Exponential
Functions  
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Principle

Introduction Do you know what exponential means? And
the question has to be asked: why is it that so few adults understand
exponential functions when a child of five can understand the basic
notions? Origins and power of the exponential symbol As always there is great, ancient and sustaining wisdom in powerful
symbols such as exponent. Its Latin origins are exponere –
to expound, to make explain, to make clear, to make manifest,
to elucidate, to clear of obscurity, to unfold or to the illustrate
the meaning of. And this is precisely what exponential functions can do. They can
unfold the story of our universe for us and provide great meaning. Change
in the universe does not happen uniformly and occurs
in differing patterns of waves, each with its unique rhythms. When the underlying patterns are revealed to us we are better able to
lead lives that are in harmony with them. We are better connected to the
universe. We can be more at one with the
greater the ebb and flow  whether it be in understanding how microbe or
human populations expand and shrink or how interest rates accumulate and
usury drives people into debt and misery or how weather systems form
and dissipate or how we
discover, extract and destroy resources such as mineral oil and gas….. This use of the function symbol is employed to convey the idea that two quantities have a relationship and changes in one affect changes in the other. They have a functioning relationship. Exponential functions connect us to our history and to our greater
environment. They can help us avoid catastrophe by teaching us how to
conserve limited resources and setting of destructive debt and weather
spirals. They can inspire us by showing us how great and sustainable ideas
can spread throughout our communities. Simple lessons for younger students Here are two examples of how a child of ten or younger can begin exploring our
exponential beings.
(Answer – the 29^{th} day and then there will be just one day to
save the pond.) (Teachers know best and can create many games based on this principle . An example might be to have a child print a message of hope 32 times on a page. They copy and paste the original thought once to give two copies. They copy these two copies to give four and so on. The student then cuts the page in two and gives each half (16 messages) to two children. They in turn cut their half page in two and give each quarter page (8 messages) to two more children. Those four children cut their quarter page in two and pass on 4 messages to two other children. At this point fifteen of a class of 32 has read the message and yet each student has only had to communicate it to two others. Played once a week students will begin to sense the power of the exponential function
"The amount of wheat is approximately 80 times what would be produced in one
harvest, at modern yields, if all of Earth's arable
land could be devoted to wheat. The total of grains is approximately
0.0031% of the number of atoms in 12 grams of carbon12
and probably more than 200,000 times the estimated number of neuronal
connections in the human brain
(see large
numbers)."
Uses of the Exponential Symbol The exponential function is invaluable for communicating how to lead sustainable lives in that it reveals how wars are born out of inflation, how usery destroys lives (as we are seeing with the current global “credit squeeze” *) and how our misuse of, for instance, our carbon potential can put us at grave risk. In the case of mineral oil and gas we
have used up half the known reserves over the centuries. However the
increase in the rate of our consumption of the resources means that
remaining supplies may last as little as a decade – assuming the
extraction rate increases with demand. *Note March 2008: The current inflation pressures and associated “credit squeeze”
or “subprime mortgage crisis” is
really a manifestation of what happens when a society confuses energy with
one of the forms it can take and does not understand simple exponential
functions.
Use this easy calculator and see how it works to answer the following questions:
Albert A Bartlett expounds in wonderful discussion the revelatory power of exponential functions using three media. He provides, in addition to much of the above, the example of bacteria multiplying in a glass jar to illustrate in a very simple and clear way what happens when exponential growth occurs in a finite environment. His thesis: "The greatest shortcoming of the human race is our inability to understand exponential functions" Article: "Forgotten Fundamentals of the Energy Crisis" Audio Dr. Albert Bartlett: Arithmetic, Population and Energy (70839 hits March 11 2008) Video (8 parts) And probably the most boring. But then again, when I told that to my students and had them give me feedback, most said that if you followed along with what the presenter (a professor emeritus of Physics at Univ of ColoradoBoulder) is saying, it's quite easy to pay attention, because it is so compelling." (43640 viewings March 11 2008) Another
useful and short video is
Are
Humans Smarter Than Yeast? An appreciation of exponential functions is invaluable for communicating climate and tectonic processes. For instance see in the footnotes the charts of the Beaufort Scale (wind speed and wind pressure) and the Richter Scale (earthquake force and acceleration.) Almost every child will have to face a world in which usury abounds and it is in the interests of money traders (banks etc) to promote ignorance of the truths that exponential functions reveal. Many householders paying interest only on their mortgage at 7% fail to understand that the mortgage doubles every seven years. Many people paying interest only on their car loans at 20% fail to appreciate that the size of the loan doubles every three and half years. The failure of our schools in countries like the USA and New Zealand to communicate the nature of exponential functions is resulting in societies that are generating massive economic/ecological debt. Check out the US National Debt Clock and be mindful this is only a fraction of the debt being generated. Check out the New Zealand household debt/disposable income graph  rising from about 50% before the "economic reforms" of the 1980s to of disposable income to Classic money riddle You are given two choices for an increasing weekly allowance: Option one: Your allowance begins with 1 cent and doubles each week. Option two: Your allowance begins with $1 and increases by $1/week Which option do you choose if you want a large allowance quickly? Check the
chart and see how wise your decision is.
With an understanding of the fundamentals of exponential functions it is
possible to enjoy glimpses into the amazing revelations of chaos
theory (an unhelpful name in that the theory actually finds order
amidst the seeming chaos of weather and climate systems, cloud, galaxy,
leaf and mountain formations, fluid dynamics, population changes, forest
fires, wars, disease, systems that follow the laws of quantum mechanics
and general relativity and the dynamics of the action potentials in our
neurons. You will probably know Edward Lorenz’s discussion "The
flapping wing represents a small change in the initial condition of the
system, which causes a chain of events leading to largescale phenomena.
Had the butterfly not flapped its wings, the trajectory of the system
might have been vastly different. The patterns of change described by chaos theory are not the same as those
produced by simply doubling a number in that they describe closed systems
in which both explosive and
implosive change occurs. However an insight into the rate at which change
in the amount of change occurs and
a sense of the wonderful complexity of change which is our universe. This is why our teachers of Mathematics are truly Environmental Educators
for they provide us with visions of change in which we can make great
connections with our universe and have a more powerful sense of its
vitality.
Return to Education Discussion
The wind pressure can be approximated by: Pressure = ½ x (density of air) x (wind speed)^{2} x (shape factor)
See
this table:
Since the
scale of Beaufort
and the wind speed are not related linearly, it is not possible to express
the wind pressure as: Earthquake force and acceleration When
trying to understand the forces of an earthquake it can help to
concentrate just upon the up and down movements. Gravity is a force
pulling things down towards the earth. This accelerates objects at 9.8 m/s^{2}.
To make something, such as a tin can, jump up into the air requires a
shock wave to hit it from underneath travelling faster than
9.8m/s^{2}. This roughly corresponds to 11 (Very disastrous) on
the Mercalli Scale, and 8.1 or above on the Richter Scale. In everyday
terms, the tin can must be hit by a force that is greater than that which
you would experience if you drove your car into a solid wall at 35 khp (22
mph).
Richter
Scale
Other NZ household debt accumulation data can be viewed in this Reserve Bank Paper (2004) Sample symbol source ex·po·nent
(
ksp
n
nt,
k
sp
n
nt)
n.
1. One that expounds or interprets. 2. One that speaks for, represents, or advocates: Our
senator is an exponent of free trade. 3. Abbr. exp
Mathematics A number or symbol, as 3 in (x + y)^{3},
placed to the right of and above another number, symbol, or expression,
denoting the power to which that number, symbol, or expression is to be
raised. Also called power. adj.
Expository; explanatory. [Latin exp
n
ns, exp
nent, present participle of exp
nere, to expound; see expound.] ex·po·nen·tial
(
k
sp
n
n
sh
l)
adj.
1. Of or relating to an exponent. 2. Mathematics a. Containing, involving, or expressed as an exponent. b. Expressed in terms of a designated power of e, the base of natural
logarithms. exponential
(
k
sp
n
n
sh
l)
Relating to a mathematical expression containing one or more
exponents.
Something
is said to increase or decrease exponentially if its rate of change
must be expressed using exponents. A graph of such a rate would appear not
as a straight line, but as a curve that continually becomes steeper or
shallower. expound [Middle
English expounden,
from AngloNorman espoundre,
from Latin exp
nere : ex, ex + p
nere, to place; see apo in IndoEuropean roots.]

 