Chapter 3.1, Problem 86E

a.

To determine

To calculate : which function is an increase at the fastest rate for large values of $x$

Explanation of Solution

Given information : The question is

Using the graphing utility sketch the graph $y_1=e^x$ , $y_2=x^2$ , $y_3=x^3$ , $y_4=\sqrt{x}$ and $y_5=\left|x\right|$

Graph : 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 expression $y_1=e^x$ , $y_2=x^2$ , $y_3=x^3$ , $y_4=\sqrt{x}$ and $y_5=\left|x\right|$ which is required to graph.

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

The graph is obtained as:

  

b.

To determine

To use : part (a) to make a conjecture

Explanation of Solution

Given information : The question is

The rate of growth of $y_1=e^x$ and $y=x^n$ where $n$ is a natural number and $x$ is large

Calculation :

To make the conjecture that for whether $y=x^n$ , where $n$ is a natural number, $y=e^x$ will eventually surpass it in rate of increase for large $x$

c.

To determine

To use : part (a) and part(b) to describe that what is implied when its stated that the quantity is growing exponentially

Explanation of Solution

Given information : The question is

The rate of growth of $y_1=e^x$ and $y=x^n$ where $n$ is a natural number and $x$ is large

It is implied that when saying growing exponentially it means an extremely high rate of growth