Chapter 1.5, Problem 65E

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 65E

When $0\le x\le 2$ then the magnitude $-9.5$ and when $x>6$ then the magnitude $-99.5$ .

Explanation of Solution

Given information: $f(x)=x^2-\frac{1}{2}\text{ , }g(x)=3x^2-1$

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 $f(x)=x^2-\frac{1}{2}$ 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 $g(x)=3x^2-1$ 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, $f(x)=x^2-\frac{1}{2}\text{ , }g(x)=3x^2-1$

  $\begin{array}{l}f(x)+g(x)=x^2-\frac{1}{2}\text{}+(-3x-1)\\ \text{ =}\frac{-4x-3}{2}=\frac{-(4x^2+3)}{2}\end{array}$ 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 $(f+g)(x)=\frac{-(4x^2+3)}{2}$ 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 $x=2$

  $\begin{array}{l}f+g=\frac{-(4\times 2^2+3)}{2}\\ \text{ =}\frac{-(16+3)}{2}\\ \text{ =}\frac{-19}{2}=-9.5\end{array}$

Hence, when $0\le x\le 2$ then the magnitude $-9.5$ .

And $x>6$

  $\begin{array}{l}f+g=\frac{-(4\times 7^2+3)}{2}\\ \text{ =}\frac{-(196+3)}{2}\\ \text{ =}\frac{-199}{2}=-99.5\end{array}$

Hence, when $x>6$ then the magnitude $-99.5$ .