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Exploring the Mean Value Theorem in Two-Hour Road Trips

Dive into the concept of the Mean Value Theorem through a scenario of a two-hour, 120-mile road trip where speed and expectations intersect. This lesson includes Rolle’s Theorem, Mean Value Theorem applications, modeling problems, and an engaging example with Geogebra. Practice exercises from Lesson 4.2 can be found on page 216 ranging from 1 to 61.

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Exploring the Mean Value Theorem in Two-Hour Road Trips

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

  2. This is Really Mean

  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. 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

  5. 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

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

  7. 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

  8. 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

  9. Assignment • Lesson 4.2 • Pg 216 • Exercises 1 – 61 EOO

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