Concavity
The graph of a function is concave up if it bends upward as we move left to right; it is concave down if it bends downward. A line is neither concave up nor concave down.
Subsection2.1Exponential Functions
Before we discuss exponential functions, it is first useful to define the notion of concavity.
An exponential function will be either be increasing or decreasing, but will always be concave up! We define such a function as follows:
Exponential Function
\begin{equation*}
P(t) = a \cdot b^t \text{, where } b \gt 0 \text{ and } b \ne 1 \text{, } a \ne 0.
\end{equation*}
The constant \(a \) is the \(y\)-value of the \(y\)-intercept of the function.
The equation
\begin{equation*}
P(t)=P_0 a^t
\end{equation*}
gives an exponential function with base \(a\text{.}\) Then
\begin{equation*}
\frac{P(t+1)}{P(t)}=\frac{P_0 a^{t+1}}{P_0 a^t}=a =\text{constant rate of growth/decay}
\end{equation*}
- Growth: \(a\gt1\text{:}\) Doubling time: the time it takes to double the initial amount
- Decay \(0\lt a\lt1\text{:}\) Half-life: the time it takes to decay to half of the initial amount
Doubling Time and Half-life
The doubling time of an exponentially increasing quantity is the time required for the quantity to double. The half-life of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half.
We will (naturally) consider exponentials (and later, logarithms) in base
\begin{equation*}
e \approx 2.718281828459... \text{ (irrational number)}
\end{equation*}
For example, the general exponential function in the natural base
\begin{equation*}
P(t)=P_0 a^t=P_0 (e^k)^t=P_0e^{kt}, \quad a=e^k.
\end{equation*}
Thus, we will have
- exponential growth if \(a\gt 1\text{,}\) which gives \(k\gt 0\)
- exponential decay if \(0\lt a\lt 1\text{,}\) which gives \(k\lt 0\)
The number \(k\) is called the continuous rate of growth/decay. To find \(k\) we will need the logarithmic function.