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Characteristics of the exponential function, y = A 0 e kt

An exponential function with the form y = A 0 e k t has the following characteristics:

  • one-to-one function
  • horizontal asymptote: y = 0
  • domain: ( ,   )
  • range: ( 0 , )
  • x intercept: none
  • y-intercept: ( 0 , A 0 )
  • increasing if k > 0 (see [link] )
  • decreasing if k < 0 (see [link] )
Two graphs of y=(A_0)(e^(kt)) with the asymptote at y=0. The first graph is of when k>0 and with the labeled points (1/k, (A_0)e), (0, A_0), and (-1/k, (A_0)/e). The second graph is of when k<0 and with the labeled points (-1/k, (A_0)e), (0, A_0), and (1/k, (A_0)/e).
An exponential function models exponential growth when k > 0 and exponential decay when k < 0.

Graphing exponential growth

A population of bacteria doubles every hour. If the culture started with 10 bacteria, graph the population as a function of time.

When an amount grows at a fixed percent per unit time, the growth is exponential. To find A 0 we use the fact that A 0 is the amount at time zero, so A 0 = 10. To find k , use the fact that after one hour ( t = 1 ) the population doubles from 10 to 20. The formula is derived as follows

  20 = 10 e k 1     2 = e k Divide by 10 ln 2 = k Take the natural logarithm

so k = ln ( 2 ) . Thus the equation we want to graph is y = 10 e ( ln 2 ) t = 10 ( e ln 2 ) t = 10 · 2 t . The graph is shown in [link] .

A graph starting at ten on the y-axis and rising rapidly to the right.
The graph of y = 10 e ( ln 2 ) t

Half-life

We now turn to exponential decay . One of the common terms associated with exponential decay, as stated above, is half-life , the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay.

To find the half-life of a function describing exponential decay, solve the following equation:

1 2 A 0 = A o e k t

We find that the half-life depends only on the constant k and not on the starting quantity A 0 .

The formula is derived as follows

1 2 A 0 = A o e k t 1 2 = e k t Divide by  A 0 . ln ( 1 2 ) = k t Take the natural log . ln ( 2 ) = k t Apply laws of logarithms . ln ( 2 ) k = t Divide by  k .

Since t , the time, is positive, k must, as expected, be negative. This gives us the half-life formula

t = ln ( 2 ) k

Given the half-life, find the decay rate.

  1. Write A = A o e k t .
  2. Replace A by 1 2 A 0 and replace t by the given half-life.
  3. Solve to find k . Express k as an exact value (do not round).

Note: It is also possible to find the decay rate using k = ln ( 2 ) t .

Finding the function that describes radioactive decay

The half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time, t .

This formula is derived as follows.

         A = A 0 e k t The continuous growth formula . 0.5 A 0 = A 0 e k 5730 Substitute the half-life for  t  and  0.5 A 0  for  f ( t ) .        0.5 = e 5730 k Divide by  A 0 . ln ( 0.5 ) = 5730 k Take the natural log of both sides .           k = ln ( 0.5 ) 5730 Divide by the coefficient of  k .          A = A 0 e ( ln ( 0.5 ) 5730 ) t Substitute for  r  in the continuous growth formula .

The function that describes this continuous decay is f ( t ) = A 0 e ( ln ( 0.5 ) 5730 ) t . We observe that the coefficient of t , ln ( 0.5 ) 5730 1.2097 × 10 −4 is negative, as expected in the case of exponential decay.

The half-life of plutonium-244 is 80,000,000 years. Find function gives the amount of carbon-14 remaining as a function of time, measured in years.

f ( t ) = A 0 e 0.0000000087 t

Radiocarbon dating

The formula for radioactive decay is important in radiocarbon dating , which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died within the last 60,000 years.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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