Please make code clear and easy to understand

PART $1$

One way to evaluate $\sqrt[6]{a}$ is to use the Newton Raphson method to find the zeros of the function

$f(x)=x^3-a,$ starting at $x_0=a.$ Use this approach to find the sixth root of $25.$

NOTE: To obtain full points for this question, you must include all code with your answer. You should include in

your code i$)$ tests that stop the iterative procedure when convergence is reached and ii$)$ a maximum number of

iterations in your code to avoid your code getting into an infinite loop. iii$)$ a check for derivatives close to zero to

avoid division by zero, so that your code exits elegantly if that happens. iv$)$ an alternative update method that is

used in case the derivative becomes close to zero.

A$.[5$ points$]$ Write Python code to show how you can implement the Newton Raphson method to find the sixth

roots of $f,$ starting at $x_0=25.$ Plot your estimate of $\sqrt[6]{25}$ as a function of iteration number. How many

iterations were needed for the sequence to converge within your specified tolerance level?

B$.[4$ points$]$ Does the process work if you start at $x_0=-25?$ Modify your code to start at this initial guess

and plot your findings as a function of iteration number. How many iterations did this take?

C$.[1$ point$]$ What happens if you specify $x_0=0?$ Why does this happen?

PART $2$

You are asked to write onboard code for a new interactive pacemaker system that monitors ECG to determine

the right time to defibrillate the heart. The monitoring part of the system requires you to analyze voltage data

gathered from an electrode placed in the right atrium and figure out the times between zero crossings in the

data to estimate the heart rate. To make sure the pacemaker pulses are applied exactly when needed it is

important to do this as efficiently as possible. The processing power of the microprocessor is very small, and

while it can perform multiplication, division, subtraction and addition, it cannot calculate derivatives or perform

Fourier transforms. The signal is $($very approximately$)$ sinusoidal.

A$.[2$ points$]$ of the four methods presented in the module $($bisection$,$ false position, secant and Newton

Raphson$)$ which would you use to try to find zero crossings, given the processor and time limitations$?$ In

your answer, briefly describe why you chose the method.

B$.[5$ points$]$ You are not able to make a plot of the data recorded by the board. Given that you know the signal

is approximately sinusoidal, and that the frequency of the signal is around $1$ Hz $,$ suggest a method you could

use for bracketing zeros $($Bisection$,$ False Position methods$)$ or choosing starting point$($s$)$ for Newton

Raphson or Secant methods.

C$.[3$ points$]$ Write pseudocode outlining the process you chose in part B$.$