AP Calculus Section 3.2
Rolle’s Theorem
and
Mean Value Theorem
Theorems and Examples
Recall:
Theorem 3.1
The Extreme Value Theorem
If f is continuous on the closed interval [a, b],
then f has both a minimum and a maximum on
the interval.
Theorem 3.3 Rolle’s Theorem
Let f be continuous on the closed interval [a, b]
and differentiable on the open interval (a, b). If
f(a) = f(b), then there exists at least one number
c in (a, b) such that f’(c) = 0.
Example 1
Find the two x-intercepts of 𝑓 𝑥 = 𝑥 2 − 𝑥 − 2 and show that f’(x) = 0 at some
point between the x-intercepts.
Example 2
Let 𝑔 𝑥 = 𝑥 4 − 5𝑥 2 − 4. Find all values of c in the interval (-2, 2) such that
g’(c)=0.
We need Rolle’s Theorem to prove
The Mean Value Theorem
If f is continuous on the closed interval [a, b] and
differentiable on the open interval (a, b), then
there exists a c in (a, b) such that
𝑓 𝑏 − 𝑓(𝑎)
′
𝑓 𝑐 =
𝑏−𝑎
′
𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎
Does 𝑓 𝑐 =
look familiar?
Where have you seen that before?
The Mean Value Theorem is most
often used to prove other theorems,
including THE FUNDAMENTAL
THEOREM OF CALCULUS. For this
reason, some people consider the
Mean Value Theorem to be the most
important theorem in calculus.
Example 3 #40
(Geometric Interpretation of the Mean Value Theorem)
Given 𝑓 𝑥 = 𝑥(𝑥 2 − 𝑥 − 2), find all values of c
in the open interval (-1, 1) such that
′
𝑓 𝑐 =
𝑓 1 −𝑓(−1)
1−−1
Read p. 175 Example 4

Example 4 #60
(Instantaneous Rate of Change = Average Rate of Change)
At 9:13 AM, a sports car is traveling 35 miles per hour. Two
minutes later, the car is traveling 85 miles per hour. Prove that at
some time during this two-minute interval, the car’s acceleration
is exactly 1500 miles per hour squared.
Alternative Form
of the Mean Value Theorem
If f is continuous on [a, b] and differentiable on (a, b),
then there exists a number c in (a, b) such that
𝑓 𝑏 = 𝑓 𝑎 + 𝑏 − 𝑎 𝑓′ 𝑐
Remember:
Polynomial functions, rational
functions, and trigonometric
functions are differentiable at
all points in their domains.
Assignment - Section 3.2:
p.176-177 # 2, 18, 22, 26, 29, 33, 34,
35, 37, 42, 51, 58