
An Intuitive Guide To Exponential Functions & e - vijaydev
http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/
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kalid
Hi all, thanks for the comments (writing from a train using a kindle so
forgive lack of links)

google "better explained math intuition" which is an article on developing the
four common definitons of e. the problem with saying "e is the function which
is its own derivative" is the same as saying "a circle is the set of points
where x squared plus y squared equals radius squared". i asks too much of a
beginner and imo is best shown after they have an idea of the basics.

ie, here is a round shape. we call it a circle. look at this neat property
where every point is the same from the middle! lets write an equation for
that...

with e. what is growth? it is like interest on your bank account. why do we
wait till the end of the year? month? second? intant? how can we write this as
an equation? (in algebra, in calculus)

the other reason to avoid calculus definitions is that 2% of students will
bother asking for clarification/more details but most high schoolers can at
least follow the algebra of compounded interest.i still don't get
limits/infinitesimals to the level i like despite writing about them.

anyway thanks for the comments, i love seeing what explanations work for
people!

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dataduck
I've liked the previous posts from betterexplained, but this one seems to be
extremely unintuitive to me. I was teaching this recently, and the simpler
idea that e is the function which is its own derivative is a much simpler way
of expressing a meaningful truth about it, and following it through, leads you
to exactly the same limit as a formal definition. It also tails very neatly
into why exponentials are used in physics for decay curves. That said, obvious
to me and one student is not necessarily obvious to everyone! Anyone else have
an opinion on this?

~~~
T-zex
Well, for me this post is the only one that made sence and thanks to it I now
have a clear understanding what e is. "e is the function which is its own
derivative" is way less intuitive.

~~~
parallel
Perhaps this would be more intuitive if shown graphically. Something like;
draw a curve where the gradient is the value. As the value goes up, so does
the gradient, so does the value and so on. What you have drawn, e^x is the
function where the value is the gradient - exponential growth.

~~~
dataduck
Thanks for this - this is exactly how you would say it to someone who doesn't
know what a derivative is. Apologies for assuming anyone interested in this
would know calculus - poor form indeed! Explaining in terms of derivatives
definitely isn't simple if you don't know what a derivative is.

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scythe
e is:

the unique number such that for all real x, e^x >= x^e

the x-coordinate of the maximum of the function x^(1/x); for all real x,
x^(1/x) < e^(1/e)

the root of the hyperbolic logarithm

...hyperbolic logarithm? Well, 1/x -- the reciprocal function -- is a
hyperbola, and the area under that function turns out to be:

<http://en.wikipedia.org/wiki/Natural_logarithm#Definitions>

Unfortunately, it's kind of hard to try to make sense of e without going into
at least some basic calculus (which betterexplained does, in the form of a
limit, but glosses over). It is commonly stated that e was discovered by
Bernoulli, but the first references are from Napier, who presented a table of
natural logarithms without any reference to e the number itself. I'm not a
terribly huge fan of the two most common explanations of e:

e = lim (1+1/n)^n as n grows

e defined such that d/dx e^x = e^x

The first definition has no obvious connection with most of the interesting
properties of e, so while it makes plenty of sense it doesn't actually clarify
anything about exponential functions or natural logarithms. The second
definition asks students to simply assume one of the most interesting
properties of e, without any clarification as to the derivation of such a
number. Worse yet, students are usually introduced to both long before they
are capable of understanding the connection between these definitions (this
requiring L'Hopital's rule to properly understand) which basically tells them
that "e is some magic that you can't understand".

My favorite treatment of the idea is the one I had the privilege of learning
as a student (perhaps I'm biased), which ignores e at first and defines the
natural logarithm as shown above, as a definite integral resulting from the
function 1/x, and then the fundamental property that d/dx e^x = e^x is proven
from the previous definition, which requires only a little basic calculus. In
fact, the text did not even say it was a logarithm, merely "Consider the
function L(x) such that L(x) = the integral of dt/t from 1 to x", and then
went on to prove that this satisfies several properties of a logarithm, that e
is the unique number such that L(e) = 1, and finally derives the fundamental
property of the exponential function (dy/dx = y) and so forth.

I think that the fear that surrounds calculus and the use of its methods does
many students a disservice by rejecting a more complete explanation in favor
of one that requires slightly fewer uses of the word "integral". Of course,
I'm a cantankerous prick.

~~~
kalid
hi, thank you for the detailed comment! i have a writeup with more details on
the common definitions (including the natural log one), if you google "better
explained math intuition" (on a kindle currently). would love to know what you
think. my ultimate goal for e elucidation is to get people to jump from any
definiton to the others as they are all linked (i.e. see them as wys to
rephrase "write down the equation for perfectly continuous growth").

~~~
scythe
Well, I have a clever way to explain the relationship between areas under a
hyperbola and continuously compounded interest:

dy/dx = 1/x

dx/dy = x

Hopefully that isn't too confusing. If you start with the second relation
(continuously compounded interest), a reciprocal gets you to the first (area
under a hyperbola) -- the connection between exponentials/logarithms and areas
under hyperbolae isn't immediately obvious to most people. Implicit
differentiation is kind of scary, but "draw a curve whose slope is its height"
requires an art degree.

I like your essay on intuition, but I don't know if your interpretation of the
factorial series is really deep enough to explain what is going on. A fully
elementary treatment of Taylor's series is a really hard thing to achieve.
This, however, could help:

<http://en.wikipedia.org/wiki/Binomial_theorem#Series_for_e>

The binomial theorem for integer exponents is easy enough to grasp, though
Wikipedia's geometrical explanation is kind of hilarious:

"[...] if one sets a = x and b = Δx, interpreting b as an infinitesimal change
in a, then this picture shows the infinitesimal change in the volume of an
n-dimensional hypercube, [...]"

I'm afraid infintesimal changes in the volume of n-dimensional hypercubes
probably aren't going to make exponential functions any easier to grasp for
epsilons...

~~~
kalid
Thanks for taking a look! My intuition for the dy/dx relations are "amount of
growth" and "time needed to grow to the next increment". So for me, e is the
amount you have after growing continuously for exactly 1 unit of time
(integrate 1/x until it is 1).

I'll have to examine that link while I have time to study (vacationing now)
and agree the wiki explanation wouldn't be the most enlightening :).

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ced
_e is like a speed limit (like c, the speed of light) saying how fast you can
possibly grow using a continuous process._

Can someone explain what he means? Doesn't that assume dx/dt <= x?

~~~
parallel
The statement is a little misleading. He's not saying you can't have a
function that grows faster than e^x as you certainly can, for example the the
gamma function.

I think he's saying that for a process where the rate of growth depends on the
current "population" such as interest in a bank or splitting bacteria then
exponential growth is limit.

