Post$-$Lab $($Homework$)$ Problems:

Problem $3$: Understanding of Errors $($using an example of the sine series$)$

Consider the Pre$-$Lab Exercise $2,$ the problem on the Taylor's series of the sine function. Modify $($slightly$)$ the MATLAB function sine$\_$taylorN so that the while$-$loop checks both the error tolerance and the maximum number of iterations. In other words, the loop terminates when either the error is sufficiently small, or the maximum number of iterations is exceeded.

a$)$ Complete the above by initializing the counter $($number of iterations$)$ as $k=0.$ This is the preferred approach as was discussed in class.

b$)$ Complete the above by initializing the counter $($number of iterations$)$ as $k=1.$

c$)$ Test your codes using the inputs $x=3,$ maxtol$=1$e$-4,$ and maxitr$=20.$ Confirm that both codes give the same results.