. Consider the IVP y 0 = 4y, y(0) = 2 0 t 0.5 (a) Determine the Improved Eulers approximation for N = 50, N = 500 and N = 5000. Fill in the following table with the values of the approximations, errors and ratios of consecutive errors at t = 0.5. One of the values has already been entered based on the computations we did above. Recall that the exact solution to the ODE is y = 2e 4t . Include the table in your report, as well as the MATLAB commands used to find the entries

N	approximation	error	ratio
5	14.2016		N/A
50
500
5000

(b) Examine the last column. How does the ratio of the errors relate to the number of steps used? Your answer to this question should confirm the fact that Improved Eulers method is of order h 2 , that is, every time the stepsize is decreased by a factor k, the error is reduced (approximately) by a factor of k 2 .