Chapter 1: Functions & Models

1.2Mathematical Models: A Catalog of

Essential Functions

Mathematical Model

• A mathematical description of a real-world phenomenon

• Uses a function or an equation

The modeling process…

Real-World Problem

Mathematical Model

Mathematical Conclusions

Real-World Predictions

Formulate

Solve

Interpret

Test

Stage One

• Formulate a mathematical model by identifying and naming the independent and dependent variables

• Make assumptions that simplify the phenomenon enough to make it mathematically tractable

• May need a graphical representation of the data

Stage Two

• Apply the mathematics we know to the model to derive mathematical conclusions

Stage Three

• Interpret the mathematical conclusions about the original real-world phenomenon by way of offering explanations or making predictions

Stage Four

• Test our predictions against new real data• If the predictions don’t compare well, we

revisit and revise

Linear Models

• Linear functions– The graph of the function is a line– Use slope-intercept form of the equation of a line– Grow at a constant rate

Example 1• (a) As dry air moves upward, it expands and cools.

If the ground temperature is 20⁰C and the temperature at a height of 1 km is 10⁰C, express the temperature T (in ⁰C) as a function of the height h (in km), assuming that a linear model is appropriate.

Example 1• (b) Draw the graph of the function in part (a). What

does the slope represent?

Example 1• (c) What is the temperature at a height of 2.5 km?

Empirical Model

• Used if there is no physical law of principle to help us formulate a model

• Based entirely on collected data

• Use a curve that “fits” the data (it catches the basic trend of the data points)

Example 2• Table 1 on page 26 lists the average carbon dioxide

level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2002. Use the data in Table 1 to find a model for the carbon dioxide level.

Example 3Use the linear model given by C = 1.55192t – 2734.55 to estimate the average CO2 level for 1987 and to predict the level for the year 2010. According to this model, when will the CO2 level exceed 400 parts per million?

Polynomials

• A function P is called a polynomial if

• Where n is a nonnegative integer• a = constants called coefficients of the polynomial

• Domain = • Degree of polynomial is n

012

21

1 ...)( axaxaxaxaxP nn

nn

Example 4• A ball is dropped from the upper observation deck of the

CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2 on pg 29. Find a model to fit the data and use the model to predict the time at which the ball hits the ground.

Power Functions

• A function of the form

• Where a is a constant

axxf )(

Power Functions, case 1

• Where and n is a positive integer

• The general shape depends on whether n is even or odd

• As n increases, the graph becomes flatter near 0 and steeper when |x| ≥ 1

nxxf )(

Power Functions, case 2

• Where and n is a positive integer

• These are root functions

• If n is even, the domain is all positive numbers

• If n is odd, the domain is all real numbers

nxxf1

)(

Power Functions, case 3

• Where

• Called the reciprocal function

• Hyperbola with the coordinate axes as asymptotes

1)( xxf

Rational Functions

• Ratio of two polynomials:

• Domain consists of all values such that Q(x) ≠ 0

)(

)()(

xQ

xPxf

Algebraic Functions

• A function constructed using algebraic operations starting with polynomials

• Any rational function is automatically an algebraic function

• Graphs can be a variety of shapes

Trigonometric Functions

• Radian measure always used unless otherwise indicated

• Domain for sine and cosine curves are all real numbers

• Range is closed interval [-1,1]• The zeroes of the sine function occur at the integer

multiples of π

Trig functions cont.• Sine and cosine are periodic functions• Period is 2π

• For all values of x, – Sin(x + 2π) = sin x– Cos(x + 2π) = cos x

• Use sine and cosine functions to model repetitive phenomena– Tides, vibrating springs, sound waves

Trig functions cont.

• Remember:

• Tangent function has period of π

• For all values of x, tan (x + π) = tan x

• Don’t forget about the reciprocal functions

x

xxcos

sintan

Exponential Functions

• Functions of the form

• The base a is a positive constant

• Used to model natural phenomena– Population growth, radioactive decay

xaxf )(

Logarithmic Functions

• Come in the form

• Base a is a positive constant

• Inverse functions of exponential functions

xxf alog)(

Transcendental Functions

• Not algebraic

• Includes trigonometric, inverse trigonometric, exponential, and logarithmic functions

• Comes back in chapter 11 (if you take calculus BC in college!)

Example 5

• Classify the following functions as one of the types of functions:

4

5

51)(

1

1)(

)(

5)(

tttu

x

xxh

xxg

xf x

Homework

• P. 34

• 1-4, 9-17 odd, 21, 23, 25