Please help with these Python problems. I really need help with this problem.

Exercise 2

a) Show that Newton's method applied to

()=^k-a

leads to fixed point iteration with function

()=((1)+(a/x^(k-1)))/k.

b) Then verify mathematically that the iteration +1=()has super-linear convergence.

Exercise 3

a) Create a Python function for Newton's method, with usage

(root, errorEstimate, iterations, functionEvaluations) = newton(f, Df, x_0, errorTolerance, maxIterations)

(The last input parameter maxIterations could be optional, with a default like maxIterations=100.)

b) based on your function bisection2 create a third (and final!) version with usage

(root, errorBound, iterations, functionEvaluations) = bisection(f, a, b, errorTolerance, maxIterations)

c) Use both of these to solve the equation

1()=102+sin()=0

i) with [estimated] absolute error of no more than 10^-6, and then

ii) with [estimated] absolute error of no more than 10^-15.

Note in particular how many iterations and how many function evaluations are needed.

Graph the function, which will help to find a good starting interval [,] and initial approximation 0.

d) Repeat, this time finding the unique real root of

2()=^33.3^2+3.631.331=0

Again graph the function, to find a good starting interval [,] and initial approximation 0.

e) This second case will behave differently than for 1 in part (c): describe the difference. (We will discuss the reasons in class.)