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\title{Math 242, Lecture 18}
\maketitle

\section{ Introduction to multivariable functions }  

The linear functions we have been studying are our
first examples of multivariable functions, which are the topic
of study for the last third of the term.     

A multivariable function describes a dependence of one
quantity on two or more others.    
In some linear programming examples we
saw, expenditures depended on the number of two different
kinds of items purchased;    temperature depends on where
you are, which is described by longitude and latitude;   
production levels depend on capital and labor.   

As we saw, the graphs of two-variable functions are 
surfaces which sit in three-dimensional space.    
We will not focus on graphing functions on our own, but 
it is helpful to look at graphs the computer generates as
we go along.   

\begin{example}
Look at the graphs of the following functions: 
\begin{itemize}
\item $f(x,y) = \frac{1}{2}(x^2 + y^2)$  
\item $f(x,y) = \frac{1}{2}(x^2 - y^2)$   
\item $f(x,y) = e^{x-y}$.   
\end{itemize}
\end{example}

We can ask many of the same kinds of questions as we did
for single-variable functions, but with some additional complications.  
Sometimes we see what are maxima and minima,    but in order
to talk about whether the function is increasing or decreasing, we
have to pick a direction.     It is helpful to draw on intuition from
hiking, thinking of the graph as a landscape.   

\subsection{ Level curves}  

We saw with linear functions that given a two-variable
function $f(x,y)$ it is helpful to look at the sets of points where
$f(x,y)$ has a fixed value like $0$, $1$, $35$, ...  
As we mentioned previously, such collections are depicted
in topographical maps as contour lines,    and on weather
maps as isotherms (lines of constant temperature).   

\begin{definition}
A {\em level curve} for a function $f(x,y)$ is the collection of all
points $(x,y)$ such that $f(x,y) = c$ for a given $c$.   
\end{definition}

\begin{example}
What are some level curves of the functions $f(x,y) = x^2 + y^2$,    
$f(x,y) = x^2 - y^2$,   and $e^{x+y}$?   
\end{example}

In some specific applications, level curves
have specialized names (e.g. contour lines for the height
function on a topographical map).     In economics, the {\em utility
function} $U$ of two (or more variables) measures ``satisfaction''
- also called {\em utility} - a consumer derives from having 
different amounts of goods.    Level curves in this setting
are called {\em indifference curves}, because
consumers are supposed to be indifferent to choices
which lead to the same level of satisfaction.  

 

\section{ Partial derivatives}  

By fixing one of the variables in a two-variable function,
we get a one-variable function.    For example, if
we take the function $x^2 + y^2$ and set $x = 2$ we
get the function $4 + y^2$.    This resulting one-variable
can then have its derivative taken;   that's the idea
behind partial derivatives.  Before seeing the general technique
we do one example ad hoc.  

\begin{example}
Find the derivative of the function $f(x,y) = x y^2$ when 
$x = 3, 5, -1, 100, \sqrt{2}, \pi$.    Notice that this derivative
is always equal to $2xy$.  
\end{example}

\begin{definition}
If $f(x,y)$ is a function of two variables, its partial 
derivative with respect to $x$,   denoted either
$\frac{\partial f}{\partial x}$ or $f_x(x,y)$,   is the function 
obtained by treating $y$ as a constant and differentiating
with respect to $x$.    Similarly, the partial derivative
with respect to $y$, denoted $\frac{\partial f}{\partial y}$ or $f_y$,
is obtained by treating $x$ as a constant and differentiating
with respect to $y$.  
\end{definition}

You will need to take partial derivatives for the quiz this week.  

\begin{example}
Find the partial derivatives with respect to $x$ and $y$ of  
\begin{itemize} 
\item $\frac{2y}{x}$.  
\item $2x^2 \sqrt{3y + 1}$.  
\item $\ln(xy)$.  
\item $e^{-(x^2 + y^2)}$.  
\end{itemize}
\end{example}



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