Thomas Calculus 13th [Solutions]

Chapter 3 Practice Exercises 209
112.
x 2 2 ( x 2 ) (2) x
2 dy 0 dy x
dy
0
e y d e y d e y e m
e 1
dx dx dx dx 2 y tan dx
(0, 1)
2(1) 2
;
m 1 2; tangent line: y 1 1 ( x 0) y 1 x ; normal line: y 1 = 2(x 0) y = 2x + 1
m
2 2
tan
113. xy 2x 5y 2 x dy
dy
2 5
dx
y dx
0 dy
dx ( x 5) 2 dy y 2 dy
y
dx x 5 dx
2
(3, 2)
line is y 2 2( x 3) 2x 4 and the normal line is y 2 1 ( x 3) 1 7
2
2 x 2 .
114.
2
dy
( y x)
2x 4 2( y x) 1 2 ( ) dy
dy 1 y x dy
y x 1 ( y x)
3
dx
dx
dx y x dx
(6, 2)
4
line is y 2 3 ( x 6) 3 x 5 and the normal line is y 2 4 ( x 6) 4 x 10.
4 4 2
3 3
the tangent
the tangent
115. x xy 6 1 dy
dy
dy 2 xy y dy
1 x y 0 x y 2 xy
5
2 xy dx
dx
dx x dx
(4,1)
4
is y 1 5 ( x 4) 5 x 6 and the normal line is y 1 4 ( x 4) 4 x 11.
4 4
5 5 5
the tangent line
3/2 3/2
1/2 1/2.
116. x 2y
3
dy dy 1/2
17 x 3y 0
x dy 1
2
dx dx 1/2
2 y dx
(1, 4)
4
y 4 1 ( x 1) 1 x 17 and the normal line is y 4 4( x 1) 4 x.
4
4 4
the tangent line is
3 3 2
3 2 dy 3 2 dy dy 3 2 dy dy dy 2 3
117. x y y x y x 3 y y (3 x ) 2y
1 3x y 2y 1 3x y
dx
dx dx dx dx dx
2 3
dy 3 2
2 3 dy 1 3x y dy
(3x y 2y
1) 1 3x y
2 , but dy is undefined. Therefore, the
dx
dx 3 2
3x y 2y
1 dx
(1,1)
4 dx
(1, 1)
curve has slope 1 at (1,1) but the slope is undefined at (1, 1).
2
dy
118. y sin( x sin x ) [cos( x sin x)](1 cos x);
y 0 sin( x sin x ) 0 x sin x k , k 2, 1, 0, 1, 2
dx
dy
(for our interval) cos( x sin x) cos( k ) 1. Therefore, 0 and y 0 when 1 cos x 0 and x k .
dx
For 2 x 2 , these equations hold when k 2, 0, and 2(since cos( ) cos 1.) Thus the curve has
horizontal tangents at the x -axis for the x -values 2 , 0, and 2 (which are even integer multiples of ) the
curve has an infinite number of horizontal tangents.
119. B graph of f , A graph of f . Curve B cannot be the derivative of A because A has only negative slopes
while some of Bs values are positive.
120. A graph of f , B graph of f . Curve A cannot be the derivative of B because B has only negative slopes
while A has positive values for x 0.
121. 122.
123. (a) 0, 0 (b) largest 1700, smallest about 1400
Copyright
2014 Pearson Education, Inc.