Math 365 Lecture Notes – J



Section 1.5 – Exponential Functions

●Introduction

The function [pic], where the exponent is a , is called an . The number 2 is called the of the exponential function. Every exponential function has rate of change.

Exponential Growth Functions. - The table below contains approximate population data for the McAllen, Texas, metropolitan area.

|Year |Population (in thousands) |

|2000 |570 |

|2001 |591 |

|2002 |613 |

|2003 |636 |

Is the population of McAllen growing linearly? Explain.

As a matter of fact, the population of McAllen is growing exponentially (increases slowly at first and eventually climbs quickly). To show this models exponential growth we divide each year’s population by the previous year’s population.

[pic]

[pic]

[pic]

Since all calculations are near , we can conclude the population grew by about between 2000 and 2001, between 2001 and 2002, and between 2002 and 2003. Whenever we have we have . The population can be modeled by the function

where the base , also called the , represents the factor by which the population grows each year.

Exponential Decay Functions - When a patient is given medication, the drug enters the bloodstream. The rate at which the drug is metabolized and eliminated depends on the particular drug. For antibiotic ampicillin, approximately 40% of the drug is eliminated every hour. A typical dose of ampicillin is 250 mg. The quantity, Q, of ampicillin (in mg) in the bloodstream, is a function of the number of hours, t, since the drug was given. Find the quantity remaining in the bloodstream at t = 0, t = 1, t = 2, and t =3.

|t |Q |

|0 | |

|1 | |

|2 | |

|3 | |

The amount of antibiotic Q can be modeled by the function .

●The General Exponential Function

We say that P is an exponential function of t with base a if where [pic] is the initial quantity (when t = 0) and a is the factor by which P changes when t increase by 1. If a > 1, we have

, if 0 < a < 1, we have . The factor a is given by where r is the decimal representation of the percent rate of change, r may be positive (for growth) or negative (for decay).

Properties of Exponential Growth Properties of Exponential Decay

1) [pic] where 1) [pic] where

2) Domain: 2) Domain:

3) Range: 3) Range:

4) Horizontal asymptote: 4) Horizontal asymptote:

5) y – intercept: 5) y – intercept:

6) Always 6) Always

7) If [pic], the graph is 7) If [pic], the graph is

Example 1: There were approximately 5.4 million farms in the US in 1950 and approximately 2.8 million in 1970.

a) Assuming that the number of farms have been decreasing exponentially, find an equation of the form [pic], where P is the number of farms (in millions) and t is the number of years since 1950.

b) Use the function found in part a) to estimate the number of farms in 2000.

●Alternate Formula for the Exponential Function

Using r (the percentage growth or decay rate where r = a – 1), is another way to write an exponential function.

[pic]

If r > 0, P is an e function. If r < 0, P is an function.

Example 2: The following functions give the population of four towns with time t in years.

i. [pic] ii. [pic] iii. [pic] iv. [pic]

a) Which town has the largest percentage growth rate? What is the percent growth rate?

b) Which town has the largest initial population? What is the initial population?

c) Are any of the towns decreasing in size? If so, which one?

Example 3: According the US Bureau of Statistics, the population of India will double in 35 years. What is the percent growth rate?

●Comparison Between Linear and Exponential Functions

Every exponential function changes at a . Every linear function changes at a .

Example 4: Find a possible formula for each statement.

a) The city’s population increases 7% each year.

b) The number of cancer cells in a tumor decays 22% per day after radiation therapy.

c) As a drug is being injected into a patient’s bloodstream, the quantity in the blood increases by 0.2 mg per minute.

d) The price of houses in a town increases 15% per year since 1990.

e) The city’s population increases by 5000 people each year.

Example 5: Climbing health care costs are a continuing concern. The table below shows the average annual health care expenditures per person for various years.

|Year |1970 |1975 |1980 |1985 |1990 |1995 |

|Health expenditures ($ per |349 |591 |1055 |1596 |2714 |3302 |

|person) | | | | | | |

a) Does a linear or an exponential model fit the data best? Explain.

b) Find a formula for the regression function that you decided is the best.

c) Graph the function with the data on the same axes.

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