Example 6.1.12. Table and graph.

Figure 6.1.13. Video Producing a graph from a table
I have a company that produces two products, widgets and gizmos. The two demand functions are:
\begin{align*} \text{PriceGizmo}\amp =10-\text{QuantityGizmo}/50\\ \text{PriceWidget}\amp =20-\text{QuantityWidget}/40\text{.} \end{align*}
Produce a table and a graph for revenue as a function of the quantity of gizmos and widgets produced.
Solution.
We need to start by producing a formula for revenue. To shorten the equations we will abbreviate the terms or use initials. We need formulas for revenue for each of our products:
\begin{align*} \text{RevG}\amp =\text{PriceG}*\text{QG}=\left(10-\frac{\text{QG}}{50}\right)\text{QG}=10\text{QG}-\frac{\text{QG}^2}{50}\\ \text{RevW}\amp =\text{PriceW}*\text{QW}=\left(20-\frac{\text{QW}}{40}\right)\text{QW}=20\text{QW}-\frac{\text{QW}^2}{40}\text{.} \end{align*}
Putting the equations together gives an equation for revenue.
\begin{equation*} \revenue=\text{RevG}+\text{RevW}=10\text{QG}-\frac{\text{QG}^2}{50}+20\text{QW}-\frac{\text{QW}^2}{40}\text{.} \end{equation*}
Next we build a build a table for the function as we have done above.
Figure 6.1.14. First table
Finally, we would like to see a graph of the function. We notice that the 3D plots in Excel have a number of drawbacks. The plots do not label the input variables. These first plots also don’t tell us what values of the variables correspond to particular points on the graph. Some of these drawbacks can be overcome, but only with more work than we wish to expend in this course. We will only add one non-intuitive option to make the graphs work better.
Figure 6.1.15. Second table
We will move the names of the variables out of the upper left corner of the chart and into the row above and to the side of the data. We leave the corner cell blank. This will let us see the values of the variables in the graphs. In the table, we select the data we would like to graph. In this example we select from cells B4 through M12. Finally, we select a chart to insert. The charts we are interested in are surface charts. The types of interest are 3-D Surface, Wireframe 3-D, and contour. Each of these chart types highlights some useful information.
The 3-D Surface gives a fast picture. It is useful in seeing local minimums and maximums.
Figure 6.1.16. Surface chart
The Wireframe 3-D chart emphasizes that we can build a reasonable picture from the curves obtained by treating either \(x\) or \(y\) as a constant. It lets us understand a function of 2 variables by putting together a collection of several functions of one variable. This point of view will be useful when we try to take derivatives.
Figure 6.1.17. Wire frame chart
The Contour chart emphasizes the level curves. The rate of change will be fastest in a direction perpendicular to the level curves.
Figure 6.1.18. Contour chart
Another alternative for seeing a graph is to use Wolfram Alpha. Unfortunately variable names in Alpha seem to be limited to a single letter, or a letter followed by a digit. Thus we change the formula to one using the names g and w.
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