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

${10}^{-4}$ of $sin(x)$ for all xin$[-1,1].$

$($b$)$ Create a MATLAB function that takes as input a positive parameter $k,$ an interval radius $($call $$ 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 xin$[-,].$ 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$).$