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Determine the value of a such that f is differentiable at x = 1. for x <1 a sin(x-1)+6 for x21 S(x) = {(x +2)² – 3

Determine the value of a such that f is differentiable at x = 1. for x <1 a sin(x-1)+6 for x21 S(x) = {(x +2)² – 3

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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Please explain as clearly as possible, thanks.

**Problem:** Determine the value of \( a \) such that \( f \) is differentiable at \( x = 1 \). **Function Definition:** \[ f(x) = \begin{cases} (x + 2)^2 - 3 & \text{for } x < 1 \\ a \sin(x - 1) + 6 & \text{for } x \geq 1 \end{cases} \] **Explanation:** The problem requires finding the value of \( a \) that ensures the function \( f(x) \) is differentiable at \( x = 1 \). This means \( f(x) \) must be continuous and have a continuous derivative at \( x = 1 \). 1. **Continuity at \( x = 1 \):** - The left-hand limit as \( x \to 1^- \) must equal the right-hand limit as \( x \to 1^+ \), which must also equal \( f(1) \). 2. **Differentiability at \( x = 1 \):** - The derivative from the left must equal the derivative from the right at \( x = 1 \). Using these conditions, you can solve for the necessary value of \( a \).
Transcribed Image Text:**Problem:** Determine the value of \( a \) such that \( f \) is differentiable at \( x = 1 \). **Function Definition:** \[ f(x) = \begin{cases} (x + 2)^2 - 3 & \text{for } x < 1 \\ a \sin(x - 1) + 6 & \text{for } x \geq 1 \end{cases} \] **Explanation:** The problem requires finding the value of \( a \) that ensures the function \( f(x) \) is differentiable at \( x = 1 \). This means \( f(x) \) must be continuous and have a continuous derivative at \( x = 1 \). 1. **Continuity at \( x = 1 \):** - The left-hand limit as \( x \to 1^- \) must equal the right-hand limit as \( x \to 1^+ \), which must also equal \( f(1) \). 2. **Differentiability at \( x = 1 \):** - The derivative from the left must equal the derivative from the right at \( x = 1 \). Using these conditions, you can solve for the necessary value of \( a \).
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