what is the true value of f’(xi ) at xi =𝟏? use basic calculus. avoid the syms x function in matlab

To find the true value of f'(xi) at xi = 1 without using the symbolic toolbox in MATLAB, we can use numerical approximation. One of the simplest methods is the finite difference method.

The finite difference approximation for the derivative of a function can be represented as:

f'(x) ≈ [f(x + h) - f(x)] / h

where h is a small step size.

To determine the value of f'(xi) at xi = 1, we need to evaluate the above expression using the given function and a suitable step size.

Here's an example of how to achieve this in MATLAB:

main.m
% Define the function
f = @(x) x^2; 

% Define the step size
h = 0.01;

% Compute the derivative at xi = 1 using finite difference approximation
f_xi_prime = (f(1 + h) - f(1)) / h;

% Display the result
disp(f_xi_prime);
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In the above code, we define the function f(x) = x^2 as an anonymous function. Then, we choose a small value for the step size h (in this case, 0.01). Finally, we compute the derivative using the finite difference approximation and output the result.

Note that this method provides an approximation of the derivative rather than the true value due to the discretization of the derivative calculation. For a more accurate result, you can decrease the step size h, albeit at the cost of increased computational time.