MAT 125 – Applied Calculus 2.3 Functions & Mathematical Models.
-
Upload
aldous-scot-wiggins -
Category
Documents
-
view
219 -
download
0
Transcript of MAT 125 – Applied Calculus 2.3 Functions & Mathematical Models.
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