Warmup 1. What is the interval [a, b] where Rolle’s Theorem is applicable? 2. What is/are the c-values? [-3, 3]

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Presentation transcript:

Warmup 1. What is the interval [a, b] where Rolle’s Theorem is applicable? 2. What is/are the c-values? [-3, 3]

Rolle’s and Mean Value Theorem 3.2 Day 2 On the agenda: Understanding and using The Mean Value Theorem HW: p. 172-3 # 8–14 even, 31-37 odd, 53-56

Mean Value Theorem for Derivatives If f (x) is continuous over [a, b] and differentiable over (a, b), then at some point c in (a, b) such that: Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope, where the average slope is . The Mean Value Theorem only applies over a closed interval.

Tangent parallel to chord. Slope of tangent: Slope of chord:

Ex. 3 Using the Mean Value Theorem Given , find all values of c in the open interval [1, 4] such that it satisfies the mean value theorem Conditions: Is f(x) continuous on [1, 4] and differentiable on (1, 4)? Then there is at least one number c, where So Next we have to find the c value(s). c-value(s) are the x values at which the derivate of f(x) has a slope of 1.

Ex. 3 Using the Mean Value Theorem cont… Which implies that c = + 2. The only possible answer for us is c = 2 since it is in the interval of (1, 4).

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