Rolle’s Theoremand Mean Value

Theorem

Theorem:

Example

Let f(x) = sin 2x. Find all values of c in the interval

such that f’(c) = 0

Does it satisfy Rolle’s Theorem?

f is continuous

f is differentiable

Find any c values:

3,6

Another Example:

Let . Find all values of c in the interval

such that f’(c) = 0

Does it satisfy Rolle’s Theorem?

Find any c values:

Theorem: Mean Value Theorem

Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there is at least one point c in (a, b) such that

ab

afbfcf

slope of tangent at c

slope of secant over the interval

[a, b]

Example

Find a value for c within the interval (-1, 1) where the tangent line at c will be parallel to the secant line through the endpoints of the interval.

Another Example

Show that the function satisfies the hypotheses of the Mean-Value-Theorem over the interval , and find all values of c in the interval (0, 2) at which the tangent line to the graph of f is parallel to the secant line joining the endpoints of the interval