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\centerline{Mathematics 41, section 1 -- Multivariable Calculus}
\centerline{Lab Day 5 -- Local Linearity for $f(x,y)$}
\centerline{October 14, 1999}
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{\it Background}
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Yesterday in class we saw that one possible way to say that a
function $f(x,y)$ has a ``total derivative'' at $(x_0,y_0)$
might be to borrow the idea of {\it local linearity\/} from 
one variable calculus.  There, for $g : {\bf R} \to {\bf R}$, 
if $g'(x_0)$ exists, then as we ``zoom in'' closer and closer to the 
point $(x_0,g(x_0))$, the portion of the graph $y = g(x_0)$ we
see becomes indistinguishable from a {\it straight line\/} of slope
$g'(x_0)$.  The analogous property for a function $f(x,y)$ would be 
that as we zoom in closer and closer to a point $(x_0,y_0,f(x_0,y_0))$,
then the graph should be become indistinguishable from a {\it plane\/}
in ${\bf R}^3$.  Moreover, we saw that if this local linearity property
is true, the plane should be defined by 
$$z = f(x_0,y_0) + {\partial f\over \partial x}(x_0,y_0) (x - x_0) + 
{\partial f\over \partial y}(x_0,y_0) (y - y_0).$$
Today, we will use Maple to 
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\item{$\bullet$} investigate several examples, 
\item{$\bullet$} determine whether they are locally linear near given points, 
and 
\item{$\bullet$} see how local linearity of the graph $z = f(x,y)$ relates to 
local linearity of the slices of the graph in vertical planes.  
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{\it Directions}
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We will be working today with the questions from Discussion 18 in 
the Collaborative Learning Handbook.  I have prepared a Maple worksheet
to get you started, including function definitions, etc. similar
to those discussed in the lab sheet.  You should begin by copying
that worksheet from the class directory to yours.  Launch a 
Terminal window, and use the command:
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\centerline{\tt cp /home/fac/little/public\_html/Mult99/LabDay5.mws LabDay5.mws}
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This will create a copy of the worksheet in your directory.  
Now, get into Maple, and read this worksheet in.  Execute the commands
one by one and read the comments to understand what is going on.
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{\it Lab Work}
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Answer the questions 1,2,3 in Discussion 18 from the CLW.  In all your
composite plots showing slices and the graph, also include the 
corresponding "zoomed" portion of the tangent plane produced using
the {\tt tplot} command from the worksheet you copied above.
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{\it Assignment}
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Lab write-ups due Wednesday, October 20 (the week after Spring Break).
\bye
