1 Functions and Models

1 Functions and Models

1.2 Mathematical Models: A Catalog of

Essential Functions

Mathematical Models: A Catalog of Essential Functions (1 of 2)

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.

Figure 1 illustrates the process of mathematical modeling.

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

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.

Linear Models

Linear Models (1 of 3)

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.

Linear Models (2 of 3)

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

x f(x) = 3x − 2

1.0 1.0

1.1 1.3

1.2 1.6

1.3 1.9

1.4 2.2

1.5 2.5

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

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?

Example 1(a) – Solution (1 of 2)

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

(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.

Example 1(b,c) – Solution (2 of 2)

T = −10h + 20 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

(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?

Linear Models (3 of 3)

If there is no physical law or principle to help us formulate a model, we

construct an empirical model (經驗模型), 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.

Polynomials

Polynomials (1 of 3)

A function P is called a polynomial if

 

P x a x a x a x a x a

where n is a nonnegative integer and the numbers a a a₀, ₁, ₂,...,a_n are constants called the coefficients of the polynomial.

The domain of any polynomial is ^{  }





 

P x x x 5 x

is a polynomial of degree 6.

Polynomials (2 of 3)

Its graph is always a parabola obtained by shifting the parabola y  ax². The parabola opens upward if a > 0 and downward if a < 0. (See Figure 7.)

Figure 7

The graphs of quadratic functions are parabolas.

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. ^{P x}

 

Polynomials (3 of 3)

A polynomial of degree 3 is of the form

 

P x ax bx cx d a

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).

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.

Time (seconds)

Height (meters)

0 450

1 445

2 431

3 408

4 375

5 332

6 279

7 216

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

Example 4 – Solution (1 of 2)

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

Time (seconds)

Height (meters)

0 450

1 445

2 431

3 408

4 375

5 332

6 279

7 216

8 143

9 61

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

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

quadratic model instead.

Example 4 – Solution (2 of 2)

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

4.90t²  0.96t  449.36  0 The quadratic formula gives

    

 

   

 

0.96 0.96 2 4 4.90 449.36 t

Power Functions

Power Functions (1 of 9)

A function of the form ^{f x}

 

We consider several cases.

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

The graphs of ^{f x}

 

(These are polynomials with only one term.)

Power Functions (2 of 9)

The general shape of the graph of ^{f x}

 

If n is even, then ^{f x}

 

If n is odd, then ^{f x}

 

 ³. y x

We already know the shape of the graphs of y = x (a line through the origin with slope 1) and y ^ x²





Power Functions (3 of 9)

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

becomes flatter near 0 and steeper when x  1. (If x is small, then x² is smaller,

3 4

is even smaller, is smaller still,

x x and so on.)

Power Functions (4 of 9)

(ii) ^{a = / n,1} where n is a positive integer

The function ^{f x}

 

function ^{f x}

 

Figure 13(a) Graph of root function

Power Functions (5 of 9)

For other even values of n, the graph of ^{y  n x} is similar to that of y  x. For n = 3 we have the cube root function f x

 

(recall that every real number has a cube root) and whose graph is shown in Figure 13(b). The graph of y  ⁿ x for n odd (n > 3) is similar to that of y  ³ x.

Power Functions (6 of 9)

(iii) a = −1

The graph of the reciprocal function ^{f x}

 

Its graph has the equation  1 ,

y x or xy = 1, and is a hyperbola with the coordinate axes as its asymptotes.

Power Functions (7 of 9)

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:

 C V 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.

Volume as a function of pressure at constant temperature

Power Functions (8 of 9)

(iv) a = −2

Among the remaining negative powers for the power function ^{f x}

 

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 f x( )  C x/ ²

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:

 C₂ I x where C is a constant.

Power Functions (9 of 9)

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.

The reciprocal of the squaring function Illumination from a light source as a function of distance from the source

Rational Functions

Rational Functions (1 of 2)

A rational function f is a ratio of two polynomials:

   

 P x

 

f x Q x

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

 

f x x whose domain is





Figure 14

The reciprocal function

Rational Functions (2 of 2)

The function

 



4 2

2

2 1

4 x x

f x x

is a rational function with domain





 ^{  }



4 2

2

2 1

4

x x

f x x

Algebraic Functions

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:

 

 







4 2

2 16 3

1 x x 2 1

f x x g x x x

x x

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

 

 



0

2 2

1 /

m f v m

v c

where m₀ is the rest mass of the particle and c  3.0 10 km/s ⁵ is the speed of light in a vacuum.

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

Trigonometric Functions

Trigonometric Functions (1 of 3)

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.

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

Trigonometric Functions (2 of 3)

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  1 sin x  1  1 cosx  1 or, in terms of absolute values, _{sinx  1 cos x  1}

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,

Trigonometric Functions (3 of 3)

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

 sin

tan cos

x x

x

and its graph is shown in Figure 20. It is

undefined whenever cos x = 0, that is, when

 

  / 2, 3 / 2, .

x Its range is (−∞, ∞).

Figure 20

y = tan x

Notice that the tangent function has period π :





tan x tanx for all x

Exponential Functions

Exponential Functions

The exponential functions are the functions of the form ^{f x}

 

The graphs of ^{y  2 and x y }

 

Figure 21

 ^{a y  2x}  ^{b y }  ^{0.5 x}

Logarithmic Functions

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

Example 6

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

(a) ^{f x}

 

(b) ^{g x}

 

(c)

 

1 h x x

x

 



(d) ^{u t}

 

Example 6 – Solution

(a) ^{f x}

 

(b) ^{g x}

 

(c)

 

1 h x x

x

 

 is an algebraic function. (It is not a rational function because the denominator is not a polynomial.)

(d) ^{u t}

 

Question:

請問y(population)和x(decades)的關係比 較接近 y=ax^n 還是 y= b^x (b>1)

Question: