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

2. 2 This is Really Mean

3. 3 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

4. 4 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) a b c

5. 5 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 a b c

6. 6 Mean Value Theorem Applied to a cubic equation Note Spreadsheet Example

7. 7 Finding c Given a function f(x) = 2x 3 – x 2  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

8. 8 Zero Derivative Theorem Consider f(x) a  Continuous function on closed interval [a, b]  Differentiable on open interval (a, b)  And … f ‘(x) = 0 on (a, b) a b What could you say about this function? f(x) = k f(x) is a constant function

9. 9 Constant Difference Theorem Given two functions f(x), g(x)  Both continuous on [a, b]  Both differentiable on (a, b) If f ‘(x) = g ‘(x)  Then there exists a constant C such that  f(x) = g(x) + C That would mean the functions differ by a constant

10. 10 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 I think he was speeding We need to prove it

11. 11 Assignment Lesson 4.2 Pg 199 Exercises 3 - 37 odd, 51