I wonder how mean this theorem really is?

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The Mean Value Theorem and Rolles Theorem Lesson 3.2 I wonder how mean this theorem really is?

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Presentation transcript:

I wonder how mean this theorem really is? The Mean Value Theorem Lesson 4.2

This is Really Mean

Think About It Consider a trip of two hours that is 120 miles in distance … You have averaged 60 miles per hour What reading on your speedometer would you have expected to see at least once? 60

Rolle’s Theorem Given f(x) on closed interval [a, b] Differentiable on open interval (a, b) If f(a) = f(b) … then There exists at least one number a < c < b such that f ’(c) = 0 f(a) = f(b) c a b

Mean Value Theorem We can “tilt” the picture of Rolle’s Theorem Stipulating that f(a) ≠ f(b) Then there exists a c such that c a b

Note Geogebera Example Mean Value Theorem Applied to a cubic equation Note Geogebera Example

Finding c Given a function f(x) = 2x3 – x2 Strategy Find all points on the interval [0, 2] where Strategy Find slope of line from f(0) to f(2) Find f ‘(x) Set equal to slope … solve for x

Modeling Problem Two police cars are located at fixed points 6 miles apart on a long straight road. The speed limit is 55 mph A car passes the first point at 53 mph Five minutes later he passes the second at 48 mph Yuk! Yuk! I think he was speeding, Enos We need to prove it, Rosco

Assignment Lesson 4.2 Pg 216 Exercises 1 – 61 EOO