Lesson 11 Exponential Functions as Business Models
Announcements
Popper 10 today
Homework
Quiz
1

Exponential models all have the same form:
( )
  x
Where I is called the initial value and is a number multiplier while b is called the base (it is never the number 1, though). The x is in the exponent place and that’s the source of the name for this family of functions.
Usually the base is some nice rational number but it can be irrational. The most common irrational base in business applications is e , the Euler number, which is very close to 2.72.
Exponential functions can be increasing or decreasing. The increasing ones are called growth functions and the decreasing ones are called decay functions.
When the Euler number is the base,
( )
 ae bx is the usual form of the function.
If b , the multiplier on the x is positive, it is called the growth constant and the function will be increasing.
If b , the multiplier on the x is negative, it is called the decay constant and the function will be decreasing.
When some rational number is the base
( )
 ab x is the usual form of the function.
If b > 1, the function is an exponential growth function.
If b is a proper fraction, then it is an exponential decay function.
We will use the usual commands in GGB to compute values at a given x or the rate of change at a given x .
Example: given  e

0.15
x growth or decay? Incr/decr? y-intercept?
Initial value?
Value at x = 20?
Rate of change at 35?
2

Example: Given ( )

26.9(3.258) x growth or decay? Incr/decr? y-intercept?
Initial value?
Value at x = 15?
Rate of change at 10?
Popper 10 Question 1
3

Example:
Given the points (0, 1000) and (3, 1095) lie on an exponential graph. Find the equation of the graph.
Popper 10 Question 2
4

When you have unrestrained or uninhibited growth…exponential functions are often the preferred model. Population work and investment growth are examples where exponential models are used.
Example
A biologist is studying a strain of bacteria. She starts with a culture of 25,000 and gives the sample ideal growing conditions. After 3 hours, the number has grown to
63,000. What is the model function?
How many will be present in her sample in 6 hours, assuming uninhibited growth?
Popper 10 Question 3
5

Example Given the following sales data in millions of dollars, find an exponential model that fits.
Year $
1998 51.4
1999 53.2
2000 55.8
2001 56.1
2002 58.1
2003 59.0
Find the rate at which the sales were changing in 2007
Popper 10 Question 4
6

The flip side to growth is decay – best recognized from Carbon 14 dating. The half-life is the time it takes for a radioactive substance to loss half its mass.
Example At the beginning of a study there are 50 grams of a substance. After
17 hours, there are 38.7 grams.
What are our two points?
Find an exponential regression model
What will be the rate of decay on the 40 th day of the study?
Popper 10 Question 5
7

Interestingly, medical applications have exponential applications, too. Suppose the drug you are taking has a half-life of 4 hours. Suppose you are given a dose of
1000mg of the drug. Find the exponential model and use it to predict the amount left in your bloodstream 24 hours later.
Popper 10 Question 6
8

And here’s an example from archeology:
The half-life of Carbon 14 is 5770 years. Find an exponential regression model for
Carbon 14. Bones found on a dig had 22% of the amount of C14 that living bones have. Find the approximate age of the bones
Popper 10 Problem 7
9

Growth is not always uninhibited. Often there are constraints that limit growth.
A function that models constrained growth is
( )
 
Ae
 kt
The t lim

( )

C and the y -intercept of the function is (0, C − A )
A learning curve is a perfect example of this type of exponential function:
Popper 10 Question 8
10

Example
An HR department determines that an employee will be able to assemble products on the line each day with the following model:
( )
  e

0.5
t after t months of employment.
How many units can an employee assemble on the first day?
How many units can an employee assemble after 3 months on the job?
How many units can a very experienced employee assemble?
At what rate is an employee’s productivity changing after 4 month on the job?
Popper 10 Problem 9
11

Logistic Function
Another type of growth is called logistic growth. The general form of the equation is

1

A
Be
 kt
With graphs looking like:
Logistic curves have features of both exponential curves and learning curves. The horizontal asymptote on the right is called the carrying capacity. It is generally the
A in the formula. Logistic curves are used to model populations where there is a limiting condition (food, water, air, predators)…they come up often in ecology and city management.
12

Example
The fisheries industry professional organization commissioned a study to determine the growth rate of a type of fish in the Pacific Northwest. The study came back with the following equation:

2.4
 e

0.338
t t is time in years; N is millions of tons of fish
What was the initial number of fish in the population?
What is the carrying capacity of the area?
What will be the fish population in 3 years?
How fast is the fish population changing in 2 years?
Popper 10 Question 10
13