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AP Calculus
WS 5.2 Mean Value Theorem
NAME: _______________________
1. Use the graph of f to estimate the numbers
in [0, 8] that satisfy the conclusion of the
Mean Value Theorem.
2. Use the graph of g to estimate the numbers
in [0, 8] that satisfy the conclusion of the
Mean Value Theorem.
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In problems 3 – 8, determine whether f satisfies the hypotheses of the Mean Value Theorem on
the interval [a, b]. If it does, find all numbers c in (a, b) such that
3.
f  x   3x2  x  4 [1,5]
5.
f  x   cos x  sin x
7.
f  x  x
1
3
  5 
 4 , 4 
[0, 27]
f c 
f b  f  a 
.
ba
2
4.
f  x  x
6.
f  x 
8.
f  x  x
3
x 1
x 1
[1, 2]
[ 3, 0]
[4, 9]
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9. The function f is continuous for  2  x  1 and differentiable for  2  x  1 . If f (2)  5 and f (1)  4 , which of
the following statements could be false?
(A) There exists c, where  2  c  1 , such that
 2  c  1 , such that
(C) There exists c, where  2  c  1 , such that
(D) There exists c, where  2  c  1 , such that
(E) There exists c, where  2  c  1 , such that
(B) There exists c, where
f (c )  0 .
f ' (c )  0 .
f (c )  3 .
f ' (c )  3 .
f (c )  f ( x )
for all x on the closed interval
 2  x 1 .
10. If
f ( x)  sin( x   )  x 2
for all values of x on the closed interval [0, 2], for what value of x is the instantaneous rate of
change of y with respect to x the same as the average rate of change in the interval [0, 2]? Use your calculator.
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11. It took 20 sec for the temperature to rise from 0°F to 212°F when a thermometer was taken from a freezer and placed in boiling
water. Explain why at some moment in that interval the mercury was rising at exactly 10.6°F/sec.
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12. A marathoner ran the 26.2-mi New York City Marathon in 2.2 h. Explain why that at least twice,
the marathoner was running at exactly 11 mph.
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13. A test plane flies in a straight line with positive velocity v (t ) , in miles per minute, where v is a differentiable function of t .
Selected values of
Time (min)
Velocity (mpm)
v (t )
for
0  t  40 are shown in the table.
0
7.0
5
9.2
10
9.5
15
7.0
20
4.5
25
2.4
30
2.4
35
4.3
40
7.3
Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero
on the open interval (0, 40)? Justify your answer.
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14. (2008B #5) Find the average rate of change g ' ( x ) on the interval
 3  x  7 . Does the Mean Value Theorem applied on the
 3  x  7 guarantee a value of c , for  3  x  7 ,
such that g ' ' ( x ) is equal to this average rate of change?
interval
Why or why not?
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15. (2002 #6) Let f be a function that is differentiable for all real numbers. The table gives the values of f and its derivative f ' for selected
points
x in the closed interval  1.5  x  1.5 . The second derivative of f
has the property that
f '' 0
for
 1.5  x  1.5
.
(a) Write an equation of the line tangent to the graph of
(b) Find a positive real number
your answer.
r
f
at the point where x = 1. Use this equation to approximate
having the property that there must exist a value
f (1.2) .
c with 0  c  0.5 and f ' ' (c)  r . Give a reason for