Let $f(x)=log(1-x)$ and $a=0.$

$($a$)$ Write down the Taylor polynomial approximation of degree $1($i$.$e$.,p_1(x))$ for the function $f(x)$ and the point of approximation $a=0.$ Graph $($by hand$)$ both $f(x)$ and $p_1(x).$

$($b$)$ What can you say about the error for this linear approximation? How would you decrease this error?

$($c$)$ Write down the Taylor polynomial of degree $n($i$.$e$.,p_n(x))$ for the function $f(x)$ and the point of approximation $a=0.$

$($d$)$ What is the approximation error when approximating $f(x)$ by $p_n(x)?$

$($e$)$ How large should $n$ be chosen to have

$|f(x)-p_n(x)|{10}^{-4}$

for $-\frac{1}{2}x\frac{1}{2}?$

$($f$)$ Why is Taylor approximation useful? Give an example.