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\title{Math 241, Lecture 7}
\maketitle

\section{Average rate of change}  

A fundamental philosophical truth is that everything changes.  
In physics, the change in position is known as velocity or speed.  
In economics, the change in price is known as   inflation.   
In business, the change in costs is sometimes known as trend.  
%and change in cost is known as marginal cost,  
In mathematics, the change in values of a function is known
as the derivative.     But to understand the derivative, which
will measure ``instantaneous'' change, you need to to 
first be comfortable with ``average'' change over some intervals.  


\subsection{Calculating average speed.  }  

If you travel 200 miles in four hours, what is your average speed?  
What about 75 miles in one and a half hours?   

$${\text{Average speed}} = \frac{{\text{distance travelled}}}{{\text{total time}}} $$  

Questions to ponder:  
Do these calculations necessarily mean you went 50mph for four hours?  

Next, what if, instead of giving you total distance travelled, you need
to calculate from a position function which describes where you are?  

\begin{example}
A ball which is dropped from the top of the Tower of Pisa has travelled down
$16 t^2$ feet after $t$ seconds.     What is its average speed over the first
three seconds?
   over the first five seconds?    between the second
and fifth second?   
\end{example}



\subsection{ Calculating slopes of secant lines to a curve. }  

Next we look at what at first appears to be unrelated to dropping a ball.

\begin{definition}
A secant line goes through two points
on the graph of the function.    In symbols, it is a line through
$(a, f(a))$ and $(b, f(b))$ for some $a$ and $b$.   
\end{definition}

\begin{example}
Find the secant lines to the graph of $f(x) = 16 x^2$ through the points
with: $a = 0, b = 3$,   $a = 0, b = 5$,   $a = 2, b = 5$.  
\end{example}

What do you notice about this and the previous problem?   

\begin{example}
Some functions will be sketched on the board; find the slopes of
secant lines as indicated.  
\end{example}

\subsection{ Formula for average rate of change}   

\begin{definition}
In general, the average rate of change of some function $f(x)$ as $x$
varies between values $a$ and $b$ is 
$$ \frac{ f(b) - f(a)}{b-a}. $$  
\end{definition}

This can be computed in any way that $f$ is presented, through a
formula, through a graph, or in a table.  

\begin{example} 
Find the average rate of change for \$1000 invested at a rate of
five percent over four years.    (Note: this is {\em not} the
interest rate).  
\end{example}

\begin{example}
Analyze different measured and predicted rates of change for world 
population according to:\\
{\tt http://www.unfpa.org/6billion/pages/worldpopgrowth.htm}  
\end{example}

\section{ The derivative}  

The derivative is the central topic of study in this class.   It measures the 
instantaneous rate of change of a function at all times.  

Before formalizing it, which is difficult, we will try to understand it
in examples parallel to those we have done for average rate of change.  

\subsection{ Instantaneous speed}  

Question to ponder: how does your speedometer calculate how 
fast you are going at one moment?  

(What does ``how fast you are going at one moment'' even mean?)  

\begin{example}
How fast was the ball falling two seconds after it was dropped
from the Tower of Pisa?   Five second after?  
\end{example}

\subsection{ Slope of tangent lines}  

\begin{definition}
A tangent line to a curve is a line which intersects the curve at
some point, but does not cross the curve.  
\end{definition}

Informally, a tangent line ``kisses'' the curve.  

\begin{question}
How could we calculate the equation of the line which is tangent to 
$f(x) = 16 x^2$ at $x = 2$?   at $x = 5$?  
\end{question}

\subsection{ Formula for instantaneous rate of change}  

We would see similarities in trying to compute average 
and instantaneous differences regardless of what our functions
are measuring.    The following definitions work in all 
such cases.  
First, for average rate of change, we re-write $a = x$ and $b = x+h$,
so that the difference quotient becomes $\frac{f(x+h) - f(x)}{h}$.  
So for example if $a = 3$ and $b = 4$,
we would instead think of $x = 3$ and $h = 1$.     The reason 
we change from $a$ and $b$ to $x$ and $h$ is to be consistent with 
the derivative.  

\begin{definition}
The derivative of $f(x)$ with respect to $x$ is the function 
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.$$  
\end{definition}

We will talk about $\lim$, which means the ``limit'', later.   
For now, we will adopt a working notion that we may compute
this for smaller and smaller $h$; if all of those computations seem
to approach a single number, for now we will call that the limit.  

\begin{example}
What is the derivative of $f(x) = 16 x^2$?    What is the value of the
derivative when $x = 2$?   What does this mean for the velocity of a
ball which is dropped from the Tower of Pisa?   or the equation
of the line tangent to $f(x)$ at $x = 2$?  
\end{example}

\begin{example}
Suppose price index, measuring the aggregate price for a large 
cross-section of household goods measured in thousands of dollars, 
has values $1 - 0.5 \sqrt{x} +  0.15 x$ over two years.  
What is the rate of price increase over two years?    What is the
``instantaneous rate of inflation'' at six months, one year,
and two years?  
\end{example}





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