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Functions and Models

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Mathematical Models: A Catalog of Essential Functions

Copyright © Cengage Learning. All rights reserved.

Mathematical Models: A Catalog of Essential Functions (1 of 3)

A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions.

The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior.

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Mathematical Models: A Catalog of Essential Functions (2 of 3)

Figure 1 illustrates the process of mathematical modeling.

Figure 1

The modeling process

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Mathematical Models: A Catalog of Essential Functions (3 of 3)

A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions.

It is important to realize the limitations of the model.

There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions.

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Linear Models

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Linear Models (1 of 4)

When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as

y = f (x) = mx + b

where m is the slope of the line and b is the y-intercept.

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Linear Models (2 of 4)

A characteristic feature of linear functions is that they change at a constant rate. For instance, Figure 2 shows a graph of the linear function f (x) = 3x − 2 and a table of sample values.

Figure 2

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Linear Models (3 of 4)

Notice that whenever x increases by 0.1, the value of f (x) increases by 0.3. So �f (x) increases three times as fast as x. This means that the slope of the graph of y = 3x − 2, namely 3, can be interpreted as the rate of change of y with respect to x.

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Example 1

(a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate.

(b) Draw the graph of the function in part (a). What does the slope represent?

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

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Example 1(a) – Solution (1 of 3)

Because we are assuming that T is a linear function of h, we can write

T = mh + b

We are given that T = 20 when h = 0, so

20 = m · 0 + b = b

In other words, the y-intercept is b = 20.

We are also given that T = 10 when h = 1, so

10 = m · 1 + 20

The slope of the line is therefore m = 10 − 20 = −10 and the required linear function is

T = −10h + 20

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Example 1(b) – Solution (2 of 3)

The graph is sketched in Figure 3.

The slope is m = −10°C/km, and this represents the rate of change of temperature with respect to height.

Figure 3

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Example 1(c) – Solution (3 of 3)

At a height of h = 2.5 km, the temperature is

T = −10(2.5) + 20 = −5°C

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Linear Models (4 of 4)

If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data.

We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points.

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Polynomials

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Polynomials (1 of 4)

A function P is called a polynomial if

where n is a nonnegative integer and the numbers

are constants called the coefficients of the polynomial.

The domain of any polynomial is

If the leading coefficient

then the degree of the polynomial is n. For example, the function

is a polynomial of degree 6.

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Polynomials (2 of 4)

A polynomial of degree 1 is of the form P (x) = mx + b and so it is a linear function.

A polynomial of degree 2 is of the form

and is called a

quadratic function.

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Polynomials (3 of 4)

Its graph is always a parabola obtained by shifting the parabola

The parabola opens upward if a > 0 and downward if a < 0. (See Figure 7.)

Figure 7

The graphs of quadratic functions are parabolas.

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Polynomials (4 of 4)

A polynomial of degree 3 is of the form

and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c).

Figure 8

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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.

Table 2

Find a model to fit the data and use the model to predict the time at which the ball hits the ground.

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Example 4 – Solution (1 of 4)

We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate.

Figure 9

Scatter plot for a falling ball

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Example 4 – Solution (2 of 4)

But it looks as if the data points might lie on a parabola, so we try a quadratic model instead.

Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model:

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Example 4 – Solution (3 of 4)

In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit.

Figure 10

Quadratic model for a falling ball

The ball hits the ground when h = 0, so we solve the quadratic equation

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Example 4 – Solution (4 of 4)

The quadratic formula gives

The positive root is t ≈ 9.67, so we predict that the ball will hit the ground after about 9.7 seconds.

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Power Functions

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Power Functions (1 of 8)

A function of the form

where a is a constant, is called a power function.

We consider several cases.

(i) a = n, where n is a positive integer

The graphs of

for n = 1, 2, 3, 4, and 5 are shown in Figure 11.

(These are polynomials with only one term.)

Figure 11

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Power Functions (2 of 8)

We already know the shape of the graphs of y = x (a line through the origin with

slope 1) and

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Power Functions (3 of 8)

The general shape of the graph of

depends on whether n is even or odd.

If n is even, then

is an even function and its graph is similar to the

parabola

If n is odd, then

is an odd function and its graph is similar to that of

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Power Functions (4 of 8)

Notice from Figure 12, however, that as n increases, the graph of

becomes flatter near 0 and steeper when

(If x is small, then

and so on.)

Figure 12

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Power Functions (5 of 8)

(ii)

where n is a positive integer

The function

is a root function. For n = 2 it is the square root

function

whose domain is [0, ∞) and whose graph is the upper half of

the parabola

[See Figure 13(a).]

Figure 13(a)

Graph of root function

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Power Functions (6 of 8)

For other even values of n, the graph of

is similar to that of

For n = 3 we have the cube root function

(recall that every real number has a cube root) and whose graph is shown in

Figure 13(b). The graph of

for n odd (n > 3) is similar to that of

Figure 13(b)

Graph of root function

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Power Functions (7 of 8)

(iii) a = −1

The graph of the reciprocal function

is shown in Figure 14.

Its graph has the equation

or xy = 1, and is a hyperbola with the

coordinate axes as its asymptotes.

Figure 14

The reciprocal function

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Power Functions (8 of 8)

This function arises in physics and chemistry in connection with Boyle’s Law, which says that when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P:

where C is a constant.

Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14.

Figure 15

Volume as a function of pressure at constant temperature

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Power Functions (9 of 8)

(iv) a = −2

Among the remaining negative powers for the power function

by far the most important is that of a = −2. Many natural laws state that one quantity is inversely proportional to the square of another quantity. In other words, the first quantity is modeled by a function of the form

and we refer to this as an inverse square law.

For instance, the illumination I of an object by a light source is inversely proportional to the square of the distance x from the source:

where C is a constant.

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Power Functions (10 of 10)

Thus the graph of I as a function of x (see Figure 17) has the same general shape as the right half of Figure 16.

Figure 16

The reciprocal of the squaring function

Figure 17

Illumination from a light source as a function of distance from the source

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Rational Functions

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Rational Functions (1 of 2)

A rational function f is a ratio of two polynomials:

where P and Q are polynomials. The domain consists of all values of x such that Q(x) ≠ 0.

A simple example of a rational function is the function

whose domain is

this is the

reciprocal function graphed in Figure 14.

Figure 14

The reciprocal function

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Rational Functions (2 of 2)

The function

is a rational function with domain

Its graph is shown in Figure 18.

Figure 18

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Algebraic Functions

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Algebraic Functions (1 of 2)

A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function.

Here are two more examples:

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Algebraic Functions (2 of 2)

An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is

where m₀ is the rest mass of the particle and

is the speed of

light in a vacuum.

​

Functions that are not algebraic are called transcendental; these include the trigonometric, exponential, and logarithmic functions.

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Trigonometric Functions

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Trigonometric Functions (1 of 6)

In calculus the convention is that radian measure is always used (except when otherwise indicated).

For example, when we use the function f (x) = sin x, it is understood that sin x means the sine of the angle whose radian measure is x.

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Trigonometric Functions (2 of 6)

Thus the graphs of the sine and cosine functions are as shown in Figure 19.

Figure 19

(a) f (x) = sin x

(b) g (x) = cos x

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Trigonometric Functions (3 of 6)

Notice that for both the sine and cosine functions the domain is (−∞, ∞) and the range is the closed interval [−1, 1].

Thus, for all values of x, we have

or, in terms of absolute values,

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Trigonometric Functions (4 of 6)

An important property of the sine and cosine functions is that they are periodic functions and have period 2π. This means that, for all values of x,

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Trigonometric Functions (5 of 6)

The tangent function is related to the sine and cosine functions by the equation

and its graph is shown in Figure 20. It is undefined whenever cos x = 0, that is, when

Its range is (−∞, ∞).

Figure 20

y = tan x

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Trigonometric Functions (6 of 6)

Notice that the tangent function has period π :

The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions.

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Exponential Functions

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Exponential Functions (1 of 2)

The exponential functions are the functions of the form

where the base

b is a positive constant.

The graphs of

are shown in Figure 21. In both cases the

domain is (−∞, ∞) and the range is (0, ∞).

Figure 21

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Exponential Functions (2 of 2)

Exponential functions are useful for modeling many natural phenomena, such as population growth (if b > 1) and radioactive decay (if b < 1).

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Logarithmic Functions

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Logarithmic Functions

The logarithmic functions f (x) = log_bx, where the base b is a positive constant, are the inverse functions of the exponential functions. Figure 22 shows the graphs of four logarithmic functions with various bases.

Figure 22

In each case the domain is (0, ∞), the range is (−∞, ∞), and the function increases slowly when x > 1.

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Example 6

Classify the following functions as one of the types of functions that we have discussed.

(a)

(b)

(c)

(d)

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Example 6 – Solution

(a)

is an exponential function. (The variable x is the exponent.)

(b)

is a power function. (The variable x is the base.) We could also

consider it to be a polynomial of degree 5.

(c)

is an algebraic function. (It is not a rational function because

the denominator is not a polynomial.)

(d)

is a polynomial of degree 4.

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