MAT 125 – Applied Calculus 2.3 Functions & Mathematical Models

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 2

Polynomial Functions  A polynomial function of degree n is a function of the form where n is a nonnegative integer and the numbers a 0, a 1, …. a n are constants called the coefficients of the polynomial function. Dr. Erickson 2.3 Functions & Mathematical Models 3

Polynomial Functions  Examples: 1.The function below is polynomial function of degree 5: 2.The function below is polynomial function of degree 3:

Modeling a Polynomial Function of Degree 2  A polynomial function of degree 2 has the form or more simply, y = ax 2 + bx + c, and is called a quadratic function.  The graph of a quadratic function is a parabola: xx Opens upwards if a > 0 Opens downwards if a < 0

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)

Rational Functions  Examples of rational functions:

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:

Rational and Power Functions  Many functions involve combinations of rational and power functions.  Examples:

Mathematical Modeling  As 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 FormulateInterpret Solve Test

Example 1 Dr. Erickson 2.3 Functions & Mathematical Models 11

Example 2 Dr. Erickson 2.3 Functions & Mathematical Models 12

Example 3 Dr. Erickson 2.3 Functions & Mathematical Models 13

Example 4 Dr. Erickson 2.3 Functions & Mathematical Models 14

Example 5 Dr. Erickson 2.3 Functions & Mathematical Models 15

Example 6 Dr. Erickson 2.3 Functions & Mathematical Models 16

Example 7 Dr. Erickson 2.3 Functions & Mathematical Models 17

Example 8 Dr. Erickson 2.3 Functions & Mathematical Models 18

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 2.3 Functions & Mathematical Models 19