Modeling and Equation Solving

Section 1.1

Mathematical structure that approximates phenomena for the purpose of studying or predicting behavior

The process of devising mathematical models is now a rich field of study itself.

We will discuss three types: numerical models, algebraic models, and graphical models

Mathematical ModelingMathematical Modeling

Numerical Models (most basic kind of model)Numbers (or data) are analyzed to gain insight into phenomena

Algebraic ModelUses formulas to relate variable quantities associated

with the phenomena being studied

Graphical ModelVisible representation of a numerical model or an

algebraic model that gives insight into the relationship between variable quantities

*Learning to interpret and use graphs is a major goal of this book

Numerical ModelNumerical ModelThe numbers in Table 1.1 show the growth of the minimum hourly wage (MHW) from 1995 to 2005. The table also shows the MHW adjusted to the purchasing power of 1996 dollars (using the CPI-U, the Consumer Price Index for all Urban Consumers).

1. In what 5 year period did the actual MHW increase the

most?

In the period 1975 to 1980 it increased by $1.00

2. In what year did a worker earning MHW enjoy the

greatest purchasing power?

19703. A worker on minimum wage in 1980 was earning nearly twice as much as a worker on minimum wage in 1970, and yet there was great pressure to raise the minimum wage again. Why?

3. Although the MHW increased from $1.60 to $3.10 in that period, the purchasing power actually dropped by $0.57 (in 1996 dollars). This is one way inflation can affect the economy.

Example 2Example 2Table 1.2 shows the growth in the number of prisoners incarcerated in state and federal prisons from 1980 to 2000. Is the proportion of female prisoners over the years increasing?

The number of female prisoners over the years is certainly increasing, but so is the total number of prisoners, so it is difficult to discern from the data whether the proportion of female prisoners is increasing. We need another column of numbers showing the ratio of female prisoners to total prisoners.

Algebraic ModelAlgebraic ModelA pizzeria sells a rectangular 18” by 24” pizza for the same

price as its large round pizza (24” diameter). If both pizzas are of the same thickness, which option gives the most pizza for the money?

Graphical ModelGraphical Model

Examples

1. Using a scatter plot to represent data

2. Fitting a curve to data

(line of best fit)

Graphical ModelingGraphical Modeling

The pattern of the data points suggests exponential growth.

Graphical ModelingGraphical Modeling

We use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model (0.008079266) (1.013731)tp

Mathematical Modeling

We see that the exponential curve fits the data reasonably well. The period of relatively slow population growth is explained by the two world wars and the Great Depressionof the 1930s.

Zero Factor Property

A product of real numbers is zero if and only if at least one of the factors in the product is zero.

ExamplesExamplesSolve 6x3 = 11x2 +10x

Solve x2 = 10 – 4x

Algebraically:

Graphically:

x = -5.742 x = 1.742

Fundamental ConnectionFundamental Connection

If a is a real number that solves the equation f(x) = 0, then these three statements are equivalent:

1. The number a is a root (or solution) of the equation f(x) = 0

2. The number a is a zero of y = f(x).

3. The number a is an x-intercept of the graph of

y = f(x).

ExampleExampleThe engineers at an auto manufacturer pay

students $0.08 per mile plus $25 per day to road test their new vehicles.

a. How much did the auto manufacturer pay Sally to drive 440 miles in one day?

b. John earned $93 test-driving a new car in one day. How far did he drive?

$60.20

850 miles

Grapher Failure and Hidden Grapher Failure and Hidden BehaviorBehavior

1. Recognize vertical asymptotes on your graphs

2. Zoom in or out to see the full graph: this helps with end behavior and x intercepts

Example: x3 – 1.1x2 – 65.4x + 229.5 = 0