MATLAB Program to Approximate Derivatives Using the Finite Difference Formula

 

Question Restated and Explained

You are asked to create a MATLAB program that computes an approximate value for the derivative of any function at a given point, using the finite difference formula:

$$f'(x) \approx \frac{f(x + h) - f(x)}{h}$$

You should then test your program with the function $f(x)=\tan (x)\; at$$x = 1$.

Step-by-Step Solution

Explanation:

• The finite difference formula gives a numerical estimate for the derivative at a point.

• The choice of $h$ (a small value, e.g., 0.0001) affects the accuracy.

• You can write a MATLAB function that takes as input any function handle, the value of $x$, and a step size $h$, and returns the approximate derivative.

• To test: use $f(x) = \tan(x)$, $x = 1$.

MATLAB Code

matlab

% Function to compute approximate derivative using finite difference
function approx_deriv = finite_difference(f, x, h)
    approx_deriv = (f(x + h) - f(x)) / h;
end

% Testing the program with tan(x) at x = 1
f = @tan;       % Function handle for tan(x)
x = 1;          % Point at which to compute the derivative
h = 1e-5;       % Small step size

approximate_derivative = finite_difference(f, x, h);

% Exact derivative for comparison: derivative of tan(x) is sec^2(x)
exact_derivative = sec(x)^2;

fprintf('Approximate derivative of tan(x) at x = 1: %f\n', approximate_derivative);
fprintf('Exact derivative (sec^2(1)) at x = 1: %f\n', exact_derivative);
fprintf('Error: %e\n', abs(approximate_derivative - exact_derivative));

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• This code defines a function finite_difference to calculate the approximate derivative.

• It then uses this function to compute the derivative of tan(x) at x = 1.

• It also compares the approximate result to the exact value, displaying both and the error.

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