We will investigate the local discretization error in applying the

We will investigate the local discretization error in applying the Euler approximation given by equations (5) and (6).
(a) If y(t) is the exact solution of y' = f(t. y) use the chain rule to calculate y "(t) and explain why it is continuous.
(b) Recall the following from calculus: Remember that y(tn + l ) = y(t" + h), and deduce that
y(r, + h) = y(t,) + y'(f,)h + y

for some t; in the interval (tn, tn+ 1 ) .
(c) Subtract equation (6) from equation (8) t o conclude that the local discretization error en+1 is given by

We will investigate the local discretization error in applying the

Where we assume that the nth approximation is exact y(tn) = yn. Hence, if |y'' (t)| ‰¤ M on [tn, tn+1] then en+1 ‰¤ M h2/2
(d) How small must h be to guarantee that this local discretization error is no greater than some prescribed É›?