Thomas Calculus 13th [Solutions]

128 Chapter 3 Derivatives
48. (a) f is differentiable on 3 x 2, 2 x 2, and 2 x 3
(b) f is continuous but not differentiable at x 2 and x 2: there are corners at those points
(c) none
49. (a) f ( x)
(b)
( ) ( )
lim f x h f x
h 0
h
2 2
( ) ( )
lim x h x
h 0
h
2 2 2
lim x 2 xh h x lim ( 2 x h ) 2x
h 0
h
h 0
(c) y 2x is positive for x 0, y is zero when x 0, y is negative when x 0
2
(d) y x is increasing for x 0 and decreasing for 0 x ; the function is increasing on intervals
where y 0 and decreasing on intervals where y 0
50. (a)
(b)
f
( x) lim
h 0
f ( x h) f ( x)
h
1 1
lim x h x
h
0
h
lim x ( x h )
h 0
x( x h)
h
lim 1 1
( ) 2
h 0 x x h x
(c) y is positive for all x 0, y is never 0, y is never negative
(d) y 1 is increasing for x 0 and 0 x
x
51. (a) Using the alternate formula for calculating derivatives: f ( x) lim
z x
2 2
( z x)( z zx x )
2 2
lim
lim z zx x 2 2
x f ( x)
x
z x
3( z x)
z x
3
(b)
f ( z) f ( x)
z x
lim
z x
z
3
x
3
3 3
z x
3 3
lim z x
z x
3( z x)
(c) y is positive for all x 0, and y 0 when x 0; y is never negative
3
(d) y x is increasing for all x 0 (the graph is horizontal at x 0 ) because y is increasing where y 0; y is
3
never decreasing
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