Chapter 1.1, Problem 49E

Solution

a)

To explain: Algebraically why $y=x$ for $x\ge 0$ .

Given Information:

The equation $y={\left(x^{200}\right)}^{\frac{1}{200}}$ .

Calculation:

Simplify $y={\left(x^{200}\right)}^{\frac{1}{200}}$ as follow.

  $\begin{array}{c}y={\left(x^{200}\right)}^{\frac{1}{200}}\\ =x^{200\cdot \frac{1}{200}}\\ =x^1\\ =x\end{array}$

Therefore, $y=x$ for $x\ge 0$ .

b)

To graph: The equation $y={\left(x^{200}\right)}^{\frac{1}{200}}$ .

Given Information:

The equation $y={\left(x^{200}\right)}^{\frac{1}{200}}$ .

Graph:

The graph of $y={\left(x^{200}\right)}^{\frac{1}{200}}$ is shown in figure 1.

  

Figure 1

c)

To state: Whether the graph of $y={\left(x^{200}\right)}^{\frac{1}{200}}$ is different from the graph of $y=x$ .

Given Information:

The equation $y={\left(x^{200}\right)}^{\frac{1}{200}}$ .

Explanation:

From figure 1 it follows that graph of $y={\left(x^{200}\right)}^{\frac{1}{200}}$ does not passes from the point 0 whereas the graph of $y=x$ is a straight line passing through 0.

Therefore, the graph of $y={\left(x^{200}\right)}^{\frac{1}{200}}$ is different from that of $y=x$ .

d)

To explain: Why the grapher failed..

Given Information:

The equation $y={\left(x^{200}\right)}^{\frac{1}{200}}$ .

Explanation:

Since $x^{200}$ is extremely very small when $x<0.3$ so, the grapher is unable to differentiate between $x^{200}$ and 0.

Thus, the grapher shows ${\left({\left(0.3\right)}^{200}\right)}^{\frac{1}{200}}=0$ .

Therefore, the grapher failed.