Perfil de casos de EDA e IRA

mining the differences among linear and polyno- mial functions and among exponential, log, and logistic functions.

cx d

y ax3_{ bx}2_

y ax2_{ bx  c}

(Source: National Vital Statistics Report, vol. 50, no. 5, February 12, 2000.)

SUMMARY

Mathematical Modeling and Functions

Mathematical modeling is the process by which we construct a mathematical framework to represent a real-life situation. In this book we often use mathemat- ical modeling to mean fitting a line or curve to data. The resulting equation, together with output label, input description, and interval description, which we refer to as the mathematical model, provides a representation of the underlying relationship between the variable quantities of interest. A function is a description of how one thing (output) changes as something else (input) changes. We encounter functions represented in four ways: tables of data, graphs, word descriptions, and equations.

The Role of Technology

In order to construct mathematical models from data, we must use appropriate tools. Normally, these tools are graphing calculators or personal computers. You should

clearly understand that we use technology simply as a tool in the service of mathematics and that no tool is a substitute for clear, effective thinking. Technology car- ries only the graphical and numerical computational burden. You yourself must perform the mathematical analyses, interpret the results, make the appropriate de- cisions, and then communicate your conclusions in a clear and understandable manner.

Function Combinations and Composition There are several ways to create new functions by com- bining two or more other functions whose input and output units are compatible. The basic construction techniques are function addition, subtraction, multipli- cation, division, and composition. In each of these con- structions, knowing the input and output units of the functions is the key to understanding how to combine the functions. Table 1.31 shows the necessary input and output compatibility.

Function Input

operation compatibility Output compatibility New input units New output units Addition Identical Same unit of measure or units Same as input Same as output unit

of measure capable of being unit of measure of measure of

combined into a larger group of original original functions

(sons_{ daughters  children)} functions

Subtraction Identical Same unit of measure or units Same as input Same as output unit

of measure capable of being unit of measure of measure of

subtracted (children  sons  of original original functions

daughters) functions

Multiplication Identical Unit of measure of one Same as input The multiplication

function should contain “per “ unit of measure (reduced if possible)

unit of measure of the other of original of the output unit

function functions of measure of the

original functions

Division Identical Same unit of measure (or) unit Same as input The numerator

of measure of the two functions unit of measure output unit of

should make sense in a phrase of original measure “per” the

containing “. . . per . . .” functions denominator output

unit of measure

Composition Output of one function (inside function) is Same as input Same as output unit

identical to the input of the second function unit of measure of measure of

(outside function) of inside function outside function

Linear Functions and Models

A linear function models a constant rate of change. Its underlying equation is that of a line: , where the constant a is called the slope of the line and is calculated as . Because the slope of a line is a measure of its rate of increase or decrease, the slope is also known as the rate of change for the linear model. The constant b appearing in the linear model is simply the output of the model when the input is zero.

Exponential Functions and Models

Second in importance to linear functions and models are exponential functions and models. Based on the fa- miliar idea of repeated multiplication by a fixed positive multiplier b (the base), the basic exponential function is of the form

The value a appearing in the equation is the output when the input is zero.

Exponential functions model constant percentage change. In terms of the function , exponen- tial growth occurs when b is greater than 1, and expo- nential decline (decay) takes place when b is between 0 and 1. The constant percentage growth or decline is

given by .

Logarithmic Functions and Models The basic form of the log function that we use is

The input of this function must be a value greater than zero. The log function is useful for situations in which the output grows or declines at an increasingly slow rate. When fitting a log equation to data, you must some- times align the input data to ensure that the input values are greater than zero or to obtain a better fit. Aligning input data has the effect of shifting the data horizontally. Logistic Functions and Models

Initial exponential growth followed by a leveling-off ap- proach toward a limiting value L is characteristic of lo- gistic growth, which is modeled by the logistic equation

If the constant B is positive, the model indicates growth. If the constant B is negative, the model indicates decline

f(x) L 1 AeBx f(x) a  b ln x (b 1)100% f(x) abx f(x) abx y ax  b rise run y ax  b

in output toward the horizontal axis as the input values increase.

When fitting exponential and logistic equations to data, it is sometimes helpful to shift the output data. This vertical shift is particularly useful when the data ap- pear to approach a value other than zero. The goal in shifting is to move the data closer to the horizontal axis. Limits and End Behavior

The idea of a limiting value of a function is a fundamen- tal theme of calculus that can be intuitively understood to be the behavior of the outputs of a function as the in- puts of the function become infinitesimally close to a specific value. Limits can also be used to describe the end behavior of a function as the magnitude of the inputs becomes infinitely large.

Polynomial Functions and Models

Polynomial functions and models have a well- established role in calculus. In this text, we consider lin- ear functions, quadratic functions, and cubic functions. Quadratic equations have graphs known as parabolas.

The parabola with equation opens

upward (is concave up) if a is a positive and opens downward (is concave down) if a is negative.

Cubic equations have graphs that show a change of concavity at an inflection point, but unlike logistic mod- els, they do not have horizontal asymptotes limiting their end behavior. In using cubic models, we must be especially careful when extrapolating beyond the range of data values from which the models are constructed. Choosing a Model

Although it is not always clear which (if any) of the functions we have discussed apply to a particular real- life situation, it helps to keep in mind a few general, common-sense guidelines: (1) Given a set of discrete data, begin with a scatter plot. The plot will often reveal general characteristics that point the way to an appro- priate model. (2) If the scatter plot does not appear to be linear, consider the suggested concavity. One-way concavity (up or down) suggests a quadratic, exponen- tial, or log model. (3) When a single change in concav- ity seems apparent, think in terms of cubic or logistic models. But remember that the graphs of logistic mod- els tend to become flat on each end, whereas the graphs of cubic models do not. Never consider using a cubic or a logistic model if you cannot identify an inflection point.