Section 1.2 Mathematical Models

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 prediction about future behavior.

Linear Models:

Linear function:  The graph of the function is a line that can be written in this form:

                                                            y = f(x) = mx + b

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

A characteristic feature of linear functions is that they grow at a constant rate

Example:

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

Solution:

            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

Polynomials:

A function P is called a polynomial if

                        P(x) = a_xxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀

Where n is a nonnegative integer and the numbers a₀, a₁, a₂,….., a_n  are constants called coefficients of the polynomial.  The domain of any is .  If the leading coefficient a_n  0, then the degree of the polynomial is n.

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

A polynomial of degree 2 is of the form P(x) ax² + bx + c and is called a quadratic function.  The graph of P is always a parabola obtained by shifting the parabola y = ax².  The Parabola opens upward if a > 0 and downward if a < 0.

A polynomial of degree 3 is of the form

                                    P(x) = ax³ + bx² + cx + d

And is called a cubic function

Example:   Find the degree of each function

a) f(x) = 5x³+4x            a) f(x) = 8x                a) f(x) = x⁶+x⁵               a) f(x) =8x⁵+12

Solution:

a) Cubic Function (3)  (b)Linear (1)               (c)6                  (d) 5

Power Functions:

A function form f(x) = x^a, where a is a constant, is called a power function.  We consider several cases.

1. a = n, where n is a positive integer

The general shape of the graph of f(x) = xⁿ depends on whether n is even or odd.  If n is even, then f(x) = xⁿ is an even function and its graph is similar to the parabola y = x².  If n is odd, then f(x) = xⁿ is an odd function and its graph is similar to y = x³.

2. a = 1/n, where n is a positive integer

The function f(x) = x^1/n =  is a root function.  For n=2 it is the square root function f(x)= whose domain is [] and whose graph is the upper half of the parabola x = y².

3. a = -1

The graph of the reciprocal function f(x) = x^-1 = 1/x.  Its graph has the equation

Y = 1/x, or xy = 1, and is a hyperbola.

Rational Functions:

A rational function f is a ratio of two polynomials:

f(x) =

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

Q(x) 0. 

Algebraic Functions:

A function f is called an algebraic function if it can be constructed using algebraic operations starting with polynomials.  Any rational function is automatically an algebraic function.  Examples:

f(x) = g(x) =

Trigonometric Functions:

In calculus the convention is that radian measure is always used.  For example, whe 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.

            Notice that for both the sine and cosine function the domain is (-) and the range is the closed interval [-1,1].  Thus for all values of x we have

Also, the zeros of the sine function occur at the integer multiples of; that is:

Sin x = 0       when       x = n       n an integer

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

sin(x + 2) = sin x          cos(x + 2) = cos x     

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

Exponential Functions:

These are the functions of the form f(x) = a^x, where the base a is a positive constant.

Logarithmic Functions:

These are the functions f(x) = log_ax. where the base a is a positive constant.  They are the inverse functions of the exponential functions.

Transcendental Functions:

These are functions that are not algebraic.  The set of transcendental functions includes the trigonometric, inverse trigonometric, exponential, and logarithmic functions, but it also include a vast number of other functions that have never been named.

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

(a)  f(x)=5^x                   (b)  g (x)=x⁵                 (c)  h(x)=         (d)  r(x)= 1 – t + 5t⁴

Solution:

Exponential Function.   (The x is the exponent)

Power Function.  (The x is the base)

Algebraic Function (Division)

Polynomial of degree 4

For Extra Help, Visit These Resources:

http://fym.la.asu.edu/~fym/mat210_web/lessons/Ch1/1_2/1_2ol.htm

http://virgil.azwestern.edu/~bk/downloads/220/mat 221 1.2 sp 00.ppt