1.1 mathematical modeling

Chapter 1

Mathematical Model

A mathematical model is a graphical, verbal, numerical, or symbolic representation of a problem situation.

Example- Page 17

15. Teacher Salary Comparison

Over 60% of men not in the teaching profession earn a higher salary than men who are teachers. The table shows how much more money the average college-educated male non-teacher makes as compared to the average male teacher.

Year Percent MoreEarned by Non-Teachers asCompared toTeachers

1940 -3.6%

1950 2.1%

1960 19.7%

1970 33.1%

1980 36.1%

1990 37.5%

2000 60.4%Source: www.nea.orgFor example, in 1990 male non-teachersmade 37.5% more than male teachers on average.

Example- Page 17

a. Describe the trend observed in these data.

b. Why was there such a big jump in the percentage of non-teachers who earn a higher salary than teachers from 1990 to 2000?

c. What does the –3.6% in 1940 indicate about salaries of male teachers?

Example- Page 19

18. Super Bowl Ticket Prices

The table shows the price of a Super Bowl ticket for selected Super Bowls.

Super Bowl Ticket

FaceValue

I (1) $10

V (5) $15

X (10) $20

XV (15) $40

XX (20) $75

XXV (25) $150

XXX (30) $300

XXXV (35) $325

XL (40) $600

(Source: www.superbowl.com.)

a. Describe the trend seen in these data. Are prices increasing or decreasing over time?

b. Compare the difference in ticket face value from one year to the next. What patterns do you notice?

c. Predict the face value of a ticket for the 60th Super Bowl.

1.2

Function Notation

y = f(x)

Input (independent variable)

Output (dependent variable)

Functions

A relation is a function if each input value has exactly one output value

Determining Functions

Determine if the relation is a function (3, 2), (4, 2), (5, 2)

(1, 2), (-1, 3), (1, 7)

Vertical Line Test

Use the vertical line test to determine whether each graph represents a function

1.3

Is it a function?

Just verify that each input value corresponds to a single output value.

Slope

Slope = rate of change

or = change in y change in x

= ∆y

or ∆x

or = Rise Run

Slope- given points

For points slope =

A positive slope goes up as you read the graph from left to right. (A negative slope goes down as you read the graph from left to right.)

As the absolute value of the slope gets larger, the steepness of the incline increases.

2 1

2 1

y y

x x

1 1 2 2( , ), ( , )x y x y

Examples

Calculate the slope between each pair of points (7, 2) and (4, -3)

(8, 2) and (-8, 6)

Average Rate of Change

= Change in function value Change in x value

= f(b) – f(a)b – a

Average Rate of Change

Page 75 # 4- Baby Girl’s Average Weight

Average Rate of Change

Page 75 #6- Divorce Rate

Average Rate of Change

Page 79 #16- Exxon Mobile

Interpreting Data

Page 81 #20-U.S. Home Sales

Functions of several variables Sometimes functions depend on more than

one variable. A pay as you go service is as follows:

Phone bill = airtime + text messages + picture messaging

M = .25a + .05t + .25p, where M is measured in dollars. M = M(a, t, p) airtime = 25 cents per minute text messages = 5 cents each picture messaging = 25 cents each

Explain what M(300,15,8) means and find its value.

Borrowing money

If you borrow P dollars at a monthly interest rate of r (as a decimal) and want to pay off the loan in t months, then the monthly payment M(P, r, t) in dollars is calculated using

P = amount borrowedr = monthly interest rate, as

a decimalt = number of monthsM = monthly payment

Mr

r

t

t

Pr( )

( )

1

1 1

Examples

What does M(6000, .035, 60) mean?

Find the monthly payment.

Suppose we borrow $5600 with a monthly interest rate of .825% and want to pay the loan off in 4 years.

Example

Page 81 #21-22

Solving Equations

Temperature Problems Page 83 #33-36

Example

Page 86 #47-48

1.4

Two-Variable Data

Ordered Pair

Domain

Range

Domain and Range Domain- the input values

In an (x, y) situation, these are the “x” values. The domain is the set of values that are

plugged into a function. When it is limited to values that make sense in

the real world it is called the practical domain. Range- the output values

In an (x, y) situation, these are the “y” values. The range is the set of solutions we get from

plugging the domain values into a function. When it is limited to values that make sense in

the real world it is called the practical domain.

State the Domain and the Range (2, 3), (-3, 7), (8,3)

Gathering Data from graphs Page 102 #1-6

Example

Page 103 #16 Cell Phone Subscribers

Examples

Page 104-105 #18-22