0.5 1 1.5 2.5 b. 572 330 h(q) 190 110 63 Estimate h' (1.5) using the table. Give your estimate rounded to one decimal place. h'(1.5) Submit Question
0.5 1 1.5 2.5 b. 572 330 h(q) 190 110 63 Estimate h' (1.5) using the table. Give your estimate rounded to one decimal place. h'(1.5) Submit Question
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
![### Understanding Derivatives from a Table
Given a table of values for a function \( h(q) \), we want to estimate \( h'(1.5) \), the derivative of \( h \) at \( q = 1.5 \), by using numerical methods. The table provided is:
| \( q \) | 0.5 | 1 | 1.5 | 2 | 2.5 |
|----------|-----|-----|-----|-----|-----|
| \( h(q) \)| 572 | 330 | 190 | 110 | 63 |
### Instructions
1. **Select Points for Approximation**: To estimate \( h'(1.5) \), we select points around \( q = 1.5 \). A natural choice is to use \( q = 1 \) and \( q = 2 \).
2. **Apply the Slope Formula**: The derivative can be approximated by the slope between two points:
\[
h'(q) \approx \frac{h(q_2) - h(q_1)}{q_2 - q_1}
\]
Substitute \( q_1 = 1 \) and \( q_2 = 2 \):
\[
h'(1.5) \approx \frac{h(2) - h(1)}{2 - 1} = \frac{110 - 330}{2 - 1}
\]
\[
h'(1.5) \approx \frac{-220}{1} = -220
\]
3. **Round Your Answer**: The estimate should be rounded to one decimal place.
### Conclusion
By using this method, we derive a numerical approximation for the derivative at a given point using a simple linear slope between adjacent data points. The estimated value of \( h'(1.5) \) is approximately \(-220.0\).
#### Input Field
- Enter your answer: \(-220.0\)
#### Action Button
- Submit your estimate using the “Submit Question” button to verify the accuracy of your calculation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F868b8534-4d89-4d63-908b-b662dc9aeef4%2F93ad3ef8-5027-43ff-91b4-c47b2af59a5c%2Fd99yub4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Derivatives from a Table
Given a table of values for a function \( h(q) \), we want to estimate \( h'(1.5) \), the derivative of \( h \) at \( q = 1.5 \), by using numerical methods. The table provided is:
| \( q \) | 0.5 | 1 | 1.5 | 2 | 2.5 |
|----------|-----|-----|-----|-----|-----|
| \( h(q) \)| 572 | 330 | 190 | 110 | 63 |
### Instructions
1. **Select Points for Approximation**: To estimate \( h'(1.5) \), we select points around \( q = 1.5 \). A natural choice is to use \( q = 1 \) and \( q = 2 \).
2. **Apply the Slope Formula**: The derivative can be approximated by the slope between two points:
\[
h'(q) \approx \frac{h(q_2) - h(q_1)}{q_2 - q_1}
\]
Substitute \( q_1 = 1 \) and \( q_2 = 2 \):
\[
h'(1.5) \approx \frac{h(2) - h(1)}{2 - 1} = \frac{110 - 330}{2 - 1}
\]
\[
h'(1.5) \approx \frac{-220}{1} = -220
\]
3. **Round Your Answer**: The estimate should be rounded to one decimal place.
### Conclusion
By using this method, we derive a numerical approximation for the derivative at a given point using a simple linear slope between adjacent data points. The estimated value of \( h'(1.5) \) is approximately \(-220.0\).
#### Input Field
- Enter your answer: \(-220.0\)
#### Action Button
- Submit your estimate using the “Submit Question” button to verify the accuracy of your calculation.
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