§3.4 Using derivatives to describe families of functions

Introduction

Describing families of functions in terms of parameters

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 $p(x) = x^3 - ax$, which pair of statements matches the two cases of the parameter?

For the same family, $p''(x) = 6x$. What does this say about concavity and the role of $a$?

For $h(x) = a(1 - e^{-bx})$ with $a, b \gt 0$, the derivative is $h'(x) = abe^{-bx}$. The function is

With $a, b \gt 0$, every member of the family $h(x) = a(1 - e^{-bx})$ is

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

Within the family $h(x) = a(1 - e^{-bx})$, changing the value of $b$ affects the graph by

For $g(x) = axe^{-bx}$ with $a, b \gt 0$, the derivative factors as $g'(x) = ae^{-bx}(1 - bx)$. A typical member

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

For the same family, $L''(t) = Ack^2e^{-kt}\frac{ce^{-kt}-1}{(1+ce^{-kt})^3}$, which vanishes only when $ce^{-kt} = 1$, that is at $t = -\frac{1}{k}\ln(\frac{1}{c})$. The curve is

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

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.