(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value Fy(0.4), namely, 7e0.4. (Round your answers to four decimal places.) h = 0.4 error = |(exact value) - (approximate value)| = h = 0.2 error = |(exact value) - (approximate value) | = h = 0.1 error = |(exact value) - (approximate value) | = What happens to the error each time the step size is halved? Each time the step size is halved, the error estimate appears to be --Select--- ✓(approximately).
(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value Fy(0.4), namely, 7e0.4. (Round your answers to four decimal places.) h = 0.4 error = |(exact value) - (approximate value)| = h = 0.2 error = |(exact value) - (approximate value) | = h = 0.1 error = |(exact value) - (approximate value) | = What happens to the error each time the step size is halved? Each time the step size is halved, the error estimate appears to be --Select--- ✓(approximately).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 44E
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I need help with C) please
![# Euler's Method Application in Solving Initial-Value Problems
Consider the initial-value problem \( y' = y \), \( y(0) = 7 \).
## Part (a)
**Objective:** Use Euler’s method with each of the following step sizes to estimate the value of \( y(0.4) \).
1. **Step Size (\(h\)) = 0.4**
\( y(0.4) = 9.8 \) ✓
2. **Step Size (\(h\)) = 0.2**
\( y(0.4) = 10.08 \) ✓
3. **Step Size (\(h\)) = 0.1**
\( y(0.4) = 10.2487 \) ✓
## Part (b)
We know that the exact solution of the initial-value problem in part (a) is \( y = 7e^x \). Draw, as accurately as you can, the graph of \( y = 7e^x \), \( 0 \leq x \leq 0.4 \), together with the Euler approximations using the step sizes in part (a). Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.
```plaintext
The estimates are [Select an option].
```
## Part (c)
The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of \( y(0.4) \), namely, \( 7e^{0.4} \).
(Round your answers to four decimal places.)
1. **Step Size (\(h\)) = 0.4**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
2. **Step Size (\(h\)) = 0.2**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
3. **Step Size (\(h\)) = 0.1**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
```plaintext
What happens to the error each time the step size is halved?
Each time the step size is halved, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad29c243-17bc-495f-9a75-f51d725c2e92%2Fb81ba7cf-48e2-496f-8b3c-70e871404b4a%2Fe50var_processed.png&w=3840&q=75)
Transcribed Image Text:# Euler's Method Application in Solving Initial-Value Problems
Consider the initial-value problem \( y' = y \), \( y(0) = 7 \).
## Part (a)
**Objective:** Use Euler’s method with each of the following step sizes to estimate the value of \( y(0.4) \).
1. **Step Size (\(h\)) = 0.4**
\( y(0.4) = 9.8 \) ✓
2. **Step Size (\(h\)) = 0.2**
\( y(0.4) = 10.08 \) ✓
3. **Step Size (\(h\)) = 0.1**
\( y(0.4) = 10.2487 \) ✓
## Part (b)
We know that the exact solution of the initial-value problem in part (a) is \( y = 7e^x \). Draw, as accurately as you can, the graph of \( y = 7e^x \), \( 0 \leq x \leq 0.4 \), together with the Euler approximations using the step sizes in part (a). Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.
```plaintext
The estimates are [Select an option].
```
## Part (c)
The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of \( y(0.4) \), namely, \( 7e^{0.4} \).
(Round your answers to four decimal places.)
1. **Step Size (\(h\)) = 0.4**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
2. **Step Size (\(h\)) = 0.2**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
3. **Step Size (\(h\)) = 0.1**
Error: \(| (\text{exact value}) - (\text{approximate value}) |\)
```plaintext
What happens to the error each time the step size is halved?
Each time the step size is halved, the
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