§3.4 Using derivatives to describe families of functions

Introduction

Describing families of functions in terms of parameters

Consider the two-parameter family of functions given by $g(x) = axe^{-bx}$, where $a$ and $b$ are positive real numbers. Fully describe the behavior of a typical member of the family in terms of $a$ and $b$, including the location of all critical numbers, where $g$ is increasing, decreasing, concave up, and concave down, and the long term behavior of $g$.

For $f(x) = a(x-h)^2 + k$ with $a \ne 0$, the vertex of the parabola is located at

For the family $p(x) = x^3 - ax$ with $a \gt 0$, solving $p'(x) = 3x^2 - a = 0$ gives critical numbers

For the logistic family $L(t) = \frac{A}{1 + ce^{-kt}}$ with $A, c, k \gt 0$, which description is correct?

For $h(x) = a(1 - e^{-bx})$ with $a = 2$ and $b = 3$, compute $h'(0)$.

For the logistic family $L(t) = \frac{A}{1 + ce^{-kt}}$ with carrying capacity $A = 100$, the inflection point occurs at height $\frac{A}{2}$. Compute that height.

With $a, b \gt 0$, every member of the family $h(x) = a(1 - e^{-bx})$ is concave down on the entire real line.