Five different methods for calculating a particular part of the Mandelbrot set using Mathematica are demonstrated in this notebook. The first method is to write a function in Mathematica and then execute that function -- this method is very slow and takes about half an hour to execute for a 128X128 resolution problem. The second and third methods are to compile the function with either the Mathematica or InterCall compilers. This speeds up execution time by a factor of 20 and 60, respectively. The fourth and fifth methods, are to either Import a C-program with InterCall, or to Install it with MathLink. In either case the speedup in execution time is a factor of 180, i.e., down to about 10 seconds.