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b) Verify the Mean Value Theorem on [1,3] when ƒ(x) = 5x² – x³. =) A function f is defined by f(x) = { sin²x 1 - cos x when x < 0, when x ≥ 0. Does the Mean Value Theorem apply to ƒ on [-, π]? d) By applying the Mean Value Theorem to a function ƒ on a suitable interval, show that ƒ(x) = f(x) + ƒ'(c)(x − xo), - where c lies between co and x. e) Suppose that f(xo) = g(xo) = 0, and g'(x) ‡ 0. Use the results of part (d) to show that ƒ(x)` lim x+xo (g(x) = f'(xo) g'(x) log(1+log x) =) Use the result of part (e) to evaluate limx→1 (log(2x²-1)

b) Verify the Mean Value Theorem on [1,3] when ƒ(x) = 5x² – x³. =) A function f is defined by f(x) = { sin²x 1 - cos x when x < 0, when x ≥ 0. Does the Mean Value Theorem apply to ƒ on [-, π]? d) By applying the Mean Value Theorem to a function ƒ on a suitable interval, show that ƒ(x) = f(x) + ƒ'(c)(x − xo), - where c lies between co and x. e) Suppose that f(xo) = g(xo) = 0, and g'(x) ‡ 0. Use the results of part (d) to show that ƒ(x)` lim x+xo (g(x) = f'(xo) g'(x) log(1+log x) =) Use the result of part (e) to evaluate limx→1 (log(2x²-1)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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can you please do c and d, can you please provide explanations

B1. (a) State the Mean Value Theorem for a function f that is continuous on [a, b] and differentiable on (a, b). (b) Verify the Mean Value Theorem on [1,3] when f(x) = 5x² – x³. (c) A function f is defined by f(x) = { sin² x when x < 0, cos x when x ≥ 0. Does the Mean Value Theorem apply to f on [-, π]? (d) By applying the Mean Value Theorem to a function f on a suitable interval, show that ƒ(x) = f(x) + ƒ'(c)(x − xo), where c lies between co and x. (e) Suppose that f(x) = g(x) = 0, and g'(x) ‡ 0. Use the results of part (d) to show that lim x-xo = f'(xo) g'(xo) log(1+log x) (f) Use the result of part (e) to evaluate limx→1 (log(2x²-1)
Transcribed Image Text:B1. (a) State the Mean Value Theorem for a function f that is continuous on [a, b] and differentiable on (a, b). (b) Verify the Mean Value Theorem on [1,3] when f(x) = 5x² – x³. (c) A function f is defined by f(x) = { sin² x when x < 0, cos x when x ≥ 0. Does the Mean Value Theorem apply to f on [-, π]? (d) By applying the Mean Value Theorem to a function f on a suitable interval, show that ƒ(x) = f(x) + ƒ'(c)(x − xo), where c lies between co and x. (e) Suppose that f(x) = g(x) = 0, and g'(x) ‡ 0. Use the results of part (d) to show that lim x-xo = f'(xo) g'(xo) log(1+log x) (f) Use the result of part (e) to evaluate limx→1 (log(2x²-1)
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