EXERCISES 1. (a) If you haven't already done so, enter the following commands: f=@(t,y) 0.5*y; = t=linspace (0,2,100); y=3* exp(0.5*t); %define exact solution of the ODE [t80, y80]=euler (f, [0, 2],3,80); %solve the ODE using Euler w/ 80 steps Determine the Euler's approximation for N = 800 and N 8000 and fill in the following ta- ble with the values of the approximations (yN (end)), errors (eN=y (end)-yN (end)) and ratios (eNprevious/eN) of consecutive errors at t = 2. Some of the values have already been entered based on the computations we did above. Include the table in your report, as well as the MATLAB commands used to find the entries. N 8 80 800 8000 approximation yN (end) 7.6974 error eN 0.4575 0.0504 ratio eNprev/eN N/A 9.0789 (b) Examine the last column. How does the ratio of consecutive errors relate to the number of steps used? Your answer to this question should confirm the fact that Euler's method is a "first-order" method. That is, every time the step size is decreased by a factor of k, the error is also reduced (approximately) by the same factor, k = k. (c) Recall the geometrical interpretation of Euler's method based on the tangent line. Using this geometrical interpretation, can you explain why the Euler approximations yn underestimate the solution y (that is, yn y, or equivalently, en y - yn > 0) in this particular example?