Final answer:
To approximate f(x)=e⁻³⁹ with a least squares polynomial, one must project the function onto a set of orthogonal polynomials and compute the coefficients using numerical integration, such as MATLAB's quad function, for both p₂(x) and p₃(x) over the interval [0,3].
Step-by-step explanation:
To find the best least squares polynomial approximation of degree at most 2 (p₂(x)) and degree at most 3 (p₃(x)) for the function f(x) = e⁻³⁹ over the interval [0,3], we use orthogonal polynomials. Since integrals in this case are complex, we use numerical methods such as MATLAB's quad function to compute the coefficients.
The process involves projecting the function onto a set of orthogonal polynomial basis functions. For the second-degree polynomial, p₂(x), we need three orthogonal polynomials (including the constant term), while for p₃(x), we need four. The orthogonal polynomials should be defined over the interval [0,3]. These polynomials are typically generated using the Gram-Schmidt process or they may be given, such as the Legendre polynomials scaled to the interval of interest.
Once the basis is established, the approximating polynomial is computed by projecting f(x) onto each basis polynomial and calculating the corresponding coefficients. These coefficients are found by integrating the product of f(x) and each basis polynomial over the interval [0,3]. This integration can be done using MATLAB's quad function.
The final polynomial p₂(x) or p₃(x), which is the sum of the weighted basis polynomials, represents the best least squares fit to the original function f(x).