Thomas Calculus 13th [Solutions]

120 Chapter 3 Derivatives
45. (a) The graph appears to have a vertical tangent
at x 1 and a cusp at x 0.
(b)
x
2/3 1/3 2/3 1/3
1: lim (1 h) (1 h 1) 1 lim (1 h) h 1
h 0
h
h 0
h
at x 1;
2/3 1/3 1/3 1/3
2/3 1/3
y x ( x 1) has a vertical tangent
f (0 h) f (0) h ( h 1) ( 1) 1 ( h 1)
x 0: lim lim lim
1 does not exist
1/3
h 0
h
h 0
h
h 0 h h h
2/3 1/3
y x ( x 1) does not have a vertical tangent at x 0.
46. (a) The graph appears to have vertical tangents
at x 0 and x 1.
(b)
x
x
1/3 1/3 1/3
(0 ) (0)
0: lim f h f
( 1) ( 1)
lim h h
h 0
h
h 0
h
1: lim (1 ) (1)
lim
1/3 1/3
f h f
(1 h) (1 h 1) 1
h 0
h
h 0
h
1/3 1/3
y x ( x 1) has a vertical tangent at x 0;
1/3 1/3
y x ( x 1) has a vertical tangent at x 1.
47. (a) The graph appears to have a vertical tangent
at x 0.
(b)
(0 ) (0)
lim f h f
h
h 0
1
hlim
0 | h |
lim h 0 lim 1 ; (0 ) (0)
lim f h f
h
x 0 h 0 h
h
h 0
y has a vertical tangent at x 0.
| h| 0 | h|
lim lim
h
| h|
h 0 h 0
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