Chapter 4.1, Problem 41E

(a)

To determine

To find: The rate of growth of the given function by the graph of both the functions in the give rectangles.

Explanation of Solution

Given:

The given functions are,

  $\begin{array}{l}\left[0,5\right]\text{by}\left[0,20\right]\\ \left[0,25\right]\text{by}\left[0,{10}^7\right]\\ \left[0,50\right]\text{by}\left[0,{10}^8\right]\end{array}$

The given functions is $f\left(x\right)=2^x$ and $g\left(x\right)=x^5$ .

Calculation:

Consider the case for the viewing rectangle $\left[0,5\right]\text{by}\left[0,20\right]$ .

From the graphical device draw the graph for the $f\left(x\right)=2^x$ and $g\left(x\right)=x^5$ in the viewing rectangle.

The required graph is shown in Figure 1

  

Figure 1

Consider the case for the viewing rectangle $\left[0,25\right]\text{by}\left[0,{10}^7\right]$ .

From the graphical device draw the graph for the $f\left(x\right)=2^x$ and $g\left(x\right)=x^5$ in the viewing rectangle.

The required graph is shown in Figure 2

  

Figure 2

Consider the case for the viewing rectangle $\left[0,50\right]\text{by}\left[0,{10}^8\right]$ .

From the graphical device draw the graph for the $f\left(x\right)=2^x$ and $g\left(x\right)=x^5$ in the viewing rectangle.

The required graph is shown in Figure 3

  

Figure 3

(b)

To determine

To find: The solution of the given equation round to one decimal place.

Answer to Problem 41E

The solution of the equation is $x=1.2,2.24$ .

Explanation of Solution

Given:

The given equation is,

  $2^x=x^5$

Calculation:

Consider the given equation.

  $2^x=x^5$

The graph of the Figure 2 and Figure 3 shows that the graph of the functions $f\left(x\right)=2^x$ and $g\left(x\right)=x^5$ intersect at $x=1.2$ and $x=2.24$ .

Thus the solution of the equation is $x=1.2,2.24$ .