2 Mean Value Theorem for Derivatives 3 Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous. 4 Mean Value Theorem for Derivatives Differentiable implies that the function is also continuous. The Mean Value Theorem only applies over a closed interval. 5 Mean Value Theorem for Derivatives 6 Tangent parallel to chord. Slope of tangent Slope of chord 7 Example
Suppose you have a function f(x) x2 in the interval 1 3.
To find c set derivative to 4 8 There can be more than 1 value of c that satisfy MVTD 9 There is no value of c that satisfies this equation. Why f(x) is not continuous in the given interval! 10 (No Transcript) 11 (No Transcript) 12 Find the value of c that satisfies the MVTD Since f(0) f(12) Rolles Theorem applies 13 Find the value of c that satisfies the MVTD Since f(0) f(1) Rolles Theorem applies 14 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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