%% Math 131, Fall, 2003
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\begin{document}

\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Professor Hwang} \\
Descriptions of Exponential Growth
\end{center}

The text has three ways of describing exponential growth. While these
three methods are mathematically equivalent, the book's terminology
about ``rate of growth/decay'' can be confusing, and demands careful
attention to wording.

\paragraph{General Base} Let $a>0$ be a positive constant. The general
exponential function with base~$a$ is a function of the form
\[
f(t) = P_0\,a^t.
\]
Usually the variable~$t$ represents time, measured in units
appropriate to a given problem. The constant~$P_0$ represents the
amount of ``stuff'' at time~0. 

For each increment of~$t$ by one unit, an exponential function's value
gets multiplied by~$a$, because
\[
f(t+1)=P_0\,a^{t+1}=P_0\,a^t\cdot a = a\cdot f(t).
\]
Consequently, if $0<a<1$, then~$f$ is decreasing (by a factor of~$a$
for each time unit), while if $a>1$, then~$f$ is increasing (by a
factor of~$a$ for each time step).

\paragraph{Growth Rate} When a population grows at a rate of~$2.6\%$
per~year, the number of individuals is described by an exponential
function of the form
\[
f(t)=P_0\,(1.026)^t=P_0\,(1+0.026)^t.
\]
Note that $2.6\%=0.026$. In general, if a quantity increases by
fraction~$r$ (that is, by~$100r\%$) in one time unit, then the
quantity is given by an exponential function
\[
P_0\,(1+r)^t,\qquad a=1+r.
\]
The number~$r$ is sometimes called the \emph{growth rate} (if $r>0$)
or the \emph{decay rate} (if $r<0$). Do not confuse the growth rate
(of a continuous process) with the ``continuous growth rate'', see
below.

For example, if a drug is removed from the body at a rate of~$15\%$
per~hour ($r=-0.15$), and if $P_0=250\,\mathrm{mg}$ are taken
initially, then the amount of drug in the body after $t$~hours is
\[
P=250\,(1-0.15)^t\,\mathrm{mg}=250\,(0.85)^t\,\mathrm{mg}.
\]

\paragraph{Continuous Growth Rate}

In order to use a uniform base for all exponential growth and decay,
it is mathematically convenient to write the general exponential
function in a third form:
\[
f(t) = P_0\,e^{kt}, \qquad e^k = a = 1+r.
\]
The constant~$k$ is called the \emph{continuous growth rate} (if
$k>0$) or the \emph{continuous decay rate} (if $k<0$). Unfortunately,
the growth rate~(``$r$'') and continuous growth rate~(``$k$'') of the
same exponential process are not identical. For example, in the drug
elimination example above, for which $r=-0.15$~and $a=0.85$, we have
$k=\ln 0.85\simeq -0.162519$, a continuous decay rate of
about~$16.25\%$. 

\paragraph{Summary} When $r$~or $k$~is close to~$0$, $r\simeq k$;
however, the two growth/decay rates are never identical (unless both
are~0). The larger the rates, the greater the disagreement between
their numerical values. For example, a growth rate $r=1$
($100\%$~growth per time unit) corresponds to $k=\ln2\simeq0.693$.

\eject

\end{document}
