Calculus and Analytical Geometry Lecture # 15 MTH 104.

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Presentation transcript:

Calculus and Analytical Geometry Lecture # 15 MTH 104

Rolle’s theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f (a)=0 and f (b)=0 then there is at least one point c in the interval (a, b) such that

When working with Rolle’s make sure 1.The function f(x) is continuous on [a, b] and differentiable on (a, b). 2.Show that f(a) =0 and f(b)=0. 3. State that there exists at least one x = c in (a, b) such that f ’(c) = 0. Example Find the two x-intercepts of the function and confirm that f’(c)=0 at some point c between these intercepts

solution

Example Does Rolle’s Theorem apply?

If so, find all the values of c. Example

Does Rolle’s Theorem apply? If so, find all values of c.

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 The slope of secant line joining (a, f(a)) and (b,f(b)) is The slope of tangent line at c is

Example Show that the function satisfies the hypotheses of the Mean Value Theorem over the interval [0, 2], 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 points (0, f(0)) and (2, f(2)).

Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval, and find all values of c in that interval that satisfy the conclusion of the theorem.