18. Consider the initial value problem = y -t, y(0) = 1 (See also exercise 5). dt a) Using the modified Euler method fill in the estimate obtained for y(3) using the number of subintervals specified in the table below. The y value should be written to the fourth decimal place. # of subintervals 100 200 400 800 y(3) b) Based on the table obtained in a), what is the smallest number of subintervals you should use to produce an approximation that has three decimal place accuracy throughout the interval [0,3]? c) Using the number of subintervals determined in b) (and the modified Euler method) use the graph of the solution y(t) to estimate a value t, in [0,3] such that y(t,) = 0. (Such a value is called a root of the solution.) Type the estimate in the comments box provided in ode.xls.
18. Consider the initial value problem = y -t, y(0) = 1 (See also exercise 5). dt a) Using the modified Euler method fill in the estimate obtained for y(3) using the number of subintervals specified in the table below. The y value should be written to the fourth decimal place. # of subintervals 100 200 400 800 y(3) b) Based on the table obtained in a), what is the smallest number of subintervals you should use to produce an approximation that has three decimal place accuracy throughout the interval [0,3]? c) Using the number of subintervals determined in b) (and the modified Euler method) use the graph of the solution y(t) to estimate a value t, in [0,3] such that y(t,) = 0. (Such a value is called a root of the solution.) Type the estimate in the comments box provided in ode.xls.
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 18: Initial Value Problem**
Consider the initial value problem given by the differential equation:
\[
\frac{dy}{dt} = y - t^2, \quad y(0) = 1
\]
(Refer to exercise 5 for additional context.)
**Tasks:**
**a)** Using the modified Euler method, calculate the estimate for \(y(3)\) using the specified number of subintervals. Record your findings in the table below with the \(y\) value rounded to four decimal places.
| # of subintervals | 100 | 200 | 400 | 800 |
|------------------|-----|-----|-----|-----|
| \(y(3)\) | | | | |
**b)** Referring to your calculations in part (a), determine the smallest number of subintervals required to achieve an approximation of \(y(3)\) with three decimal place accuracy over the interval \([0,3]\).
**c)** Utilizing the number of subintervals determined in part (b) and the modified Euler method, plot the graph of the solution \(y(t)\). Estimate a value \(t_0\) within \([0,3]\) such that \(y(t_0) = 0\). This value, where the solution equals zero, is a root of the solution. Document your estimate in the provided comments box in the 'ode.xls' file.
**d)** From the output table, identify the smallest possible interval that encloses the root \(t_0\). If required to estimate the root, determine the most precise estimate possible based on this data. Additionally, calculate the largest potential discrepancy between your estimate and the exact root. Clearly state your estimate and provide an explanation in the comments box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F700d529c-a1d7-4c33-8497-c808e2f00779%2F02a92ff3-b383-4423-9304-51b2edc61d94%2F1s52yue_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 18: Initial Value Problem**
Consider the initial value problem given by the differential equation:
\[
\frac{dy}{dt} = y - t^2, \quad y(0) = 1
\]
(Refer to exercise 5 for additional context.)
**Tasks:**
**a)** Using the modified Euler method, calculate the estimate for \(y(3)\) using the specified number of subintervals. Record your findings in the table below with the \(y\) value rounded to four decimal places.
| # of subintervals | 100 | 200 | 400 | 800 |
|------------------|-----|-----|-----|-----|
| \(y(3)\) | | | | |
**b)** Referring to your calculations in part (a), determine the smallest number of subintervals required to achieve an approximation of \(y(3)\) with three decimal place accuracy over the interval \([0,3]\).
**c)** Utilizing the number of subintervals determined in part (b) and the modified Euler method, plot the graph of the solution \(y(t)\). Estimate a value \(t_0\) within \([0,3]\) such that \(y(t_0) = 0\). This value, where the solution equals zero, is a root of the solution. Document your estimate in the provided comments box in the 'ode.xls' file.
**d)** From the output table, identify the smallest possible interval that encloses the root \(t_0\). If required to estimate the root, determine the most precise estimate possible based on this data. Additionally, calculate the largest potential discrepancy between your estimate and the exact root. Clearly state your estimate and provide an explanation in the comments box.
![e) The solution \( y(t) \) also has a maximum in the interval \([0, 3]\). Using the graph, determine an estimate for this maximum and type your estimate in the comments box.
f) Use the output table to locate the maximum as accurately as you can. State a single estimate for this maximum. Explain how you obtained this estimate.
g) Print the first sheet of the output file containing the solution graph and your answers to parts b) - f).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F700d529c-a1d7-4c33-8497-c808e2f00779%2F02a92ff3-b383-4423-9304-51b2edc61d94%2Fn2fz0e4_processed.png&w=3840&q=75)
Transcribed Image Text:e) The solution \( y(t) \) also has a maximum in the interval \([0, 3]\). Using the graph, determine an estimate for this maximum and type your estimate in the comments box.
f) Use the output table to locate the maximum as accurately as you can. State a single estimate for this maximum. Explain how you obtained this estimate.
g) Print the first sheet of the output file containing the solution graph and your answers to parts b) - f).
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