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13. The graph of y = f(x) and y = g(x) are shown below. g(x) 6 (a) (b) f(x) 2 10 -2 Find h'(9) if h(x) = g(f(x)) Find an equation for the tangent line to h(x) = g(f(x)) when x = 9.

13. The graph of y = f(x) and y = g(x) are shown below. g(x) 6 (a) (b) f(x) 2 10 -2 Find h'(9) if h(x) = g(f(x)) Find an equation for the tangent line to h(x) = g(f(x)) when x = 9.

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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**Graph of Functions \( y = f(x) \) and \( y = g(x) \) - Educational Explanation** ### Problem Statement: The following graph shows the plots of functions \( y = f(x) \) (solid line) and \( y = g(x) \) (dashed line). **Graph Details:** - **Axes:** The graph is plotted with the x-axis and y-axis extending from -2 to 10 and -2 to 4, respectively. - **Function \(y = g(x)\):** This function appears as a dashed line. - From \( x = 0 \) to \( x = 4 \), \( g(x) \) remains constant at \( y = 4 \). - From \( x = 4 \) to \( x = 7 \), \( g(x) \) declines linearly, reaching \( y = 0 \). - From \( x = 7 \) to \( x = 10 \), \( g(x) \) declines further, reaching \( y = -2 \) at \( x = 10 \). - **Function \(y = f(x)\):** This function appears as a solid line. - From \( x = 0 \) to \( x = 4 \), \( f(x) \) increases linearly from \( y = -2 \) to \( y = 2 \). - From \( x = 4 \) to \( x = 7 \), \( f(x) \) remains constant at \( y = 2 \). - From \( x = 7 \) to \( x = 10 \), \( f(x) \) increases linearly, passing through \( y = 2 \) but not reaching any visible endpoint. ### Questions: **(a)** Find \( h'(9) \) if \( h(x) = g(f(x)) \). **Solution Steps to consider:** 1. Determine \( f(9) \) from the graph. 2. Use this value to find \( g(f(9)) \). 3. Differentiate \( h(x) = g(f(x)) \) using the chain rule to find \( h'(9) \). **(b)** Find an equation for the tangent line to \( h(x) = g(f(x)) \) when \( x = 9 \). **Solution Steps to consider:**
Transcribed Image Text:**Graph of Functions \( y = f(x) \) and \( y = g(x) \) - Educational Explanation** ### Problem Statement: The following graph shows the plots of functions \( y = f(x) \) (solid line) and \( y = g(x) \) (dashed line). **Graph Details:** - **Axes:** The graph is plotted with the x-axis and y-axis extending from -2 to 10 and -2 to 4, respectively. - **Function \(y = g(x)\):** This function appears as a dashed line. - From \( x = 0 \) to \( x = 4 \), \( g(x) \) remains constant at \( y = 4 \). - From \( x = 4 \) to \( x = 7 \), \( g(x) \) declines linearly, reaching \( y = 0 \). - From \( x = 7 \) to \( x = 10 \), \( g(x) \) declines further, reaching \( y = -2 \) at \( x = 10 \). - **Function \(y = f(x)\):** This function appears as a solid line. - From \( x = 0 \) to \( x = 4 \), \( f(x) \) increases linearly from \( y = -2 \) to \( y = 2 \). - From \( x = 4 \) to \( x = 7 \), \( f(x) \) remains constant at \( y = 2 \). - From \( x = 7 \) to \( x = 10 \), \( f(x) \) increases linearly, passing through \( y = 2 \) but not reaching any visible endpoint. ### Questions: **(a)** Find \( h'(9) \) if \( h(x) = g(f(x)) \). **Solution Steps to consider:** 1. Determine \( f(9) \) from the graph. 2. Use this value to find \( g(f(9)) \). 3. Differentiate \( h(x) = g(f(x)) \) using the chain rule to find \( h'(9) \). **(b)** Find an equation for the tangent line to \( h(x) = g(f(x)) \) when \( x = 9 \). **Solution Steps to consider:**
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