Functions f(x) and g(x) satisfy the following conditions. f(4) = –2, f'(4) = 3, g(1) = 4, g(1) = 11 Define function h(x) as h(x) = fog(x). a. Compute h'(1).
Functions f(x) and g(x) satisfy the following conditions. f(4) = –2, f'(4) = 3, g(1) = 4, g(1) = 11 Define function h(x) as h(x) = fog(x). a. Compute h'(1).
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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1.Functions f(x) and g(x) satisfy the following conditions.
f(4)=-2, f'(4)=3, g(1)=4, g'(1)=11
Define function as h(x)=f o g(x)
a. Compute h'(1)
![**Problem Statement:**
1. Functions \( f(x) \) and \( g(x) \) satisfy the following conditions:
\[
f(4) = -2, \quad f'(4) = 3, \quad g(1) = 4, \quad g'(1) = 11
\]
Define function \( h(x) \) as \( h(x) = f \circ g(x) \).
a. Compute \( h'(1) \).
---
**Explanation:**
The problem involves understanding the composition of functions and the application of the chain rule for differentiation. Given the functions \( f(x) \) and \( g(x) \), and their specific values and derivatives at certain points, the task is to find the derivative of the composed function \( h(x) = f(g(x)) \) at \( x=1 \).
To solve it, use the chain rule:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Substitute \( x = 1 \) into the equation:
\[ h'(1) = f'(g(1)) \cdot g'(1) \]
Use the given values:
- \( g(1) = 4 \)
- \( f'(4) = 3 \)
- \( g'(1) = 11 \)
Finally, calculate using these values:
\[ h'(1) = f'(4) \cdot g'(1) = 3 \cdot 11 = 33 \]
Therefore, \( h'(1) = 33 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd76d30b-2160-4217-8ed1-5f85016c97ed%2F54345803-79cc-4817-88d9-5a1249c33755%2F2rt8hz_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1. Functions \( f(x) \) and \( g(x) \) satisfy the following conditions:
\[
f(4) = -2, \quad f'(4) = 3, \quad g(1) = 4, \quad g'(1) = 11
\]
Define function \( h(x) \) as \( h(x) = f \circ g(x) \).
a. Compute \( h'(1) \).
---
**Explanation:**
The problem involves understanding the composition of functions and the application of the chain rule for differentiation. Given the functions \( f(x) \) and \( g(x) \), and their specific values and derivatives at certain points, the task is to find the derivative of the composed function \( h(x) = f(g(x)) \) at \( x=1 \).
To solve it, use the chain rule:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
Substitute \( x = 1 \) into the equation:
\[ h'(1) = f'(g(1)) \cdot g'(1) \]
Use the given values:
- \( g(1) = 4 \)
- \( f'(4) = 3 \)
- \( g'(1) = 11 \)
Finally, calculate using these values:
\[ h'(1) = f'(4) \cdot g'(1) = 3 \cdot 11 = 33 \]
Therefore, \( h'(1) = 33 \).
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