91Ó°ÊÓ

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 2+x, & -2

Verify by direct integration that the functions are orthogonal with respect to the indicated weight function on the given interval. $$ \begin{aligned} &L_{0}(x)=1, L_{1}(x)=-x+1, L_{2}(x)=\frac{1}{2} x^{2}-2 x+1 ; \quad w(x)=e^{-x}, \\ &{[0, \infty)} \end{aligned} $$

Expand the given function in an appropriate cosine or sine series. $$ f(x)=x,-\pi

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=e^{x}, \quad-\pi

Expand the given function in an appropriate cosine or sine series. $$ f(x)=x^{2},-1

Write out the first five nonzero terms in the Fourier-Legendre expansion of the given function. If instructed, use a CAS as an aid in evaluating the coefficients. Use a CAS to graph the partial sum \(S_{5}(x)\). $$ f(x)=\left\\{\begin{array}{lr} 0, & -1

Let \(\left\\{\phi_{n}(x)\right\\}\) be an orthogonal set of functions on \([a, b]\) such that \(\phi_{0}(x)=1\). Show that \(\int_{a}^{b} \phi_{n}(x) d x=0\) for \(n=1,2, \ldots\)

Expand the given function in an appropriate cosine or sine series. $$ f(x)=x|x|,-1

Let \(\left\\{\phi_{n}(x)\right\\}\) be an orthogonal set of functions on \([a, b]\) such that \(\phi_{0}(x)=1\) and \(\phi_{1}(x)=x\). Show that \(\int_{a}^{b}(\alpha x+\beta) \phi_{n}(x) d x=0\) for \(n=2,3, \ldots\) and any constants \(\alpha\) and \(\beta\).

Write out the first five nonzero terms in the Fourier-Legendre expansion of the given function. If instructed, use a CAS as an aid in evaluating the coefficients. Use a CAS to graph the partial sum \(S_{5}(x)\). $$ f(x)=e^{x},-1