Question $1.$ Taylor Polynomials

Engineers and physicists frequently use the approximation sin$($x$)$ x for x small, which is a first$-$order Taylor

polynomial approximation for sin$($x$)$ centered at x $=0.$ In this exercise we will use Taylor$$s theorem to create a

simple MATLAB function capable of telling us what degree of Taylor polynomial approximation is needed to obtain

a desired level of accuracy on a specified interval.

$($a$)$ Let Pn$($x$)$ be the degree n Taylor Polynomial for f$($x$)=$ sin$($x$)$ about x $=0.$ Find n so that Pn$($x$)$ is within

$104$ of sin$($x$)$ for all x in $[1,1].$

$($b$)$ Create a MATLAB function that takes as input a positive parameter k$,$ an interval radius $\backslash$delta $($call $\backslash$delta something

like delta in MATLAB$),$ and a desired error threshold err, and returns a value n for which you can prove that

the n

th$-$order Taylor Polynomial for f$($x$)=$ sin$($kx$)$ about x $=0$ is within err of sin$($kx$)$ for all x in $[\backslash$delta $,\backslash$delta $].$ Call

your function with suitable input values to reproduce your answer to question $1($a$).$

HINT: Your code does not need to be complicated! By mimicking your theoretical analysis from $1($a$),$ you

should only $$need$$ MATLAB at the very end of this problem, where you can write some code to find the first

n satisfying a certain inequality, rather than looking for it by hand $($as we did in class$).$