Chapter 14.CR, Problem 23CR

To determine

a.

To find:

The equilibrium points $x,0\le x\le 1$ for the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

b.

To find:

The value of $\left|f^{\prime }\left(x\right)\right|$ at all equilibrium points for the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

c.

The stability of all equilibrium points in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

d.

To find:

The next four values of the sequence, starting with $x_1=0.15$ in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

e.

To find:

The next four values of the sequence, starting with $x_1=0.4$ in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

f.

To find:

The next four values of the sequence, starting with $x_1=0.65$ in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

g.

To find:

The next four values of the sequence, starting with $x_1=0.85$ in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

h.

To draw:

A cobweb diagram, starting with $x_1=0.4$ and iterating four times in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.

To determine

i.

To discuss:

The results of part d through g compare with the results of part c times in the expression $f\left(x\right)=\frac{1}{4}-\left|\frac{x}{2}-\frac{1}{4}\right|$.