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

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

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›?

y(r, + h) = y(t,) + y'(f,)h + y"()h %3D