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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Rates of Change in the Natural and Social Sciences">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{\label{Rates of change}\label{Applications of derivative}Rates of
Change in the Natural and Social Sciences}

Whenever the function $y=f(x)$ has a specific interpretation in one of the
sciences, its derivative will have a specific interpretation as a rate of
change. Velocity, density, current, power, and temperature gradient in
physics; rate of reaction and compressibility in chemistry; rate of growth
and blood velocity gradient in biology; marginal cost and marginal profit in
economics; rate of heat flow in geology; rate of improvement of performance
in psychology; rate of spread of a rumor in sociology---these are all
special cases of a single mathematical concept, the derivative.

In the first example that follows, the derivative is interpreted as
velocity, and in the second example the derivative is used to find an
approximation to the rate of growth of a population.

\begin{example}
If the position of a particle is given by the equation $4^{3}-6\left(
4^{2}\right) +36=\allowbreak 4=\allowbreak 4$ 
\[
s=f(t)=t^{3}-6t^{2}+9t
\]
at time $t$, where $t$ is measured in seconds and $s$ in meters,

\begin{enumerate}
\begin{enumerate}
\item  The velocity at time $t$ is 
\[
v(t)=\frac{ds}{dt}=3t^{2}-12t+9
\]

\item  The velocity after $2$ seconds is the instantaneous velocity when $t=2
$, that is, 
\[
v(2)=\left[ \frac{dx}{dt}\right] _{t=2}=3\left( 2\right) ^{2}-12\left(
2\right) +9=-3\unit{m}/\unit{s}
\]

\item  The particle is at rest when $v(t)=0$, that is, 
\[
3t^{2}-12t+9=3\left( t-1\right) \left( t-3\right) =0
\]
and this is true when $t=1$ or $t=3$. Thus the particle is at rest after $1%
\unit{s}$ and after $3\unit{s}$.

\item  The particle moves in the positive direction when $v(t)>0$, that is 
\[
3t^{2}-12t+9=3\left( t-1\right) \left( t-3\right) >0 
\]
This inequality is true when both factors are positive ($t>3$) or when both
factors are negative ($t<1$). Thus the particle moves in the positive
direction in the time intervals $t<1$ and $t>3$. It moves in the negative
direction when $1<t<3$.

\item  The motion of the particle plotted against time is depicted in the
following graph.\FRAME{dtbpF}{2.0314in}{2.348in}{0pt}{}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 2.0314in;height
2.348in;depth 0pt;display "PICT";function
\TEXUX{$t^{3}-6t^{2}+9t$};linecolor "magenta";linestyle 1;linethickness
1;pointstyle "point";xmin "0";xmax "4.5";xviewmin "-0.25";xviewmax
"4.75";yviewmin "-1.302";yviewmax "10";viewset"XY";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{t};var1name
\TEXUX{$x$};}}The particle starts at position $0$ at time $t=0$. When $t=1$,
the particle has moved $4\unit{m}$ from the starting position. At time $t=3$
the particle has returned to its original position.

\item  To find the total distance traveled by the particle in the first $4$
seconds, find the sum of the distances it travels between changes of
direction:

\begin{enumerate}
\item  The distance traveled in the first second is 
\[
\left| f(1)-f(0)\right| =\left| 4-0\right| =4\unit{m} 
\]

\item  The distance traveled in the next two seconds is 
\[
\left| f(3)-f(1)\right| =\left| 0-4\right| =4\unit{m} 
\]
\end{enumerate}

The distance traveled in the fourth and fifth seconds is 
\[
\left| f(5)-f(3)\right| =\left| 20-0\right| =20\unit{m} 
\]
The total distance is $4+4+20\unit{m}$.
\end{enumerate}
\end{enumerate}
\end{example}

\begin{example}
Let $n=f(t)$ be the number of individuals in an animal or plant population
at time $t$. The change in the population size between the times $t=t_{1}$
and $t=t_{2}$ is $\Delta n=f(t_{2})-f(t_{1})$ and so the \textit{average
rate of growth} during the time period $t_{1}\leq t\leq t_{2}$ is 
\[
\frac{\Delta n}{\Delta t}=\dfrac{f(t_{2})-f(t_{1})}{t_{2}-t_{1}}
\]
The \textit{instantaneous rate of growth} is obtained from this average rate
of growth by letting the time period $t_{2}-t_{1}$ approach $0$. 
\[
\lim_{\Delta t\rightarrow 0}\frac{\Delta n}{\Delta t}=\frac{dn}{dt}
\]
Of course the actual growth is not continuous because populations increase
and decrease by integer multiples. However, for a large animal or plant
population, you can replace the graph by a smooth approximating curve as
depicted in the following plot.\FRAME{dtbpF}{3.0441in}{2.0384in}{0pt}{}{}{%
growth.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
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144.5625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'growth.wmf';file-properties "XNPEU";}}With a smooth approximating curve you
can use the power of calculus to study population growth.
\end{example}

\subsection{Related Reference Topics}

{\small \hyperref{Compound interest}{}{}{Al05_02.tex#Compound Interest}}

{\small \hyperref{Radioactive decay}{}{}{Al05_06.tex#Radioactive Decay}}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Evaluating Calculus Expressions}{}{}{%
../../Help/DM7-1.tex#Calculus expressions}}

{\small \hyperref{Limits}{}{}{../../Help/DM7-2.tex#Limit}}

{\small \hyperref{Differentiation}{}{}{../../Help/DM7-3.tex#Derivative}}

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\noindent \hrulefill

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