Assignment #5



Examples:
Figure out what the following commands do.

> with(plots):
> pointplot({[0,1],[1,-1],[3,0],[4,-3]},axes=boxed,view=[-5..5,-5..5]);
> implicitplot( y^2 - x^3 + x, x = -5..5 , y=-5..5);
> polarplot([2*cos(t), sin(t), t = -Pi .. Pi], numpoints = 50);
> plot([x^2-x, sqrt(x), x = 0 .. 3*Pi], -8 .. 8, -10 .. 10);
> ?plot[parametric]


The Folium of Descartes (see the wikipedia page about this curve) is an algebraic curve defined by the equation $$x^3 + y^3 - 3 axy = 0.$$ You will need the information on this wikipedia page in order to plot the following graphs. I'll just ask you for this assignment to plot them for $a=1$, but you should try changing the value of $a$ to see how this changes the graph.

Question #1:
Graph the curve with $a=1$ using the command implicitplot in the range of $-3 \leq x \leq 3$ and $-3 \leq y \leq 3$. Add the plot of the line $x+y+a=0$ on the same graph but dashed and with a different color.

Question #2:
Graph the same curve using the command polarplot (again with the plot of the line for comparison).

Question #3:
Graph the same curve using the command plot in its rational parametric equations form (again with the plot of the line for comparison).

Question #4:
Graph the same curve rotated 45 degrees as is described on the wikipedia page. Use this equation to find the area enclosed by the loop.

wikipedia says: The curve was first proposed by Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

Question #5:
Write a function that given a point $p$ in its rational parameterization, returns a formula for the tangent line to the curve. Plot this line on the same graph as the Folium of Descartes for at least one point.


You should open up a new worksheet and start from scratch. You will have to save your work in a file and upload that file on to the course moodle. Your solution should be a sequence of commands where it is easy to change the input string and after you execute the sequence of commands you should have the correct output string. Please add documentation to your worksheet to explain how it works. Just a few sentences is sufficient, but imagine that you were opening up the worksheet for the first time and wanted to know what it did.

You should finish your assignment by October 16. Please document your answers and ensure that the worksheet is self contained and readable.