Tutorial: Linear functions and models
This tutorial: Part C: Applications: Linear models
(This topic is also in Section 1.3 in Finite Mathematics and Applied Calculus)
Linear modeling
Using linear functions to describe or approximate relationships in the real world is called linear modeling.
A linear model that tells us how the value of the variable $y$ depends on the value of the variable $x$ has the form $y = mx + b$, where $m$ and $b$ are constants. $y$ is the dependent variable and $x$ is the independent variable.
The slope $m$ is the rate at which $y$ is increasing per unit increase in $x,$ while the $y$-intercept $b$ is the value of $y$ that corresponds to $x = 0.$ The slope $m$ is measured in units of $y$ per unit of $x,$ while the intercept $b$ is measured in units of $y.$
#[In this tutorial we will focus on the following specific kinds of linear models:][En este tutorial nos centraremos en los siguientes tipos específicos de modelos lineales:]#
#[Linear Cost functions][Funciones lineales de costos lineal]#
\\ #[Linear demand functions][Funciones lineales de demanda lineal]#
\\ #[Linear time-change functions][Funciones lineales de cambio en el tiempo]#
We already studied these concepts in the %%modelsstut, but here we focus on constructing linear models for them.
#[Linear cost functions][Funciones lineales de costos]#
We saw in the %%modelsstut that a cost function specifies the cost $C$ as a function of the number of items $x,$ and we also saw that a cost function has the general form
$C(x) = {}$ Variable Cost + Fixed Cost
where the variable cost depends on $x$ and the fixed cost is a constant.
Linear cost function
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
It costs your organic donut service $\$100$ per day to rent display space at the local OrganicMart, plus and additional $\$2$ to prepare each box of organic donuts. The cost function is therefore
$C(x) = 2x + 100\qquad$ \t #[$C$ = daily cost, $x$ = number of boxes prepared][$C$ = costo diario, $x$ = número de cajas preparados]#
The fixed cost is $\$100$, the variable cost is $2x,$ and the marginal cost is $2.$
Some for you
Units of measurement
Look again at the organic donuts cost function. Recall that the cost to make $x$ boxes of donuts was
%%A Let's look at the terms one by one:
$C(x) = 2x+100$ #[Dollars][Dólares]#
%%Q What are the units of measurement of the terms in a linear cost function like this one? %%A Let's look at the terms one by one:
- Units of C: First, the cost itself $C(x) = mx+b = 2x+100$ is measured in dollars as stated.
- Units of mx and b: It follows that the variable cost, $mx=2x$ and fixed cost $b=100$ are also measured in dollars.
- Units of x: We stated above that a cost function specifies the cost $C$ as a function of the number of items x, so $x$ is the number of items; in this case is the number of boxes (of donuts), and so its units of measurement are boxes (or perhaps "boxes of donuts").
- Units of m: As $mx = 2x$ is the cost to prepare $x$ boxes, it must be that $m = 2$ by itself is the cost per box. So, its units of measurement are dollars per box (often written as dollars/box).
#[Your turn][Tu turno]#
#[Linear demand functions][Funciones lineales de demanda]#
As we saw in the %%modelsstut, a demand function expresses demand $q$
(the number of items demanded--for example the number of items sold per month) as a function of the unit price $p$ (the price per item).
Linear demand function
A linear demand function has the form
The unit price, $p$, is measured in units of currency (for instance dollars) and $q$ is measured in units of demand (for instance items sold per month). The slope $m$ is measured in units of $q$ per unit of $p$; that is, units of demand per unit price (for instance, monthly sales per \$1 increase in the price). The intercept is measured in the same units as $q$ (units of demand). #[Interpretation of $m$][Interpretación de $m$]#
The (usually negative) slope $m$ measures the change in demand per unit change in price. Thus for instance, if $p$ is measured in dollars and $q$ in monthly sales, and $m = -400$, then each \$1 increase in the price per item will result in a drop in sales of $400$ items per month. #[Interpretation of $b$][Interpretación de $b$]#
The quantity $b$ gives the intercept on the vertical ($q$) axis; which gives the demand when $p = 0$; that is, the demand if the items were given away.
A linear demand function has the form
$q(p) = mp+b$ \gap[40] \t #[Function form][Forma función]#
\\ $q = mp+b$ \gap[40] \t #[Equation form][Forma ecuación]#
#[Note: $p$ plays the role of $x$ and $q$ plays the role of $y$][Nota: $p$ juega el papel de $x$ y $q$ juega el papel de $y$]#
#[Units of measurement][Unidades de medida]#
The unit price, $p$, is measured in units of currency (for instance dollars) and $q$ is measured in units of demand (for instance items sold per month). The slope $m$ is measured in units of $q$ per unit of $p$; that is, units of demand per unit price (for instance, monthly sales per \$1 increase in the price). The intercept is measured in the same units as $q$ (units of demand). #[Interpretation of $m$][Interpretación de $m$]#
The (usually negative) slope $m$ measures the change in demand per unit change in price. Thus for instance, if $p$ is measured in dollars and $q$ in monthly sales, and $m = -400$, then each \$1 increase in the price per item will result in a drop in sales of $400$ items per month. #[Interpretation of $b$][Interpretación de $b$]#
The quantity $b$ gives the intercept on the vertical ($q$) axis; which gives the demand when $p = 0$; that is, the demand if the items were given away.
%%Example
If the demand for T-shirts, measured in daily sales, is given by
If the demand for T-shirts, measured in daily sales, is given by
$q = -4p + 90\qquad$
where $p$ is the sale price in dollars, then daily sales drop by four T-shirts for every \$1 increase in price. It the T-shirts were given away, the demand would be 90 T-shirts per day.
#[Linear time-change functions][Funciones lineales de cambio en el tiempo]#
In %%modelsPtBstut we saw examples of functions used to model quantities $q$ that are functions of time $t$. When $q(t)$ happes to be a linear function, we are modeling linear change over time.
Linear change over time
If a quantity $q$ is a linear function of time $t$, so that
The units of measurement of $m$ are units of q per unit of time; for instance, if $q$ is income in dollars and $t$ is time in years, then the rate of change $m$ is measured in dollars per year.
If a quantity $q$ is a linear function of time $t$, so that
$q(t) = mt + b,$
then the slope $m$ measures the rate of change of $q$, and $b$ is the quantity at time $t = 0$, the initial quantity. If $q$ represents the position of a moving object, then the rate of change is also referred to as the velocity.
Units of m
The units of measurement of $m$ are units of q per unit of time; for instance, if $q$ is income in dollars and $t$ is time in years, then the rate of change $m$ is measured in dollars per year.
Examples
- If the accumulated revenue from sales of your video game app is given by $R(t) = 2000t + 500$ dollars, where $t$ is time in years from now, then you have earned \$500 in revenue so far, and the accumulated revenue is increasing at a rate of \$2000 per year.
- You are riding down the Ohio Turnpike on your motorcycle such that the total number of kilometers you have traveled $t$ hours after midnight is given by $s(t) = 100t + 20.$ Then your speed is 100 km per hour, and at time $t = 0$ (midnight) you had already traveled 20 km.
Now try the exercises in Section 1.3 in Finite Mathematics and Applied Calculus.
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