Reproduce Figure 7.6 by performing the following

Chapter , Problem 7

(choose chapter or problem)

Reproduce Figure 7.6 by performing the following calculations: (a) Calculate the expressions M1 = f t + f y f (t, y) and M2 = f y for f (t, y) = 2t y2. (b) Given the initial-value problem dy dt = 2t y2, y(0) = 1, the interval 0 t 2, and n = 100 steps, calculate the point (t1, y1) that corresponds to the first step of Eulers method. Evaluate the quantity M1 at the point (t0 +t/2, (y0 + y1)/2), and use this value to estimate the error e1. How does your estimate compare with the true error e1? (c) Calculate the point (t2, y2) that corresponds to the second step of Eulers method. Evaluate M1 at the point (t1 + t/2, (y1 + y2)/2) and M2 at the point (t1, (y1 + y2)/2). Use these values to estimate the error e2. How does your estimate compare with the true error e2? (d) Repeat part (c) for the remaining 98 steps. (e) Make two plotsone that plots the true errors ek versus k and the other that plots the estimated error versus k.