Feel free to work on the assignment with other students in the class, however please write up your own work and code

Feel free to work on the assignment with other students in the class, however please write up your own work and code. All code must be turned in as a .IPYNB file AND a link to your Google Collab notebook. Code will not be graded without both of these. When you are asked to create a function to solve some problem, do not use some built-in function. Non-coding solutions may either be turned in as a PDF of handwritten work or a PDF of a INIBdocument. No file types other than PDF will be accepted. Please show all of your work. All problems or parts of problems are worth 10 points.

1. (Coding) Make a table of the error of the three-point centered difference formula (f/ (x)

   

   f (x — h))/2h) where f (a;) — cos(x) — sin(x) where h 10—1 10—12 at 0. Draw a plot of the results. Does the minimum error correspond with the theoretical expectation?
2. Show that

 + 0(h² )

3. Consider the integral e^m (IT.

By hand, approximate the integral with composite Trapezoid rule with m How does this compare with the true value? By hand, approximate the integral with composite Simpson's rule with m How does this compare with the true value?

1, 2, and 4 panels.      1, 2, and 4 panels.

3. 3. By hand, approximate the integral with composite Romberg integration up to R3,3. How does this compare with the true value.
   4. By hand, approximate the integral with Gaussian Quadrature with 2, 3, and 4 points. How does this compare with the true value?
4. (Coding) Create a Python function which takes as inputs a function f, numbers a, b, and an integer k and outputs composite Trapezoid rule with k points to approximate fa It can be helpful to check your code against 3a to make sure you get the same results.
5. (Coding) Create a Python function which takes as inputs a function f, numbers a, b, and an integer k and outputs composite Simpson's rule with k points to approximate fa It can be helpful to check your code against 3b to make sure you get the same results.
6. (Coding) Modify my code from lecture 5.4 to run with adaptive Simpson's rule rather than adaptive trapezoid rule.
7. Show that if (x) — 5:r³ — 3m, then for any polynomial of the form q(x)           a:r ² + bx + c that

1

—1

Extra Credit (Some Coding): Using techniques from this chapter computationally approximate to 10 decimal places using only addition, subtraction, multiplication, division, or powers. (Hint: Think geometrically).