When a quantity of something
changes with time, we often make the simplifying assumption that it changes
in a "uniform" way. This means that if 
are two intervals of equal time duration ( 
)
then the amount of change over the first interval is the same as the amount
of change over the second interval.
We assume this for example
when we say a car is traveling at a "constant speed" and write 
where R is the speed, T is the time, and D is the distance
traveled. Galileo's law for uniformly accelerating bodies is a similar assertion.
He says that over equal intervals of time, the speed of the body suffers equal
changes. We write that law
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where 
is the speed at the start of the interval of duration T, and 
is the speed at the end of that interval. The number g is of course the
constant gravitational acceleration.
Uniform change such as this
is the easiest type of change to understand and to describe. But how does one
pass from the statement that: "v changes by equal amounts in equal
times." to a statement like 
?. It is not obvious, and in fact it is something that Galileo was unwilling
to do. Let us see what we do that Galileo would not.
We imagine that there is
a quantity of some substance that varies with time. Call that quantity 
. Then our assertion is, when we choose a duration 
, we must observe
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whatever the starting times 
and 
are.
This means that the change
in Q that we might call
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does not depend on t at all, but only depends on 
.
We write
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for some function f to express this idea. What is f ? We do not
know, and did not say in the first formulation of the principle of uniform change.
Certainly, 
but if it does not depend on 
also, then Q itself will be constant.
But there is more. If n
is any positive integer, then
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This is because for any t and 
, it must be true that
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but, using the properties of measurement that we introduced in the Polar Coordinates Section we see that
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and each
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By a simple reversal of this
reasoning, we see also that for any positive integer n,
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and then for any positive rational number 
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Galileo was willing to go
this far. Our next step (using algebra) is to extend the conclusion to all rational
numbers (fractions) 
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I cannot speak for Galileo. But the final step for
us is to write the equation, for all real numbers 
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Irrational numbers in measurement
This, Galileo and the Greeks
would not do. They did not think of real numbers as "numbers" but
only as relationships. In any case, we write such equations using Algebra, and
so we can conclude that if for example, we say that there is a real number m
such that
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then
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Or
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Or, finally, letting t be 0, and 
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This is the standard equation
for a straight line. The number 
is the "slope" of the line, and the number 
is
the "y-intercept" of the line. The equation is usually written in
Cartesian coordinates (replacing 
with x)
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A linear relationship like this is always implied when a quantity y varies with another quantity x in such a way that equal changes in x give equal changes in y. Our job is usually to determine the m and the b.
Now, what does this have
to do with exponential functions ? The answer is that, just as a linear relationship
exists between variables x and y when quantity y varies
with quantity x in such a way that equal changes in x give equal
changes in y, there is another form of co-variation between positive
quantities that characterizes exponential relationships. That form of co-variation
is every bit as simple as linear co-variation.
Suppose then that we return
to time variation and imagine that there is a positive quantity of some substance
that varies with time. Call that quantity 
again. Suppose now that when we choose a duration 
,
we must observe
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whatever the starting times 
and 
are. In other words, for equal durations, we observe equal ratios (instead of
equal changes) of quantity Q. Such quantities exist and arise in a natural
way.
For example, suppose Q
is a quantity of radioactive substance composed of many atoms. Suppose for each
atom of Q there is a definite probability p that it will transform
into some other type of atom (it will decay) in a given interval of time
.
If there are 
atoms at the start then the number of atoms of type Q at the end of the interval
would be approximately, given the law of large numbers,
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Put another way,
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This relationship, since it
is assumed in Quantum Mechanics to be a matter of pure chance, should be independent
of t for any fixed interval 
.
Boltzmann distribution of energy states example
Another example is given
by the Boltzmann distribution, which is also treated probabilistically in Quantum
Mechanics. Suppose given a large ensemble A of atoms where each atom
can be in one of a large but finite number of discrete energy states:
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At any given time, let the
number of atoms in the 
energy state be 
. Suppose the total number of atoms in A is N. Then we may say
that the probability of finding an atom of A in the 
energy state is 
.
Now we may define the frequency functions 
of the energy 
of any amount of substance A as the value of 
after it is placed in the presence of a "heat bath" at fixed temperature
T and then allowed to settle to equilibrium at that temperature.
Given
two quantities of the substance, say A and B, at energies 
and 
,
we may call the composite quantity 
.
Then we may ask what the relation is between 
.
We assume that A and B do not interact, and so the probabilities
are independent. We conclude that (assuming all probabilities are positive),
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since independent probabilities multiply. Put another way,
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This will tell us that the
form of the function 
as function of fixed bath temperature T, is exponential.
What does it mean to say that 
depends only on 
and not on t ?
It means that we may write
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for some function f. In this case, 
, f is positive, and if it does not depend on 
also, then Q itself will be constant.
But now, if n is any
positive integer, then
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This
is because for any t and 
,
it must be true that
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as we saw before,
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and each ratio
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Now,
we apply the same reasoning that we did above (remembering that Q is positive)
to write the equation, for all real numbers 
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And so we can conclude that
if for example, we say that there is a real number m such that
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then
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Or
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Or, finally, letting t
be 0, and 
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Standard form of exponential function
This is the standard form
of an exponential function. The number 
is the "ratio" of change on a unit interval, and the number 
is the "initial value at time 0" of the function. The equation is
usually written in Cartesian coordinates (replacing 
with x)
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An exponential relationship
like this is always implied when a quantity y varies with another quantity
x in such a way that equal changes in x give equal ratios of change
in y. We will devote the next two parts to explaining what a real number
exponent can mean.
We can, if we wish, allow C to be 0 or negative, with appropriate reinterpretation of its meaning.
Exploration: A Discrete Exponential Function and Compound Interest
Suppose
that one of your ancestors, a lawyer, was in Philadelphia on July 4, 1776. He
became so enthusiastic about the prospects for a new nation and his hopes for
its long life that he invested $1 with a local banker to gain interest at the
rate of 3% per year, compounded annually, until July 4, 2000. At that time the
entire sum would go to one of his descendants. The particular formula for who
is to receive this money is intricate, but the upshot is that you are the lucky
one.
1. Finding a Formula. We'll let k be a variable that designates, on any given July 4, the number of years the account has been in existence. Define k to vary between 0 and an appropriate upper limit n.
Find the value of n
and enter it in the field:
We will let A
be the amount of money in the account. Define 
,
the amount in the account at the beginning, to be 1. Enter that amount in the
field:
Find a formula that describes
how to obtain 
,
the amount at the end of 
years, from 
, the amount in the fund after k years. Your formula should have the
following form:
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Of course, you have to replace
"something involving "
with the appropriate formula. For example,if you wrote 
,
then at the end of 10 years, you would have $11.00. Your definition of 
is an example of a recursive formula, one that allows you to carry out a calculation
in a number of steps, one step at a time. Enter your formula in the field:
2. Finding the Final Amount (compounded yearly). When you press the button
the system reads the value for n from the corresponding input text field, and displays the values of all the
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in the MathEdit labeled "Amount".
It also displays the amounts graphically in the Graph2D gadget labeled Graph:
On the horizontal scale is
the year, and on the vertical scale, the amounts. You can read the extents of
these variables (they will change from calculation to calculation) at the left-right
sides for the year, and at the bottom-top sides for the amounts.
Find the amount in the account
in the year 2000.
What does all of this mean?
Well, if you answered parts 1 and 2, then you learned that
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was the proper form for the recursion that produced the amount at time k+1 given the amount at time k. For 3% yearly interest the formula is
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. But another way to write
that equation is:
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(2) |
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Since h is constant,
this says that the "average" rate of change of A from time
= k to time = k+1 is proportional to 
. We are not calculating a derivative, but we are calculating a difference quotient.
This says that 
is a discrete exponential function. It is discrete because it is only defined
for integer points k. We call it exponential, not because of the
equation (2) but because of another equation that follows immediately from it.
Let P be any positive
integer whatsoever, and let j and k be given. Then we have the
remarkable fact (compare with the definition of continuous exponential functions
above):
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(3) |
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That is, over equal (integral) intervals of time P, the ratio of the ending amount to the starting amount is the same, wherever you start. This follows from the facts that, from (1) above,
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(4) |
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As we saw, this condition
(3) of equal ratios over equal intervals, properly characterizes exponential
functions, whether discrete or continuous, and establishes a fundamental analogy
with linear functions, for which the differences, not the ratios, are equal.
These exponential functions
are determined by two quantities: 
and the ratio 
in the expression:
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which follows easily from (1).
Question 1: What if 
had been doubled? Suppose your ancestor had been twice as enthusiastic and had
invested $2. Make a conjecture about the amount your account would be worth
in the year 2000. Modify your entry for 
in the field:
(Delete the older one) and press the button "Calculate Amount" to check your conjecture.
Question 2: What if the interest rate had been doubled? Suppose your
ancestor had only invested $1, but had obtained a rate of 6%. Conjecture how
much your account would be worth in the year 2000. Modify your formula for 
in terms of 
and enter your modified formula in the field:
(Delete the older one) and press the button " Yearly Compounded Amount " to check your conjecture.
Question 3: What if the interest is compounded twice a year? Return to the assumption of $1 invested at a rate of 3% per year. Now assume that the interest is compounded twice a year. So, on January 4, 1777, 1.5% interest is added to the amount, and on July 4, 1777, 1.5% of this amount is added.
Modify your formula for 
in terms of 
. Here, we have to interpret k in a slightly different way than
we did. It should now represent the number of half-year periods form the beginning,
instead of the number of years. This is because the amount will be recomputed
each half-year from July 4, 1776. Thus A(1) is the amount on Jan 4, 1777,
and A(2) is the amount on July 4, 1777, et cetera.
Now enter your modified formula
in the field (delete the older one) and press the button "Yearly Compounded
Amount " to find out how much this would produce by July 4, 2000 with that
of your first calculation, then calculate the amount that would have been obtained
by 2000 if, instead of 3% interest compounded annually, the annual interest
rate was 3.0225 %. This is not a coincidence. Try to understand what you observed.
Let us consider more frequent
compounding. We call this monthly compounding, although the interest may be
calculated every 6 months, 2 months, or even a fraction of a month. In any case,
we would like to investigate compounding rates more frequent than 2 times per
year. However, the computer must perform a large number of computations - more
as 2 gets larger! - and it would be tedious to compute all of the intermediate
values. For reasons that follow from equation (1), there is a simple explicit
(i.e., non-recursive) formula that will do the same job much more quickly.
Suppose we write 
for the amount in the account after n years if interest is compounded
m times per year at rate r per year. Actually, this
program does not use a built-in exponentiation rule, but calculates the result
below using multiplications only, each done with 12-place precision, and reported
with the precision set with the "Set Precision" button:
Then
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To do semi-yearly compounding,
instead of using the "Yearly Compounded Amount" button to display
the whole list of amounts, just fill in the fields as you did before with the
number of years in the top field, the starting amount in the second, and the
formula for the amount after each year as if the interest were not compounded
in the year - that is, if it was simple interest for the year - in the third
field.
In fact, if you pressed the
" Yearly Compounded Amount " button now, you would get the amounts
after each year from 0, ..., n with the interest being computed once
each year. Now take care of the compounding in this way: enter the number of
times each year that you want the interest to be computed in the field:
and then press the "Monthly Compounded Amount " button.
It may be surprising to learn
that there is an upper limit to the amount you could receive as you compound
the interest more and more frequently. We cannot prove that here, but you may
find that limit by experiment.
Question 4: Find the largest amount you could have received in the year 2000 (to the nearest cent) if the original amount was $1 and the rate of interest was 3% under any compounding scheme. What is the smallest number of times a year would it be necessary to compound in order to obtain this amount? What is the length of time between compoundings to obtain this amount? (Use an approximate measure of time.)
We now perform an experiment
that will motivate the term "natural exponential" that we discuss
on the next page. It will also give an interpretation of Euler's number: e.
Suppose that we invest $1.00 at 100% interest for 1 year. Then at the end of
that year we will have $2.00
Now suppose we compound twice
in that year. Then according to our formula, we would expect to have $2.25 at
the end of the year. That is, 
Now, as it happens, there is an upper limit to the amount that can accumulate
over 1 year, even if we allow arbitrary precision, and we compound the interest
more and more frequently, that is, we compound the interest m times in the year
as m becomes arbitrarily large. This may not be surprising in light of
the previous experiment, but it will lead us to some startling conclusions.
First, set up the experiment.
Type the appropriate expressions in the fields, as follows:
The calculations, as mentioned, are done to 12-place accuracy, but the results are reported to two decimal places by default. We may increase the precision of the reported answer by entering the number of places we want in the field:
Enter 4 as indicated for a start. Next, type the number of times to compound the interest in the year in the field:
To get the final amount to 4 places, press the "Monthly Compounded Amount " button.
You will see a message like:
The system does not show
2.0000, but shows 2 because it does not report trailing 0s. Now, try to find
the upper limit to 3 decimal place accuracy to the amount after 1 year, letting
the number of compoundings get large. What is that limit to 3 places ? Roughly
how many times must you compound in the year to attain, without rounding in
the third place, that 3-place number ?
To prove that these three
places are correct, and no further amount of compounding will change them is
difficult - and we have not done that.
The actual upper limit is
what is called a "transcendental number." It is an infinite decimal
expression that does not repeat itself in any simple way. It is called Euler's
number: e, and, because it plays an important part in many fields
of Mathematics, it has its own name, like 
; it is denoted e which is approximately 2.718281828459045
What we claim, and will prove
in the next part, Infinite Series and Natural Exponentials, is that as
m gets large without bound,
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We are now ready to explore
a remarkable property of this number. Recall that we began with a rate of interest
of 100%. Writing that as an absolute number, instead of a percent, the rate
of interest initially was 1. We could have entered as the recursive formula
in the field:
to make it look more like our other interest calculations. Now, suppose the rate of interest was 200% or 2, instead of 1. Enter in the field:
and calculate the limiting value of this amount!
Use the number of iterations
you used to get e to three-place accuracy. When you get your answer, go to the
symbolic calculator, press
and type in the yellow command field calc e^2; then press Enter.
You see: 7.389056098930649 How do the numbers compare? They are close, but not the same. Now, see if, by increasing the number of iterations, you believe the conjecture that as m increases without bound,
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This is only the beginning
of the story. For any number x, the sequence of numbers:
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approaches a limit as m increases without bound. Suppose we call that
limit 
(for colorful reasons). We will show in the next part: Infinite Series and
Natural Exponentials that the limit e exists, and following
statements are true:
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As strange as the Euler number
e is, it has the virtue that it can be exponentiated (raised to
a power) rather easily using the limiting expression, or, better, one derived
from it. In fact, for the practical construction of tables of logarithms, it
is, for the reasons indicated above, a "natural" choice for base.
You will see much more about that on the next page.
We conclude this part by
reviewing our discussion of the properties of "exponential functions"
since they will arise in many contexts as models for phenomena. The basic properties
were described at the beginning of the section (equal ratios over equal intervals
of the independent variable), but we repeat them here for emphasis.
It can easily be shown that
functions defined in the more colloquial way as exponential: that is:
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always satisfy the conditions stated below. And only those functions do. But this way of defining them brings their essential nature to the surface and makes it easy to understand why they behave the way they do.
Definition: A function A is an exponential function if it satisfies the conditions:
1) A is defined for all real numbers.
2) Either 
for all x, or 
is never 0,
and for any number h that is chosen, for any x and y in the domain of A,
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That is, over equal intervals
h, the ratio of the ending amount to the starting amount is the same, wherever
you start. This condition properly characterizes exponential functions. We showed
at the beginning of this section that such functions can always be written in
the form:
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for 
and 
, the basic ratio. This establishes a fundamental analogy with linear functions,
for which the differences, not the ratios, are equal.
Also it follows that
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and so,
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Letting h go to 0,
we see that the general derivative relation:
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must hold for any exponential function. That is, the derivative is proportional to the value. If we call this (constant) ratio k, then we have the more familiar
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Now consider the same exponential
function A defined this time on the discrete values:
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for some fixed positive h. In fact, let the discrete function be called B, and write
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for any integer n.
The relation above tells
us that
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Read this: The slope of the segment connecting the 
to the 
point is proportional to the value at the 
point. The constant of proportionality is:
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Thus, any discrete exponential
function constructed in this way from an exponential function A also
has the property that the average rate of change over the nth interval is proportional
to the value at the left endpoint of that interval. The proportionality constant
is, however, not the same as that of the original continuous exponential function.
It approaches that constant as h, the length of the discrete interval
approaches 0.
Now, we shall construct a
"discrete recursive model" of the equiangular spiral
in the last part of this section. The construction is geometric and easy to
visualize. For that, we will use a discrete exponential function such as the
one described above. This strategy often gives insight into the differentiable
exponential analog, and in fact plays a powerful heuristic role in the task
of "solving" a differential equation, like the one we will analyze,
and like the one that Nature solves so effortlessly when it builds a Nautilus
shell.
All of this is the introduction
to another "Art of Approximation," the solution (integration) of ordinary
differential equations. This, we will see, is what Newton did when he solved
Kepler's problem, and the approach taken here will guide us in that fateful
deduction.
