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Mean Value Theorem

Mean Value Theorem. Do Now. When you’re finished, enter your answer in Socrative – Rm : # 487130. What gives?. After I started driving this morning, I passed a speed trap going 65 mph (the legal speed limit).

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Mean Value Theorem

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  1. Mean Value Theorem

  2. Do Now • When you’re finished, enter your answer in Socrative – Rm: # 487130

  3. What gives? • After I started driving this morning, I passed a speed trap going 65 mph (the legal speed limit). • A half hour later, I passed another speed trap 40 miles down the same highway, still going 65 mph (the legal speed limit). • The cop pulls me over and issues me a ticket for speeding. What gives?

  4. Mean Value Theorem • If a function f(x) is continuous on [a , b] and differentiable on (a , b), then, for some c on [a , b]:

  5. Looking at the MVT graphically:

  6. Why not the “Nice” Value Theorem?

  7. A typical problem: • Find a number “c” on the interval [-1 , 3] that satisfies the Mean Value Theorem for the function

  8. A typical problem:

  9. A typical problem:

  10. A typical problem:

  11. Why doesn’t this work? • Can you find a value “c” on the interval [1 , 4] that satisfies the Mean Value Theorem for the function ?

  12. Why doesn’t this work?

  13. Why doesn’t this work?

  14. Rolle’s Theorem • If f(x) is continuous on [a , b] and differentiable on (a , b) and f(a) = f(b), then for some c on [a , b], f’(c) = 0.

  15. Rolle’s Theorem

  16. Three Theorems: • The Intermediate Value Theorem (IVT) • The Extreme Value Theorem (EVT) • The Mean Value Theorem (MVT)

  17. Practice AP Problems: Multiple Choice • Practice with the following 3 AP multiple choice problems

  18. Answers: • 2003 #80 – B • 1998 #4 – B • 1998 #91 - E

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