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\begin{document}

\title{Math 241, Lecture 8}
\maketitle

\section{ The derivative}  

\subsection{ Slopes 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.    
The derivative at $x$ which measures instantaneous change at $x$, is the
slope of the tangent line at $x$. 

We already know one point on the tangent line, namely the point
on the curve which it is kissing.  
What we can do to find the slope of the tangent line
is to look at slopes of secant lines for points close to
the point of tangency.  

\begin{example}
Estimate the slope of the line which is tangent to 
$f(x) = 16 x^2$ at $x = 2$?   at $x = 5$?  
\end{example}

\begin{example}
Calculate the equation of the line tangent to $f(x) = \frac{1}{x}$
when $x = 1$.  
\end{example}

\subsection{ The sign of the derivative}  

Knowledge of the derivative sheds a lot of light on the original function.  
One of the most basic facts about the derivative, since it
measures the rate of change of a function, is based
on simply seeing whether the derivative is greater than or less
than zero.  

\begin{theorem}
A function is increasing at $x$ if its derivative there is positive.    
A function is decreasing at $x$
if the derivative at $x$ is negative.  
\end{theorem}

\begin{example}
Find where the function $f(x) = 2x - x^2$ is increasing or
decreasing.     Check the answer against the graph of the function.  
\end{example}

The next example might require some audience participation.  

\begin{example}
The derivative game: given some graphs of  derivative functions, sketch possible
graphs for the original functions.  
\end{example}


\subsection{ Formula for the derivative - the algebraic viewpoint}  

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, given an algebraic way to compute the derivative.  
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.  
We will often leave $h$ in the equation ``until the very end'', at which
point we ``let it become very small - so small it is negligible''.  

\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 $2 - \frac{1}{1+x}$ 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}

\end{document}
 