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4.2 Practice
Name___________________________________
Determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given interval.
1) f(x) = x1/3 on [-2, 5]
2) g(x) = x3/4 on [0, 3]
Give an appropriate answer.
3) Find the value or values of c that satisfy
48
f(b) - f(a)
on the interval [3, 16].
= f′(c) for the function f(x) = x +
x
b- a
Answer the question.
4) A marathoner ran the 26.2 mile New York City Marathon in 2.7 hrs. Did the runner ever exceed a speed of 9
miles per hour?
Use analytic methods to find the local extrema.
x-1
5) h(x) =
2
x + 5x + 10
Use analytic methods to find those values of x for which the given function is increasing and those values of x for which
it is decreasing.
6) f(x) = 7x2 - 5x
Find all possible functions with the given derivative.
7) f'(x) = 5e5x
Find the function with the given derivative whose graph passes through the point P
8) f'(x) = x2 + 5, P(3, 51)
Sketch a graph of a function y = f(x) that has the given properties.
9) a) Differentiable everywhere except x = 0
b) Continuous for all real numbers
c) f'(x) < 0 on (-∞, 0)
d) f'(x) > 0 on (0, ∞)
e) f(-2) = f(2) = 5
f) y-intercept and x-intercept at (0, 0)
1
Answer Key
Testname: APCALCULUS4_2PRACTICE
1)
2)
3)
4)
No
Yes
4 3
Yes, the Mean Value Theorem implies that the runner attained a speed of 9.7 mph, which was her average speed
throughout the marathon.
5) Local minimum at x = -3; local maximum at x = 5
5
5
6) Increasing on
, ∞ , decreasing on -∞,
14
14
7) e5x + C
8) f(x) =
x3
+ 5x + 27
3
9) Possible Answer:
2