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Consider the initial-value problem y'= y, y(0) - 7. (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4). (1) h=0.4 y(0.4) - (i) = 0.2 y(0.4) = (iii) = 0.1 y(0.4) =

Consider the initial-value problem y'= y, y(0) - 7. (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4). (1) h=0.4 y(0.4) - (i) = 0.2 y(0.4) = (iii) = 0.1 y(0.4) =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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Consider the initial-value problem y' = y, y(0) = 7. (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4). (i) h = 0.4 y(0.4) = (ii) h = 0.2 y(0.4) = (iii) h = 0.1 y(0.4) = (b) We know that the exact solution of the initial-value problem in part (a) is y = 7ex. Draw, as accurately as you can, the graph of y = 7ex, 0 ≤ x ≤ 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. The estimates are ---Select--- (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, 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).
Transcribed Image Text:Consider the initial-value problem y' = y, y(0) = 7. (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4). (i) h = 0.4 y(0.4) = (ii) h = 0.2 y(0.4) = (iii) h = 0.1 y(0.4) = (b) We know that the exact solution of the initial-value problem in part (a) is y = 7ex. Draw, as accurately as you can, the graph of y = 7ex, 0 ≤ x ≤ 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. The estimates are ---Select--- (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, 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).
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