<< Chapter < Page Chapter >> Page >
  • Use the exponential growth model in applications, including population growth and compound interest.
  • Explain the concept of doubling time.
  • Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling.
  • Explain the concept of half-life.

One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications.

Exponential growth model

Many systems exhibit exponential growth. These systems follow a model of the form y = y 0 e k t , where y 0 represents the initial state of the system and k is a positive constant, called the growth constant . Notice that in an exponential growth model, we have

y = k y 0 e k t = k y .

That is, the rate of growth is proportional to the current function value. This is a key feature of exponential growth. [link] involves derivatives and is called a differential equation. We learn more about differential equations in Introduction to Differential Equations .

Rule: exponential growth model

Systems that exhibit exponential growth    increase according to the mathematical model

y = y 0 e k t ,

where y 0 represents the initial state of the system and k > 0 is a constant, called the growth constant .

Population growth is a common example of exponential growth. Consider a population of bacteria, for instance. It seems plausible that the rate of population growth would be proportional to the size of the population. After all, the more bacteria there are to reproduce, the faster the population grows. [link] and [link] represent the growth of a population of bacteria with an initial population of 200 bacteria and a growth constant of 0.02 . Notice that after only 2 hours ( 120 minutes), the population is 10 times its original size!

This figure is a graph. It is the exponential curve for y=200e^0.02t. It is in the first quadrant and an increasing function. It begins on the y-axis.
An example of exponential growth for bacteria.
Exponential growth of a bacterial population
Time (min) Population Size (no. of bacteria)
10 244
20 298
30 364
40 445
50 544
60 664
70 811
80 991
90 1210
100 1478
110 1805
120 2205

Note that we are using a continuous function to model what is inherently discrete behavior. At any given time, the real-world population contains a whole number of bacteria, although the model takes on noninteger values. When using exponential growth models, we must always be careful to interpret the function values in the context of the phenomenon we are modeling.

Population growth

Consider the population of bacteria described earlier. This population grows according to the function f ( t ) = 200 e 0.02 t , where t is measured in minutes. How many bacteria are present in the population after 5 hours ( 300 minutes)? When does the population reach 100,000 bacteria?

We have f ( t ) = 200 e 0.02 t . Then

f ( 300 ) = 200 e 0.02 ( 300 ) 80,686 .

There are 80,686 bacteria in the population after 5 hours.

To find when the population reaches 100,000 bacteria, we solve the equation

100,000 = 200 e 0.02 t 500 = e 0.02 t ln 500 = 0.02 t t = ln 500 0.02 310.73.

The population reaches 100,000 bacteria after 310.73 minutes.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
explain for me
Usman
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how en where can u apply it
Migos
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier Reply
Practice Key Terms 4

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask