Functions and Models

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Functions that we use to represent real life situations, like the function we used in the preceding tutorial to talk about Facebook membership, are mathematical models.
Mathematical Models

To mathematically model a situation means to represent it in mathematical terms. The particular representation used is called a mathematical model of the situation.

Examples
Situation Model
1. There are currently 50 movies on your hard drive, and this number is growing by 2 per week. Model the size of your collection as a function of time.
N(t) = 50 + 2t
t = time in weeks, N = number of movies
2.
t = time in days, E = number of e-mails
3.
x = number of sodas, C = total cost
4. I invest $1000 at 5% interest compounded quarterly. Find the value of the investment after t years.
A(t) = 1000(1 + 0.0125)^{4t}
From the compound interest formula (see Part B)
5. Facebook membership
n(t) = 4t if 0 ≤ t ≤ 3       million members
50t-138     if 3 < t ≤ 5

t = time in years since the start of 2004, n = membership in millions

Note
Examples 1–4 are analytical models, obtained by analyzing the situation being modeled, whereas Example 4 is a curve-fitting model, obtained by finding a mathematical formula that approximates the observed data.

Cost, Revenue and Profit Models

Take a look at the third example above, where the total cost of purchasing a number of items (sodas) was expressed as a function of the number of items x. This function is an example of a cost function. The rest of this explanation will appear here only after you correctly entered the function in the third example above.
Cost Function

A cost function specifies the cost C as a function of the number of items x. Thus, C(x) is the cost of x items, and has the form

    Cost = Variable cost + Fixed cost
where the variable cost is a function of x and the fixed cost is a constant. A cost function of the form
    C(x) = mx + b
is called a linear cost function; the variable cost is mx and the fixed cost is b. The slope m, the marginal cost, measures the incremental cost per item.
Example

The daily cost to your company 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 = and the fixed cost is b =

Revenue Function

The revenue resulting from one or more business transactions is the total payment received, sometimes called the gross proceeds. If R(x) is the revenue from selling x items at a price of m each, then R is the linear function R(x) = mx and the selling price m can also be called the marginal revenue.

Example

Suppose that your publishing company sells sci-fi paperbacks to a large retailer for per book. Then

    dollars.
The marginal revenue is m = per book.
Profit Function

The profit is 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.

    Profit = Revenue − Cost
    P = RC
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
    R = C Break Even
The break even point is the number of items x at which break even occurs.
Example

Going back to the sci-fi novels, we already have the cost and revenue functions:

    Daily cost to make x books
    Revenue from the sale of x books
Thus, you should sell ??? books per day to break even, and more than that to make a profit of $??? per additional book. ($??? is the marginal profit.)

Sometimes if it more convenient to express models in equation form:
Function and Equation Form of Mathematical Models

As an example, take a look at the above cost and revenue functions again:

    Cost function
    Revenue function
Instead of using function notation, we could express the cost and revenue functions using equation notation:
    Cost equation
    Revenue equation

Here, the independent variable is x, and the dependent variables are C and R. Function notation and equation form, using the same letter for the function name and the dependent variable, are often used interchangeably, so we can say, for example, that the cost equation above specifies C as a function of x.

The next quiz example is similar to Example 2 in Section 1.2 of :

You are the manager of Sassy Surf Creations, a new trend-setting clothing manufacturer. The cost function for your very exclusive Tai Kwon Do Dragon T shirts is dollars, and you sell the shirts for each. Take a look at the graphs of cost and revenue below the questions to assist you in answering them, but you need to use the formulas to give the exact answers.

        Revenue and Cost Functions

Last Updated: November, 2009
Copyright © 2009 Stefan Waner