Rolle’s Theorem Section 3.2

Michel Rolle (1652-1719)

Rolle’s Theorem For a continuous function on the closed interval [a,b] that is differentiable on (a,b), if f(a) = f(b), then there is at least one number c in (a,b) such that f ‘(c)=0.

Visual Interpretation f(2) = f(6) Function is continuous and differentiable on (2,6). Is there an x-value in (2,6) where there is a horizontal tangent line?

Does Rolle’s Theorem Apply? NO!!!!! The function is NOT differentiable at x = -3.

Does Rolle’s Theorem Apply? YES!!! f(0)=f(π), and the function is continuous and differentiable on (0,π).

Determine whether Rolle’s Theorem applies on [a,b] Determine whether Rolle’s Theorem applies on [a,b]. If so, find all values c in (a,b) such that f ‘(c) = 0. f(x) is a polynomial…continuous everywhere. Rolle’s Theorem applies!

Assignment p.172: 1,2,3-15 odd

Mean Value Theorem Section 3.2

Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that We’ll use this to prove the Fundamental Theorem of Calculus! (coming this spring…)

Visual Interpretation of the Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b), then…

Visual Interpretation of the Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b), then… …there is some c in (a,b) such that the tangent line of the graph at c is parallel to the secant line through a and b.

Consider on the interval [1,4]. Does the Mean Value Theorem apply? If so, find all values for c guaranteed by the theorem. MVT applies

Assignment p.172: 29-34