Mean Value Theorem

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• 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).

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?

Mean Value TheoremIf a function f(x) is continuous on [a , b] and differentiable on (a , b), then, for some c on [a , b]:

Looking at the MVT graphically:

Why not the “Nice” Value Theorem?

A typical problem:

Find a number “c” on the interval [-1 , 3] that satisfies the Mean Value Theorem for the function

A typical problem:

y

A typical problem:

y

A typical problem:

y

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 ? f (x)2 x 3 4

Why doesn’t this work?

x

y

Why doesn’t this work?

x

y

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.

Rolle’s Theorem

Three Theorems:

• The Intermediate Value Theorem (IVT)

• The Extreme Value Theorem (EVT)

• The Mean Value Theorem (MVT)

Practice AP Problems: Multiple Choice

Practice with the following 3 AP multiple choice problems

Answers:

2003 #80 – B

1998 #4 – B

1998 #91 - E