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(4-х, XS0 10) Determine if the function y(X) = 3x, is differentiable at x =0. x>0

(4-х, XS0 10) Determine if the function y(X) = 3x, is differentiable at x =0. x>0

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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**Problem 10: Differentiability of a Piecewise Function** Determine if the function \( y(x) \) is differentiable at \( x = 0 \). \[ y(x) = \begin{cases} 4 - x, & x \leq 0 \\ 3x, & x > 0 \end{cases} \] **Explanation:** - The function \( y(x) \) is defined piecewise with two expressions: - \( 4 - x \) when \( x \leq 0 \) - \( 3x \) when \( x > 0 \) To determine if \( y(x) \) is differentiable at \( x = 0 \), we need to check: 1. The continuity of \( y(x) \) at \( x = 0 \). 2. The left-hand derivative and right-hand derivative at \( x = 0 \) are equal. For \( x = 0 \): - The limit from the left (\( x \to 0^- \)) and the limit from the right (\( x \to 0^+ \)) must be evaluated for continuity. - The left-hand derivative of \( 4 - x \) and the right-hand derivative of \( 3x \) at \( x = 0 \) must be calculated and compared to check for differentiability.
Transcribed Image Text:**Problem 10: Differentiability of a Piecewise Function** Determine if the function \( y(x) \) is differentiable at \( x = 0 \). \[ y(x) = \begin{cases} 4 - x, & x \leq 0 \\ 3x, & x > 0 \end{cases} \] **Explanation:** - The function \( y(x) \) is defined piecewise with two expressions: - \( 4 - x \) when \( x \leq 0 \) - \( 3x \) when \( x > 0 \) To determine if \( y(x) \) is differentiable at \( x = 0 \), we need to check: 1. The continuity of \( y(x) \) at \( x = 0 \). 2. The left-hand derivative and right-hand derivative at \( x = 0 \) are equal. For \( x = 0 \): - The limit from the left (\( x \to 0^- \)) and the limit from the right (\( x \to 0^+ \)) must be evaluated for continuity. - The left-hand derivative of \( 4 - x \) and the right-hand derivative of \( 3x \) at \( x = 0 \) must be calculated and compared to check for differentiability.
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