Module 13  Implicit Differentiation  
Introduction  Lesson 1  Lesson 2  SelfTest  
Lesson 13.1: Graphs of Implicit Relations  
This lesson investigates graphing implicit relations. The expression x^{3} + y^{3} – 9xy = 0 implies a relation between x and y but the relation may not be obvious. A graph will help to illustrate the relation. Graphing an Implicit Relation The TI89 can graph an implicit relation that has two variables x and y, but the Graph mode needs to be set properly. Setting the Mode
Entering the Relation In FUNCTION graphing mode, you define a set of functions of the form y(x), such as y1, but in 3D Graph mode, each function has the form z(x, y). That is, z is a function of two variables x and y and the graph of the function is a surface in threespace. When the Graph mode is 3D the names of the functions shown in the Y= Editor are z1, z2, etc. On the TI89, an implicit relation is graphed by plotting all the points where z(x, y) = 0. The relation under consideration is x^{3} + y^{3} – 9xy = 0 , so let z = x^{3} + y^{3} – 9xy.
Setting the Graph Format Have the calculator display the axes, which are probably turned off in the 3D graphing mode.
Selecting Implicit Plot Style
Setting the Viewing Cube Values For 3D graphs the Viewing Window is called the Viewing Cube, and it is defined by the values of xmin, xmax, ymin, ymax, zmin, and zmax. The viewing angle is determined by the three components shown below. eye is the angle in the x y  plane from the positive xaxis. eye is the angle from the positive zaxis. eye is the angle by which the graph is rotated counterclockwise around the line of sight set by eye and eye . The three viewing angle components are always entered as DEGREE measures regardless of the currently selected Angle mode. Setting the three viewing angle components as shown below displays the graph of the implicit relation as seen from above the xy  plane with the x and y axis in the standard orientation on the screen. Whenever one changes into IMPLICIT PLOT mode, the three viewing angles are set to these values automatically.
Be patient. It will take a minute or two for the graph to appear.
This graph is called a folium. It dates to Descartes in 1638.
13.1.1 Find the points on the graph of x^{3} + y^{3} – 9xy = 0 when x = 4 by using the solve command. 

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