CABChp4-2NTscomplt - CalcBC Chp 4-2 Rolles Theorem and Mean...

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CalcBC Chp 4- 2 Rolle’s Theorem and Mean Value Theorem NOTES 1 ROLLE’S TH EOREM A set of conditions used to determine if there is a turning point within the given interval. Let f be a function that satisfies the following three hypotheses: (1) f is CONTINUOUS on the (closed) interval [a, b] (2) f is DIFFERENTIABLE on the (open) interval (a, b) (3) f(a) = f(b) (the endpoints have the same y-value ) Then, there is at least one number c in the (open) interval (a, b) such that f’(c) = 0 Visual example of how Rolle's involves extreme value points: f(a) f(a) f(a) f(b) f(b) f(b) f(c) f(c) f(c) Why deriv.= 0? Because the segment connecting the endpoints is horizontal and at x = c there may be an EVP.
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2. Ex(Apply Rolle’s Thm)— Verify the function satisfies the three hypotheses of Rolle’s Thm on the given interval. If the conditions are met, find all numbers c that satisfy the conclusions of Rolle’s Thm. f(x) = [ 6, 0] <Vocabulary assistance> hypotheses are the three conditions: continuous, differentiable, f(a) = f(b) conclusion: f '(c) = 0 <Answer> continuous (no undefined y-values) function does not negative value under the square-root for the given interval it is all continuous on the interval [ 6, 0] (check) differentiable (no undefined slopes) could be a problem where the derivative has variable in denominator get derivative to see what it looks like (product and chain rules used) f '(x) = + (fast add) f'(x) = f '(x) = 0 (use numerator=0)
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CABChp4-2NTscomplt - CalcBC Chp 4-2 Rolles Theorem and Mean...

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