Chapter 3.1, Problem 82E

a.

To determine

To find which function increases at faster rate for “large’ values of $x$ .

Explanation of Solution

Given information:

The given statement is:

“Use a graphing utility to graph $y_1=e^x$ and each of the functions $y_2=x^2,y_3=x^3,y_4=\sqrt{x}\text{ and }{y}_5=\left|x\right|$ in the same viewing window.”

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 $y_1=e^x,y_2=x^2,y_3=x^3,y_4=\sqrt{x}\text{ and }{y}_5=\left|x\right|$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

For large values of $x$ , the function $y_5=\left|x\right|$ increases at the fastest rate.

b.

To determine

To make a conjecture about the $y_1=e^x\text{ and }y=x^n$ growth rates, where $n$ is a natural number and $x$ is large.

Answer to Problem 82E

For large values of $x$ , the rate of growth of $y_1=e^x$ is more.

Explanation of Solution

Given information:

The given statement is:

“Use a graphing utility to graph $y_1=e^x$ and each of the functions $y_2=x^2,y_3=x^3,y_4=\sqrt{x}\text{ and }{y}_5=\left|x\right|$ in the same viewing window.”

For large values of $x$ , the rate of growth of $y_1=e^x$ is more.