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 ) show the code for the true value of f’(xi ) as well? in matlab

To estimate the true value of f’(xi) using centered finite difference, we can use the following formula:

f'(xi) ≈ (f(xi + h) - f(xi - h)) / (2h)

where h is a small step size.

In this case, we need to estimate f’(𝜋/2). Let's assume h = 0.01 for this example.

The true value of f’(𝜋/2) can be calculated analytically for f(x) = cos(x) by taking the derivative of f(x):

f'(x) = -sin(x)

Thus, the true value of f’(𝜋/2) is:

f'(𝜋/2) = -sin(𝜋/2) = -1

Here's the code in MATLAB to calculate the true value of f’(𝜋/2):

main.m
xi = pi/2;
f_prime_true = -sin(xi);
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The variable f_prime_true will contain the true value of f’(𝜋/2), which is -1.