3.2 Rolle's Theorm and
Mean Value Theorem
Objective: You will be able to:
• understand and use Rolle's Theorem
• understand and use the Mean Value Theorem
Nov 8­10:35 AM
Rolle's Theorem: If f is continuous and differentiable on [a, b]
and f(a) = f(b),
then there is at least one place on (a, b) where
f' = 0
Illustrate Rolle's Theorem
f(a)=f(b)
Nov 8­10:37 AM
1
Ex. 1 Find all values of c in the interval
such that f'(c) = 0
f (x) = x2 ­ 5x + 4
[1,4]
Sep 16­1:56 PM
Find all values of c in the interval (-2, 2) such
that f'(c) = 0.
f(x) = x4 - 2x2.
Sep 17­10:18 AM
2
Stand and Deliver
Mean Value Theorem for Derivatives:
3.2
If f is continuous and differentiable on [a, b]
then there exists some number c such that
f'(c) = f(b) - f(a)
b-a
slope of the tangent line = slope of the secant line
instantaneous velocity
= average velocity
calculus slope
= algebra slope
Nov 8­10:40 AM
Rolle's Theorem leads us to a very powerful theorem in Calculus, the Mean Value Theorem
Theorems.gsp
Sep 17­10:19 AM
3
Mean Value Theorem for Derivatives
Sep 18­9:47 AM
Find all values of c in (a, b) such that f'(c) = f(b) - f(a)
b-a
f(x) = x + 1
[1/2, 2]
x
Nov 8­11:02 AM
4
Consider the graph of the function f(x) = -x2 - x + 6.
a. Find the equation of the secant line joining (-2, 4) and (2, 0)
b. Use the Mean Value Theorem to determine a point c in the
interval [-2, 2] such that the tangent line is parallel to the secant
c. Find the equation of the tangent line through c.
d. Use a graphing utility to graph f, the secant, and the tangent.
Nov 8­10:50 AM
Oct 2­8:42 PM
5
Attachments
Theorems.gsp