$($Please explain both A and B$)$ I think A is not coding and B is$.$ 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$).$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$).$

$(($dddddestion $1.$ 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$).$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$).$

$(($ddddd