\documentclass[30pt,landscape]{foils}

\usepackage{amsmath}

\usepackage{pause}               \usepackage{background}
\usepackage{graphicx}           \usepackage{geometry}
\usepackage{hyperref}

\usepackage{pp4slide}
\geometry{headsep=3ex,hscale=0.9}

\newtheorem{example}{Example} 
\newtheorem{question}[example]{Question}
\newtheorem{definition}[example]{Definition}
\newtheorem{theorem}[example]{Theorem}

\newcommand{\pz}{\pause}
\newcommand{\pb}{\pagebreak}

\definecolor{bgblue}{rgb}{0.04,0.39,0.53}
\vpagecolor{bgblue}

\hypersetup{pdftitle={lecture},
  pdfsubject={A mathematics lecture},
  pdfauthor={Dev Sinha, University of Oregon},
  pdfkeywords={pdftex, acrobat},
  pdfpagemode={FullScreen},
  linkcolor={red}
}

\begin{document}

{\Large Quadratic functions} \pause

After linear functions, the next simplest kinds of functions are quadratic
functions.  \pause We will review basic properties
of these functions now, so they are familiar when we 
apply calculus to them later. \pause

\begin{theorem}
A parabola intersects the $y$-axis at the point $(0, c)$. \pause
A parabola intersects the $x$-axis at points given by the quadratic
equation. \pause
A parabola reaches its maximum or minimum when
$x = \frac{-b}{2a}$. \pause
\end{theorem}

We can easily verify the first two statements.  \pause
The third statement can also be verified through the algebra
of ``completing the square'',
but will be very easy to see with a little calculus. \pz

\begin{example}
Graph the function $f(x) = -\frac{1}{2} x^2 + 3x - 4$, indicating
the places where the graph crosses the axes, as well as the minimum
or maximum.  \pause
\end{example} 

\begin{example}
Suppose the total profit made by manufacturing $x$ thousands of units of Freshies
doggie biscuits is given by $P(x) = -0.4 x^2 + 80 x - 20$ (measured
in tens of thousands of dollars). \pause
\begin{itemize}
\item Explain why this is a reasonably-shaped profit function. \pause
For example, $P(0) = -20$; does that make sense? \pause
\item Find the optimum number of Freshies to produce. \pb
\end{itemize}
\end{example}


{\Large Exponential functions} \pz

Our next goal is to increase our toolbox of functions beyond polynomials, etc. \pz
Our first new functions will be the exponential functions.  \pz  These functions
arise in population modeling, finance, the study of heat transfer and rates of 
reaction in physics, chemistry and biology. \pz 

Exponential functions arise by taking one number to the power of another, just
as polynomial functions like $x^3$ do. \pz  For exponential functions, the variable
occurs as the power, not as the base. \pb

It is an interesting story to see how the exponential function is defined for different
kinds of numbers; \pz its careful development required many of the fundamental
ideas in calculus. \pz  

We develop the function $4^x$, to be concrete and to be clear that for 
exponential functions, the variable appears in the power. \pb

\begin{definition}
\begin{itemize}
\item $4^0 = 1$. \pz
\item If $x$ is a positive integer, the value of $4^x$ is $4 \times 4 \times \cdots \times 4$,
where $4$ is multiplied by itself $x$ times. \pz For example $4^3 = 4 \times 4 \times 4 = 64$. \pz
\item The two laws of exponents state that $4^x 4^y = 4^{x+y}$ and $(4^x)^y = 4^{xy}$.  
\pz  For example
$4^2 \times 4^3 = 16 \times 64 = 1024 = 4^5 = 4^{2+3}$ \pz and 
$(4^2)^2 = 16^2 = 256 = 4^4 = 4^{2 \times 2}$.  \pz  These laws of exponents
determine what $4^x$ must be for other kinds of $x$. \pz 
\item If $x$ is a positive integer, $4^{-x} = \frac{1}{4^x}$.  \pz  For example $4^{-2} = \frac{1}{4^2} =
\frac{1}{16}$.  \pz  This rule must hold since $4^x \times 4^{-x} = 4^{x + (-x)} = 4^0 = 1$ by
the law of exponents, \pz so that $4^x$ and $4^{-x}$ are reciprocals. \pz
\item If $x$ is a positive integer $4^{\frac{1}{x}} = \sqrt[x]{4}$.  \pz For example, 
$4^{\frac{1}{2}} = \sqrt{4} = 2$. \pz  This rule must hold because $(4^{\frac{1}{x}})^x = 
4^{\frac{1}{x} x} = 4^1 = 4$.  \pz (Interesting question: why must there be a $\sqrt[3]{4}$?) \pz
\item If $x$ is a fraction $x = \frac{p}{q}$ then $4^x = (4^{\frac{1}{q}})^p$, 
as must be the case immediately from the laws of exponents. \pz
For example $4^{\frac{-3}{2}} = (4^{\frac{1}{2}})^{-3} = \sqrt{4}^{-3} = 2^{-3} =
\frac{1}{2^3} = \frac{1}{8}.$ \pz
\item Finally, note that any real number $x$ is approximated arbitrarily closely
by rational numbers $\frac{p}{q}$.  \pz If $x$ is an arbitrary real number
we define $4^x$ as the {\em limit} of $4^{\frac{p}{q}}$ as the fraction
$\frac{p}{q}$ approaches $x$ . \pz For example $4^\pi$
is the limit of the sequence $4^3, 4^{\frac{31}{10}}, 4^{\frac{314}{100}}, \ldots$. \pz
We will develop the notion of limit more fully later. \pz
\end{itemize}
%Note that we need this discussion when talking about 
%functions like $x^{\frac{5}{7}}$ and $x^{\sqrt{2}}$. \pz
\end{definition}

Fortunately, our calculators can compute any of these values for us. \pz

Important fact: if $4^x = 4^y$ then $x = y$.  \pz In other words, the function 
$4^x$ doesn't repeat any values (in contrast with most other functions \pz
-- for example, $x^2 + 2$ has the same value when $x = 2$ or $x = -2$.) \pz

The graph of an exponential function always starts or ends close to the
$x$-axis \pz (why?) \pz and then gets far away from the axis very quickly. \pz

\begin{example}
Sketch the graphs of the exponential functions $f(x) = -3 \cdot 2^x$ and 
$y = (\frac{2}{3})^x$. \pb
\end{example}



\end{document}
 