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
![### Initial Value Problem Exploration
Consider the initial value problem:
\[ y' + \frac{2}{5}y = 1 - \frac{1}{2}t, \quad y(0) = y_0. \]
---
#### Question 1: Touching the t-axis
**What equation expresses the requirement that the solution touches the t-axis?**
- [ ] \( y(t) = 0 \)
- [ ] \( y'(t) = 0 \)
- [x] \( y''(t) = 0 \)
---
#### Question 2: Not Crossing the t-axis
**What additional equation expresses the requirement that the solution does not cross the t-axis?**
- [ ] \( y''(t) = 0 \)
- [ ] \( y'(0) = 0 \)
- [x] \( y'(t) = 0 \)
---
#### Combining Equations to Solve for r
**The equations found in (a) and (b) form a system of two equations for \( t \) and \( y_0 \). Show how the DE can be used to solve directly for \( t \).**
1. \( y' + \frac{2}{5}y = 0 \) but also \( 1 - \frac{1}{2}t = 0 \), so \( t = 5 \).
- [x] True
2. \( y'' + \frac{2}{5}y = 0 \) but also \( 1 - \frac{1}{5}t = 0 \), so \( t = 5 \).
- [ ] True
3. \( y' + \frac{2}{5}y = 0 \) but also \( 1 - \frac{1}{2}t = 0 \), so \( t = 2 \).
- [ ] True
4. \( y' + \frac{5}{2}y = 0 \) but also \( 1 - \frac{1}{3}t = 0 \), so \( t = 2 \).
- [ ] True
---
#### Final Solution
**Find the solution for the initial value problem with initial condition \( y(0) = y_0 \).**
\[ y = \frac{5}{4} + \left( y_0 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc262d38-8642-4292-8f86-e10c24c29ddb%2F0ab0baf4-4da9-4078-9da8-f10540ff255e%2F17a0tk8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Initial Value Problem Exploration
Consider the initial value problem:
\[ y' + \frac{2}{5}y = 1 - \frac{1}{2}t, \quad y(0) = y_0. \]
---
#### Question 1: Touching the t-axis
**What equation expresses the requirement that the solution touches the t-axis?**
- [ ] \( y(t) = 0 \)
- [ ] \( y'(t) = 0 \)
- [x] \( y''(t) = 0 \)
---
#### Question 2: Not Crossing the t-axis
**What additional equation expresses the requirement that the solution does not cross the t-axis?**
- [ ] \( y''(t) = 0 \)
- [ ] \( y'(0) = 0 \)
- [x] \( y'(t) = 0 \)
---
#### Combining Equations to Solve for r
**The equations found in (a) and (b) form a system of two equations for \( t \) and \( y_0 \). Show how the DE can be used to solve directly for \( t \).**
1. \( y' + \frac{2}{5}y = 0 \) but also \( 1 - \frac{1}{2}t = 0 \), so \( t = 5 \).
- [x] True
2. \( y'' + \frac{2}{5}y = 0 \) but also \( 1 - \frac{1}{5}t = 0 \), so \( t = 5 \).
- [ ] True
3. \( y' + \frac{2}{5}y = 0 \) but also \( 1 - \frac{1}{2}t = 0 \), so \( t = 2 \).
- [ ] True
4. \( y' + \frac{5}{2}y = 0 \) but also \( 1 - \frac{1}{3}t = 0 \), so \( t = 2 \).
- [ ] True
---
#### Final Solution
**Find the solution for the initial value problem with initial condition \( y(0) = y_0 \).**
\[ y = \frac{5}{4} + \left( y_0 -
![Finish solving the equations found in (a) and (b).
Select the correct value for \( y_0 \) from the following options:
1. \[ y_0 = \frac{45}{8} e^{\frac{25}{2}} + \frac{25}{8} \]
2. \[ y_0 = \frac{25}{8} e^{\frac{4}{3} \cdot 5} + \frac{45}{8} \]
3. \[ y_0 = \frac{25}{8} e^{\frac{4}{5}} - \frac{45}{8} \]
4. \[ y_0 = \frac{4}{25} e^{\frac{2}{3} \cdot 5} + \frac{45}{8} \]
5. \[ y_0 = \frac{45}{8} e^{\frac{4}{5}} + \frac{25}{8} \]
Please choose the correct option by evaluating the given equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc262d38-8642-4292-8f86-e10c24c29ddb%2F0ab0baf4-4da9-4078-9da8-f10540ff255e%2Fyn5kkhf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Finish solving the equations found in (a) and (b).
Select the correct value for \( y_0 \) from the following options:
1. \[ y_0 = \frac{45}{8} e^{\frac{25}{2}} + \frac{25}{8} \]
2. \[ y_0 = \frac{25}{8} e^{\frac{4}{3} \cdot 5} + \frac{45}{8} \]
3. \[ y_0 = \frac{25}{8} e^{\frac{4}{5}} - \frac{45}{8} \]
4. \[ y_0 = \frac{4}{25} e^{\frac{2}{3} \cdot 5} + \frac{45}{8} \]
5. \[ y_0 = \frac{45}{8} e^{\frac{4}{5}} + \frac{25}{8} \]
Please choose the correct option by evaluating the given equations.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.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