1. 4.2 The Mean Value Theorem

2. 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) .
Then there is a number c in (a,b) such that f ′(c)=0.
Examples on the board.

3. If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b:
Mean Value Theorem for Derivatives
The Mean Value Theorem says that at some point in the interval, the actual slope equals the average slope.
Note: The Mean Value Theorem only applies
over a closed interval.

4. An illustration of the Mean Value Theorem.
Tangent parallel to chord.
Slope of tangent:
Slope of chord:

5. Corollaries of the Mean Value Theorem
Corollary 1: If f ′(x)=0 for all x in an interval (a,b), then f is constant on (a,b) .
Corollary 2: If f ′ (x) = g′ (x) for all x in an interval (a,b), then f - g is constant on (a,b) , that is f(x)= g(x) + c where c is a constant.
(see the next slide for an illustration of Corollary 2)

6. These two functions have the same slope at any value of x.
Functions with the same derivative differ by a constant.