\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} a) Suppose that $m$ and $n$ are
integers. Compute ${\int_0^{2\pi}
\bigl(\cos (mx)\bigr) \bigl(\cos (nx)\bigl) \, dx}$.

\noindent (Be careful:
there will be two different results, one when $m=n$ and one when $m\ne
n$.)

\medskip

\noindent b) Suppose $f(x)= A \cos(x) + B \cos(2x) + C \cos(3x)$, and
that you also know

\medskip

\centerline{$\int_0^{2\pi}  f(x) \cos(x) \, dx = 5\, ; \quad
\int_0^{2\pi}  f(x) \cos(2x) \, dx = 6\, ; \quad
\int_0^{2\pi}  f(x) \cos(3x) \, dx = 7\, .$}

\medskip

\noindent Find $A$ and $B$ and $C$.

\medskip

\noindent {\bf Note} The ideas of this computation are used often with
Fourier series, a standard method of analyzing periodic phenomena.

\vfil\eject\end