91Ó°ÊÓ

Find the complex Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} -1, & -2

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -\pi

Determine whether the function is even, odd, or neither. $$ f(x)=\sin 3 x $$

In Problems 1 and 2, find the eigenfunctions and the equation that defines the eigenvalues for the given boundary-value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4} .\) Give the eigenfunctions corresponding to these approximations. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0, y(1)+y^{\prime}(1)=0\)

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=x, f_{2}(x)=x^{2} ; \quad[-2,2] $$

Determine whether the function is even, odd, or neither. $$ f(x)=x \cos x $$

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} -1, & -\pi

Show that the given functions are orthogonal on the indicated interval. $$ f_{1}(x)=x^{3}, f_{2}(x)=x^{2}+1 ; \quad[-1,1] $$

Find the eigenfunctions and the equation that defines the eigenvalues for the given boundary-value problem. Use a CAS to approximate the first four eigenvalues \(\lambda_{1}, \lambda_{2}, \lambda_{3}\), and \(\lambda_{4} .\) Give the eigenfunctions corresponding to these approximations. \(y^{\prime \prime}+\lambda y=0, y(0)+y^{\prime}(0)=0, y(1)=0\)

Determine whether the function is even, odd, or neither. $$ f(x)=x^{2}+x $$