The Mean Value Theorem and Rolle's Theorem
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Homework
1) State the hypotheses of Rolle’s Theorem.
2)
State the conclusion of Rolle’s Theorem.
3) State the hypotheses of the Mean
Value Theorem.
4) State the conclusion of the Mean Value Theorem.
5) Consider
the function f in the figure below.
a) Does f satisfy the hypotheses of the Mean Value Theorem on [5, 7]?
b) If the answer to (a) is no, state why. If the answer to (a) is yes, find
all values of c in [5, 7] that satisfy the conclusion of the Mean Value Theorem.
6) Consider the function f in Figure 4.3.7b on p. 261 of your text.
a) Does
f satisfy the hypotheses of the Mean Value Theorem on [-1, 1]?
b) If the
answer to (a) is no, state why. If the answer to (a) is yes, find all values
of c in [-1, 1] that satisfy the conclusion of the Mean Value Theorem.
c)
Does f satisfy the hypotheses of the Mean Value Theorem on [0.5, 1.5]?
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7)
Consider the function f in the figure above.
a) Does f satisfy the hypotheses
of the Mean Value Theorem on [0, 3]?
b) If the answer to (a) is no, state
why. If the answer to (a) is yes, find all values of c in [0, 3] that satisfy
the conclusion of the Mean Value Theorem.
8) Give an example of a function
that shows that the hypothesis f(a) = f(b) = 0 is necessary for Rolle’s Theorem
to be true.
9) Give an example of a function that shows that the hypothesis
that f must be continuous on [a, b] is necessary for Rolle’s Theorem to be true.
10) Give an example of a function that shows that the hypothesis that f must be
differentiable on (a, b) is necessary for Rolle’s Theorem to be true.
11)
Give an example of a function that shows that the hypothesis that f must be continuous
on [a, b] is necessary for the Mean Value Theorem to be true.
12) Give an
example of a function that shows that the hypothesis that f must be differentiable
on (a, b) is necessary for the Mean Value Theorem to be true.
13) True or
False? Rolle’s Theorem is a special case of the Mean Value Theorem.
14) True
or False? Rolle’s Theorem is a generalization of the Mean Value Theorem.
15) Is the following analogy true or false? Rolle’s Theorem is to the Mean Value
Theorem as the Pythagorean Theorem is to the Law of Cosines. (If you forgot what
the Law of Cosines says, it is stated on p. A20 in appendix B of your text.)
16) Let f satisfy the hypotheses of the Mean Value Theorem on [a, x]. Show that
f (x) = f (a) + f ´(c) (x – a)
where c is some number that satisfies a <
c < x.
17) Assume f satisfies the hypotheses of the Mean Value Theorem
on [x1, x2]. Show that
where
c is in [x1, x2].
18) Suppose that f satisfies the hypotheses
of the Mean Value Theorem and that f ¢ (c) is negative where a < c < b.
Show that f (b) < f (a).
19) Let x0 < c1 <
x1 < c2 < x2 < c3 < x3
and assume f satisfies the hypotheses of the Mean Value Theorem for all x. Simplify:
f ´(c1) (x1 – x0) + f ´(c2) (x2
- x1) + f ´(c3) (x3 – x2)
Answers:
1) f is differentiable on (a, b), f is continuous on [a, b], f (a) = 0, and f
(b) = 0
2) There is at least one point c in (a, b) where f ´(c) = 0
3)
f is differentiable on (a, b), f is continuous on [a, b]
4) There is at least
one point c in (a, b) where 
5a) Yes 5b) c = 6
6a) No 6b) f is
not differentiable at x = 0 6c) Yes
7a) No 7b) f is not continuous at x
= 1(or f is not differentiable at x = 1)
8) Answers vary. Any non-horizontal,
non-vertical line on any interval will work.
9) Answers vary. One possible
answer is f (x) = tanx on [0, pi].
10) Answers vary. One possible answer
is given in figure 3.2.10 on p. 183 of your text.
11) Answers vary. One possible
answer is 
on [-1, 1].
12) Answers vary. One possible answer is f (x) = |x| on [-1,
1]
13) True
14) False, the Mean Value Theorem is a generalization of Rolle’s
Theorem.
15) True
16) 
f (x)
– f (a) = f ´(c) (x – a)
f (x) = f (a) + f ´(c) (x – a)
17)
18)
f (b)
– f (a) = f ´(c) (b – a)
f (b) – f (a) < 0
f (b) < f (a)
19)
f (x1) – f (x0) + f (x2) – f (x1)
+ f (x3) – f (x2) = f (x3) – f (x0)