% Implicit Differentiation
% MATH266A Exercise 


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\begin{center}
{\Large Implicit Differentiation
\footnote{Copyright \copyright 2002 Winfried Just, Department of Mathematics,
Ohio University.  All rights reserved.}}
\end{center}

\bigskip

In this exercise, we will graphically explore the idea behind implicit differentiation.  
If this \textsc{MatLab} exercise is being counted in your grade, 
then be sure to print the figures for submission.

%%%%You may be asked to submit your pictures for this exercise as part of Quiz 4.

Let us use the \verb$ ezplot$ command to draw the graph of the equation

\begin{equation}\label{circle}
x^2 + y^2 = 25
\end{equation}

You do this by entering:

\smallskip

\verb$>> ezplot('x^2 + y^2 = 25', [-6, 6], [-6, 6])$

\smallskip

Look at your picture.  It doesn't look like a circle, does it?  You can improve its look by entering:

\smallskip

\verb$>> axis square$

\smallskip

Clearly, the graph you see is not the graph of a function.  However, let us see what happens if we zoom in on a neighborhood of the point $(3,4)$.  Enter:

\smallskip

\verb$>> axis([1 5 2 6])$

\smallskip

As you can see, the graph now looks pretty much like the graph of a function.  In class we found the equation of the tangent line to the graph of this function.  It is:

\begin{equation}\label{tangentcircle}
y = -{3 \over 4}(x - 3) + 4
\end{equation}

Equivalently, this equation can be written as:

\begin{equation}\label{tangentcircle1}
-{3 \over 4}(x - 3) + 4 - y = 0
\end{equation}

Let us plot the tangent line on the same graph.  Enter:

\smallskip

\verb$>> hold on$

\verb$>> ezplot('-(3/4)*(x-3) + 4 - y = 0', [-6, 6], [-6, 6])$

\smallskip

You should see a nice tangent line to the graph of a function.  Let us zoom out again to see how the tangent line is related to the whole graph of equation~(\ref{circle}).  Enter:

\smallskip

\verb$>> axis([-6 6 -6 6])$

\smallskip

Now give a descriptive title to your picture and print it if
submission is required.

\medskip

It is not the case that for \emph{all} points $(x_0,y_0)$ on the graph of equation~(\ref{circle}) the equation defines a function $y(x)$ in a neighborhood of $(x_0, y_0)$.  Please mark on your printout all points $(x_0, y_0)$ where equation~(\ref{circle}) \emph{does not} define such a function.

\medskip

Now let us explore the equation:\nopagebreak

\begin{equation}\label{descartes}
x^3 + y^3 = 4.5xy
\end{equation}

To clean your graphics window, enter:

\smallskip

\verb$>> hold off$

\smallskip

Now enter:

\smallskip

\verb$>> ezplot('x^3 + y^3 - 4.5*x*y = 0', [-2, 4], [-2, 4])$

\smallskip

The shape you see in the picture is called the "folium of Descartes."  
The whole figure clearly is not the graph of a function.  To see what happens in a neighborhood of the point $(1,2)$, enter:

\smallskip

\verb$>> axis([0.5 1.5 1.5 2.5])$

\smallskip

Again, you can see the graph of a function.  In the handout, we computed the equation of the tangent line to the graph of this function at $(1,2)$.  It is:

\begin{equation}\label{tangentdescartes}
y = {6 \over 7.5}(x-1) + 2
\end{equation}

\smallskip

Add the graph of this tangent line to the picture by entering:

\smallskip

\verb$>> hold on$

\verb$>> ezplot('(6/7.5)*(x-1) + 2 - y = 0', [-2, 4], [-2, 4])$

\smallskip

Now zoom out again to look at the whole picture.  Enter:

\smallskip

\verb$>> axis([-2 4 -2 4])$

\smallskip

Add a suitable title to your picture and print it if submission is required.
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