When you write the Taylor series of a function about 0, also it is called Maclau

Question

When you write the Taylor series of a function about 0, also it is called Maclaurin Series. Here we have the Maclaurin series of sin(x). sin(x) = sigma^k = infinity_N = 0 (-1)^N/(2N + 1)! x^2N + 1 Please write a MATLAB function to calculate the Maclaurin Series of sin(x) with the number of the series terms (k) and the vector array x as the input arrays. By using the function in question 3, with the k = 1, please plot the exact and approximate value of sin(x) for [-pi lessthanorequalto x lessthanorequalto pi]. In addition, plot the error bar figure. Use the step value "pi/8".

Explanation / Answer

function smp = maclaurin_sin(x, n)
% x = a vector with values around 0
% n = number of terms in the series

% Start the series with 0
smp = 0;

% Consider all the possible derivatives
deriv = [0 1 0 -1]';

% Iterate n times (from 0 to n-1)
for i = 0 : n-1

    % Implement the Maclaurin expansion
    t(i+1, :) = deriv(1) * x.^(i) / factorial(i);

    % Find the derivative for the next term
    deriv = circshift(deriv, -1);

end

% Add-up the calculated terms
smp = sum(t);   

clear, clc, close all

% Work with two full periods
x = -2*pi : .1 : 2*pi;
y = sin(x);

% Plot the goal
plot(x, y, 'g', 'Linewidth', 3)
title('Study of Maclaurin series')
xlabel('x')
ylabel('sin(x) with different number of terms')
axis([-2*pi 2*pi -3 3])
grid on
hold on

% Consider 4 terms
smp = maclaurin_sin(x, 4);
plot(x, smp, 'ro')

% Consider 6 terms
smp = maclaurin_sin(x, 6);
plot(x, smp, 'b-.')

% Consider 10 terms
smp = maclaurin_sin(x, 10);
plot(x, smp, 'k', 'linewidth', 3)

% Label the calculated lines
legend('sin(x)', '4-term series', ...
       '6 terms', '10 terms')

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