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When you work out the slope of the straight line joining
`(a,f(a))` to `(b,f(b))`, it
comes out to be

It's no surpise that this is the quantity from the
statement of the theorem, because that's really exactly what the mean value
theorem
**says**:

Providedfis suitably continuous and differentiable, there is a point in(a,b)at which the instantaneous slope offis the same as the slope of the straight line joining the values offataandb.