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
![### Problem Statement
Let \( F(x) = f(f(x)) \) and \( G(x) = (F(x))^2 \).
You also know that \( f(5) = 3 \), \( f(3) = 2 \), \( f'(3) = 8 \), \( f'(5) = 14 \).
### Objective
Find \( F'(5) = \) [______] and \( G'(5) = \) [______]
### Solution Steps
To find \( F'(5) \) and \( G'(5) \), follow these steps:
1. **Find \( F'(x) \):**
- Use the chain rule on \( F(x) = f(f(x)) \).
\[ F'(x) = f'(f(x)) \cdot f'(x) \]
2. **Calculate \( F'(5) \):**
- Substitute \( x = 5 \):
\[ F'(5) = f'(f(5)) \cdot f'(5) \]
- From the given data:
\[ f(5) = 3 \]
\[ f'(5) = 14 \]
\[ f'(f(5)) = f'(3) = 8 \]
- Therefore:
\[ F'(5) = 8 \cdot 14 = 112 \]
3. **Find \( G'(x) \):**
- Use the chain rule on \( G(x) = (F(x))^2 \).
\[ G'(x) = 2F(x) \cdot F'(x) \]
4. **Calculate \( G'(5) \):**
- Substitute \( x = 5 \):
\[ G'(5) = 2F(5) \cdot F'(5) \]
- From the given data:
\[ F(5) = f(f(5)) = f(3) = 2 \]
- We already calculated \( F'(5) = 112 \).
- Therefore:
\[ G'(5) = 2 \cdot 2 \cdot 112 = 4 \cdot 112 = 448 \]
### Conclusion
\[
\begin{aligned}
F'(5) &= 112 \\
G](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2Fd7fe1c84-2268-420f-ab98-ed700cc17b1f%2F6ocv0de_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Let \( F(x) = f(f(x)) \) and \( G(x) = (F(x))^2 \).
You also know that \( f(5) = 3 \), \( f(3) = 2 \), \( f'(3) = 8 \), \( f'(5) = 14 \).
### Objective
Find \( F'(5) = \) [______] and \( G'(5) = \) [______]
### Solution Steps
To find \( F'(5) \) and \( G'(5) \), follow these steps:
1. **Find \( F'(x) \):**
- Use the chain rule on \( F(x) = f(f(x)) \).
\[ F'(x) = f'(f(x)) \cdot f'(x) \]
2. **Calculate \( F'(5) \):**
- Substitute \( x = 5 \):
\[ F'(5) = f'(f(5)) \cdot f'(5) \]
- From the given data:
\[ f(5) = 3 \]
\[ f'(5) = 14 \]
\[ f'(f(5)) = f'(3) = 8 \]
- Therefore:
\[ F'(5) = 8 \cdot 14 = 112 \]
3. **Find \( G'(x) \):**
- Use the chain rule on \( G(x) = (F(x))^2 \).
\[ G'(x) = 2F(x) \cdot F'(x) \]
4. **Calculate \( G'(5) \):**
- Substitute \( x = 5 \):
\[ G'(5) = 2F(5) \cdot F'(5) \]
- From the given data:
\[ F(5) = f(f(5)) = f(3) = 2 \]
- We already calculated \( F'(5) = 112 \).
- Therefore:
\[ G'(5) = 2 \cdot 2 \cdot 112 = 4 \cdot 112 = 448 \]
### Conclusion
\[
\begin{aligned}
F'(5) &= 112 \\
G
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Follow-up Question
The answer was correct but I don't understand, how you got the number 2 on the last line.
G'(5) = 2 F(5) . F'(5)
= (2) (2) (112) = 448
Because F(5)=3. Could you explain it, please? Thanks.
Solution
by Bartleby ExpertFollow-up Question
The answer was correct but I don't understand, how you got the number 2 on the last line.
G'(5) = 2 F(5) . F'(5)
= (2) (2) (112) = 448
Because F(5)=3. Could you explain it, please? Thanks.
Solution
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