(Solved): In this problem, we approximate f(x)=cos(x) using Taylor series. (a) How many terms in the Taylor s ...

In this problem, we approximate $f(x)=\cos (x)$ using Taylor series. (a) How many terms in the Taylor series approximation to $f(x)=\cos (x)$ do I need to make sure that my error is less than $2\times 10^{?8}$ for $x?[0,?/2]$ ? Use the Taylor approximation error of $E_n(x)=\frac{(x?c{)}^{n+1}}{(n+1)!}{f}^{(n+1)}(?)$ to derive your answer, or another another theorem from calculus. If you expand $\cos (x)$ around $c=0$, then the Taylor series will have some zero terms. You should count these terms towards the $n$ in your answer. (b) Write a function called my_cos which takes in two arguments: a number $x$ and a positive integer $n$. Your function should approximate $\cos (x)$ using a Taylor series approximation of order $n$. The function should return the approximate value. c) Now we experimentally verify the bound from part (a) in Matlab using your function from part (b). Define a vector $\mathbf{x}$ of 100 equally spaced points between $[0,?/2]$. For each of these points, compute the error, $?\cos (\mathbf{x})?m_{?}\_\cos (\mathbf{x},n)?$, using two different values of $n$ : - $n=3$ - Value of $n$ that you compute in part (a) Plot the errors (I recommend a logarithmic vertical axis) for both these values of $n$ against the vector $\mathbf{x}$. Is your error below the bound in part (a) at all these points for the two values of $n$ ?