Exercise 11.  Use Theorem 4.11 and the paragraph immediately before it to show that the point    belongs to the Mandelbrot set.  

Solution 11.

See text and/or instructor's solution manual.

If we let  ,  then  

                      

                     

Hence, if   ,   then   .  

Therefore, by Theorem 4.11, the point    belongs to the Mandelbrot set.  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can let Mathematica explore what is happening with  .  

The orbit of  0  determined by     is bounded, hence      is in the Mandelbrot set.    

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell