Question

Q.3 Consider the function f(x) = x^2– 2x + 4 on the interval [-2, 2] with h = 0.25. Write the MATLAB function file to find the first derivatives in the entire interval by all three methods i.e., forward, backward, and centered finite difference approximations.

Answer 1

Answer :-

%% Function f and its derivatives
f = @(x) x.^3-2*x+4;
f_1 = @(x) 3*x.^2 -2; % first derivative
f_2 = @(x) 6*x; % second derivative

h = 0.25; 
%% First derivative finite difference approximations
f_1b = @(x) (f(x)-f(x-h))/h; % backward 
f_1f = @(x) (f(x+h)-f(x))/h; % forward
f_1c = @(x) (f(x+h)-f(x-h))/(2*h); % centered

%% Second derivative finite difference approximations
f_2b = @(x) (f(x)-2*f(x-h)+f(x-2*h))/(h^2); % backward 
f_2f = @(x) (f(x+2*h)-2*f(x+h)+f(x))/(h^2); % forward
f_2c = @(x) (f(x+h)-2*f(x)+f(x-h))/(h^2); % centered

%% Plotting
x = -2:0.1:2;

subplot(1,3,1)
plot(x, f(x))
legend('f(x)')
title('Function f')

subplot(1,3,2)
plot(x, f_1(x), 'r-*', x, f_1b(x), 'b', x, f_1f(x), 'm', x, f_1c(x) , 'g')
legend('df/dx', 'backward finite difference approximation of df/dx', 'forward finite difference approximation of df/dx', 'centered finite difference approximation of df/dx'  )
title('First derivative of f and its approximations')

subplot(1,3,3)
plot(x, f_2(x), 'r-*', x, f_2b(x), 'b', x, f_2f(x), 'm', x, f_2c(x) , 'g')
legend('d^2f/dx^2', 'backward finite difference approximation of d^2f/dx^2', 'forward finite difference approximation of d^2f/dx^2', 'centered finite difference approximation of d^2f/dx^2'  )
title('Second derivative of f and its approximations')

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