Tutorial: Functions and models

This tutorial: Part A: Cost, revenue, and profit models

Go to Part B: Demand, supply and time-change models

(This topic is also in Section 1.2 in Finite Mathematics and Applied Calculus)

#[I don't like this new tutorial. Take me back to the older tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#

Resources

Function evaluator and grapher

Excel grapher

What is a mathematical model?

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

Mathematical models

To model a situation mathematically means to represent it in mathematical terms, and the particular representation we use is called a mathematical model of the situation.

%%Examples

Situation	Model
1. It costs your donut delivery service an average of $\$2$ to deliver each box of organic donuts, plus an additional $\$100$ per day for associated overheads (wages, rent, utilities, etc.) Model the daily cost of delivering organic donuts as a function of the number of boxes delivered.	$C(x) = 2x + 100$ #[$C$ = daily cost to deliver organic donuts, $x$ = number of boxes delivered][$C$ = costo diario entregar donas rgánicas, $x$ = número de cajas entregadas]#
2. 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.	$N(t) = 2t + 50$ #[$N$ = number of movies, $t$ = time in weeks][$N$ = número de películas, $t$ = tiempo en semanas]#
3. I invest $\$900$ at 4% annual interest compounded quarterly. Find the value of the investment after $t$ years	$V(t) = 900(1.01)^{4t}$ #[$V$ = value of investment, $t$ = time in years][$V$ = valor de la inversión, $t$ = tiempo en años]# (#[See][Mira]# %%partBtut.)
4. Facebook membership model (2004–2009)	$\displaystyle n(t) = \begin{cases} {4t} & \text{if } 0 \leq t \leq 3 \\{50t-138}& \text{if } 3 \lt t \leq 5 \end{cases}$ #[$n$ = number of members in millions, $t$ = time in years since 2004][$n$ = número de miembros en millones, $t$ = tiempo en años desde 2004]# (#[See the][Mira el]# %%prevtut.)

%%Note
Examples 1–3 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 first example above, where the daily cost to deliver $x$ boxes of organic donuts is is expressed as a function of $x.$ This function is an example of a cost function. Notice that this particular cost function,][Eche un vistazo al primer ejemplo anterior, donde el costo entregar $x$ cajas de donas orgánicas se expresa como una función de $x.$ Esta función es un ejemplo de una función de costo. Observe que esta función de costo particular,]#

$C(x) = 2x + 100,$

(which happens to be a linear function) is a sum of two parts: a constant, or fixed cost, $\$100$, which is the same regardless of the number $x$ of boxes being delivered, or "items," and a variable cost, $2x,$ which does depend on the number of items:

Cost = Variable Cost + Fixed Cost

The quantity 2 by itself is the incremental cost per call; you might recognize it as the slope of the given linear function. In this context we call 2 the marginal cost. You might also recognize the fixed cost 100 as the $C$-intercept of the linear cost function.

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 $m$ in a linear cost function is the marginal cost, and measures the incremental cost per item.

Examples

The example we considered above: The daily cost to your donut service to prepare $x$ boxes of organic donuts is

$C(x) = 2x + 100\qquad$ \t #[$C$ = daily cost, $x$ = number of boxes prepared][$C$ = costo diario, $x$ = número de cajas preparados]#

(which happens to be a linear function). The fixed cost is $\$100$, the variable cost is $2x,$ and the marginal cost is $2.$

An example for you

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

Your donut service sells organic donuts for $\$4.50$ per box. Thus, the revenue from the sale of $x$ boxes is

$R(x) = 6.50x$ $\qquad$ \t $R$ =revenue, $x$ = number of boxes sold

The marginal revenue is $m = \$4.50$ per box.

#[Profit Function][Función de ganancia]#

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][Ganancia]# \t ${}={}$ #[Revenue − Cost][Ingreso − Costo]# \\ #[$P$][$G$]# \t #[${}= R - C$][${}= I - C$]#

#[If the profit is negative, say &minus\$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][Si la ganancia es negativa, digamos &menos\$500, nos referimos a una pérdida (de \$500 en este caso). Cubrir gastos significa no obtener ni ganancias ni pérdidas. Por lo tanto, el

punto de equilibrio

se produce cuando $G = 0,$ o]#

#[$R = C \iff P = 0$][$I = C \iff G = 0$]# \t \t #[Break even][Equilibrio]#

The break even point is the number of items $x$ at which break even occurs.

Example

Continuing with the donut scenario: Given the above cost and revenue functions, the profit function is

#[$P(x){}$][$G(x){}$]# \t #[${}= R(x) - C(x)$][${}= I(x) - C(x)$]# \\ \t ${}= 4.5x - (2x + 100)$ \\ \t ${}= 2.5x - 100$

For break even, we set $P = 0:$.

$P {}= 2.5x - 100 = 0$${}\ \implies x = \dfrac{100}{2.5} = 40$ #[boxes][cajas]#

So, to avoid a loss, you would need to sell at least 40 boxes of organic donuts.

An example for you

Functions versus equations

%%Q #[In my business and finance courses, I learn about "demand equations, cost equations" etc, but here we appear to be talking about "demand functions, cost functions," and so on. What is the difference?][En mis cursos de negocios y finanzas, aprendo sobre "ecuaciones de demanda, ecuaciones de costos", etc. pero aquí parece que estamos hablando de "funciones de demanda, funciones de costos" y así. ¿Cuál es la diferencia?]#
%%A #[It's just really a question of how we express things. As mathematicians we ike to write equations in function form when possible, but we could equally have been talking about equations rather than functions. Here is a brief guide as to how to translate from the language of functions to that of equations and vice-versa:][En realidad es sólo una cuestión de cómo expresamos las cosas. Como matemáticos nos gusta escribir ecuaciones en forma de función cuando posible, pero igualmente podríamos haber estado hablando de ecuaciones en lugar de funciones. Aquí hay una breve guía sobre cómo traducir del lenguaje de funciones al de ecuaciones y viceversa:]#

#[Function and equation form of mathematical models][Forma de función y de ecuación de los modelos matemáticos]#

#[As an example, take another look at the cost and revenue functions for your donut operation:][Como ejemplo, echa otro vistazo a las funciones de costos e ingresos de su operación de donas:]#

$C(x) = 2x + 100$ $\qquad$ \t #[Cost function][Función de costos]# \\ #[$R(x) = 4.5x$][$I(x) = 4.5x$]# \t #[Revenue function][Función de ingresos]#

#[Instead of using function notation, we could express the cost and revenue functions using equation notation:][En vez de usar notación de función, podemos expresar las funciones de costos y ingresos en la forma de ecuaciones:]#

$C = 2x + 100$ $\qquad \quad$ \t #[Cost equation][Ecuación de costos]# \\ #[$R = 4.5x$][$I = 4.5x$]# \t #[Revenue equation][Ecuación de ingresos]#

#[Here, the independent variable is $x,$ and the dependent variables are $C$ and $R.$ Function form 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 $x.$][Aquí, la variable independiente es $x,$ y las variables dependientes son $C$ y $I.$ La forma de función y la forma de ecuación, que usan la misma letra para el nombre de la función y la variable dependiente, a menudo se usan indistintamente, por lo que podemos decir, por ejemplo, que la ecuación de costos anterior especifica $C$ como función de $x.$]#

Now try the exercises in Section 1.2 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.