Exercises

Section 2.12 Exercises

1

Let \(f:\mathbb R\to \mathbb R\) be continuously differentiable. We say that \(x_0\) is a simple root of \(f\) if \(f(x_0)=0\) and \(f'(x_0)\neq0\text{.}\) Show that if \(x_0\) is a simple root of \(f\text{,}\) then \(x_0\) is a super-attracting fixed point of the Newton's method iteration function \(N\) for \(f\text{.}\)

2

Find an example of a continuously differentiable function \(f:\mathbb R\to \mathbb R\) that attracts no critical point.

Hint

Draw a graph. Of course, you can't violate theorem Theorem 2.6.2.

3

Let \(f(x)=x^2-4x+5\text{.}\) Show that \(f\) has a super-attractive orbit of period 2.

4

Let \(f(x)=3 x^2-6 x+3.415\text{.}\) Find all attractive orbits of \(f\text{.}\)

5

Use software to find all orbits of period 2, 3, and 4 for \(f(x)=x^2-2\text{.}\)

6

Use software to find one orbit of period 11 for \(f(x)=x^2-2\text{.}\)

7

Let \(f_{\lambda}(x)=\lambda x(1-x)\text{.}\) Find the value of \(\lambda\) such that \(f_{\lambda}\) has a super-attractive orbit of period 3.

8

Find an orbit of period 11 for the doubling map.

9

Show that there is no orbit of the doubling map that is dense in some proper subinterval of \(H\) but not dense in \(H\) itself.