§3.4 Using derivatives to describe families of functions

Introduction

Describing families of functions in terms of parameters

Let $a$, $h$, and $k$ be arbitrary real numbers with $a \ne 0$, and let $f$ be the function given by the rule $f(x) = a(x-h)^2 + k$.

What familiar type of function is $f$? What information do you know about $f$ just by looking at its form? (Think about the roles of $a$, $h$, and $k$.)

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 $g$.

Find $p'(x)$ and determine the critical numbers of $p$. How many critical numbers does $p$ have?

Construct a first derivative sign chart for $p$. What can you say about the overall behavior of $p$ if the constant $a$ is positive? Why? What if the constant $a$ is negative? In each case, describe the relative extremes of $p$.

Find $p''(x)$ and construct a second derivative sign chart for $p$. What does this tell you about the concavity of $p$? What role does $a$ play in determining the concavity of $p$?

Without using a graphing utility, sketch and label typical graphs of $p(x)$ for the cases where $a\gt 0$ and $a \lt 0$. Label all inflection points and local extrema.

Find the first derivative and the critical numbers of $h$. Use these to construct a first derivative sign chart and determine for which values of $x$ the function $h$ is increasing and decreasing.

Find the second derivative and build a second derivative sign chart. For which values of $x$ is a function in this family concave up? concave down?

What is the value of $\displaystyle \lim_{x \to \infty} a(1-e^{-bx})$?$\displaystyle \lim_{x \to -\infty} a(1-e^{-bx})$?

How does changing the value of $b$ affect the shape of the curve?

Without using a graphing utility, sketch the graph of a typical member of this family. Write several sentences to describe the overall behavior of a typical function $h$ and how this behavior depends on $a$ and $b$.

Observe that we can equivalently write $L(t) = A(1+ce^{-kt})^{-1}$. Find $L'(t)$ and explain why $L$ has no critical numbers. Is $L$ always increasing or always decreasing? Why?

It turns out that

. Given this fact, find all values of $t$ such that $L''(t) = 0$ and hence construct a second derivative sign chart. For which values of $t$ is a function in this family concave up? concave down?

What is the value of $\displaystyle \lim_{t \to \infty} \frac{A}{1+ce^{-kt}}$?$\displaystyle \lim_{t \to -\infty} \frac{A}{1+ce^{-kt}}$?

Find the value of $L(x)$ at the inflection point found in (b).

Without using a graphing utility, sketch the graph of a typical member of this family. Write several sentences to describe the overall behavior of a typical function $L$ and how this behavior depends on $A$, $c$, and $k$ number.