§3.4 Using derivatives to describe families of functions (continued)

Summary

For $p(x) = x^3 - ax^2$ with $a \gt 0$, the derivative factors as $p'(x) = x(3x - 2a)$ and $p''(x) = 6x - 2a$. The critical numbers and inflection point are

For $q(x) = \frac{e^{-x}}{x - c}$, the function has a vertical asymptote at $x = c$ because

For $q(x) = \frac{e^{-x}}{x - c}$, the function has a vertical asymptote at $x = c$ because the denominator vanishes there while the numerator equals $e^{-c} \ne 0$.

For the same family, the derivative simplifies to $q'(x) = -\frac{x - c + 1}{(x-c)^2e^x}$. The only critical number and its location relative to the pole are

For the bell shaped family $E(x) = e^{-\frac{(x-m)^2}{2s^2}}$ with $s \gt 0$, the unique critical number is

For the same family $E(x) = e^{-\frac{(x-m)^2}{2s^2}}$, the inflection points occur at

For the same family, the limits $\lim_{x \to \infty}E(x)$ and $\lim_{x \to -\infty}E(x)$ are

For $E(x) = e^{-\frac{(x-m)^2}{2s^2}}$, compute $E'(m)$.

For the family $p(x) = x^3 - ax^2$ with $a = 6$, one critical number is $x = 0$. Compute the other one.