What Good is Calculus: Well You Can Use it Tune Your Radio
Abstract: The signals that local radio stations broadcast can be expressed
as functions f(t), where t represents time and f is roughly a voltage. Many
radio stations are broadcasting simultaneously and it is the job of your
radio to pick out the signal of the specific station to which you tune your
radio by adjusting the frequency dial. How does the radio do that? Well,
the answer is by means of Fourier Series, which can be understood as a
simple but elegant application of integral calculus applied to
trigonometric functions. The process is intimately tied up with the
concept of a Fourier Series. The latter were discovered by the French
mathematician Joseph Fourier in the nineteenth century. By his era,
scientists understood very well how to express the functions of calculus
in terms of power or Taylor series. Fourier demonstrated that those
functions could also be expressed as infinite series involving sinusoidal
functions instead of monomials. People doubted the validity of his work
for a long time – in part, because the convergence of a Fourier Series is
not as “strong” as is that of a power series. But time and science
vindicated Fourier and today, Fourier Series plays an integral role in
modern mathematics. In this lecture, we will see how Fourier Series arise
in your efforts to tune your radio.