The Exp onential Function
Supp ose the newspap er headlines read The cost of living went up this year Can we translate
this information into an equation Let V denote the value of a dollar in terms of the real go o ds
it can buy whatever economists mean by that Let the elapsed time t b e measured in years y
Then supp ose that V is a function of t V t which function we would like to know explicitly
Call now t and let the initial value of the dollar now b e V which we could take to b e
if we disregard in ation at earlier times
Then our news item can b e written
V V whereas V y V V
This formula can b e rewritten in terms of the changes in the dep endent and indep endent variables
V V y V and t y
V
V
t
where it is now to b e understo o d that V is measured in dollars and t is measured in years
That is the average time rate of change of V is prop ortional to the value of V at the b eginning of
the time interval and the constant of prop ortionality is y By y or inverse years we
mean the p er year rate of change
This is almost like a derivative If only t were in nitesimally small it would b e a derivative Since
we re just trying to describ e the qualitative b ehaviour let s make an approximation assume that
t year is close enough to an in nitesimal time interval and that the ab ove formula for
the in ation rate can b e turned into an instantaneous rate of change
dV
V
dt
This means that the dollar in your p o cket right now will b e worth only in one
second
Well this is interesting but we cannot go any further with it until we ask a crucial question What
will happ en if this go es on That is supp ose we assume that equation is not just a temp orary
situation but represents a consistent and ubiquitous prop erty of the function V t the real value
of your dollar bill as a function of time
Applying the d dt op erator to b oth sides of Eq gives
d d V dV dV d
V or
dt dt dt dt dt
1
Since our dollar will b e worth less a year from now we should really call it de ation
2
The error intro duced by this approximation is not very serious
3
Banks insurance companies trade unions and governments all pretend that they don t assume this but they would all
go bankrupt if they didn t assume it
But dV dt is given by If we substitute that formula into the ab ove equation we get
d V
V V
dt
That is the rate of change of the rate of change is always p ositive or the negative rate of change
is getting less negative all the time In general whenever we have a p ositive second derivative of
a function as is the case here the curve is concave upwards Similarly if the second derivative
were negative the curve would b e concave downwards
So by noting the initial value of V which is formally written V but in this case equals and
by applying our understanding of the graphical meaning of the rst derivative slop e and the
second derivative curvature we can visualize the function V t pretty well It starts out with a
maximum downward slop e and then starts to level o as time increases This general trend con
tinues indefinitely Note that while the function always decreases it never reaches zero This is
b ecause the closer it gets to zero the slower it decreases see Eq This is a very cute feature
that makes this function esp ecially fun to imagine over long times
We can also apply our analytical understanding to the formulas and for the derivatives
every time we take still another derivative the result is still prop ortional to V the constant of
prop ortionality just picks up another factor of This is a really neat feature of this function
namely that we can write down all its derivatives with almost no e ort
dV
V
dt
d V
V V
dt
d V
V V
dt
d V
V V
dt
n
V d
n
V for any n
n
dt
This is a pretty nifty function What is it That is can we write it down in terms of familiar things
like t t t and so on
First note that Eq can b e written in the form
n
d V
n
k V where k
n
dt
A simpler version would b e where k giving
n
d W
W
n
dt
4
A p olitician trying to obfuscate the issue might say The rate of decrease is decreasing
W t b eing the function satisfying this criterion We should p erhaps try guring out this simpler
problem rst and then come back to V t
Let s try expressing W t then as a linear combination of such terms For starters we will try a
third order p olynomial i e we allow terms up to t
W t a a t a t a t Then
dW
a a t a t
dt
follows by simple di erentiation a single word for taking the derivative Now these two
equations have similar lo oking right hand sides provided that we pretend not to notice that term
in t in the rst one and provided the constants a ob ey the rule a na i e a a a a
n n n
and a a But we can t really neglect that t term To b e sure its co e cient a is smaller
than any of the rest so if we had to neglect anything it might b e the b est choice but we re trying
to b e precise right How precise Well precise enough In that case would we b e precise enough
if we added a term a t preserving the rule ab out co e cients a a No Then how ab out
a t And so on No matter how precise an agreement with Eq we demand we can always
take enough terms using this pro cedure to achieve the desired precision Even if you demand
in nite precision we just just take an in nite numb er of terms
1
X
a
n
n
a t where a n a or a W t
n n n n
n
n
Now supp ose we give W t the initial value If we want a di erent initial value we can just
multiply the whole series by that value without a ecting Eq Well W tells us that
a In that case a also and a and a and a and so on If we
de ne the factorial notation
n n n n n
read n factorial and de ne we can express our function W t very simply
1
n
X
t
W t
n
n
We could also write a more abstract version of this function in terms of a generalized variable x
1
n
X
x
W x
n
n
Let s do this and then de ne x k t and set V t V W x Then by the Chain Rule for
derivatives
dW dx dV
V
dt dx dt
5
Linear combination means we multiply each term by a simple constant and add them up
6
The Chain Rule for derivatives says that if z is an explicit function of y z y and y is an explicit function of x
y x then z is an implicit function of x and its derivative with resp ect to x is given by
dz dy dz
dx dy dx
d
and since k t k we have
dt
dV
k V W k V
dt
By rep eating this we obtain Eq Thus
1
n
X
k t
V t V W k t V
n
n
where k in the present case
This is a nice description we can always calculate the value of this function to any desired degree
of accuracy by including as many terms as we need until the change pro duced by adding the next
term is to o small to worry us But it is a little clumsy to keep writing down such an unwieldy
formula every time you want to refer to this function esp ecially if it is going to b e as p opular as we
claim After all mathematics is the art of precise abbreviation So we give W x from Eq a
sp ecial name the exp onential function which we write as either
x
exp x or e
In FORTRAN it is represented as EXP X It is equal to the numb er
e
th
raised to the x p ower In our case we have x t so that our answer is
t
V t V e
which is a lot easier to write down than Eq
x
Now the choice of notation e is not arbitrary There are a lot of rules we know how to use on a
numb er raised to a p ower One is that
x
e
x
e
You can easily determine that this rule also works for the de nition in Eq
The inverse of this function the p ower to which one must raise e to obtain a sp eci ed numb er
is called the natural logarithm or ln function We write
x
if W e then x ln W
or
x
x ln e
A handy application of this de nition is the rule
x x ln y x
y e or y exp x ln y
7
This is exactly what a scienti c hand calculator do es when you push the function key whose name will b e revealed
momentarily
8
Now you know which key it is on a calculator
Before we return to our original function is there anything more interesting ab out the natural
logarithm than that it is the inverse of the exp onential function And what is so all red sp ecial
ab out e the base of the natural log Well it can easily b e shown that the derivative of ln x
is a very simple and familiar function
d ln x
dx x
This is p erhaps the most useful feature of ln x b ecause it gives us a direct connection b etween
the exp onential function and a function whose derivative is x The handy and versatile rule
r
d x
r
r x is valid for any value of r including r but it do esn t help us with this task
dx
Why It also explains what is so sp ecial ab out the numb er e
Summary The Exp onential Function s
x x
Figure The functions e e ln x and x plotted on the same graph over the range from
x to x Note that ln is unde ned There is no nite p ower to which we can raise
e and get zero Similarly x is unde ned at x while x x Also ln
b ecause any numb er raised to the zeroth p ower equals you can easily check this against the
de nitions and ln where any p ositive numb er less than is negative However there is no
such thing as the natural logarithm of any negative numb er
Our formula for the real value of your dollar as a function of time is the only function which
will satisfy the di erential equation from which we started The exp onential function is one of
the most useful of all for solving a wide variety of di erential equations For now just rememb er
this
9
Watch for this phrase Whenever someone says It can easily b e shown they mean This is p ossible to prove but I
haven t got time b esides I might want to assign it as homework
dy
k x
Whenever you have k y you can b e sure that y x y e where y is the
dx
initial value of y when x Note that k can b e either p ositive or negative
Finally note the prop erty of the second derivative
d y
k y
dx
We will see another equation almost like this when we talk ab out Simple Harmonic Motion
An Example from Mechanics Damping
We should really work out at least one example applying the exp onential function to a real Mechanics
problem The classic example is where an ob ject mass m is moving with an initial velo city v
starting from an initial p osition x and exp eriences a frictional damping force F which is
d
prop ortional to the velo city and as always for frictional forces in the direction opp osite to the
velo city F v The equation of motion then reads a m v or
d
d x dx
k
dt dt
where we have combined and m into the constant k m This can also b e written in the
form
dv
k v
dt
which should ring a b ell The solution for the velo city v is
k t
v t v e
To obtain the solution for x t we switch back to the notation
Z Z
t x
dx
k t k t
e dt dx v v e )
dt
x
0
k t k t
and note that the function whose time derivative is e is e giving
k
i h
t
v
k t
e x x
k
t
where the notation means that the expression in the square brackets is to b e evaluated
b etween and t i e plug in the upp er limit just t itself for t in the expression and then
subtract the value of the expression with the lower limit substituted for t In this case the lower
limit gives e e anything to the zeroth p ower gives one so the result is
v
k t
e x t x
k
The qualitative b ehaviour is plotted in Fig Note that x t approaches a xed asymptotic
k t
is another useful addition to value x x v k as t ! 1 The generic function e
max
your pattern recognition rep ertoire
Figure Solution to the damping force equation of motion in which the frictional force is pro
p ortional to the velo city