Download Honors Precalculus Modeling Using Exponential and Logarithmic

Honors Precalculus
Modeling Using Exponential and Logarithmic Functions
Many real-world phenomena can be modeled using exponential functions, specifically those types of
populations which grow as a proportion of itself over time. In fact, that is the hallmark of an exponential
growth or decay function. Examples include.
Compound interest – Calculating the amount of money in an interest-bearing account over time.
Population growth – of a town, a colony of rabbits, bacteria in a petri dish, etc.
Radioactive decay – Determining the age of an object by examining the proportion of a radioactive isotope
(such as carbon-14) which remains in a sample when the half-life of the isotope is known.
Logistic growth – Calculating the number of people who have heard a rumor or who have been infected by a
particular disease.
Let’s examine some of these in more detail. Rather than just give you the formulas (that’s no fun!) that will
help you, I’d like you to think about some of the factors that will affect a population over time.
First, let’s look specifically at compound interest. What are some of the factors you need to know in order to
calculate the future value of an investment today?
Now, let’s look at the difference between “discrete” and continuous growth. Specifically, what happens as
?
What about populations that grow or decay in set multiples over time? For example, a colony of bacteria which
doubles every 8 days? Or a company in bankruptcy whose sales are expected to halve every month for the next
few months? What factors are important here? How can we relate them to an exponential model?
A word about “half-life”. The meaning of half-life is the amount of time it takes for a quantity to decrease to
half of its present value. Theoretically, according to this model, a quantity can never really decay to zero, but it
can get close enough to zero for many of us to be satisfied.
The confusing thing for people to understand is: Suppose I tell you that the half-life of an isotope is 20 years.
Well, if half of the object is gone in 20 years, isn’t the other half gone in the next 20 years? No, because over
the span of the next 20 years, half of the REMAINING quantity is gone, leaving one-quarter of the original
object.
Just for fun, let’s say you started with a sample which contains 240 grams of a radioactive isotope whose halflife is 20 years. Without using a formula, how many grams of the isotope will be present in 40 years? 60 years?
100 years? It’s an easy calculation, right? It gets complicated when the number of half-life spans is not an
integer. For example, how many grams remain after 25 years?
Logistic Growth Curves
This one is really interesting because it does model
(approximately, of course), some really interesting
phenomena. However, this type of curve deals with
“saturation”. In other words, not all things can grow
forever—at some point, the saturation point of the
population is reached, and growth slows, despite the initial
exponential growth of the function. Although the domain
of this function is technically all real numbers, it is
particularly useful for values of x within a few units of
zero.