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Subsection2.1Exponential Functions

Before we discuss exponential functions, it is first useful to define the notion of concavity.

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.

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.