Implicit Curves and Mathematica

Unique curves can be generated by graphing implicit equations in Mathematica.

1. a. Look up ImplicitPlot in the Master Index. Notice that use the function ImplicitPlot, a graphics package must first be loaded. This is done by either entering << Graphics`ImplicitPlot` by typing it in or copying and pasting from the Master Index. (Also notice that the "apostrophe" is actually a backwards single quote - the key is on the upper left of your keyboard.) b. Use ImplicitPlot to plot the curve y(y2 -1)(y- 2) = x(x -1)(x - 2) over the x- range of -2 to 5.

dy c. Use implicit differentiation (by hand) to find and then use that to help find the dx equation of the line tangent to the curve at the point (0,1). Use the ImplicitPlot function to plot both the curve and this tangent line on the same set of axes. d. At how many points does the curve appear to have horizontal tangents? Use the Solve function in Mathematica to help you solve for the exact x-values at which the curve has horizontal tangents. e. Use the Solve function and copying and pasting to find the exact y-values at which the horizontal tangents occur. f. On the same set of axes, graph the curve and all of the horizontal tangents.

2. Starting with the implicit curve in question 1, experiment with the equation to find other graphs which you find interesting or unique. Include at least 3 in your report.

3. The curve with the equation 2y3 + y2 - y5 = x4 - 2x3 + x2 has been likened to a bouncing wagon. Use Mathematica to graph the curve and see why. Then repeat parts c through f (using the point (1,0) for part c) from question 1 with this curve.