1.7 Exponential Growth and Decay

Math 150

Spring 2005

Lab: Exponential Growth and Decay (2)

This lab is a guided exercise. It consists of an exercise in Vista titled as above

with the notes contained in this presentation. You should read these notes and work the exercises

when told to do so. You may use a graphing calculator, your text and

your notes as needed. You must work this lab individually but you may ask

the mentors or your instructor for help.

Growth and Decay (Decrease) A quantity that changes over time may either increase or

decrease. A quantity Q that increases by a fixed percent every year is said

to have “exponential growth” A quantity Q that decreases by a fixed percent every year is said

to have “exponential decay” In general, we say that a function Q(t) is exponential if it has a relative (percent) rate of change that is constant.

Another way of saying this is that the amount by which the function Q(t) changes at time t depends on the actual value of the function at time t

You should work exercises 1 and 2 from the lab now.

A Familiar Example In the example to the right,

the population is increasing at a constant rate of .042 (i.e. 0.042P(t) people are being added at time t)

You need to know what the population is at a given time in order to determine how many people are being added, but it will always be 4.2% of the population at that time.

You should work exercise 3 from the lab now.

Doubling and Halving

Quantities that get smaller over time will at some point be half of where they started

Quantities that get larger will at some point be twice their original size

The Half Life of Ozone

You should work exercise 4 from the lab now.

Financial Applications

Compound Interest Exponential growth Investment value goes up (hopefully) Property value Equipment and facilities depreciation (value goes

down a fixed percent every year) Future and Present Value

Two Kinds of Compounding

Compounding can occur a fixed number of times per year We will focus on compounding once per year

Compounding can occur continuously In this case, use e

An Example or Two

Exercises

Work exercises 5 and 6 from the lab now.

Present and Future Value The value of money now will differ from the value years from now (due to

inflation, investment growth, etc.) To find the value B of a sum P in some number of years, we will only consider

interest accumulations For example, $100 deposited in an account paying 7% interest compounded

annually will be worth $107 in 1 year. $100 is the present value and $107 is the future value.

Computations

In order to apply these ideas, we need some formulas.

Future and Present values depend on the type of interest that is accumulating

An Example: Lottery Winnings

Exercises

You should work exercise 7 from the lab now.

Same Example: The Other Way Around