the goal of this assignment is to investigate the rate of convergence

Question

The goal of this assignment is to investigate the rate of convergence of this rule depending on the
functions we want to integrate. To do this, first download from Canvas the MATLAB function
that implements the trapezoidal rule. It has the following inputs: lower and upper bounds of the
interval, number of points in the partition, and the function you want to integrate. To test the
function, compute the integral of the function f(x)=³ over [0, 1] with N = 100 points. What is
the value you get?
Consider the intervals I₁ = [0,] and 1₂ := [0, 2x]. Approximate the integrals over both I₁ and 1₂
of the functions fi(r) = sin(r), f2(2):= |sin(2x)], and fa(z) := cos(x) by using the Trapezoidal
rule. Then plot the absolute error of the computed integral and its corresponding true value. Note
that the true values can be easily computed by hand. Also, find the rate of convergence at which
each error is going to zero. Which one is the fastest? Note that one error plot is always close to
machine epsilon, what do you think is causing this extreme accuracy? This last question is difficult
- you should use an internet search to try and get some information on what is happening and
perhaps why.
Your conclusions should be explained in a one-page report. Your report must include the following:
(a) Output (value of the integral) of your test case f(x) = 2³ on [0, 1] with N = 100 points.
(b) Two loglog plots of the absolute errors, one for I and one for I₂.
(c) Rate of convergence of each error (6 in total). Why is one of the plots around machine epsilon
for almost any values of N?
(d) Make sure you answer all the questions in the document.