Exercise (a)
Use Euler's method with each of the following step sizes toestimate the value of y(1.6), where y is thesolution of the initial-value problem y' = y,y(0) = 6.
(i)Â Â Â Â h = 1.6
(ii)Â Â Â Â h = 0.8
(iii)Â Â Â Â h = 0.4
Exercise (b)
We know that the exact solution of the initial-value problem inpart (a) is y = 6ex. Draw, asaccurately as you can, the graph of y =6ex, 0 ≤ x ≤ 1.6, togetherwith the Euler approximations using the step sizes in part (a).(Your sketches should resemble the figures for the first Eulerapproximation, Euler approximation with step size 0.5, and Eulerapproximation with step size 0.25.) Use your sketches to decidewhether your estimates in part (a) are underestimates oroverestimates.
Exercise (c)
The error in Euler's method is the difference between the exactvalue and the approximate value. Find the errors made in part (a)in using Euler's method to estimate the true value ofy(1.6), namely, 6e1.6. What happens tothe error each time the step size is halved?
