## Section2.12Exercises

###### 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.