Mean Value Theorem - Calculus I - Lecture Slides, Slides of Calculus

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4.2 The Mean Value Theorem

Rolle’s Theorem

Let f be a function that satisfies the following three conditions:

1. f is continuous on the closed interval [a,b].
2. f is differentiable on the open interval (a,b).
3. f(a) = f(b).

Then there is a number c in (a,b) such that f ′ (c)=0.

Examples on the board.

y

x 0

A

B

a (^) b

Slope of chord:

f ( b ) f ( a)

b a

− −

Slope of tangent:

f ′ ( )c

y = f ( x)

Tangent parallel to chord.

c

An illustration of the Mean Value Theorem.

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)