Problem $1$

Develop an efficient program that creates a plot of the function

$f(x)=\frac{1}{2}+\frac{1}{\sqrt[\phantom{2}]{}}{\displaystyle \underset{0}{\overset{x}{}}}{e}^{-t^2}dt$

for xin$[0,10].$ The integral should be approximated using

Q$1$: the Trapezoidal Rule,

Q$2$: the Simpson's $3/8$ Rule,

and use as few function evaluations of $e^{-t^2}$ as possible.

Problem $2$

Consider the function

$f(x)=sin(\frac{1}{x+})$

for $x$ ranging from $0$ to $1,$ and the derivative

$f^{\text{'}}(x)=\frac{-cos(\frac{1}{x+})}{(x+{)}^2}$

where $$ is a given input parameter.

Q$1$: Develop a program that creates a plot of the derivative of $f=f(x)$ based on a finite difference approximation using $n$ computational nodes. The program should also graph the exact derivative given by $f^{\text{'}}=f^{\text{'}}(x)$ above.

Q$2$: Test the program using $n=10$ and $=\frac{1}{5}.$

Q$3$: How large do you have to choose $n$ in order for the difference between these two functions to be less than $0\mathrm{.}1?$