10) The coordinates s of a moving body for various values of t are given in the table below. t (sec) 0.5 1 1.5 2.0 s (feet) 12 26 36 44 48 Estimate the value of s'(1). State the unit of measure.
10) The coordinates s of a moving body for various values of t are given in the table below. t (sec) 0.5 1 1.5 2.0 s (feet) 12 26 36 44 48 Estimate the value of s'(1). State the unit of measure.
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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![**Problem Statement:**
The coordinates \( s \) of a moving body for various values of \( t \) are given in the table below.
| \( t \) (sec) | 0 | 0.5 | 1 | 1.5 | 2.0 |
|---------------|----|-----|----|-----|-----|
| \( s \) (feet) | 12 | 26 | 36 | 44 | 48 |
Estimate the value of \( s'(1) \). State the unit of measure.
---
**Solution Explanation:**
To estimate the value of \( s'(1) \), we need to approximate the derivative of \( s \) at \( t = 1 \). The derivative \( s'(t) \) represents the velocity of the moving body at time \( t \), and it is measured in feet per second.
Since we are given discrete points, we can use the difference quotient to approximate the derivative at \( t = 1 \).
The difference quotient for \( t = 1 \) can be approximated using the data points surrounding \( t = 1 \). Therefore, we look at the interval \([0.5, 1]\) and \([1, 1.5]\).
1. Calculate the average rate of change between \( t = 0.5 \) and \( t = 1 \):
\[
\frac{s(1) - s(0.5)}{1 - 0.5} = \frac{36 - 26}{0.5} = \frac{10}{0.5} = 20 \text{ feet/second}
\]
2. Calculate the average rate of change between \( t = 1 \) and \( t = 1.5 \):
\[
\frac{s(1.5) - s(1)}{1.5 - 1} = \frac{44 - 36}{0.5} = \frac{8}{0.5} = 16 \text{ feet/second}
\]
3. The best estimate for \( s'(1) \) will be the average of these two rates:
\[
s'(1) \approx \frac{20 + 16}{2} = 18 \text{ feet/second}
\]
Therefore, the estimated](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa55013e1-91aa-48be-9c4b-bf8a0e47de22%2F52e0cc46-121f-46a7-bd86-77f2494baf78%2Fy7apmdk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The coordinates \( s \) of a moving body for various values of \( t \) are given in the table below.
| \( t \) (sec) | 0 | 0.5 | 1 | 1.5 | 2.0 |
|---------------|----|-----|----|-----|-----|
| \( s \) (feet) | 12 | 26 | 36 | 44 | 48 |
Estimate the value of \( s'(1) \). State the unit of measure.
---
**Solution Explanation:**
To estimate the value of \( s'(1) \), we need to approximate the derivative of \( s \) at \( t = 1 \). The derivative \( s'(t) \) represents the velocity of the moving body at time \( t \), and it is measured in feet per second.
Since we are given discrete points, we can use the difference quotient to approximate the derivative at \( t = 1 \).
The difference quotient for \( t = 1 \) can be approximated using the data points surrounding \( t = 1 \). Therefore, we look at the interval \([0.5, 1]\) and \([1, 1.5]\).
1. Calculate the average rate of change between \( t = 0.5 \) and \( t = 1 \):
\[
\frac{s(1) - s(0.5)}{1 - 0.5} = \frac{36 - 26}{0.5} = \frac{10}{0.5} = 20 \text{ feet/second}
\]
2. Calculate the average rate of change between \( t = 1 \) and \( t = 1.5 \):
\[
\frac{s(1.5) - s(1)}{1.5 - 1} = \frac{44 - 36}{0.5} = \frac{8}{0.5} = 16 \text{ feet/second}
\]
3. The best estimate for \( s'(1) \) will be the average of these two rates:
\[
s'(1) \approx \frac{20 + 16}{2} = 18 \text{ feet/second}
\]
Therefore, the estimated
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