(This topic is also in Section 1.3 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)
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To use this tutorial, you should be familiar with methods for finding the equation of a straight line with specified data. See the preceding tutorial to learn about some methods.
Using linear functions to describe or approximate relationships in the real world is called linear modeling. Here, we study several kinds of linear model:
A cost function gives us the cost as a function of the number of items. Here, we are interested in linear cost functions.
Linear Cost Function
A linear cost function expresses cost as a linear function of the number of items. In other words,
Here, C is the total cost, and x is the number of items. In this context, the slope m is called the marginal cost and b is called the fixed cost.
The daily cost to print x paperback sci-fi novels is
Note that C is measured in dollars, and x is measured in books (paperback sci-fi novels, to be precise).
The marginal cost is m = 3.5, and the fixed cost is b = 1000.
To understand what these quantities tell us, let us compute some costs:
Units of Measurement.
As we saw above, x is measured in items (books in the above example) and C is measured in units of currency (dollars in the above example). Also:
Revenue results from the sale of items, and profit is the excess of revenue over costs. Both the revenue and profit depend on the number of items, x, we buy and sell, and so, like the cost function, they too are functions of x.
Linear Revenue and Profit Functions
The marginal revenue is m = $4.75 per book. Notice again that the units of measurement of the slope are units of y (dollars) per unit of x (books).
The profitis the net proceeds, or what remains of the revenue when costs are subtracted. If the profit depends linearly on the number of items, the slope m is called the marginal profit. Profit, revenue, and cost are related by the following formula.
P = R - C.
If the profit is negative, say -$500, we refer to a loss (of $500 in this case). To break even means to make neither a profit nor a loss. Thus, break-even occurs when P = 0, or
The break-even point is the number of items x at which break-even occurs.
Going back to the sci-fi books, we already have the cost and revenue functions:
Thus, you should sell 800 books per day to break even, and more than that to make a profit of $1.25 per additional book. ($1.25 is the marginal profit.)
Now go on to Part B: Linear Demand, Supply, and Time-Change Models to see other important applications of linear models.
Alternatively, try some of the exercises in Section 1.3 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.
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