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\title{ Math 242:   Calculus for Business and
the Social Sciences,   II}  
\maketitle

\section{ Outline of the course}  

We will first continue with one of the greatest mathematical
inventions of all time,   the calculus,   now in
its {\em integral} (as opposed to differential) version,  
based on the {\em fundamental theorems of calculus.}  

We will next lay the algebraic foundation for solving
problems involving many variables  
and in particular development of multivariable calculus.  
Recalling how important lines are in calculus,  
it is no surprise that we will study
{\em linear algebra} in many variables.  

Finally, we will develop optimization techniques, 
  as were the main focus at the end of last term, 
  in the setting of multiple variables.  

\section{ Why are we studying these topics?} 

The study of mathematics places stringent demands on
one's ability to reason clearly and completely   --
this is why the university has a general requirement in mathematics.  

The integral calculus is the second,   crucial,   half 
of one of the greatest intellectual stories of all time,  
namely the calculus.    The fundamental theorems,  
which relate integral to differential calculus,   are two
of the most important, often used theorems in all of 
mathematics.  

Most often in real-world modeling it is not practical or useful
to isolate one or two variables.    Starting to work on 
many-variable problems adds a key tool to our
mathematical modeling toolbox.  

Multivariable optimization through calculus is ubiquitous
in economics and other social sciences.  
Graduate programs in economics often give crash courses
in this material to their incoming students.   


\section{ Practicalities} 

\begin{itemize}
\item Course materials, including a complete syllabus, and announcements
are at: \\
http://noether.uoregon.edu/$\sim$dps/242/  
\item Lectures will be projected in outline with details worked out in writing.  
The outlines will be available online before class.  
\item Homework is due on Fridays at the beginning of class.   
All assignments are posted on the syllabus.  
\item Sections are on Wednesdays or Thursdays.     There will often
be quizzes (as posted on syllabus).    
\item There will be three fifty-minute exams, each covering one third of the
material.    They will be non-cumulative, as a significant fraction of the
material is non-cumulative.    Accordingly, there will be no final exam.  
\end{itemize}

\section{ First (half) lecture:   Discrete calculus,   a chance to both review
and to look forward}  

The functions with which we worked in differential calculus depended on
a {\em continuous} variable   - one which could take on any value
  (such as $\frac{1}{2}$   or $\sqrt{2}$   or $\pi$   or $e$).  
This variable was often measuring time.    For some problems it is more
natural to have a discrete variable   (for example, when measuring financials
every quarter as opposed to continuously).  

\begin{definition}
A discrete function is one whose domain consists of integers  
(but whose range consists of any kind of number)  
\end{definition}

Any of the familiar examples of functions (polynomials,
exponentials) can be 
{\em restricted} to be discrete functions.   We often
use $n$ to name the variable instead of $x$,  
so that the squaring function is named $f(n) = n^2$.  

\subsection{ Difference functions}  

\begin{definition}
Given a discrete function $f(n)$ define its difference function
$Df(n)$ by $Df(n) = f(n) - f(n-1)$.  
\end{definition}

\begin{example}
\begin{itemize}
\item If $f(n) = n^2$,   then $Df(n) = n^2 - (n-1)^2  
= n^2 - (n^2 -2n + 1)   = n^2 - n^2 + 2n - 1   = 2n -1$.  
\item If $f(n) = n^3$,   then $Df(n) = n^3 - (n-1)^3  
= n^3 - n^3 + 3n^2 - 3n + 1   = 3n^2 - 3n + 1.$  
\item If $f(n) = 2^n$, then $Df(n) = 2^n - 2^{n-1}  
 = 2^{n-1}(2 - 1)   = 2^{n-1}.$  
\end{itemize}
\end{example}

Notice the similarities with the derivative!  

One can use the difference function much as we used the
derivative.    In particular, a function achieves a relative 
maximum or minimum where its difference function changes
sign.  

\subsection{ Aggregate functions}  

Aggregate functions measure total amounts   (for example,
total sales when given monthly sales).  

\begin{definition}
Given a discrete function $f(n)$ define the aggregate function
$Af(n)$ to be $f(1) + f(2) + \cdots + f(n)$.  
\end{definition}

\begin{example}
\begin{itemize}
\item If $f(n) = n$, $Af(n) = \frac{n(n+1)}{2}$   (as Gauss figured
out when he was eight years old!)  
\item If $f(n) = n^2$, $Af(n) = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}$.  
\item If $f(n) = 2^{n-1}$, $Af(n) = 2^n -1$.  
\end{itemize}
\end{example}

Notice that the derivative of $Af(n)$ looks related to $f(n)$ itself. 

\subsection{ The fundamental theorems of discrete calculus}

If one were given a total sales function, the way one would
find the sales just for the fifth month would be to subtract
the total for the first four months from the total for the first
five months.    This simple observation leads to the first fundamental
theorem of discrete calculus.  

\begin{theorem}[First fundamental theorem] 
If $f(n)$ is a discrete function, then $D(Af(n)) = f(n)$.  
\end{theorem}

Proof:   We compute 
\begin{align*}
D(Af(n)) &= Af(n) - Af(n-1)    \\
&= \{ f(1) + \cdots + f(n) \} - \{ f(1) + \cdots + f(n-1) \}    \\
& = f(n).   
\end{align*}

Along similar lines, we have the following.

\begin{theorem}[Second fundamental theorem]
If $f(n)$ is a discrete function then $A(Df(n)) = f(n) - f(0)$.  
\end{theorem}

Proof:   
\begin{align*}
A(Df(n)) &= Df(1) + Df(2) + \cdots + Df(n)   \\
&= (f(1) -f(0)) + (f(2) - f(1)) + \cdots  \\ & \; \; \; \; + (f(n) - f(n-1))   \\
&= -f(0) + f(1) - f(1) + f(2) - f(2) + \cdots  \\ & \; \; \; \;  + f(n-1) - f(n-1) + f(n)   \\
& = f(n) - f(0)  
\end{align*}

 

For discrete functions, these theorems are the result of
simple arithmetic.     Their continuous analogues are much more
subtle.   The difference function is the discrete analogue of the
derivative,   and the aggregate function is the discrete analogue
of something we will call the (definite) integral.     Both of the
former have to do with differences and change.    Both of the
latter have to do with total and accumulation.     Their
interrelationship through fundamental theorems 
will be of both conceptual and computational use.

\end{document}
 