Problem 1 With constant parameters $\tau > 0$ and $\omega_0$, define a function $S$ by
$$S(\omega) = \frac{(1/\tau)^2}{(\omega - \omega_0)^2 + (1/\tau)^2}$$
(1.1) Find a value of $\omega$ where $S(\omega) = 1/2$.
(1.2) Calculate the derivative, $S'$, of the function $S$: find a formula for $S'(\omega)$.
(1.3) Identify all the stationary points of $S$: solve the equation $S'(\omega) = 0$.
(1.4) Calculate the value, $S(\omega)$, at each stationary point of $S$.
(1.5) Calculate the second derivative, $S''$, of $S$: find a formula for $S''(\omega)$.
(1.6) Identify all the flexes of $S$: solve the equation $S''(\omega) = 0$.
(1.7) Calculate the value, $S(\omega)$, at each flex of $S$.
(1.8) Determine whether the flexes of $S$ are points of inflection.
(1.9) Identify the horizontal asymptotes of $S$: find $\lim_{\omega \to \infty} S(\omega)$ and $\lim_{\omega \to -\infty} S(\omega)$.
(1.10) Make a plot of $S$ that includes the stationary points and the flexes.
