suppose f(x) = cos(x), and we want to estimate f’(x) at xi=𝝅/2, (so f(xi )=0) using centered finite difference.what is the true value of f’(xi )? in matlab

To estimate the value of f'(xi) using centered finite difference, we need to compute the difference between f(xi+h) and f(xi-h), and divide it by 2h, where h is a small value.

In this case, xi = pi/2, so we can choose a small value for h, such as h = 0.01. We can then calculate the value of f'(xi) using the following formula:

f'(xi) = (f(xi + h) - f(xi - h)) / (2 * h)

Let's write the code in MATLAB:

main.m
% Define the function
f = @(x) cos(x);

% Define the value of xi
xi = pi/2;

% Choose a small value for h
h = 0.01;

% Calculate the value of f'(xi) using centered finite difference
f_prime = (f(xi + h) - f(xi - h)) / (2 * h);

% Display the result
disp(f_prime);
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The true value of f'(xi) can be calculated analytically by taking the derivative of f(x) = cos(x). In this case, the derivative is f'(x) = -sin(x). Thus, the true value of f'(xi) at xi = pi/2 is -sin(pi/2) = -1.

So, in MATLAB, the true value of f'(xi) is -1.