Rolles Theorem and The Mean Value Theorem

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Rolles Theorem and The Mean Value Theorem

• 4.2

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Mean Value Theorem for Derivatives
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Mean Value Theorem for Derivatives
Differentiable implies that the function is also
continuous.
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Mean Value Theorem for Derivatives
Differentiable implies that the function is also
continuous.
The Mean Value Theorem only applies over a closed
interval.
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Mean Value Theorem for Derivatives
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Tangent parallel to chord.
Slope of tangent
Slope of chord
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Example

• Suppose you have a function f(x) x2 in the
  interval 1 3.

To find c set derivative to 4
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There can be more than 1 value of c that satisfy
MVTD
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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
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Find the value of c that satisfies the MVTD
Since f(0) f(1) Rolles Theorem applies
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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