The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function D(t) - π 5 sin -t + 6 7π 7T) + 6 +4 where t is the number of hours after midnight. Find the rate at which the depth is changing at 5 a.m. Round your answer to 4 decimal places. ft/hr
The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function D(t) - π 5 sin -t + 6 7π 7T) + 6 +4 where t is the number of hours after midnight. Find the rate at which the depth is changing at 5 a.m. Round your answer to 4 decimal places. ft/hr
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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![The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function
\[ D(t) = 5 \sin \left( \frac{\pi}{6} t + \frac{7\pi}{6} \right) + 4 \]
where \( t \) is the number of hours after midnight. Find the rate at which the depth is changing at 5 a.m. Round your answer to 4 decimal places.
\[ \boxed{\hspace{2cm}} \text{ ft/hr} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e51674d-3d29-4cb4-94ad-529601d10286%2F62c11849-db73-4707-9377-49d374c2da74%2F89mp35m_processed.png&w=3840&q=75)
Transcribed Image Text:The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function
\[ D(t) = 5 \sin \left( \frac{\pi}{6} t + \frac{7\pi}{6} \right) + 4 \]
where \( t \) is the number of hours after midnight. Find the rate at which the depth is changing at 5 a.m. Round your answer to 4 decimal places.
\[ \boxed{\hspace{2cm}} \text{ ft/hr} \]
![## Problem Statement
Find the following using the table below.
### Table of Values
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( f(x) \) | 4 | 1 | 2 | 3 |
| \( f'(x) \) | 4 | 1 | 2 | 3 |
| \( g(x) \) | 4 | 1 | 3 | 2 |
| \( g'(x) \) | 4 | 1 | 2 | 3 |
### Questions
1. \( h'(3) \) if \( h(x) = f(x) \cdot g(x) \)
\[ \text{{Answer: }} \_\_\_\_\_\_\_ \]
2. \( h'(3) \) if \( h(x) = \frac{f(x)}{g(x)} \)
\[ \text{{Answer: }} \_\_\_\_\_\_\_ \]
3. \( h'(3) \) if \( h(x) = f(g(x)) \)
\[ \text{{Answer: }} \_\_\_\_\_\_\_ \]
### Explanation
For each case, use the appropriate rules of differentiation:
1. **Product Rule:** When \( h(x) = f(x) \cdot g(x) \), use \( h'(x) = f'(x)g(x) + f(x)g'(x) \).
2. **Quotient Rule:** When \( h(x) = \frac{f(x)}{g(x)} \), use \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).
3. **Chain Rule:** When \( h(x) = f(g(x)) \), use \( h'(x) = f'(g(x)) \cdot g'(x) \).
Fill in the answers using the values from the table at \( x = 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e51674d-3d29-4cb4-94ad-529601d10286%2F62c11849-db73-4707-9377-49d374c2da74%2F0zwj1y_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Find the following using the table below.
### Table of Values
| \( x \) | 1 | 2 | 3 | 4 |
|---------|---|---|---|---|
| \( f(x) \) | 4 | 1 | 2 | 3 |
| \( f'(x) \) | 4 | 1 | 2 | 3 |
| \( g(x) \) | 4 | 1 | 3 | 2 |
| \( g'(x) \) | 4 | 1 | 2 | 3 |
### Questions
1. \( h'(3) \) if \( h(x) = f(x) \cdot g(x) \)
\[ \text{{Answer: }} \_\_\_\_\_\_\_ \]
2. \( h'(3) \) if \( h(x) = \frac{f(x)}{g(x)} \)
\[ \text{{Answer: }} \_\_\_\_\_\_\_ \]
3. \( h'(3) \) if \( h(x) = f(g(x)) \)
\[ \text{{Answer: }} \_\_\_\_\_\_\_ \]
### Explanation
For each case, use the appropriate rules of differentiation:
1. **Product Rule:** When \( h(x) = f(x) \cdot g(x) \), use \( h'(x) = f'(x)g(x) + f(x)g'(x) \).
2. **Quotient Rule:** When \( h(x) = \frac{f(x)}{g(x)} \), use \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).
3. **Chain Rule:** When \( h(x) = f(g(x)) \), use \( h'(x) = f'(g(x)) \cdot g'(x) \).
Fill in the answers using the values from the table at \( x = 3 \).
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