MAT 125 – Applied Calculus2.3 Functions & Mathematical Models

2.3 Functions & Mathematical Models

2 Today’s Class We will be learning the following concepts in Section 2.2:

The Sum, Difference, Product, and Quotient of Functions

Composition of Functions

We will be learning the following concepts in Section 2.3: Mathematical Models

Rational and Power Functions

Economic Models

Dr. Erickson

2.3 Functions & Mathematical Models

3 Polynomial Functions

A polynomial function of degree n is a function of the form

where n is a nonnegative integer and the numbers a0, a1, …. an are constants called the coefficients of the polynomial function.

Dr. Erickson

1 21 2 1 0 ( 0) )( n n

n n nf x a x a x a x a x a a

Polynomial FunctionsExamples:

1. The function below is polynomial function of degree 5:

2. The function below is polynomial function of degree 3:

5 4 3 212( ) 2 3 2 6f x x x x x

3 2( ) 0.001 0.2 10 200g x x x x

Modeling a Polynomial Function of Degree 2

A polynomial function of degree 2 has the form

or more simply, y = ax2 + bx + c, and is called a quadratic function.

The graph of a quadratic function is a parabola:

x x

Opens upwards

if a > 0

Opens downwards

if a < 0

22 1 0 2( ) ( 0)y f x a x a x a a

Rational Functions

A rational function is simply the quotient of two polynomials. In general, a rational function has the form

where f(x) and g(x) are polynomial functions.

Since the division by zero is not allowed, we conclude that the domain of a rational function is the set of all real numbers except the zeros of g (the roots of the equation g(x) = 0)

( )( )

( )

f xR x

g x

Rational Functions

Examples of rational functions:

3 23 1( )

2

x x xF x

x

2

2

1( )

1

xG x

x

Power Functions

Functions of the form where r is any real number, are called power functions.

We encountered examples of power functions earlier in our work.

Examples of power functions:

( ) rf x x

1/2 22

1( ) ( )f x x x g x x

x and

Rational and Power Functions

Many functions involve combinations of rational and power functions.

Examples:2

2

2

1/22 3/2

1( )

1

( ) 3 4

1( ) (1 2 )

( 2)

xf x

x

g x x x

h x xx

Mathematical ModelingAs we have seen, mathematics can be used to solve real-world problems.

Regardless of the field from which the real-world problem is drawn, the problem is analyzed by using a process called mathematical modeling.

The four steps in this process are:Real-world problem Mathematical

model

Solution of real-world problem

Solution of mathematical model

Formulate

Interpret

SolveTest

2.3 Functions & Mathematical Models

11 Example 1

Dr. Erickson

2.3 Functions & Mathematical Models

12 Example 2

Dr. Erickson

2.3 Functions & Mathematical Models

13 Example 3

Dr. Erickson

2.3 Functions & Mathematical Models

14 Example 4

Dr. Erickson

2.3 Functions & Mathematical Models

15 Example 5

Dr. Erickson

2.3 Functions & Mathematical Models

16 Example 6

Dr. Erickson

2.3 Functions & Mathematical Models

17 Example 7

Dr. Erickson

2.3 Functions & Mathematical Models

18 Example 8

Dr. Erickson

2.3 Functions & Mathematical Models

19 Next Class

We will discuss the following concepts: Introduction to Calculus

Intuitive Definition of a Limit

Evaluating the Limit of a Function

Indeterminate Forms

Limits at Infinity

Please read through Section 2.4 – Limits in your text book before next class.

Dr. Erickson