5.7 rolle's thrm & mv theorem
TRANSCRIPT
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