CALCULUS PROJECT:
Implicit Differentiation
of the Eight Curve
Implicit Differentiation of the Eight Curve
The graph of the eight curve, 
,
is shown below.
1. Solve the equation for y and use the results to graph the curve using your graphing calculator, letting a = 0.5, 1, 2, 3, 4, and 5. [You need NOT draw by hand the results.] Describe the results as a changes.
2. Use WinPlot to sketch the curve as it is implicitly defined. Print the results for four different values of a. Indicate scale on the axes and display the equation with each curve.
3. Find the derivative of the explicit versions of the equation.
4. Show by implicit differentiation that results in 
.
Show that this is the same as the result from part 3.
5. Show that
can be simplified further so that 
.
6. Using ,
determine the points on the curve where there are horizontal
tangent lines and vertical tangent lines.
7. Use the results from part 5 to determine the domain and range of the eight curve in terms of a.
8. Show that the slope of a line tangent to the curve at the
origin is always ±1. Use the implicit derivative for this
activity.
This page was created on December 11, 2003.
Last Updated on August 25, 2004.