7. Newton's Method: a. Use the Newton's method to manually approximate the solution to the equation: 1 = x' + x. Complete 3 iterations 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 1 = x + x, and the derivative of f (x). b. Write a program in Python that estimates the solutions of x + x = 1 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* - 6x2 + x + 2. 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. d. Write a paragraph explaining the changes you had to do 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) = x3 with xo = 1 and calculate X1, X2, X3, and X4. Find a formula for |X, |. Consider lim |x, | and explains what this limit means in words. Draw a picture that shows what is going on. b. Write a Python program to estimate it up to a prescribed precision using the Newton's method to solve the equation tan x = 0 with x, = 3. Include a screenshot of your program along with its output estimating it 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 e 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 its applications to your particular major based on this project