Chapter 9  Modeling Our World

Unit 9A Functions:  The Building Blocks of Mathematical Models

Unit 9B  Linear Modeling

Unit 9C Exponential Modeling

Unit 9 Functions:  The Building Blocks of Mathematical Models

Function:  relates one or more quantities (independent variables) to unique quantity (dependent variable).

When there is only one dependent variable, the function can be considered to be set of ordered pair of quantities:
(independent variable, dependent variable)

Generally denote the independent variable by x and the dependent variable by y;  the function f is the function that associates with each x the value y defined by y = f(x).

Example:  The function y = x², that is f(x) = x².   

We represent functions in several ways:

1. Data Table (suitable if there are a reasonable small number of values in the domain of the function).
2. Graph. See the Brief Review page 563.

Coordinate plane: coordinates (x,y) with x-coordinate giving directed horizontal distance from origin, and y coordinate giving directed vertical distance from the origin.

3. Equation, when there is a mathematical formula expressing the relationship between the independent and dependent variables.

Domain and Range of a function:  The domain is the set of values that the independent variable can have.  The range of a function is the set of values that the dependent variable takes on corresponding to the values in the domain. 

Examples of functions that model real world phenomena:

Example (Figure 9.3, page 564) Temperature as a function of time of day.
Record temperature data at certain times during the day.  Along the horizontal axis plot time.  Domain is time in hours after 6 a.m. (from 0 to 12).     Along the  vertical axis plot temperature at that time of data.  For this observed data range varies from about 40 degrees to 74 degrees..  Data points are (time, temperature).  Fill in the data points with a smooth curve.  Use the curve to estimate temperature at times of day between those observed times.

 Example 2:  Pressure as a function of altitude.
See sample data page 565, table 9.2  and corresponding graph in figure 9.5.  Pressure decreases as altitude increases.

Unit 9 B  Linear Modeling

A linear function has a constant rate of change, and therefore a straight line graph. 
Slope of linear graph represents this rate of change and  is calculated as "the rise over the run".  That is,

slope = change in dependent variable

change in independent variable 

See the three figures 9.8 a,b,c, page 572,  representing rain depth at three different rates of rain fall --
.5 in/hr;   1.5 in/hr;  and 2 in/hr.

The algebraic form for the equation of a line is

y = mx + b (or y = b + mx)

Here y is the dependent variable, x is the independent variable, m is the rate of change, and b is the y intercept  (the value of y when x is equal to 0).

In other words:
y = b + mx
dependent variable = initial value + (rate of change) * (independent variable).

Example 2:  Price - Demand Function.
A small stores sells fresh pineapples, which vary in price between 2 and 7 dollars.  They determine that demand  (in terms of numbers of pineapples sold) varies with price linearly.   At price of 2 dollars the demand is 80 pineapples.  At a price of 5 dollars the demand if 50 pineapples.  Here price is the independent variable and demand is the dependent variable.

Creating a linear equation for this model. 
Here we have two data points  (2, 80) and (5, 50). 

Letting x be the independent variable (price in dollars) and y be the dependent variable (demand in pineapples) we calculate the slope m as change in y / change in x or rate of change is change  in independent variable / change in independent variable  =  (50 - 80 ) pineapples / (5 - 2) dollars or -10 pineapples / dollar. 

Next plug in one of the data points (x,y) and the calculated slope m into the equation y = mx+b and solve for b:

80 = (-10)*2 + b, so b = 100.

This means that the demand for pineapples if free (price is $0) would be 100. Thus the equation for demand is:

y = -10*x + 100. 

Note the Change in demand = Slope X Change in price. (See Example 3, page 575).

Unit 9C Exponential Modeling

• Exponential growth occurs when a quantity grows by the same percentage (proportion or rate) each time period.
• Exponential decay occurs when a decreases by the same percentage each time period.

Exponential Growth Law

Initial quantity Q₀

Time t

r = fractional growth rate per unit of time (must be same units used for a t)

Q= Q₀ * (1+ r )^t

---

Exponential Decay Law

We have exponential decay when the rate is negative  9-r).

• Initial quantity Q₀
• Time t
• r = fractional decay rate per unit of time (must be same units used for a t)
•  Formula:  Q= Q₀ * (1- r )^t

Doubling Time

• Doubling time:  The time required for an exponentially growing quantity to double.
• Alternate formula for calculating exponential growth from the doubling time:
• Let T_double be the doubling time for the quantity.
• After time t, the original quantity will have increased by a factor of 2^t/T_double.

i.e. Quantity at time t = Initial Quantity *2^t/T_double.

• Example: If time to double population is T_double = 10 years.   After 30 years the population will increase by a factor of 2^30/10=2³= 8.

Exponential Decay and Half-Life

• Quantity is decreasing by same percentage each unit of time.
• Half life:  Time it takes for the quantity to decrease to half of its original amount.
• Formula calculating the amount remaining after time t:   If T_half is the half life

Quantity at time t = Initial Quantity * (1/2)^t/T_half .

Examples:

• Radioactive decay. 
• Assume a half life of 40 years.  What fraction remains after 200 years?  (1/2)^200/40. = 1/32.
• Plutonium:  Half life of 24,000 years.  100 pounds deposited.  After 100,000 years amount remaining is

100*(1/2)^{1000000/240000}= approx. 5.6 pounds.

• Example 5: Carbon emissions in China and the U.S. (page 592)
• Example 6: Drug Concentrations. (page 594)

Approximate and Exact Formulas for Doubling Time and Half Life

Assuming a fractional growth rate of r,  or a percentage growth rate of P  (r = .01*P)
or  a fractional decay rate of r (as a negative number) or a percentage decay rate of P  (r = .01*P)
The approximate formulas only work well if | P |  (or r) is small -- < 15% or .15

Approximate Doubling Time Tdouble = 70/P	Exact Doubling Time Tdouble = log102/log10(1+r)
Approximate Half Life Thalf  = -70/P	Exact Half Life Thalf  = - log102/log10(1+r)

Review of Logs (page 588)

log₁₀x  means the power to which must raise 10 to obtain x.    10^? = x

In general,  log_bx  means the power to which must raise b to obtain x.    b^? = x

Properties:

1.  log₁₀(10x) = x
2.  10^(log₁₀^x) = x
3.  log₁₀( x * y ) = log₁₀( x )+ log₁₀( y )
4.  log₁₀( a^x ) = x * log₁₀( a^x)
5.  log(x)  is an increasing function.  But it increases very slowly for x >1.
6. To solve an equation b^x = c for x where b and c are constants:

x = log(c)/log(b)

Radiometric dating: See Example 7 page 595

We solve the equation Quantity at time t = Initial Quantity * (1/2)^t/T_half for t.