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of xo. You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. Let f be continuous at xo and sup- pose that limf'(x) exists. Then f is differentiable at xo, and f'(xo) = limf(x). 67. Show that '+x+1, *SI * > I f(x) = 3x₂ is continuous at x = 1. Determine whether f is differen- tiable at x = 1. If so, find the value of the derivative there. Sketch the graph of f.

of xo. You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. Let f be continuous at xo and sup- pose that limf'(x) exists. Then f is differentiable at xo, and f'(xo) = limf(x). 67. Show that '+x+1, *SI * > I f(x) = 3x₂ is continuous at x = 1. Determine whether f is differen- tiable at x = 1. If so, find the value of the derivative there. Sketch the graph of f.

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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67-70 You are asked in these exercises to determine whether a piecewise-defined function ƒ is differentiable at a value x = x₂ where f is defined by different formulas on different sides of xo. You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. Let f be continuous at xo and sup- pose that limf'(x) exists. Then f is differentiable at xo, and f'(xo) = limxf(x). 67. Show that f(x) = [x²+x+1, 3x, is continuous at x = 1. Determine whether f is differen- tiable at x = 1. If so, find the value of the derivative there. Sketch the graph of f. X x
Transcribed Image Text:67-70 You are asked in these exercises to determine whether a piecewise-defined function ƒ is differentiable at a value x = x₂ where f is defined by different formulas on different sides of xo. You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. Let f be continuous at xo and sup- pose that limf'(x) exists. Then f is differentiable at xo, and f'(xo) = limxf(x). 67. Show that f(x) = [x²+x+1, 3x, is continuous at x = 1. Determine whether f is differen- tiable at x = 1. If so, find the value of the derivative there. Sketch the graph of f. X x
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