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 Analysis and Derivative Calculation**
Below we have two graphs representing functions \(g(x)\) and \(h(x)\):
1. **Graph of \(g(x)\)**:
- This graph is a straight line.
- The line decreases from \( (1, 8) \) to \( (4, 1) \).
2. **Graph of \(h(x)\)**:
- This graph is a piecewise linear function.
- The function increases from \( (1, 2) \) to \( (3, 5) \), and then decreases from \( (3, 5) \) to \( (5, 2) \).
### Function Definition
If \( f(x) = \frac{g(x)}{h(x)} \), then the derivative \( f'(3) \) can be calculated using the quotient rule for derivatives.
The quotient rule states:
\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \]
### Task
Calculate \( f'(3) \):
\[ f'(3) = \]
**Steps to Calculate \( f'(3) \)**:
1. **Determine \( g(x) \) and \( h(x) \) at \( x = 3 \):
- Find \( g(3) \) and \( h(3) \).
- For the graph \( g(x) \), at \( x = 3 \), \( g(3) = ... \).
- For the graph \( h(x) \), at \( x = 3 \), \( h(3) = ... \).
2. **Find the Derivatives \( g'(x) \) and \( h'(x) \) at \( x = 3 \):
- Calculate the slopes of the linear functions at the given point.
- For \( g(x) \), calculate the slope \( g'(3) = ... \).
- For \( h(x) \) at \( x = 3 \), determine if we are on an increasing or decreasing segment and calculate \( h'(3) = ... \).
3. **Apply the Quotient Rule**:
- Plug in the values into the quotient rule formula to get \( f'(3) \).
This transcription provides an outline for understanding how to interpret the graphs and apply the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf6bcca1-d77f-469d-8067-ac9f27c3130a%2Fb843588b-48b6-4c72-9825-84a37b59a03a%2F3p609zq_processed.png&w=3840&q=75)
Transcribed Image Text:**Graph Analysis and Derivative Calculation**
Below we have two graphs representing functions \(g(x)\) and \(h(x)\):
1. **Graph of \(g(x)\)**:
- This graph is a straight line.
- The line decreases from \( (1, 8) \) to \( (4, 1) \).
2. **Graph of \(h(x)\)**:
- This graph is a piecewise linear function.
- The function increases from \( (1, 2) \) to \( (3, 5) \), and then decreases from \( (3, 5) \) to \( (5, 2) \).
### Function Definition
If \( f(x) = \frac{g(x)}{h(x)} \), then the derivative \( f'(3) \) can be calculated using the quotient rule for derivatives.
The quotient rule states:
\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \]
### Task
Calculate \( f'(3) \):
\[ f'(3) = \]
**Steps to Calculate \( f'(3) \)**:
1. **Determine \( g(x) \) and \( h(x) \) at \( x = 3 \):
- Find \( g(3) \) and \( h(3) \).
- For the graph \( g(x) \), at \( x = 3 \), \( g(3) = ... \).
- For the graph \( h(x) \), at \( x = 3 \), \( h(3) = ... \).
2. **Find the Derivatives \( g'(x) \) and \( h'(x) \) at \( x = 3 \):
- Calculate the slopes of the linear functions at the given point.
- For \( g(x) \), calculate the slope \( g'(3) = ... \).
- For \( h(x) \) at \( x = 3 \), determine if we are on an increasing or decreasing segment and calculate \( h'(3) = ... \).
3. **Apply the Quotient Rule**:
- Plug in the values into the quotient rule formula to get \( f'(3) \).
This transcription provides an outline for understanding how to interpret the graphs and apply the
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