Total sales, Q, of a product are a function of its price and...

Total sales, $Q$, of a product are a function of its price and the amount spent on advertising. Figure 12.57 shows a contour diagram for total sales. Which axis corresponds to the price of the product and which to the amount spent on advertising? Explain.
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Key Concepts

Interpretation of Independent Variables

In this context, the product's total sales are modeled as a function of price and advertising spending. Recognizing which axis corresponds to which variable is fundamental. By the standard convention, the price is taken as the first variable plotted on the horizontal axis, while advertising spending, being the second variable, is plotted on the vertical axis.

Assignment of Axes in a Function of Two Variables

In functions of two variables, each variable is typically assigned to one axis on the coordinate plane. The convention is to place the first variable on the horizontal axis and the second on the vertical axis. This assignment helps in maintaining consistency when reading and interpreting graphs, such as contour diagrams.

Contour Diagrams

A contour diagram is a graphical tool used to represent a function of two variables by plotting curves on which the function has a constant value. These curves, known as level curves, help visualize how changes in the two input variables affect the output of the function without showing a full three-dimensional surface.

Level Curves

Level curves are drawn on a two-dimensional plane and each represents all the combinations of the input variables that yield the same output. They allow one to see the gradient or rate of change of the function in different regions of the domain, which is key in understanding how the independent variables interact.

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