Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Understanding Velocity: Average and Instantaneous**
Let's explore how to determine the velocity of an object moving along the y-axis.
Given:
- The position of the object along the y-axis is defined by the equation: \( y = 2x^2 + 8 \)
- Here, \( y \) represents the location in meters and \( x \) is the time in seconds.
### (a) Finding the Average Velocity
To find the average velocity of the object over a time interval, we use the formula:
\[ \text{Average Velocity} = \frac{\Delta y}{\Delta x} \]
where:
- \( \Delta y \) is the change in position
- \( \Delta x \) is the change in time
For \( x \) changing from 1 to 5 seconds:
1. Calculate \( y \) at \( x = 1 \):
\[ y(1) = 2(1)^2 + 8 = 10 \]
2. Calculate \( y \) at \( x = 5 \):
\[ y(5) = 2(5)^2 + 8 = 58 \]
3. Compute \( \Delta y \) and \( \Delta x \):
\[ \Delta y = y(5) - y(1) = 58 - 10 = 48 \]
\[ \Delta x = 5 - 1 = 4 \]
4. Find the average velocity:
\[ \text{Average Velocity} = \frac{48}{4} = 12 \text{ meters/second} \]
### (b) Finding the Instantaneous Velocity
Instantaneous velocity is determined by finding the derivative of the position function with respect to time \( x \).
Given the position function:
\[ y = 2x^2 + 8 \]
The derivative with respect to \( x \) is:
\[ \frac{dy}{dx} = 4x \]
To find the instantaneous velocity at \( x = 2 \) seconds:
\[ \left. \frac{dy}{dx} \right|_{x=2} = 4(2) = 8 \text{ meters/second} \]
Thus, the instantaneous velocity of the object at 2 seconds is 8 meters/second.](https://content.bartleby.com/qna-images/question/3c9b7cf8-bc31-43fb-8f2d-f6ce11207b4a/e39b5124-bef7-4730-a72f-a79bda2582a9/39ahbsg_processed.jpeg)
Transcribed Image Text:**Understanding Velocity: Average and Instantaneous**
Let's explore how to determine the velocity of an object moving along the y-axis.
Given:
- The position of the object along the y-axis is defined by the equation: \( y = 2x^2 + 8 \)
- Here, \( y \) represents the location in meters and \( x \) is the time in seconds.
### (a) Finding the Average Velocity
To find the average velocity of the object over a time interval, we use the formula:
\[ \text{Average Velocity} = \frac{\Delta y}{\Delta x} \]
where:
- \( \Delta y \) is the change in position
- \( \Delta x \) is the change in time
For \( x \) changing from 1 to 5 seconds:
1. Calculate \( y \) at \( x = 1 \):
\[ y(1) = 2(1)^2 + 8 = 10 \]
2. Calculate \( y \) at \( x = 5 \):
\[ y(5) = 2(5)^2 + 8 = 58 \]
3. Compute \( \Delta y \) and \( \Delta x \):
\[ \Delta y = y(5) - y(1) = 58 - 10 = 48 \]
\[ \Delta x = 5 - 1 = 4 \]
4. Find the average velocity:
\[ \text{Average Velocity} = \frac{48}{4} = 12 \text{ meters/second} \]
### (b) Finding the Instantaneous Velocity
Instantaneous velocity is determined by finding the derivative of the position function with respect to time \( x \).
Given the position function:
\[ y = 2x^2 + 8 \]
The derivative with respect to \( x \) is:
\[ \frac{dy}{dx} = 4x \]
To find the instantaneous velocity at \( x = 2 \) seconds:
\[ \left. \frac{dy}{dx} \right|_{x=2} = 4(2) = 8 \text{ meters/second} \]
Thus, the instantaneous velocity of the object at 2 seconds is 8 meters/second.
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