Thomas Calculus 13th [Solutions]

150 Chapter 3 Derivatives
64.
cos( x h) cos x
As h takes on the values of 1, 0.5, 0.3, and 0.1 the corresponding dashed curves of y get closer
h
cos( ) cos
and closer to the black curve y sin x because d
x h x
(cos x) lim sin x . The same is true as h
dx
h 0
h
takes on the values of 1, 0.5, 0.3, and 0.1.
65. (a)
(b)
sin( x h) sin( x h)
The dashed curves of y are closer to the black curve y cos x than the corresponding
2h
dashed curves in Exercise 63 illustrating that the centered difference quotient is a better approximation of
the derivative of this function.
cos( x h) cos( x h)
The dashed curves of y are closer to the black curve y sin x than the corresponding
2h
dashed curves in Exercise 64 illustrating that the centered difference quotient is a better approximation of
the derivative of this function.
66.
67.
|0 h| |0 h| | h| | h|
h 0
2h
x 0
2h
h 0
lim lim lim 0 0
derivative of f ( x) | x | does not exist at x 0.
2
y tan x y sec x,
so the smallest value
2
y sec x takes on is y 1 when x 0; y has no
2
maximum value since sec x has no largest value
2
on , ; y is never negative since sec x 1.
2 2
the limits of the centered difference quotient exists even though the
68.
2
y cot x y csc x so y has no smallest
2
value since csc x has no minimum value on
(0, ); the largest value of y is 1, when x 2 ;
the slope is never positive since the largest value
2
y csc x takes on is 1.
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