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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 ). If Then there is at least one number c in ( a,b ) such that f’(c)=0. What does Rolle’s Thrm do?.

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Presentation Transcript
rolle s theorem
Rolle’s Theorem

Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If

Then there is at least one number c in (a,b) such that f’(c)=0

what does rolle s thrm do
What does Rolle’s Thrmdo?
  • 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)

Horizontal Tangent Line

ie: f’(c)=0

f(a)=f(b)

a

c

b

notes about rolle s thrm
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 (Extremaon a closed Interval)
example 1 of rolle s thrm
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
Example 1 Continued
  • Therefore by Rolle’s Thrm,
example 2 of 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
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!
mean value theorem
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