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\Large
\rm{Math 111}
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\rm{Chapter 2.7:  Derivatives \& Rates of Change}
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(DEFINITION) The {\bf derivative of a function $f$ at a number $a$} is:
\begin{displaymath}
f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}
  \end{displaymath}


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The meaning of $f'(a)$ is
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\item{}
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\item{}
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(EXAMPLE)  Find $f'(9)$ if $f(x) = 4-\sqrt{x}$.  Sketch the graph of $f$ and the tangent line at $x=9$.


\pagebreak


(EXAMPLE)  The height of a falling object in meters is given by $y(t) = 150 - 4.9t^2$, where time $t$ is measured
in seconds.  Find the velocity of the object at the time $t=5$.  What are the units of $y'(5)$?


\vspace{3.5in}


(EXAMPLE)  Find the derivative $p'(2)$ if $p(x) = 3/x$.  Sketch the graph of $p$, the line tangent to the
graph at $(2, 1.5)$, and give the equation of the line.


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(INTERPRETATIONS)\\

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In certain circumstances, the volume of a gas is inversely proportional to the pressure.  We could write
volume as a function of pressure $V(P) = k/P$ where $k$ is a constant, $V$ is measured in cm$^3$ and pressure measured
in kPa.
\begin{enumerate}
\item{What does $V'(100)$ measure?}
\item{What is the sign of $V'(100)$ ?}
\item{What are the units $V'(100)$ ?}
\item{Which do you think is larger, $V'(100)$ or $V'(500)$ ?}
\end{enumerate}

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The garbage $G$ produced by a city is a function of its population $P$.  Suppose $P$ is measured in thousands and
$G$ is measured in tons.  
\begin{enumerate}
\item{What does $G'(250)$ measure?}
\item{What is the sign of $G'(250)$ ?}
\item{What are the units $G'(250)$ ?}
\end{enumerate}


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Let $A(t)$ represent the total mass of apples (in kg) you gather as a function of time you spend in an orchard (in hours).
\begin{enumerate}
\item{What is the sign of $A'(6)$ ?}
\item{What are the units $A'(6)$ ?}
\item{Which do you think is larger $A'(6)$ or $A'(200)$?  Why? }
\end{enumerate}


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Suppose we know that $g$ is a continuous function with $g(1) = 2$, $g'(1) = 0$, $g(4) = 0$, and $g'(4) = 1 $.
Sketch a possible graph of $g$.




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