PROFESSOR: Hi.
Well, this is exponential day,
the day for the function that
only calculus could create,
y is e to the x.
And it couldn't have come from
algebra because, however we
approach e to the x, there's
some limiting step.
Something goes to 0.
Something goes to infinity.
I've got different ways to reach
e to the x, but all of
them involve that limiting
process, which we haven't
discussed in full.
Let me come back at a later
time to the whole theory,
discussion of limits and just
go forward here with this
highly important function.
And I'd like to start with
its most important
property, which is--
so it has this remarkable
property that its slope is
equal to itself.
That's what is special
about e to the x.
The slope is equal
to the function.
Now, I have to admit that if we
had a function like that, y
equals e to the x, then 2e to
the x, x would work just as
well, or 10e to the x.
Those would--
the factor 2 or the factor 10
would be in y and it would
also be in the slope and
it would cancel and--
this is a differential
equation, our first
differential equation.
A differential equation is an
equation that involves, as
this one does, the function
and the slope.
It connects them.
And that's the fantastic
description of nature, is by
differential equations.
So it's great to see this one
early and it's the most
important one.
When you get this one,
you've got a whole
lot of others solved.
OK.
But I needed to give it a
starting point so that the
solution would be e to the
x and not 10e to the x.
So where should I start it?
Well if I want it to be an e to
the x, then when x is 0, e
to the 0 power, some number to
the 0 power, is always 1.
So let me start this y equals
1 at x equals 0.
Differential equations, you have
to tell where they begin.
So that's our starting point.
And do you see what
this means?
This means that it
starts at 1.
And what's it's slope at
the starting point?
The slope is also 1.
So it's climbing.
As it climbs--
so y gets larger because it's
got a positive slope.
As y gets larger, the
slope gets larger.
So it climbs faster.
And then it's gone higher, y is
bigger, the slope is equal,
so the slope is also bigger,
so it climbs even faster.
It just takes off.
It climbs much faster than
x to the 100th power.
You might think x to
the 100th, that's
climbing pretty well.
2 to the 100th, 10 to the
100th, but now way.
It doesn't come close
to keeping up with y
equals e to the x.
OK.
I've got several things to do.
And one more thing I have to
do, this is a key property,
but there's another key property
that is true for any
2 to the x, 3 to the
x, e to the x.
And that key property
is also to show--
I have to show this, that my
function, e to the x, times e
to the possibly a different
x is equal to--
do you know what we want here?
This has got to come out of the
construction, out of this property.
It's got to come--
but we want this to deserve,
to be called some number to
the x power.
If we take some number x times
multiplied by that same number
capital x times, then
we've got that
number how many times?
x plus capital x.
So that's a key property
to be proved.
So what will I do?
Let me summarize in advance,
outline in advance.
I'm going to construct this
function from its property.
Then I'm going check that it's
got this property, that
important equality there.
Then, of course,
I'll graph it.
I'll figure out what e is, and
I'll say something about cases
where this comes up.
I could even say something right
away about, where does
this happen that growth is equal
or proportional to the
function itself?
It happens with interest,
with money in a bank.
When you get interest, the
interest is proportional, of
course, the amount there.
And if they add that interest
in, if you don't take it out
and spend it but you compound
it, put it in there, then you
have more money.
When they compute the interest
again, it's computed on that
larger amount and is more
interest than the first time.
And so it goes.
So money in the bank is a case
of exponential growth.
A hedge fund grows faster than
our bank account does, but all
following e to the x.
If you just hang on long enough,
you're way up there.
OK.
So here's my job.
Follow this rule and start
at y equals 1.
So can I just do it this way?
Here is my function, y of x,
that I want to construct.
I want to build that function.
And I know that it
starts at 1.
But it's going to have
some more things.
Now, this has to equal dy dx.
These have to be the same.
That's my rule.
So dy dx is going to
start with a 1.
But now I can't stop because
if the derivative is a 1, I
better put--
I have to put an x up
here so that its
derivative will be 1, right?
Its slope will be 1.
That's that steadily climbing
x whose slope is 1.
But now, these are supposed
to be equal again.
So I have to put this
x also here.
But now, I've got to add
something more on the top so
that the slope will
be 1 plus x.
The slope of the x was 1.
What do I need here to give
the slope to be x?
Remember, x squared had the
slope 2x, so I need half of x
squared so that I'll have 1x.
So I need a half of x squared.
Good.
The slope of that is this.
But I'm also trying to get
the 2 to be equal.
So I better--
I have no choice.
I have to put in the 1/2
x squared there.
You see, I'm never going
to catch up.
Or only if I go forever.
That's the point.
I'll have to go forever.
And what will the next one be?
Oh yeah.
If you see the next one, then
we can see the pattern.
Now what am I doing?
This one has to have
this slope.
I'm fixing the top line now.
If I'm aiming for a slope of
x squared, then I need some
number of x cubes.
So how many x cubes do I need?
Well, I need to know, what's
the slope of x cube?
The rule for powers of
x, x to the n, is n
times one smaller power.
The slope of x cube is
3 times x squared.
So I had better divide by that 3
so that the 3 cancels the 3.
Now the slope of that, the 3x
squared, the threes would
cancel and I would
get x squared.
But I'm looking for
1/2 x squared.
I need also a 2.
Do you see that it's 1/6
of x cube that's
going to do the job.
1/6 of x cubed because
the slope--
the 3 cancels the 3 and I wanted
to end up with a 2.
And now, do you know
what's coming?
These are supposed
to be equal.
I have to have this 1/6
x cubed down here too.
And I never get to stop.
We have to see, OK, what
is a typical--
after I've done this, say, n
times, I'd like to have some
idea of what is it when I get
up to x to some nth power,
then it's multiplied by some
fraction and I'm looking to
see, what is that fraction?
What is that fraction?
And then, of course, they'll all
show up down there again.
Well, if you see this pattern,
this was 3 times 2-- you could
say 3 times 2 times 1.
This one was 2 times 1.
This one was just 1.
It's n factorial.
n factorial is what I need.
I need n times n minus 1.
I need all these numbers all the
way and I'll throw in the
1 at the end.
And I have to put the
mathematicians take it away
symbol, the little three
dots that mean
don't stop, keep going.
But do you see that
this will be OK?
This is called n factorial, x
to the nth over n factorial,
because when I take the
slope of x to the nth,
an n will come down.
Cancel that n.
x, I'll have one lower power.
You see, when I take the slope
of this, I'll have the n will
cancel the n.
So I'll still have these
other guys down below.
And I'll have x to
the n minus 1.
And that will be x
to the n minus 1
over n minus 1 factorial.
That will be the previous one.
But now I have to add in the x
to the nth over n factorial
because y and dy to the x have
to be the same, so I have to
keep going.
OK.
So you might say, well, you're
going to blow up.
Not personally, the series.
But what saves you?
What saves you is the fact that
these n factorials, those
fractions, that n factorial gets
to be really large really
fast, faster than this
x to nth could grow.
So altogether, these terms, x
to the nth over n factorial,
they get extremely,
extremely small.
And then this series of things,
it comes to a limit.
It doesn't keep going, getting
bigger, and bigger, and bigger
as I had more terms, because
what I'm adding is
so small, so small.
And that's the point where we
have to discuss limits later.
OK.
So that's my construction.
Construction complete.
The exponential function
e to the x--
this is e to the x--
is being defined by 1 plus x
plus 1/2 x squared plus 6 x
cubed, and so on.
OK.
I've got a function.
Now, its property.
And the key property
is this one.
Can I move to the next board?
So the next step is, check--
well, I've asked you.
I've got e to the x.
And let me write again
what it is.
1 plus x plus 1/2 x squared plus
1/6 x cubed and so on.
And then I've got e to the any
other power, or even the same
power, 1 plus--
I'll just use capital
x four this power.
1/6 of capital x cubed
plus so on.
And I want to multiply those
and see what I get.
OK.
I apologize.
Here I ask you to believe in
this infinite series, and
yeah, a little dodgy,
but it works.
And now I ask you to multiply
two of the things.
You might say, OK, you're
asking a lot here.
But just hang on.
Let's multiply these.
e to the x times e to the
capital x, because that's what
I'm interested in knowing.
OK.
Just do all the multiplications.
And we'll see what we get.
OK, so 1 times 1 is 1.
No problem.
1 times x is the x.
1 times this x is the big x.
Now can I keep going?
All right, well, 1 times
1/2 x squared is--
and now I have x
times a big x.
And now I have a 1 times
1/2 big x squared.
And more, of course.
Notice the way I'm doing is
like I'm keeping all the
things that have two
x's together.
And then I would keep all the
things that have three x's
together, and so on.
Now what is it that
I'm hoping?
I'm hoping that this is the same
as the series for x plus
capital x, OK?
What's that?
That's my exponential series.
And every time, I have to
put in x plus capital x.
In other words, of course,
it starts with 1.
Then it has the x
plus capital x.
And then it has the 1/2 of
x plus capital x squared.
And it keeps going.
And I just wanted
you to say, yes.
I guess I hope you say yes
when I ask, is this big
multiplication the
same as this one?
Well, I think it is.
Let's just start to
check, anyway.
The ones are good.
The x and the x--
I'm really just putting
parentheses around all the--
now I'm going to put parentheses
around all the
second degree terms and say,
is that the same as that?
Yeah.
This is the critical
point here.
Do we, at least, start
out correctly?
So we have to remember, how do
you do-- but, of course, you
do remember how to multiply x
plus capital x by itself.
You just do the multiplications.
x, when I multiply that by
itself, I get x squared.
With 1/2, I get that.
And then, you remember?
How many x times x's do I get?
Little x times big x, there'd
be two of those.
But then the 1/2 factor
leaves me with 1, and
that's what I want.
And then, finally, this guy by
himself squared is the 1/2
capital x squared that
I also want.
So far, so good.
Do you want to see
the cubed terms?
Well, I'd rather you did it,
but I should at least show
that I'm willing to try.
So what do I mean by
the cubed terms?
I mean that here, I want to
get-- the next one should be
1/6 of x plus x cubed.
And from the multiplication,
I get some separate pieces.
I get 1 times--
when I do that multiplication,
I get 1/6 x cubed.
And then I maybe get some
1/2 x squared times x.
You see why I would rather
you did this.
But I'll finish this
little line.
There's also an x times
1/2 x squared.
So that's 1/2 of x times
the big x squared.
And then there is the 1
times the 1/6 x cubed.
So those are the four pieces
that come, third degree, when
I do the big multiplication.
And they have to match
the third degree
term in the last line.
And they do match.
Do you remember the right
words to say now?
Binomial theorem.
The binomial theorem tells you
how to take the nth power all
a sum like x plus capital
x to the nth power.
It tells you all the many
pieces you get.
And those many pieces are
exactly the pieces that we get
directly by multiplying that
line by that line.
So the binomial theorem, at
long last, pays off and
confirms our great
property here.
So this is a big deal.
OK.
So let me now come back here,
having checked that.
I wanted to say something about
this series, 1 plus x
plus 1/2 x squared, where the
typical term is x to the nth
over n factorial.
This is the, I would say, the
second most important infinite
series in mathematics, the
exponential series.
And it's the way I wanted to
construct e to the x by
matching term by term and
seeing that these n
factorials show up.
You might want to know, what's
the most important series?
Reasonable question.
For me, the most important
series would be the one
looking like this, except it
doesn't have the fractions.
For me, the most important
series would be the one--
I'll slip it up here--
1 plus x plus x squared, without
the 1/2, plus x cubed,
without the 1/6, plus so on,
plus x to the n without this n
factorial that's making
it so small.
Can you see this 1 plus x plus
x squared plus x cubed
plus x to the n?
That, I think it's called
the geometric series.
Powers of x.
Now, it's simpler because it
doesn't have these fractions.
But it's riskier because those
fractions were making the
exponential series succeed.
Whereas here, with the geometric
series, well, look
what happens when x is 1.
When x is 1, we have 1 plus 1
plus 1 plus 1 plus 1 forever.
All ones.
It blows up.
And when x is bigger
than 1, that series
blows up even faster.
So in this series, the geometric
series, this most
important one, does succeed but
only when x is below 1.
x equal 1 is the cutoff and
it fails after that.
There is no cutoff for the
exponential series because of
dividing by these bigger
and bigger numbers.
This works for all x.
OK, so those are
the two series.
OK.
So let me ask you, what happens
if I put x equal 1 in
the exponential series?
That gives me e to the first
power, which is e.
So finally, you may say, it's
rather late in the day.
i'm going to figure out what
e is from this series.
Put in set x equal 1 and you
learn that e to the first
power, which is e, is--
can I just put it in?
1 plus x is 1 plus 1/2 of 1
squared plus 1/6 of 1 cubed.
What's the next term in this?
So these are numbers now, and
I'm getting a number.
I'm getting this incredible
number e, named after Euler.
Euler was a fantastic
mathematician.
I think he wrote more important
papers then any
mathematician in history.
So he was allowed to name this
number after himself, e.
E-U-L-E-R, his name
is spelled.
OK, what's the next term?
This is 3 factorial, right?
3 times 2 times 1.
The next term will
be 4 factorial.
I'll multiply that by 4.
It'll be 1/24.
And then times 5.
1/120, and so on.
They're getting small.
What can I tell you
about this number?
It will be a definite number.
And is more than--
well, it's certainly more
than 2 1/2, because I
start with 2 1/2 here.
And then I add these.
Well, I could even
throw in 1/6.
That's more than 2 2/3,
would that be?
If I quit here, I'd
have 2 2/3.
And then I get a little more.
It's easy to show.
No way you would reach
as far as 3.
These later terms are dropping
too fast. And actually, the
number turns out to be--
so it's 2 point something.
2 point--
let's see, a little more than
2 2/3, so it's around 2.7.
But it's it's not exactly 2.7.
In fact, it's not exactly
any fraction
or any finite decimal.
It goes on and on.
1, 8, 2, 8, something.
I think there are more eights
than you'd expect right here
at the beginning, but then,
in the long run, not.
So that's the number, e.
OK.
Oh, so now we know e.
We know e to the x.
We know e.
We know this thing.
I should draw a graph, right?
That's the other thing
you do with a
function is draw a graph.
OK.
So here's a graph.
This is x.
Let me put in x equals 0 here
and x equal 1 here.
And this is going to be
a graph of e to the x.
And at x equals 0, what is it?
We started with that.
It should be--
so this is y.
I'm graphing y.
And it starts at 1.
That's what we said.
At x equals 0, I've started
at 1 with a slope of 1.
So I have a slope of 1, but
the slope, the slope, the
slope is climbing up.
And it reaches here.
That height is what--
e.
That height is e.
Because when we said x equal
1 here, we got e.
So it's climbing, climbing,
climbing.
And now what about on
the other side?
That had a slope of 1, so
it was more like that.
Now what about when
x is negative?
When x is negative, this is
a highly useful fact.
Suppose I want to think about
e to the minus x.
Well now, let me just
take capital x to be
minus little x.
So I get e to the x times
e to the minus x.
What is that?
What does that equal if I
multiply e to the x times e to
the minus x?
As usual, I'm supposed
to add these.
I get 0, so I get e to
the 0, which is 1.
In other words, e to the minus
x is 1 over e to the x, which
we fully expected.
So that at x equal minus 1 here,
I'm down to 1 over e,
1/3, approximately.
So it's going down.
In this way, it's decaying very
fast. It almost touches
that line, but never quite.
This way, it's climbing.
It's growing, growing really--
well, it's growing
exponentially.
And that's what this
graph looks like.
And now I would like to connect
back, at the end of
this lecture, to the insurance
business--
sorry, the interest business,
the bank compounding interest.
Can I take your time with that
important example of the
exponential function?
And we'll see a new
way to reach e.
I like this way.
I like the way we did it with
the infinite series.
But here's another way.
So suppose you're getting 100%
interest. Generous bank.
OK.
And you start with
$1 at 100% now.
It's 100%.
And the bank gives you
interest at the
end of every year.
So at the end of the first year,
you had $1 dollar in the
bank, it adds in 100%.
It adds in another dollar.
So now you've got $2 in the
bank after the first year.
At the end of the second year,
it gives you 100% of what
you've got in the bank.
So it gives you 2 more.
It give you 4.
At the end of the third year, it
gives you an additional 4.
You're up 50 to 8.
And you see what's happening.
It's the powers of two.
Well, that's pretty
good growth.
But it's not calculus.
Calculus doesn't do things
in steps of a year.
Calculus says cut
that step down.
You would want to ask your bank,
couldn't you just, like,
figure the interest a little
more often and put it in
there-- like, figure
it every month?
So what would happen
if you figured the
interest every month?
Of course, you wouldn't get
100% interest in a month.
You'd get 100% divided by 12,
because we're only talking
about one month.
So if it was months,
you start with 1.
You have 1 plus 1/12.
That's what you'd have
after a month.
Now, what would you have after
2 months and what would you
have after 12 months?
Well, we're going to
follow the rule.
They gave you the 1/12 at
the end of January.
So through all of February,
you've got 1
plus 1/12 in there.
At the end of February, they
take 1/12 of that, add it in.
What you get the next time
is 1 plus 1/12 squared.
That's what you have.
Essentially every time, they're
going to multiply what
you've got by this number
1 plus 1/12.
1 to give you--
leave the money in.
You have to leave your money.
I'm sorry.
Plus 1/12 of it for the
interest. And then twice, and
after 1 year, it's done this.
You see what happens
after 1 year, it's
multiplied 12 times.
1 plus 1/12 to the 12th power.
And that's better
than 2, right?
You've got the 2 only when they
put the interest in just
once a year.
Now we're speeding up the bank
and getting more out of it.
So I don't know exactly what 1
plus 1/12 to the 12th power
is, but I know it's
more than 2.
And actually, I'm sure
it's not more than 3.
In fact, yeah, I'm claiming
that it's not
as much as e, 2.7.
But it was worth doing, to get
them to compound every month.
But, of course, you think, okay,
I'm on to a good thing.
Every day.
Why not?
So what would every day be?
1 plus 1/365.
That's the interest you would
get for just that day.
But then they would compound
it 365 times.
So that would be a little more
than this because they're
adding the interest in
more frequently.
And, in general, I'm
going to divide the
year up into n pieces.
In every piece, they multiply my
wealth by 1 plus 1 over n.
And they do it n times
in a year.
And the beautiful thing is that
as n goes to infinity,
and calculus comes in, because
we're asking them to compound
interest continuously, not just
every month, not every
day, every second even,
but all the time.
You don't get an infinite
amount out of this.
You get e.
As n gets bigger, that
approaches this number e.
That's another way to construct
e, as the limit--
you see, as n gets bigger, it's
like 1 to the infinity,
which is kind of meaningless.
I don't want to say
that 1 to the--
I had an email the other day
that said, well, 1 to the
infinity is e.
What's happening?
That's not true.
It's this thing that's going to
1, this thing that's going
to infinity.
Then the combination
goes to e.
OK.
So that's the application
that shows the
number e appearing again.
OK.
You've got the essence
of e to the x.
I just would like to say one
thing, coming back to the very
beginning here.
The great differential equation,
dy dx equal y.
That was beautiful.
Which we've now solved.
Now I want to ask, what if the
differential equation was dy
dx is some multiple of y?
How would that come up?
Well, up to now, c was 1.
We were getting 100%
interest per year.
But now, if c is sort of the
interest rate, the growth
rate, or the decay rate of c is
negative, we may be losing
money in this bank.
So can I just tell you what
is the solution to this
differential equation?
When I tell you, and we
learned about taking
derivatives, you'll see, of
course, that's all it is.
It's just the solution
to this one.
I'll also start at one.
The solution to that one
is y of x is e--
e is coming in again--
to the cx.
What I'm doing is like changing
the rate at--
I've made the rate
of chance c.
And then that c is going to
come up there and in the
derivative, the slope of this
guy, that c will come down.
The slope of this will be c e to
the cx, which is cy, which
is what that second
differential
equation tells us.
So that's just a comment looking
ahead, that we've
solved not only the most
important differential
equation with the most important
function that
calculus creates but a whole
collection of related
equations in which the rate can
be any fixed number, c.
OK.
Thank you.
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