Find the following using the table below. 1 2 3 4 12 x f(x) 3 4 f'(x) 2 3 1 g(x) 2 1 3 g'(x) 2 3 1 h'(1) if h(x) = f(g(x)) Question Help: 4 4 4 Message instructor
Find the following using the table below. 1 2 3 4 12 x f(x) 3 4 f'(x) 2 3 1 g(x) 2 1 3 g'(x) 2 3 1 h'(1) if h(x) = f(g(x)) Question Help: 4 4 4 Message instructor
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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Question
![The task is to find \( h'(1) \) if \( h(x) = f(g(x)) \), using the provided table.
### Table Data:
The table provides values for the functions at given points:
| \( x \) | \( 1 \) | \( 2 \) | \( 3 \) | \( 4 \) |
|---------|---------|---------|---------|---------|
| \( f(x) \) | \( 3 \) | \( 4 \) | \( 1 \) | \( 2 \) |
| \( f'(x) \) | \( 2 \) | \( 3 \) | \( 1 \) | \( 4 \) |
| \( g(x) \) | \( 2 \) | \( 1 \) | \( 3 \) | \( 4 \) |
| \( g'(x) \) | \( 2 \) | \( 3 \) | \( 1 \) | \( 4 \) |
### Explanation:
To find \( h'(1) \), use the chain rule, which states:
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
For \( x = 1 \):
- Find \( g(1) \).
- Then find \( f'(g(1)) \) and \( g'(1) \).
- Multiply these values to find \( h'(1) \).
### Interactive Element:
There is a text box for users to input their answer. Additionally, there is an option to message the instructor for help, indicating an interactive learning environment.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b9a25ce-7950-429e-b571-3c0f034f6fd8%2Fd237ba05-3e71-4c90-aad4-5e670b304cc1%2Ffl8aljn_processed.png&w=3840&q=75)
Transcribed Image Text:The task is to find \( h'(1) \) if \( h(x) = f(g(x)) \), using the provided table.
### Table Data:
The table provides values for the functions at given points:
| \( x \) | \( 1 \) | \( 2 \) | \( 3 \) | \( 4 \) |
|---------|---------|---------|---------|---------|
| \( f(x) \) | \( 3 \) | \( 4 \) | \( 1 \) | \( 2 \) |
| \( f'(x) \) | \( 2 \) | \( 3 \) | \( 1 \) | \( 4 \) |
| \( g(x) \) | \( 2 \) | \( 1 \) | \( 3 \) | \( 4 \) |
| \( g'(x) \) | \( 2 \) | \( 3 \) | \( 1 \) | \( 4 \) |
### Explanation:
To find \( h'(1) \), use the chain rule, which states:
\[
h'(x) = f'(g(x)) \cdot g'(x)
\]
For \( x = 1 \):
- Find \( g(1) \).
- Then find \( f'(g(1)) \) and \( g'(1) \).
- Multiply these values to find \( h'(1) \).
### Interactive Element:
There is a text box for users to input their answer. Additionally, there is an option to message the instructor for help, indicating an interactive learning environment.
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