Project In the Calculus we study the Newton$-$Raphson method for solving equations. This technique uses the value of a function and its derivative to estimate where the function crosses the $x$ axis. If necessary it repeats the process until the amount of change in the estimate is smaller than some predetermined amount.

The formula is

$x_{n+1}=x_n-\frac{f(x_n)}{f^{\text{'}}(x_n)}$

You will write a program that uses a recursive function to perform this operation estimating where a function crosses the x $-$axis.

Methodology

Prompt the user for the mathematical function that you want to solve.

Prompt the user for the starting value. Call this $0.$

Call the get$\_$zero function. You will pass the function and the starting value for the zero of the function. The function will return a new value for the zero.

In the get$\_$zero function you will set a stopping criteria. Call this eps. It should be a small value, say eps $=0\mathrm{.}001.$ This is done first, before the calculations or the selection.

Then calculate a value for $x$ by evaluating the function and an estimate of the derivative at $x.$ Use the derivative function that you wrote for project $6.$

Compare $|f(x)|$ to eps. This would be if$($abs$(f(x)$

If $|f(x)|>$ eps then call $x=$ get$\_$zero$($f$,x$ from the selection structure. You pass the same function and the same variable, but the current $x$ contains the new predicted value for the intercept.

If If $|f(x)|$ eps the function returns the calculated intercept value.

Call print$\_$results to show the function, the starting value, and the estimated zero. An example of what the output might look like is

$\backslash$table$[[$Function$,$Start,Zero$],[x.{?}^{{?}^{?}}2-7,3,2\mathrm{.}6459]]$

Call the plot function to see the results graphically

You may create any function to integrate that you would like. I suggest that you start with simple curves to allow you to check that your results are accurate. Just be sure to choose a function that does cross the $x-$axis. As an example, try $x^2-2.$ The correct result will be $\sqrt[\phantom{2}]{2}.$ Remember that every program that you submit must include an introduction consisting of your name, the date, and a description of the program as both comments at the beginning of

the program and as text that is printed when you call print$\_$header function. The program must implement top$-$down design in that you should be using functions through out your code.