Suppose an object moves along the y-axis so that its location is y = 2x² +8 at time (where y is measured in meters and is in seconds). (a) Find the average velocity of the object, for a changing from 1 to 5 seconds. (b) Find the instantaneous velocity of the object at 2 seconds.
Suppose an object moves along the y-axis so that its location is y = 2x² +8 at time (where y is measured in meters and is in seconds). (a) Find the average velocity of the object, for a changing from 1 to 5 seconds. (b) Find the instantaneous velocity of the object at 2 seconds.
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...
Related questions
Question
100%
![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c9b7cf8-bc31-43fb-8f2d-f6ce11207b4a%2Fe39b5124-bef7-4730-a72f-a79bda2582a9%2F39ahbsg_processed.jpeg&w=3840&q=75)
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning