The Mean Value Theorem

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

c 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 numbera < c < b such that f ’(c) = 0 f(a) = f(b) b a

c 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 b a

Mean Value Theorem • Applied to a cubic equation Note Spreadsheet Example

Finding c • Given a function f(x) = 2x3 – x2 • 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

f(x) is a constant function What could you say about this function? f(x) = k 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

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

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 We need to prove it I think he was speeding

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

The Mean Value Theorem