61. One of the following graphs is of , and the other is of . Is the first graph or the second graph?
62. Given the following data about a function f, estimate the rate of change of the derivative at x = 4.5.
.
x 
3 
3.5 
4 
4.5 
5 
5.5 
6 

f(x) 
10 
8 
7 
4 
2 
0 
1 

63. A function defined for all x has the following properties:
• f is increasing
• f is concave down
• f (3) = 2
• f ?(3) = 1/2
How many zeros does f(x) have in the interval ?
64. A function defined for all x has the following properties:
• f is increasing
• f is concave down
• f (4) = 2
•
Is it possible that ?
65. Assume that f is a differentiable function defined on all of the real line. Is it possible that f > 0 everywhere, f ? > 0 everywhere, and f ? < 0 everywhere?
66. Assume that f and g are differentiable functions defined on all of the real line. Is it possible that f ?(x) > g?(x) for all x and f(x) < g(x) for all x?
67. Assume that f and g are differentiable functions defined on all of the real line. If f ?(x) = g?(x) for all x and if for some , then must f(x) = g(x) for all x?
68. Assume that f and g are differentiable functions defined on all of the real line. If f ‘ > 0 everywhere and f > 0 everywhere then must?
69. Suppose a function is given by a table of values as follows:
x 
1.1 
1.3 
1.5 
1.7 
1.9 
2.1 

f(x) 
14 
17 
23 
25 
26 
27 

Give your best estimate of .
70. If the Figure 1 is, could Figure 2 be?
Figure 1 Figure 2
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