Activity 3.4.4.
Let \(L(t) = \frac{A}{1+ce^{-kt}}\text{,}\) where \(A\text{,}\) \(c\text{,}\) and \(k\) are all positive real numbers.
(a)
Observe that we can equivalently write \(L(t) = A(1+ce^{-kt})^{-1}\text{.}\) Find \(L'(t)\) and explain why \(L\) has no critical numbers. Is \(L\) always increasing or always decreasing? Why?
(b)
It turns out that
\begin{equation*}
L''(t) = Ack^2e^{-kt} \frac{ce^{-kt}-1}{(1+ce^{-kt})^3}\text{.}
\end{equation*}
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?
(c)
What is the value of \(\displaystyle \lim_{t \to \infty} \frac{A}{1+ce^{-kt}}\text{?}\) \(\displaystyle \lim_{t \to -\infty} \frac{A}{1+ce^{-kt}}\text{?}\)
(d)
Find the value of \(L(x)\) at the inflection point found in (b).
(e)
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 \(A\text{,}\) \(c\text{,}\) and \(k\) number.
