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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![### Calculus Problem on Derivatives Using a Table
#### 3) Using the table below:
| x | f(x) | g(x) | f'(x) | g'(x) |
|-----|------|------|-------|-------|
| 0 | -1 | 3 | 0 | 2 |
| 1 | 3 | 2 | -1 | 1 |
---
**a) If \( h(x) = f(x) \cdot g(x) \), find \( h'(0) \).**
**b) If \( h(x) = \frac{f(x)}{g(x)} \), find \( h'(1) \).**
---
### Solution:
**a) Finding \( h'(0) \) when \( h(x) = f(x) \cdot g(x) \):**
To find \( h'(x) \) for the function \( h(x) = f(x) \cdot g(x) \), use the product rule for derivatives, which states:
\[ h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \]
Substitute \( x = 0 \) and use the values from the table:
- \( f(0) = -1 \)
- \( g(0) = 3 \)
- \( f'(0) = 0 \)
- \( g'(0) = 2 \)
Now, calculate \( h'(0) \):
\[ h'(0) = f'(0) \cdot g(0) + f(0) \cdot g'(0) \]
\[ h'(0) = 0 \cdot 3 + (-1) \cdot 2 \]
\[ h'(0) = 0 - 2 \]
\[ h'(0) = -2 \]
**Answer:** \( h'(0) = -2 \)
---
**b) Finding \( h'(1) \) when \( h(x) = \frac{f(x)}{g(x)} \):**
To find \( h'(x) \) for the function \( h(x) = \frac{f(x)}{g(x)} \), use the quotient rule for derivatives, which states:
\[ h'(x) = \frac{f'(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa55013e1-91aa-48be-9c4b-bf8a0e47de22%2F7370ba32-9b89-4cda-8dfb-e33da876d1c3%2F2fhvb52_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Problem on Derivatives Using a Table
#### 3) Using the table below:
| x | f(x) | g(x) | f'(x) | g'(x) |
|-----|------|------|-------|-------|
| 0 | -1 | 3 | 0 | 2 |
| 1 | 3 | 2 | -1 | 1 |
---
**a) If \( h(x) = f(x) \cdot g(x) \), find \( h'(0) \).**
**b) If \( h(x) = \frac{f(x)}{g(x)} \), find \( h'(1) \).**
---
### Solution:
**a) Finding \( h'(0) \) when \( h(x) = f(x) \cdot g(x) \):**
To find \( h'(x) \) for the function \( h(x) = f(x) \cdot g(x) \), use the product rule for derivatives, which states:
\[ h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x) \]
Substitute \( x = 0 \) and use the values from the table:
- \( f(0) = -1 \)
- \( g(0) = 3 \)
- \( f'(0) = 0 \)
- \( g'(0) = 2 \)
Now, calculate \( h'(0) \):
\[ h'(0) = f'(0) \cdot g(0) + f(0) \cdot g'(0) \]
\[ h'(0) = 0 \cdot 3 + (-1) \cdot 2 \]
\[ h'(0) = 0 - 2 \]
\[ h'(0) = -2 \]
**Answer:** \( h'(0) = -2 \)
---
**b) Finding \( h'(1) \) when \( h(x) = \frac{f(x)}{g(x)} \):**
To find \( h'(x) \) for the function \( h(x) = \frac{f(x)}{g(x)} \), use the quotient rule for derivatives, which states:
\[ h'(x) = \frac{f'(x)
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