Section 2 rolle s theorem the mean value theorem
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Section 2: Rolle’s Theorem & The Mean Value Theorem. I. 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 is at least one number, c, in (a, b) such that f’(c) = 0.

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Section 2 rolle s theorem the mean value theorem

Section 2: Rolle’s Theorem & The Mean Value Theorem


I rolle s theorem
I. 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 is at least one number, c, in (a, b) such that f’(c) = 0.

  • Rolle’s Theorem guarantees an _________ _________ inside of the interval where the Extreme Value Theorem can have them on the endpoints.


Ex 1 illustrating rolle s theorem
Ex. 1 Illustrating Rolle’s Theorem

  • Find the two x-intercepts of f(x) = x² - 3x + 2 and show that f’(x) = 0 at some point between the intercepts.


Section 2 rolle s theorem the mean value theorem
Ex 2

  • Let f(x) = . Find all values of c on the interval (-2, 2) such that f’(c) = 0.


Homework
HOMEWORK

  • Pg 172 #1-20 odds, 26



Review
Review

  • Describe the Extreme Value Theorem.

  • Describe Rolle’s Theorem.


Ii mean value theorem mvt
II. Mean Value Theorem (MVT)

  • If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number, c, in (a, b) such that

  • The MVT says that the slope of a tangent line on a curve is equal to the slope of the secant line on the same curve at a particular point. 


Ex 1 slope of the tangent line
Ex 1: Slope of the Tangent Line

  • What value of c in the open interval (0, 4) satisfies the MVT for ?

  • Given , find all values of c in the open interval (1,4) such that


Ex 2 finding an instantaneous rate of change
Ex 2: Finding an Instantaneous Rate of Change

  • Two stationary patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first car, its speed is clocked at 55 mph. Four minutes later, the truck passes the 2nd patrol car at 50 mph. Prove that the truck must have exceeded the speed limit (55 mph) at some time during the 4 minutes.


Homework1
HOMEWORK

  • Pg 172 #27 – 38 odds, 53 - 56