Question & Answer: Write a function/script file that solves this problem on MATLAB…..

Write a function/script file that solves this problem on MATLAB

We will work with the function f(x) = e^-2x sin (x). (a) Find the 3rd degree Taylor polynomial for f (x) centered at x_0 = 0. Show your work. (b) Use this to approximate f(0.3). (c) What is the actual value of f(0.3)? How much (actual) error is present in our polynomial approximation at this point? (d) Plot f(x) and the Taylor polynomial on the same axes from about -1 to 1.

Expert Answer

Matlab Code

clc
clear vars
syms x
fx = exp(-2*x)*sin(x);
taylor_poly = taylor(fx,x, 0, ‘Order’, 3); % Calculating taylor series
fprintf(‘(a) %sn’, char(taylor_poly)); % Printing the series
b = double(subs(taylor_poly, 0.3)); % Calculating the series for x = 0.3
fprintf(‘(b) f(0.3) = %fn’, b);
c = double(subs(fx, 0.3)); % Calculating the actual value of f(0.3)
fprintf(‘(c) Actual value of f(0.3) = %fn’, c);
Actual_error = double(abs(b – c)); % error calculating
fprintf(‘(c) Actual Error = %fn’, Actual_error);
interval = [-1 : 0.1: 1];
fx_points = subs(fx, interval);
taylor_points = subs(taylor_poly, interval);

% Plotting the functions
plot(fx_points, ‘k’, ‘LineWidth’, 1.5)
hold on
plot(taylor_points, ‘b’, ‘LineWidth’, 1.5)
title(‘Plot of Function vs its Taylor series’);
l = legend(‘f(x) = e^(-2x)sin(x)’, ‘Taylor Polynomial of f(x)’);
set(l,’Location’,’southeast’)

Command Window Output

Output Plot

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