Section 4.2 Rolle’s Theorem and The Mean Value Theorem

Section 4.2 Rolle’sTheorem andThe Mean Value Theorem PhongChau

Explore • Draw a graph of f(x) on an interval [-4,7] • Draw a secant line that passes through the two endpoints • Can you find a tangent line to the graph of f(x) that is parallel to the secant line in step 2? • 4) Can you modify the graph of f(x) so that there is no way to find a tangent line as described in step 3?

The Mean Value Theorem • Let f be a function that satisfies the following two hypotheses: • f is continuous on the closed interval [a, b]. • f is differentiable on the open interval (a, b). • Then there is a number c in (a, b) such that

Example: MVT Verify that the function satisfies the 2 hypotheses of MVT on the given interval. Then find all numbers c that satisfy the conclusion of MVT.

Example: A special case of MVT Determine whether MVT can be applied. If it can be applied, find all numbers c that satisfy the conclusion of MVT.

Rolle’s Theorem • Let f be a function that satisfies the following three hypotheses: • 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.

Example: Rolle’s Theorem Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.

Example Use IVT and Rolle’s Theorem to show that the equation 2x – 1 – sin x = 0 has exactly one real solution.

Section 4.2 Rolle’s Theorem and The Mean Value Theorem