1. Be able to apply The Mean Value Theorem to various functions. Objectives: 1. Be able to apply The Mean Value Theorem to various functions. Critical Vocabulary: The Mean Value Theorem Warm Up: Determine if Rolle’s Theorem can be applied. If it can, find all values of c such that f’(c) = 0. f(x) = x2 - 5x + 4; [1, 4]

If f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that Secant Line Tangent Line Tangent Line Secant Line (b, f(b)) (a, f(a)) a c b

Example 1: Given f(x) = 5 - (4/x), find all values of c in the open interval (1,4) such that 1st: Find f’(c): f(4) = 5 - 1 f(1) = 5 - 4 f’(c) = 1 f(4) = 4 f(1) = 1 2nd: Set f’(x) = f’(c) to determine the x values determined by the Mean Value Theorem f(x) = 5 – 4x-1 f(2) = 5 - 2 f’(x) = 4x-2 f(2) = 3 f’(x) = 4/x2 (2, 3) is the point where the tangent is parallel to the secant

(-1/3, 14/27) is the point where the tangent is parallel to the secant Example 2 (page 326 #24): Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f(x) = x(x2 - x - 2) ;[-1, 1] 1st: Secant Line f(1) = -2 f’(c) = -1 f(-1) = 0 2nd: Tangent Line f(-1/3) = -1/3(1/9 + 1/3 - 2) f(x) = x3 – x2 - 2x 3x2 – 2x - 2 = -1 f(-1/3) = 14/27 f(x) = 3x2 – 2x - 2 3x2 – 2x - 1 = 0 (3x + 1)(x - 1) = 0 (-1/3, 14/27) is the point where the tangent is parallel to the secant 3x + 1 = 0 x - 1 = 0 x = -1/3 x = 1

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