a$.$ Use the Newton's method to manually approximate the solution to the equation: $x=3x^3+1.$ Complete $3$ lterations by hand and include all your organized work in your report. You should Include a clear definition of the function $f(x)$ you used to find the solutions to $x=3x^3+1,$ and the derlvative of $f(x).$

b$.$ Write a program in Python that estimates the solutions of $3x^3+1=x$ using $10$ Iterations. In your report, you should include a screenshot of your program along with Its output and a brief description of how your program works.

c$.$ Use a copy of your program in $($b$),$ adapting it as appropriate to find approximate all the zeros of $g(x)=x^4-6x^2-x+1.$ Include a screenshot of the adapted program along with its output for each zero of $g(x).$ Note that $g(x)$ has $4$ real roots, you may use Desmos to visualize the graph and obtain good guess points for each zero.

d$.$ Write a paragraph explaining the changes you had to make to your program in part $($b$)$ for each zero in $($c$)$ and why these changes were necessary.

$8.$ Further Theory:

a$.$ Apply the Newton's method to $f(x)=x^{\frac{1}{5}}$ with $x_0=1$ and calculate $x_1,x_2,x_3,$ and $x_4.$ Find a formula for $|x_n|.$ Consider $\underset{n}{lim}|x_n|$ and explains what this limit means in words. Draw a picture of the graph of $y=x^{\frac{1}{4}}$ along with the first $3$ tangent lines assoclated with the Newton's method, make sure to include the $x-$intercept of each of these lines in the drawing:

b$.$ Write a Python program to estimate $$ up to a prescribed precision using the Newton's method to solve the equation $tanx=0$ with $x_0=3.$ Include a screenshot of your program along with its output estimating $$ up to $10$ decimal places.

c$.$ Explain how you would use the Newton's method to approximate the value of $e$ then write a Python program that uses the Newton's method to estimate the value of up to $10$ decimal places. Include a screenshot of your program along with its output.

$9.$ Conclusion:

a$.$ Write a paragraph of what you have learned about the Newton's method and some of