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Rolles Theorem and The Mean Value Theorem

- 4.2

Mean Value Theorem for Derivatives

Mean Value Theorem for Derivatives

Differentiable implies that the function is also

continuous.

Mean Value Theorem for Derivatives

Differentiable implies that the function is also

continuous.

The Mean Value Theorem only applies over a closed

interval.

Mean Value Theorem for Derivatives

Tangent parallel to chord.

Slope of tangent

Slope of chord

Example

- Suppose you have a function f(x) x2 in the

interval 1, 3.

To find c, set derivative to 4

There can be more than 1 value of c that satisfy

MVTD

There is no value of c that satisfies this

equation. Why?

f(x) is not continuous in the given interval!

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Find the value of c that satisfies the MVTD

Since f(0) f(12), Rolles Theorem applies

Find the value of c that satisfies the MVTD

Since f(0) f(1), Rolles Theorem applies

Example

- A trucker handed in a ticket at a toll booth

showing that in 2 hours she had covered 159 miles

on a toll road with a speed limit of 65 mph. Can

the trucker be cited for speeding?

MVTD tells us at some point in the interval, the

instantaneous velocity is equal to the average

velocity! Thus she must have been speeding at

some point!

Average velocity is 159/2 79.5

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