Question If an object is travelling in a one dimensional path, i.e. along a straight line, is the instantaneous velocity ever equal to the average velocity?

Answer If the motion is smooth, i.e. s=f(t), and f and f' are both continuous for t in (a,b), then the answer is yes.

Mean Value Theorem If s=f(t), and

1. f is continuous on [a,b]
2. f' is continous on (a,b)

then there is a point t=c when the average velocity ( f(b)-f(b) ) / (b-a) is equal to the instantaneous velocity f'(c).

f'(c) = ( f(b)-f(b) ) / (b-a)

A special case of this is :

Rolle's Theorem If a function f satisfies the three properties:

1. f is continuous on [a,b]
2. f' is continous on (a,b) f(a) = f(b)

then for some number c in (a,b), f'(c)=0!

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The Mean Value Theorem has two important consequences

Theorem If f'(x)=0 for all x in (a,b), then f is constant on (a,b).

Theorem If f'(x)=g'(x) for all x in (a,b), then f(x)=g(x)+c, where c is an arbitrary constant.

Last modified Wed Nov 6 23:05:08 CST 1996