Chapter 1.5, Problem 37E

To determine

To find: The function contribute most to the magnitude of the sum when $0\le x\le 2$ and $x>6$ and use a graphing utility to graph f, g, f+g.

Answer to Problem 37E

When $0\le x\le 2$ then the magnitude $5.2$ and when $x>6$ then the magnitude $13.3$ .

Explanation of Solution

Given information: $f(x)=3x\text{ , }g(x)=-\frac{x^3}{10}$

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 $\text{f(x)=3x}$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

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 $g(x)=-\frac{x^3}{10}$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

Now, we find $f+g$

Given, $f(x)=3x\text{ , }g(x)=-\frac{x^3}{10}$

  $\begin{array}{l}f(x)+g(x)=3x-\frac{x^3}{10}\\ \text{ =}\frac{30x-x^3}{10}=\frac{x(30-x^2)}{10}\end{array}$ Then, 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 $(f+g)(x)=\frac{x(30-x^2)}{10}$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

now, the magnitude of the sum when $0\le x\le 2$

Putting $x=2$

  $\begin{array}{l}f+g=\frac{2(30-4)}{10}\\ \text{ =}\frac{2\times 26}{10}=5.2\end{array}$

Hence, when $0\le x\le 2$ then the magnitude $5.2$ .

And $x>6$

  $\begin{array}{l}f+g=\frac{7(30-49)}{10}\\ \text{ =}\frac{-7\times 19}{10}=13.3\end{array}$

Hence, when $x>6$ then the magnitude $13.3$ .