Chapter 2.3, Problem 28E

(a)

To determine

To sketch:The graph of the continuous function.

Explanation of Solution

Given information:

A continuous function that is $-1\text{for}x\le -0.01,1\text{for }x\ge 0.01$ and is linear for $-0.01<x<\mathrm{0.01.}$

Graph:

The sigma function is defined as the function which has the value -1for the negative argument, 0 when the argument is 0 and 1 for the positive argument.

The graph for the given continuous function $-1\text{for}x\le -0.01,1\text{for }x\ge 0.01$ and is linear for $-0.01<x<\mathrm{0.01.}$ is as follows,

  

Interpretation:

From the graph it is obtained that the value -1 has negative argument and 1 has positive argument. The graph is increasing and decreasing with respect to the argument value

(b)

To determine

To find:The formula for the function.

Answer to Problem 28E

The formula for the function is given below as,

  $S(x)=\left\{\begin{array}{l}-1\text{if}x\le -0.01\\ -1+100(x+0.01)\text{if}\text{-0}\text{.01<x<0}\text{.01}\\ \text{1}\text{if }x\ge 0.01\end{array}\right\}$

Explanation of Solution

Given information:

A continuous function that is $-1\text{for}x\le -0.01,1\text{for }x\ge 0.01$ and is linear for $-0.01<x<\mathrm{0.01.}$

Calculation:

To determine the formula for the function as follows,

Consider the slope for (0.01.l) as 100, and the linear function for $-0.01<x<0.01$ .Linear function is written in the form of $f(x)=mx+b$ .Here the value of $m=100$ and as we want the linear for $-0.01<x<0.01$ .Hence the function as $-1+100(x+0.01)$ .

The formula for the function is given below as,

  $S(x)=\left\{\begin{array}{l}-1\text{if}x\le -0.01\\ -1+100(x+0.01)\text{if}\text{-0}\text{.01<x<0}\text{.01}\\ \text{1}\text{if }x\ge 0.01\end{array}\right\}$

(c)

To determine

The inputs have to be to 0 for the output to be within 0.1 for 0.

Answer to Problem 28E

The input should be within -0.001 to 0.001 for the output to be within 0.1 to 0.

Explanation of Solution

Given information:

A continuous function that is $-1\text{for}x\le -0.01,1\text{for }x\ge 0.01$ and is linear for $-0.01<x<\mathrm{0.01.}$

Calculation:

The given function is $f(x)=-1+100(x+0.01)$ .Since it is polynomial, it is continuous.

To determine the input at $x=0$ for the output within 0.1 to 0, first suppose that $f(x)=0.1$ implies $x=0.001$ .

Similarly, consider $f(x)=-0.1$ implies $x=-0.001$

From the two values of $f(x)$ we have,

  $\begin{array}{l}-0.01\le -1+10(x+0.01)\le 0.01\\ -0.01\le 100x\le 0.01\\ -0.001\le x\le 0.001\end{array}$

Hence, the input should be within -0.001 to 0.001 for the output to be within 0.1 to 0.