244
Chapter 3 | Derivatives
⎧ ⎩ ⎨ 2 x , x ≤1 2 x , x >1
67.
77. f ( x ) =
For the following graphs, a. determine for which values of x = a the lim x → a f ( x ) exists but f is not continuous at x = a , and b. determine for which values of x = a the function is continuous but not differentiable at x = a . 78.
For the following exercises, the given limit represents the derivative of a function y = f ( x ) at x = a . Find f ( x ) and a .
(1+ h ) 2/3 −1 h ⎣ 3(2+ h ) 2 +2 ⎤ cos( π + h )+1 h (2+ h ) 4 −16 h h
68. lim h →0
69. lim h →0 ⎡
⎦ −14
70. lim h →0
79.
71. lim h →0
[2(3+ h ) 2 −(3+ h )]−15 h
72. lim h →0
e h −1 h
73. lim h →0
For the following functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at x =1.
⎧ ⎩ ⎨ 2 x , 0≤ x ≤1 3 x −1, x >1
74. f ( x ) =
⎧ ⎩ ⎨ 3, x <1
75. f ( x ) =
3 x , x ≥1
⎧ ⎩ ⎨ − x 2 +2, x ≤1 x , x >1
76. f ( x ) =
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