Chapter 3.2, Problem 124E

a.

To determine

To use : a graphing utility to graph g (for $n=2$ )

Explanation of Solution

Given information : The question is

Let $f\left(x\right)=\ln x\&g\left(x\right)=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$

Graph : Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the data $\ln x\&x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$ which is required to graph.

Step 4: Press $\boxed{\text{GRAPH}}$ button to graph the function.

The graph is obtained as:

  

Interpretation:

From the above graph it is observed that the graph shows between $f\left(x\right)$ and $g\left(x\right)$ at $n=2$

b.

To determine

which function is increasing at a greater rate as $x$ approaches infinitely

Answer to Problem 124E

The function is increasing at a greater rate as $f\left(x\right)$

Explanation of Solution

Given information : The question is

Let $f\left(x\right)=\ln x\&g\left(x\right)=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$

Calculation :

From the above graph the function is increasing at a greater rate as $f\left(x\right)$

c.

To determine

To repeat : part (a) and (b) for $n=3,4\&5$

Answer to Problem 124E

From part (a) and (b) for $n=3,4\&5$ the changes shows that function is increasing

Explanation of Solution

Given information : The question is

Let $f\left(x\right)=\ln x\&g\left(x\right)=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$

Calculation :

From part (a) and (b) the number of $n$ increases than the $f\left(x\right)$ is now increasing at a rate greater than $g\left(x\right)$