In many of the following problems, it will be essential to have a calculator or comp...

In many of the following problems, it will be essential to have a calculator or computer available. You may use a software package† or write a program for solving initial value problems using the improved Euler’s method algorithms on pages 127 and 128. (Remember, all trigonometric calculations are done in radians.)

In Example 1 the improved Euler’s method approximation to e with step size h was shown to be

First prove that the error approaches zero as h 0. Then use L’Hôpital’s rule to show that

Compare this constant with the entries in the last column of Table 3.5.