Question

Problem 3 In this problem you add some random noise to the sine function and then calculate the derivative. • First, generate a vector (the same length as x, above) which consists of random numbers in the range (-0.02) to (0.02). Use the MATLAB function rand. rand will return a random number between 0 and 1. For this problem, you actually need random numbers between +/- 0.02. How will you convert the random numbers you obtain from rand to be within the correct range? To learn more about rand, type help rand in the Command Window. • In one figure window, plot sin(x) and sin(x) with added random noise on the same graph. • In the second figure window, plot the known (analytical) derivative, and the calculated derivative, using both the forward and central difference formulas, on the same graph. • In the third figure window, plot the error (difference between the calculated derivative and analytical derivative). Hand in: Your graphs for different methods and step sizes. Comment on the effects of method, step size and how the noise in your original (sine) function compares to the noise in the numerical derivative.

Answer 1

As per the HomeworkLib policy on multipart questions, answer to first part is being given here:

As the x vector is not given in the context, we are assuming a sample x values here as:

%===================================================

clc;
clear all;

x = [0:pi/4:2*pi]; % Sample x values
f1 = sin(x);
noise = -0.2 + 0.4*rand(1,length(x));
f2 = sin(x)+noise;

figure
plot(x,f1,'-+');
hold on
plot(x,f2,'-o');
legend('Noiseless sin(x)','Noisy sin(x)');
xlabel('x');
ylabel('y');

%===================================================

Output:

Note: The output plot will be different for each run as the program will likely generate different random number on each run.