 1
2 dt 2 2 2
1 t t 1 t 2t
60. 2 1 dy
y (1 t ) cot 2t 2t cot 1 2 t (1 t
2 ) 2
dt
2
1 4t
61.
1 2 1 2 1/2
y z sec z z 1 z sec z ( z 1)
dy 1 2 1/2 1 1
z 1 (sec z )(1) 1 1
2 2
( z 1) (2 z ) z z sec z z sec z , z
dz
1
2 2 2
z z 1 z z 1 z 1 z 1
62.
1 1/2 1 1/2
y 2 x 1sec x 2( x 1) sec ( x )
dy 2 1 1
1 1/2 1 1/2 1/2 sec 1 sec 1
2 ( 1) sec ( ) ( 1) x
2 x x
x x x
dx x x 1 2 x 1 x 2 x 1 x
1
2
1/2
63.
y
csc 1 (sec ) dy sec tan tan
d
2
sec sec 1
tan
1,
0
2
64.
1 1 tan
1
x
1 1
2 tan x tan x 2 tan x tan x
y (1 x ) e y 2 xe (1 x ) e 2xe e
2
1 x
65. xy 2x 3y 1 ( xy y) 2 3y 0 xy 3y 2 y y ( x 3)
2
y
y
y
x
2
3
66.
2 2
x xy y 5x 2 2 x x dy
dy
2 5
dx
y y dx
0 x dy
2 dy
dx
y dx
5 2 dy
x y
dx
( x 2 y ) 5 2x y
dy 5 2x y
dx x 2 y
67.
2 dy
1/3 dy
2x 3x 4x 4y 4y
dx
dx
2
1/3 2 dy 2 3x 4 y
(4x
4 y ) 2 3x
4y
dx 1/3
4 x 4 y
3 4/3
x 4xy 3y
dy
dx
dy 1/3 dy
2 4x
4y
dx dx
2
2 3x
4y
68.
4/5 6/5
5x
10y
1/5 1/5 dy
15 4x
12y
dx
1/5 dy
0 12y
dx
4
1/5
dy
dx
1 1/5 1/5 1
3 3( xy)
x x y
1/5
69.
1/2 1 1/2 dy
( xy) 1 ( xy)
x y
2
dx
0
1/2 1/2 dy
x y
dx
1/2 1/2 dy 1 dy y
x y x y
dx dx x
70.
2 2 2 dy 2
x y 1 x 2 y y (2 x)
dx
2 dy
0 2x y
dx
2
2xy
dy
dx
y
x
2 dy ( x 1)(1) ( x)(1)
dy
71. y x 2y
1
x 1 dx 2 2
( x 1) dx 2 y( x 1)
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