4.2 The Mean Value Theorem. Rolle’s Theorem. Let f be a function that satisfies the following three conditions: f is continuous on the closed interval [a,b] . f is differentiable on the open interval (a,b) . f(a) = f(b) .
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Let f be a function that satisfies the following three conditions:
Then there is a number c in (a,b) such that f ′(c)=0.
Examples on the board.
If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b:
The Mean Value Theorem says that at some point in the interval, the actual slope equals the average slope.
Mean Value Theorem for Derivatives
Note: The Mean Value Theorem only applies
over a closed interval.
Tangent parallel to chord.
Slope of tangent:
Slope of chord:
(see the next slide for an illustration of Corollary 2)
These two functions have the same slope at any value of x.