# Section2.13Exercises

##### 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
##### 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

Find a value of $c$ such that $f_c(x)=x^2+c$ is affinely conjugate to $g(x)=(x-1)(x+2)\text{.}$ Show that both functions have neutral fixed points.

##### 6

Find an orbit of period 11 for the function $g(x)=4x(1-x)$

##### 7

We wish to find a number $x_0 \in I=[-2,2]$ whose orbit is dense in $I$ under iteration of $f(x)=x^2-2\text{.}$

1. Outline a strategy for finding $x_0\text{.}$
2. Find a decimal approximation to $x_0$ that is valid to 10 decimal places.
##### 8

We wish to find a number $x_0 \in I=[0,1]$ whose orbit is dense in $I$ under iteration of $g(x)=4x(1-x)\text{.}$

1. Outline a strategy for finding $x_0\text{.}$ Express it exactly in a form that uses a sum and, possibly, a conjugating function.
2. Find a decimal approximation to $x_0\text{.}$