3.2 Rolles & Mean Value

Theorem

Rolle’s Theorem

Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). IfThen there is at least one number c in (a,b) such that f’(c)=0

What does Rolle’s Thrm do? Rolle’s theorem states some x value

exists (x=c) so that the tangent line at that specific x value is a horizontal tangent (f’(c)=0)

a b

f(a)=f(b)

c

Horizontal Tangent Lineie: f’(c)=0

Notes about Rolle’s Thrm

It is an EXISTENCE Theorem, it simply states that some c has to exist. It does NOT tell us exactly where that value is located.

In order to find the location x=c, we would take f’(x)=0 and find critical numbers like in section 3.1 (Extrema on a closed Interval)

Example 1 of Rolle’s Thrm Determine whether Rolle’s thrm can be applied. If

it can be applied, find all values of c such that f’(c)=0

Ex:

Since f is a polynomial, it is continuous on [1,4] and differentiable (1,4).

Therefore, Rolle’s Theorem can be applied and states there must be some x=c on [1,4] such that f’(c)=0.

Lets find those x values!

Example 1 Continued

Therefore by Rolle’s Thrm,

Example 2 of Rolle’s Thrm Determine whether Rolle’s thrm can be applied. If

it can be applied, find all values of c such that f’(c)=0

Ex:

f is continuous on [-2,3]

f is not differentiable on (-2,3)

ROLLE’s cannot be used!

Example 3 of Rolle’s Thrm Determine whether Rolle’s thrm can be applied. If

it can be applied, find all values of c such that f’(c)=0

Ex:

f is continuous on

f is differentiable on

ROLLE’S APPLIES!

Example 3 Continued

Therefore

Mean Value Theorem If f is continuous on [a,b] and differentiable on the

open interval (a,b), then there exists a number c in (a,b) s.t.

𝑓 (𝑏 )− 𝑓 (𝑎)𝑏−𝑎

f(b)

f(a)

a b

𝑓 ′ (𝑐 )

MVT Examplepg 177 #37 parts a,b,c,d