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Date: May14, 2012
OUTLINE
Title: Applications of calculus in Economic
Introduction (2 Paragraphs)
Thesis Statement
Preview topic (Integral, optimization, and Exponential functions)
Who was the founder? Who combine this application to economic?
People who make use of it
Examples
Closing sentence
Body 1 (3 Paragraphs)
Integral
The useful role this application played in economics in the America society.
Credible source (Quotation, supporting details)
Examples (Problem and solution, innovation of this application)
Statistics (study done)
Closing Sentence
Body 2 (3 Paragraphs)
Optimization
The useful role this application played in economics in the America society.
Credible source (Quotation, supporting details)
Examples (Problem and solution, innovation of this application)
Statistics (study done)
Closing Sentence
Body 2 (3 Paragraphs)
Exponential functions
The useful role this application played in economics in the America society.
Credible source (Quotation, supporting details)
Examples (Problem and solution, innovation of this application)
Statistics (study done)
Closing Sentence
Conclusion (2 Paragraphs)
Restate thesis statement
Sum up the ideas (Integral, optimization, and Exponential functions)
Close it up.
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Exponential functions play tremendous roles not only in a person academic life, but also
in the real word; so therefore, fully understanding exponential functions, the different types of
exponential functions, their rules, and usefulness in the America society will help solve and
prevent problems resulting to complex situations. After much argument by scholars, exponential
functions are widely known as one of the most important types of functions in science. This
function is useful in different aspects of life for example; they are very meaningful when it
comes to estimating population growth and decay. Another area of life where this function play a
vital role is the banking and finance profession.
In addition, exponential functions have a wide range of impact in finance, economics,
biology, ecology, physics, and many other sciences fields. A simple example of exponential
functions, in which a certain amount of money is deposited on a bank account at a given interest
rate, will follow shortly as well as a track of growth needed over a specified time period. For
example, if $200 is deposited in an account earning 2% interest compounded annually, then the
exponential function f(t) = 200(1.02)t (where t stands for time), given the amount of money in
the account after t years. So at the end of 2 years it would be calculated as 200(1.02)2 = $208.08.
This means that the money is compounded continuously at every instant of time.
The first recorded modern use of exponent in mathematics was in a book called
"Arithemetica Integra" written in 1544 by English author and mathematician Michael Stifel.”
Furthermore, Stifel coined the word "Exponent" using words from Latin. One of the words used
was “Expo” meaning "out of" and “Ponenere”, which stands for "place". However, both
combinations made the word exponent and it true meaning as “to move out of place" (Ayres 9).
Because it is moving out of place, in the real word, this refers to the way the number is lifted
above other number in the text line.
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Nevertheless, bookmen also placed a definition and meaning for this function that made it
last overtime. According to them, “Exponential function is defined as a function in which an
independent variable appears as an exponent.” (Wordwebonline.com). An illustration of the
function tells that exponential functions have the form f(x) = abx, where a is a nonzero real
number b is a positive real number not equal to 1 and x can take on all real number values.
Examples of exponential functions include f(x) = 2x, or g(x) = 3(0.5)x or h(t) = 1300(1.02)t.
Scientists from various fields used exponential functions as tool for measure in which a
quantity is assumed to increase or decrease at a rate which is proportional to the amount of the
quantity at any given time. The power of exponential growth can be exemplified by the
following example. Suppose Tom is taking on a certain job for one month and Tom is given the
choice of being paid a fee of $20,000 for the month or being paid 1 penny on the first day, 2
pennies on the second day, 4 pennies on the third day, 8 pennies on the fourth day, and so on
with the number of pennies earned doubling each day through the end of the month.
If Tom does not understand exponential growth he will prefer the flat $20,000 fee. But,
the exponential function y = 2x-1 will give the number of pennies earned on day x. So, on the 31st
day of the month, Tom will get paid 230 pennies or $1.07374. Exponential functions are currently
being used to study different phenomena of life such as, the growth of the internet, the spread of
disease, the projected increase and decrease of the national debt, and much more. Above all,
wherever growth and decay are studied, exponential models acts as a fundamental part of it.
In an example adapted from the famous story of the Emperor of China who wanted to
thank the inventor of the game of chess by giving him anything he wanted in the kingdom. The
inventor only wanted rice, which sounded like a simple gift, but it was said in a particular
manner: “I would like one grain of rice on the first square of the chessboard, two grains on the
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second, four grains on the third and so forth. I would like all of the grains of rice that are placed
on the chessboard in this way.” To the emperor, this didn’t sound like much, perhaps a bushel or
two of rice, just as my students assuming that placing pennies on a checkerboard in the same
manner as the rice wouldn’t have a greater value than $100 per day for 1 year. Let’s look at the
growth of this exponential function:
X 1 2 3 4 5 6 7 8 9 10 11
F(x) 1 2 4 8 16 32 64 128 256 512 1024
This is similar to the previous function of f(x) = 2x, where a doubling process is
witnessed. However, this function is slightly different because the first number in the range is 1,
not 2. This function can be stated as f(x) = 2x-1. Looking at the first square on the chessboard and
placing only one grain of rice on it, signifies the starting point for this exponential growth. This
is important because these eventually large numbers will have begun from the most natural of
numbers, 1. We can see that this function holds true: f (1) = 21-1 = 20 =1, f(2) = 22-1 = 21 = 2, and
so forth. This function allows for the previous table’s data to be used, simply shifting the
information down to the next value of x.
According to this table, we only have $20.47 (sum of f(1) through f(11)). At this point,
students (and possibly the emperor) still stick with their initial thought of $100 a day for a year.
They compare this value to the $1,100 that they could have after 11 days and believe it to be
impossible for this doubling process to overtake the linear equation of a set rate of money per
day. This exponential function, as do others, may grow slowly at first, but there is a turning point
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where the exponential function will grow rapidly. Let’s continue the previous chart out a little
further: (Shawn A. Mousel).
X 12 13 14 15 16 17 18 19 20
F(x) 2048 4096 8192 16384 32768 65536 131,072 262,144 524,288
In mathematics and the real world, exponential function has relatively close meaning and
interpretation with multiplication. When a number is multiplied by itself, the expression can be
written just like this (2 x 2 x 2 x 2) or it can be shortened like this (24). The 2 in this case stands
for the base, while 4 is known as the exponential function. This means, multiply the base by
itself 4 times. Exponential functions can also be called powers. In this case, it is translated as
“two to the fourth power”. This study is so much important because many events in the real
world follow a pattern of some base number raised to a fractional exponent.
An important example is earthquakes, earthquakes cannot be predicted in general. The
number and size of earthquakes worldwide in a year follows a pattern. The pattern is given by a
law for example, 10(5 - M) where the exponent is (5 - M). M stands for magnitude, which is a
measurement of an earthquake's power. Using this formula and putting a 3 in for magnitude
gives 10(5 - 3) and since 5 - 3 = 2, the formula tells that there will be 100 earthquakes of
magnitude 3 expected in a year. But, 100 is the same as 10 x 10, or 102. The important thing is
that, as the magnitude of the earthquakes becomes smaller and smaller, the number becomes
bigger and bigger.
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Despite the fact that exponential function helps in different aspect of life, one must fully
understand the rules and law governing it. Since exponential functions are functions of the form
f(x) = bx for a fixed base b which could be any positive real number. All the same, exponential
functions are characterized by the fact that their rate of increase depends on their value.
Basically, the main rules used in manipulating exponential functions are bx+y = (bx)(by), bxy =
(bx)y, b0 = 1, b-x = 1/(bx), an/m = m√an = ( m√a)n.
Emphasizing on one of the rules and using it’s example to prepare an informal graph will
play a significant role in this paper. The rule that says b0=1 will be detailed as follow.
Psychologically speaking from a stand point of view, this law or rule is kind of confusing
because of its broad meaning. However, many students fail to understand and prove how b0=1.
To understand the rule b0=1, one must understand that as the values keep decreasing the numbers
are divide by their base for example, 25 = 32, 24 = 16, 23 = 8, 22 = 4, 21 = 2 20 = 1 or 35 = 243,
34 = 81, 33 = 27, 32 = 9, 31 = 3 30 = 1. This automatically leads to the facts that 20 = 1 and 30 = 1.
The same method could be done for other numbers too and it would work just exactly the same
way. So at least for all positive whole numbers b it is true that b0 = 1
Table Graph
x F(x) = 2x
5 32
4 16
3 8
2 4
1 2
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Again, to clearly understand this function's help, usefulness, and impact in the America
society enough analysis needs to be done. However, if plenty of examples that connect this
function to the real word are done then one will easily see how valuable this function is in
America society. In pre-school, the arithmetic at that time always had topics related to the
exponential function. It now appears conniving that exponential functions appear so frequently in
different area of mathematics and the activities one perform in the real world.
According to Albert A. Bartlett in his article "The exponential function-Part I”, “While
the advertising agencies continue to pour forth unparalleled calls for more growth, a few
thoughtful people are beginning to ask "Can we continue in the future to grow as we have grown
in the past?" The answer to this vital question can be found in the mathematics of the exponential
function. The mathematics of growth is the mathematics of the exponential function. Growth is
one of the greatest cornerstones of most business and economic system.”(1). He then added that,
“The greatest shortcoming of the human race is man's inability to understand the exponential
function.”
In conclusion, exponential functions are part of the day to day activity the society runs; so
therefore, fully understanding exponential function, the types of exponential function and their
rules positively will help America society. However, these functions also contribute to productive
decision making. As a matter of fact, given the pertinence of exponential functions in practical and
theoretical ways help not only individuals in the society, but also the America economy as a whole
because it study factors such as national debt and finance that can destabilize the economy. As
exponential functions help predict the behavior of such processes more rigorously, its Theorem also
helps for the discovery and understanding of different aspects of human nature.
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Work Cited
Ayers, Chuck. “The History of Exponents.” Math.unt.edu. eHow.com, 25 Jan. 2012. Web. 7
May. 2012. PDF.
Bartlett, Albert A. “The Exponential Function-Part I.” Colorado: Department of Physic,
University of Colorado. 17 July. 2010. Web. 7 May. 2012. PDF.
Shawn, Mousel A. “The Exponential Function Expository Paper.” Nebraska: Department of
Mathematics, Columbia University, May. 2006. Web. 6 May. 2012. PDF.
Weisstein, Eric W. "Exponent Laws.” Mathworld.wolfram.com. MathWorld. Wolfram Research,
Inc. 26 Apr. 2012. Web. 7 May. 2012.
