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PROFESSOR: Hi.
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Well, this is exponential day,
the day for the function that
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only calculus could create,
y is e to the x.
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And it couldn't have come from
algebra because, however we
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approach e to the x, there's
some limiting step.
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Something goes to 0.
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Something goes to infinity.
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I've got different ways to reach
e to the x, but all of
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them involve that limiting
process, which we haven't
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discussed in full.
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Let me come back at a later
time to the whole theory,
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discussion of limits and just
go forward here with this
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highly important function.
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And I'd like to start with
its most important
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property, which is--
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so it has this remarkable
property that its slope is
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equal to itself.
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That's what is special
about e to the x.
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The slope is equal
to the function.
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Now, I have to admit that if we
had a function like that, y
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equals e to the x, then 2e to
the x, x would work just as
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well, or 10e to the x.
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Those would--
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the factor 2 or the factor 10
would be in y and it would
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also be in the slope and
it would cancel and--
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this is a differential
equation, our first
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differential equation.
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A differential equation is an
equation that involves, as
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this one does, the function
and the slope.
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It connects them.
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And that's the fantastic
description of nature, is by
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differential equations.
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So it's great to see this one
early and it's the most
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important one.
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When you get this one,
you've got a whole
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lot of others solved.
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OK.
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But I needed to give it a
starting point so that the
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solution would be e to the
x and not 10e to the x.
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So where should I start it?
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Well if I want it to be an e to
the x, then when x is 0, e
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to the 0 power, some number to
the 0 power, is always 1.
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So let me start this y equals
1 at x equals 0.
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Differential equations, you have
to tell where they begin.
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So that's our starting point.
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And do you see what
this means?
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This means that it
starts at 1.
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And what's it's slope at
the starting point?
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The slope is also 1.
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So it's climbing.
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As it climbs--
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so y gets larger because it's
got a positive slope.
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As y gets larger, the
slope gets larger.
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So it climbs faster.
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And then it's gone higher, y is
bigger, the slope is equal,
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so the slope is also bigger,
so it climbs even faster.
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It just takes off.
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It climbs much faster than
x to the 100th power.
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You might think x to
the 100th, that's
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climbing pretty well.
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2 to the 100th, 10 to the
100th, but now way.
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It doesn't come close
to keeping up with y
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equals e to the x.
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OK.
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I've got several things to do.
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And one more thing I have to
do, this is a key property,
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but there's another key property
that is true for any
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2 to the x, 3 to the
x, e to the x.
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And that key property
is also to show--
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I have to show this, that my
function, e to the x, times e
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to the possibly a different
x is equal to--
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do you know what we want here?
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This has got to come out of the
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construction, out of this property.
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It's got to come--
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but we want this to deserve,
to be called some number to
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the x power.
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If we take some number x times
multiplied by that same number
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capital x times, then
we've got that
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number how many times?
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x plus capital x.
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So that's a key property
to be proved.
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So what will I do?
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Let me summarize in advance,
outline in advance.
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I'm going to construct this
function from its property.
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Then I'm going check that it's
got this property, that
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important equality there.
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Then, of course,
I'll graph it.
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I'll figure out what e is, and
I'll say something about cases
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where this comes up.
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I could even say something right
away about, where does
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this happen that growth is equal
or proportional to the
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function itself?
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102
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It happens with interest,
with money in a bank.
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When you get interest, the
interest is proportional, of
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course, the amount there.
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And if they add that interest
in, if you don't take it out
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and spend it but you compound
it, put it in there, then you
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have more money.
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When they compute the interest
again, it's computed on that
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larger amount and is more
interest than the first time.
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And so it goes.
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So money in the bank is a case
of exponential growth.
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A hedge fund grows faster than
our bank account does, but all
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following e to the x.
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If you just hang on long enough,
you're way up there.
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OK.
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So here's my job.
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Follow this rule and start
at y equals 1.
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So can I just do it this way?
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Here is my function, y of x,
that I want to construct.
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I want to build that function.
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And I know that it
starts at 1.
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But it's going to have
some more things.
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Now, this has to equal dy dx.
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These have to be the same.
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That's my rule.
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So dy dx is going to
start with a 1.
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But now I can't stop because
if the derivative is a 1, I
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better put--
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I have to put an x up
here so that its
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derivative will be 1, right?
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Its slope will be 1.
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That's that steadily climbing
x whose slope is 1.
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But now, these are supposed
to be equal again.
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So I have to put this
x also here.
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But now, I've got to add
something more on the top so
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that the slope will
be 1 plus x.
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The slope of the x was 1.
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What do I need here to give
the slope to be x?
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Remember, x squared had the
slope 2x, so I need half of x
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squared so that I'll have 1x.
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So I need a half of x squared.
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Good.
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The slope of that is this.
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But I'm also trying to get
the 2 to be equal.
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So I better--
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I have no choice.
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I have to put in the 1/2
x squared there.
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You see, I'm never going
to catch up.
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Or only if I go forever.
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That's the point.
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I'll have to go forever.
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And what will the next one be?
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Oh yeah.
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If you see the next one, then
we can see the pattern.
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Now what am I doing?
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This one has to have
this slope.
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I'm fixing the top line now.
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If I'm aiming for a slope of
x squared, then I need some
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number of x cubes.
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So how many x cubes do I need?
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Well, I need to know, what's
the slope of x cube?
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The rule for powers of
x, x to the n, is n
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times one smaller power.
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The slope of x cube is
3 times x squared.
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So I had better divide by that 3
so that the 3 cancels the 3.
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Now the slope of that, the 3x
squared, the threes would
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cancel and I would
get x squared.
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But I'm looking for
1/2 x squared.
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I need also a 2.
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Do you see that it's 1/6
of x cube that's
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going to do the job.
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1/6 of x cubed because
the slope--
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the 3 cancels the 3 and I wanted
to end up with a 2.
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And now, do you know
what's coming?
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These are supposed
to be equal.
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I have to have this 1/6
x cubed down here too.
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And I never get to stop.
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We have to see, OK, what
is a typical--
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after I've done this, say, n
times, I'd like to have some
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00:11:03,260 --> 00:11:10,660
idea of what is it when I get
up to x to some nth power,
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then it's multiplied by some
fraction and I'm looking to
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see, what is that fraction?
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What is that fraction?
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And then, of course, they'll all
show up down there again.
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Well, if you see this pattern,
this was 3 times 2-- you could
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say 3 times 2 times 1.
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This one was 2 times 1.
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This one was just 1.
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It's n factorial.
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n factorial is what I need.
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I need n times n minus 1.
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I need all these numbers all the
way and I'll throw in the
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1 at the end.
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00:11:51,420 --> 00:11:58,580
And I have to put the
mathematicians take it away
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00:11:58,580 --> 00:12:01,870
symbol, the little three
dots that mean
201
00:12:01,870 --> 00:12:04,230
don't stop, keep going.
202
00:12:04,230 --> 00:12:07,090
But do you see that
this will be OK?
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This is called n factorial, x
to the nth over n factorial,
204
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because when I take the
slope of x to the nth,
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an n will come down.
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00:12:17,220 --> 00:12:19,810
Cancel that n.
207
00:12:19,810 --> 00:12:23,180
x, I'll have one lower power.
208
00:12:23,180 --> 00:12:26,720
You see, when I take the slope
of this, I'll have the n will
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00:12:26,720 --> 00:12:27,960
cancel the n.
210
00:12:27,960 --> 00:12:33,900
So I'll still have these
other guys down below.
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00:12:33,900 --> 00:12:36,990
And I'll have x to
the n minus 1.
212
00:12:36,990 --> 00:12:39,470
And that will be x
to the n minus 1
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00:12:39,470 --> 00:12:40,960
over n minus 1 factorial.
214
00:12:40,960 --> 00:12:42,750
That will be the previous one.
215
00:12:42,750 --> 00:12:50,860
But now I have to add in the x
to the nth over n factorial
216
00:12:50,860 --> 00:12:54,850
because y and dy to the x have
to be the same, so I have to
217
00:12:54,850 --> 00:12:55,610
keep going.
218
00:12:55,610 --> 00:12:56,640
OK.
219
00:12:56,640 --> 00:13:03,010
So you might say, well, you're
going to blow up.
220
00:13:03,010 --> 00:13:06,590
Not personally, the series.
221
00:13:06,590 --> 00:13:09,830
But what saves you?
222
00:13:09,830 --> 00:13:15,690
What saves you is the fact that
these n factorials, those
223
00:13:15,690 --> 00:13:19,840
fractions, that n factorial gets
to be really large really
224
00:13:19,840 --> 00:13:25,650
fast, faster than this
x to nth could grow.
225
00:13:25,650 --> 00:13:31,370
So altogether, these terms, x
to the nth over n factorial,
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00:13:31,370 --> 00:13:36,020
they get extremely,
extremely small.
227
00:13:36,020 --> 00:13:42,790
And then this series of things,
it comes to a limit.
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00:13:42,790 --> 00:13:46,270
It doesn't keep going, getting
bigger, and bigger, and bigger
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00:13:46,270 --> 00:13:48,620
as I had more terms, because
what I'm adding is
230
00:13:48,620 --> 00:13:50,935
so small, so small.
231
00:13:50,935 --> 00:13:53,810
232
00:13:53,810 --> 00:13:59,200
And that's the point where we
have to discuss limits later.
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00:13:59,200 --> 00:13:59,980
OK.
234
00:13:59,980 --> 00:14:04,600
So that's my construction.
235
00:14:04,600 --> 00:14:05,850
Construction complete.
236
00:14:05,850 --> 00:14:09,050
237
00:14:09,050 --> 00:14:11,380
The exponential function
e to the x--
238
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this is e to the x--
239
00:14:13,390 --> 00:14:19,470
is being defined by 1 plus x
plus 1/2 x squared plus 6 x
240
00:14:19,470 --> 00:14:21,350
cubed, and so on.
241
00:14:21,350 --> 00:14:22,600
OK.
242
00:14:22,600 --> 00:14:24,430
243
00:14:24,430 --> 00:14:26,480
I've got a function.
244
00:14:26,480 --> 00:14:30,510
Now, its property.
245
00:14:30,510 --> 00:14:34,950
And the key property
is this one.
246
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Can I move to the next board?
247
00:14:37,780 --> 00:14:40,580
So the next step is, check--
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00:14:40,580 --> 00:14:43,370
well, I've asked you.
249
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I've got e to the x.
250
00:14:44,620 --> 00:14:46,562
251
00:14:46,562 --> 00:14:48,715
And let me write again
what it is.
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00:14:48,715 --> 00:14:57,810
1 plus x plus 1/2 x squared plus
1/6 x cubed and so on.
253
00:14:57,810 --> 00:15:01,880
And then I've got e to the any
other power, or even the same
254
00:15:01,880 --> 00:15:03,990
power, 1 plus--
255
00:15:03,990 --> 00:15:08,970
I'll just use capital
x four this power.
256
00:15:08,970 --> 00:15:15,240
1/6 of capital x cubed
plus so on.
257
00:15:15,240 --> 00:15:18,620
And I want to multiply those
and see what I get.
258
00:15:18,620 --> 00:15:21,200
259
00:15:21,200 --> 00:15:21,580
OK.
260
00:15:21,580 --> 00:15:23,840
I apologize.
261
00:15:23,840 --> 00:15:29,360
Here I ask you to believe in
this infinite series, and
262
00:15:29,360 --> 00:15:34,560
yeah, a little dodgy,
but it works.
263
00:15:34,560 --> 00:15:38,730
And now I ask you to multiply
two of the things.
264
00:15:38,730 --> 00:15:41,380
You might say, OK, you're
asking a lot here.
265
00:15:41,380 --> 00:15:45,030
But just hang on.
266
00:15:45,030 --> 00:15:47,430
Let's multiply these.
267
00:15:47,430 --> 00:15:52,850
e to the x times e to the
capital x, because that's what
268
00:15:52,850 --> 00:15:55,560
I'm interested in knowing.
269
00:15:55,560 --> 00:15:56,810
OK.
270
00:15:56,810 --> 00:15:59,760
271
00:15:59,760 --> 00:16:01,470
Just do all the multiplications.
272
00:16:01,470 --> 00:16:05,590
273
00:16:05,590 --> 00:16:07,750
And we'll see what we get.
274
00:16:07,750 --> 00:16:09,350
OK, so 1 times 1 is 1.
275
00:16:09,350 --> 00:16:10,580
No problem.
276
00:16:10,580 --> 00:16:13,000
1 times x is the x.
277
00:16:13,000 --> 00:16:16,210
1 times this x is the big x.
278
00:16:16,210 --> 00:16:17,490
Now can I keep going?
279
00:16:17,490 --> 00:16:22,640
All right, well, 1 times
1/2 x squared is--
280
00:16:22,640 --> 00:16:27,620
and now I have x
times a big x.
281
00:16:27,620 --> 00:16:30,870
And now I have a 1 times
1/2 big x squared.
282
00:16:30,870 --> 00:16:34,050
283
00:16:34,050 --> 00:16:36,570
And more, of course.
284
00:16:36,570 --> 00:16:42,370
Notice the way I'm doing is
like I'm keeping all the
285
00:16:42,370 --> 00:16:45,600
things that have two
x's together.
286
00:16:45,600 --> 00:16:48,020
And then I would keep all the
things that have three x's
287
00:16:48,020 --> 00:16:50,160
together, and so on.
288
00:16:50,160 --> 00:16:54,140
Now what is it that
I'm hoping?
289
00:16:54,140 --> 00:17:04,890
I'm hoping that this is the same
as the series for x plus
290
00:17:04,890 --> 00:17:08,240
capital x, OK?
291
00:17:08,240 --> 00:17:09,750
What's that?
292
00:17:09,750 --> 00:17:12,099
That's my exponential series.
293
00:17:12,099 --> 00:17:16,030
And every time, I have to
put in x plus capital x.
294
00:17:16,030 --> 00:17:19,180
In other words, of course,
it starts with 1.
295
00:17:19,180 --> 00:17:22,210
Then it has the x
plus capital x.
296
00:17:22,210 --> 00:17:27,380
And then it has the 1/2 of
x plus capital x squared.
297
00:17:27,380 --> 00:17:28,630
And it keeps going.
298
00:17:28,630 --> 00:17:32,030
299
00:17:32,030 --> 00:17:35,170
And I just wanted
you to say, yes.
300
00:17:35,170 --> 00:17:40,920
I guess I hope you say yes
when I ask, is this big
301
00:17:40,920 --> 00:17:44,080
multiplication the
same as this one?
302
00:17:44,080 --> 00:17:45,760
Well, I think it is.
303
00:17:45,760 --> 00:17:48,530
Let's just start to
check, anyway.
304
00:17:48,530 --> 00:17:50,020
The ones are good.
305
00:17:50,020 --> 00:17:50,890
The x and the x--
306
00:17:50,890 --> 00:17:56,680
I'm really just putting
parentheses around all the--
307
00:17:56,680 --> 00:17:59,440
now I'm going to put parentheses
around all the
308
00:17:59,440 --> 00:18:05,360
second degree terms and say,
is that the same as that?
309
00:18:05,360 --> 00:18:06,330
Yeah.
310
00:18:06,330 --> 00:18:09,080
This is the critical
point here.
311
00:18:09,080 --> 00:18:13,690
Do we, at least, start
out correctly?
312
00:18:13,690 --> 00:18:16,730
So we have to remember, how do
you do-- but, of course, you
313
00:18:16,730 --> 00:18:22,000
do remember how to multiply x
plus capital x by itself.
314
00:18:22,000 --> 00:18:24,790
You just do the multiplications.
315
00:18:24,790 --> 00:18:29,990
x, when I multiply that by
itself, I get x squared.
316
00:18:29,990 --> 00:18:32,060
With 1/2, I get that.
317
00:18:32,060 --> 00:18:33,900
And then, you remember?
318
00:18:33,900 --> 00:18:37,360
How many x times x's do I get?
319
00:18:37,360 --> 00:18:41,380
Little x times big x, there'd
be two of those.
320
00:18:41,380 --> 00:18:44,700
But then the 1/2 factor
leaves me with 1, and
321
00:18:44,700 --> 00:18:47,030
that's what I want.
322
00:18:47,030 --> 00:18:51,940
And then, finally, this guy by
himself squared is the 1/2
323
00:18:51,940 --> 00:18:54,520
capital x squared that
I also want.
324
00:18:54,520 --> 00:18:55,770
So far, so good.
325
00:18:55,770 --> 00:18:59,430
326
00:18:59,430 --> 00:19:03,530
Do you want to see
the cubed terms?
327
00:19:03,530 --> 00:19:07,150
328
00:19:07,150 --> 00:19:12,550
Well, I'd rather you did it,
but I should at least show
329
00:19:12,550 --> 00:19:16,260
that I'm willing to try.
330
00:19:16,260 --> 00:19:19,230
So what do I mean by
the cubed terms?
331
00:19:19,230 --> 00:19:24,800
I mean that here, I want to
get-- the next one should be
332
00:19:24,800 --> 00:19:29,360
1/6 of x plus x cubed.
333
00:19:29,360 --> 00:19:35,650
And from the multiplication,
I get some separate pieces.
334
00:19:35,650 --> 00:19:38,750
I get 1 times--
335
00:19:38,750 --> 00:19:43,690
when I do that multiplication,
I get 1/6 x cubed.
336
00:19:43,690 --> 00:19:49,750
And then I maybe get some
1/2 x squared times x.
337
00:19:49,750 --> 00:19:56,040
You see why I would rather
you did this.
338
00:19:56,040 --> 00:19:58,210
But I'll finish this
little line.
339
00:19:58,210 --> 00:20:04,500
There's also an x times
1/2 x squared.
340
00:20:04,500 --> 00:20:08,780
So that's 1/2 of x times
the big x squared.
341
00:20:08,780 --> 00:20:12,775
And then there is the 1
times the 1/6 x cubed.
342
00:20:12,775 --> 00:20:19,270
343
00:20:19,270 --> 00:20:23,560
So those are the four pieces
that come, third degree, when
344
00:20:23,560 --> 00:20:25,360
I do the big multiplication.
345
00:20:25,360 --> 00:20:27,570
And they have to match
the third degree
346
00:20:27,570 --> 00:20:30,820
term in the last line.
347
00:20:30,820 --> 00:20:32,270
And they do match.
348
00:20:32,270 --> 00:20:35,200
349
00:20:35,200 --> 00:20:40,530
Do you remember the right
words to say now?
350
00:20:40,530 --> 00:20:42,970
Binomial theorem.
351
00:20:42,970 --> 00:20:48,680
The binomial theorem tells you
how to take the nth power all
352
00:20:48,680 --> 00:20:54,180
a sum like x plus capital
x to the nth power.
353
00:20:54,180 --> 00:20:58,200
It tells you all the many
pieces you get.
354
00:20:58,200 --> 00:21:02,960
And those many pieces are
exactly the pieces that we get
355
00:21:02,960 --> 00:21:10,490
directly by multiplying that
line by that line.
356
00:21:10,490 --> 00:21:15,920
So the binomial theorem, at
long last, pays off and
357
00:21:15,920 --> 00:21:19,780
confirms our great
property here.
358
00:21:19,780 --> 00:21:22,020
So this is a big deal.
359
00:21:22,020 --> 00:21:25,800
360
00:21:25,800 --> 00:21:27,050
OK.
361
00:21:27,050 --> 00:21:30,560
362
00:21:30,560 --> 00:21:33,880
So let me now come back here,
having checked that.
363
00:21:33,880 --> 00:21:36,760
364
00:21:36,760 --> 00:21:41,950
I wanted to say something about
this series, 1 plus x
365
00:21:41,950 --> 00:21:45,430
plus 1/2 x squared, where the
typical term is x to the nth
366
00:21:45,430 --> 00:21:46,900
over n factorial.
367
00:21:46,900 --> 00:21:53,740
This is the, I would say, the
second most important infinite
368
00:21:53,740 --> 00:21:58,510
series in mathematics, the
exponential series.
369
00:21:58,510 --> 00:22:06,010
And it's the way I wanted to
construct e to the x by
370
00:22:06,010 --> 00:22:11,460
matching term by term and
seeing that these n
371
00:22:11,460 --> 00:22:13,260
factorials show up.
372
00:22:13,260 --> 00:22:17,900
You might want to know, what's
the most important series?
373
00:22:17,900 --> 00:22:20,220
Reasonable question.
374
00:22:20,220 --> 00:22:25,380
For me, the most important
series would be the one
375
00:22:25,380 --> 00:22:30,060
looking like this, except it
doesn't have the fractions.
376
00:22:30,060 --> 00:22:33,230
For me, the most important
series would be the one--
377
00:22:33,230 --> 00:22:35,410
I'll slip it up here--
378
00:22:35,410 --> 00:22:43,500
1 plus x plus x squared, without
the 1/2, plus x cubed,
379
00:22:43,500 --> 00:22:49,570
without the 1/6, plus so on,
plus x to the n without this n
380
00:22:49,570 --> 00:22:52,445
factorial that's making
it so small.
381
00:22:52,445 --> 00:22:55,310
382
00:22:55,310 --> 00:23:00,200
Can you see this 1 plus x plus
x squared plus x cubed
383
00:23:00,200 --> 00:23:02,640
plus x to the n?
384
00:23:02,640 --> 00:23:06,560
That, I think it's called
the geometric series.
385
00:23:06,560 --> 00:23:08,620
Powers of x.
386
00:23:08,620 --> 00:23:14,260
Now, it's simpler because it
doesn't have these fractions.
387
00:23:14,260 --> 00:23:19,320
But it's riskier because those
fractions were making the
388
00:23:19,320 --> 00:23:22,420
exponential series succeed.
389
00:23:22,420 --> 00:23:26,160
Whereas here, with the geometric
series, well, look
390
00:23:26,160 --> 00:23:29,330
what happens when x is 1.
391
00:23:29,330 --> 00:23:35,550
When x is 1, we have 1 plus 1
plus 1 plus 1 plus 1 forever.
392
00:23:35,550 --> 00:23:36,270
All ones.
393
00:23:36,270 --> 00:23:38,710
It blows up.
394
00:23:38,710 --> 00:23:40,960
And when x is bigger
than 1, that series
395
00:23:40,960 --> 00:23:44,130
blows up even faster.
396
00:23:44,130 --> 00:23:47,350
So in this series, the geometric
series, this most
397
00:23:47,350 --> 00:23:54,260
important one, does succeed but
only when x is below 1.
398
00:23:54,260 --> 00:23:58,050
x equal 1 is the cutoff and
it fails after that.
399
00:23:58,050 --> 00:24:03,760
There is no cutoff for the
exponential series because of
400
00:24:03,760 --> 00:24:07,940
dividing by these bigger
and bigger numbers.
401
00:24:07,940 --> 00:24:09,910
This works for all x.
402
00:24:09,910 --> 00:24:12,290
OK, so those are
the two series.
403
00:24:12,290 --> 00:24:13,540
OK.
404
00:24:13,540 --> 00:24:15,410
405
00:24:15,410 --> 00:24:21,470
So let me ask you, what happens
if I put x equal 1 in
406
00:24:21,470 --> 00:24:25,550
the exponential series?
407
00:24:25,550 --> 00:24:31,630
That gives me e to the first
power, which is e.
408
00:24:31,630 --> 00:24:35,770
So finally, you may say, it's
rather late in the day.
409
00:24:35,770 --> 00:24:41,100
i'm going to figure out what
e is from this series.
410
00:24:41,100 --> 00:24:47,770
Put in set x equal 1 and you
learn that e to the first
411
00:24:47,770 --> 00:24:51,770
power, which is e, is--
412
00:24:51,770 --> 00:24:53,060
can I just put it in?
413
00:24:53,060 --> 00:25:02,180
1 plus x is 1 plus 1/2 of 1
squared plus 1/6 of 1 cubed.
414
00:25:02,180 --> 00:25:06,470
What's the next term in this?
415
00:25:06,470 --> 00:25:09,160
So these are numbers now, and
I'm getting a number.
416
00:25:09,160 --> 00:25:14,290
I'm getting this incredible
number e, named after Euler.
417
00:25:14,290 --> 00:25:16,360
Euler was a fantastic
mathematician.
418
00:25:16,360 --> 00:25:21,110
I think he wrote more important
papers then any
419
00:25:21,110 --> 00:25:22,360
mathematician in history.
420
00:25:22,360 --> 00:25:25,450
421
00:25:25,450 --> 00:25:31,620
So he was allowed to name this
number after himself, e.
422
00:25:31,620 --> 00:25:34,890
E-U-L-E-R, his name
is spelled.
423
00:25:34,890 --> 00:25:36,180
OK, what's the next term?
424
00:25:36,180 --> 00:25:38,930
425
00:25:38,930 --> 00:25:40,910
This is 3 factorial, right?
426
00:25:40,910 --> 00:25:41,990
3 times 2 times 1.
427
00:25:41,990 --> 00:25:43,920
The next term will
be 4 factorial.
428
00:25:43,920 --> 00:25:45,570
I'll multiply that by 4.
429
00:25:45,570 --> 00:25:47,140
It'll be 1/24.
430
00:25:47,140 --> 00:25:48,750
And then times 5.
431
00:25:48,750 --> 00:25:51,090
1/120, and so on.
432
00:25:51,090 --> 00:25:54,310
They're getting small.
433
00:25:54,310 --> 00:25:58,090
What can I tell you
about this number?
434
00:25:58,090 --> 00:25:59,355
It will be a definite number.
435
00:25:59,355 --> 00:26:02,300
436
00:26:02,300 --> 00:26:04,230
And is more than--
437
00:26:04,230 --> 00:26:06,840
well, it's certainly more
than 2 1/2, because I
438
00:26:06,840 --> 00:26:08,250
start with 2 1/2 here.
439
00:26:08,250 --> 00:26:09,720
And then I add these.
440
00:26:09,720 --> 00:26:12,510
Well, I could even
throw in 1/6.
441
00:26:12,510 --> 00:26:16,350
That's more than 2 2/3,
would that be?
442
00:26:16,350 --> 00:26:19,090
If I quit here, I'd
have 2 2/3.
443
00:26:19,090 --> 00:26:21,460
And then I get a little more.
444
00:26:21,460 --> 00:26:22,740
It's easy to show.
445
00:26:22,740 --> 00:26:25,790
No way you would reach
as far as 3.
446
00:26:25,790 --> 00:26:30,450
These later terms are dropping
too fast. And actually, the
447
00:26:30,450 --> 00:26:32,590
number turns out to be--
448
00:26:32,590 --> 00:26:34,850
so it's 2 point something.
449
00:26:34,850 --> 00:26:36,360
2 point--
450
00:26:36,360 --> 00:26:39,820
let's see, a little more than
2 2/3, so it's around 2.7.
451
00:26:39,820 --> 00:26:43,690
452
00:26:43,690 --> 00:26:47,290
But it's it's not exactly 2.7.
453
00:26:47,290 --> 00:26:50,530
In fact, it's not exactly
any fraction
454
00:26:50,530 --> 00:26:53,690
or any finite decimal.
455
00:26:53,690 --> 00:26:54,980
It goes on and on.
456
00:26:54,980 --> 00:26:59,800
1, 8, 2, 8, something.
457
00:26:59,800 --> 00:27:04,280
I think there are more eights
than you'd expect right here
458
00:27:04,280 --> 00:27:10,370
at the beginning, but then,
in the long run, not.
459
00:27:10,370 --> 00:27:13,010
So that's the number, e.
460
00:27:13,010 --> 00:27:14,025
OK.
461
00:27:14,025 --> 00:27:16,330
Oh, so now we know e.
462
00:27:16,330 --> 00:27:17,480
We know e to the x.
463
00:27:17,480 --> 00:27:18,340
We know e.
464
00:27:18,340 --> 00:27:19,810
We know this thing.
465
00:27:19,810 --> 00:27:21,750
I should draw a graph, right?
466
00:27:21,750 --> 00:27:23,080
That's the other thing
you do with a
467
00:27:23,080 --> 00:27:25,860
function is draw a graph.
468
00:27:25,860 --> 00:27:26,610
OK.
469
00:27:26,610 --> 00:27:29,740
So here's a graph.
470
00:27:29,740 --> 00:27:31,410
This is x.
471
00:27:31,410 --> 00:27:36,040
Let me put in x equals 0 here
and x equal 1 here.
472
00:27:36,040 --> 00:27:39,460
And this is going to be
a graph of e to the x.
473
00:27:39,460 --> 00:27:44,820
And at x equals 0, what is it?
474
00:27:44,820 --> 00:27:46,060
We started with that.
475
00:27:46,060 --> 00:27:46,950
It should be--
476
00:27:46,950 --> 00:27:48,090
so this is y.
477
00:27:48,090 --> 00:27:50,060
I'm graphing y.
478
00:27:50,060 --> 00:27:53,430
And it starts at 1.
479
00:27:53,430 --> 00:27:54,210
That's what we said.
480
00:27:54,210 --> 00:27:58,280
At x equals 0, I've started
at 1 with a slope of 1.
481
00:27:58,280 --> 00:28:00,920
So I have a slope of 1, but
the slope, the slope, the
482
00:28:00,920 --> 00:28:02,260
slope is climbing up.
483
00:28:02,260 --> 00:28:05,530
And it reaches here.
484
00:28:05,530 --> 00:28:06,890
That height is what--
485
00:28:06,890 --> 00:28:09,490
486
00:28:09,490 --> 00:28:10,830
e.
487
00:28:10,830 --> 00:28:12,080
That height is e.
488
00:28:12,080 --> 00:28:14,410
489
00:28:14,410 --> 00:28:18,140
Because when we said x equal
1 here, we got e.
490
00:28:18,140 --> 00:28:20,410
So it's climbing, climbing,
climbing.
491
00:28:20,410 --> 00:28:23,130
And now what about on
the other side?
492
00:28:23,130 --> 00:28:28,470
That had a slope of 1, so
it was more like that.
493
00:28:28,470 --> 00:28:32,890
Now what about when
x is negative?
494
00:28:32,890 --> 00:28:38,770
When x is negative, this is
a highly useful fact.
495
00:28:38,770 --> 00:28:44,010
Suppose I want to think about
e to the minus x.
496
00:28:44,010 --> 00:28:47,240
Well now, let me just
take capital x to be
497
00:28:47,240 --> 00:28:48,420
minus little x.
498
00:28:48,420 --> 00:28:52,410
So I get e to the x times
e to the minus x.
499
00:28:52,410 --> 00:28:54,220
What is that?
500
00:28:54,220 --> 00:28:57,110
What does that equal if I
multiply e to the x times e to
501
00:28:57,110 --> 00:29:00,090
the minus x?
502
00:29:00,090 --> 00:29:03,310
As usual, I'm supposed
to add these.
503
00:29:03,310 --> 00:29:07,100
I get 0, so I get e to
the 0, which is 1.
504
00:29:07,100 --> 00:29:13,030
In other words, e to the minus
x is 1 over e to the x, which
505
00:29:13,030 --> 00:29:15,670
we fully expected.
506
00:29:15,670 --> 00:29:22,430
So that at x equal minus 1 here,
I'm down to 1 over e,
507
00:29:22,430 --> 00:29:24,770
1/3, approximately.
508
00:29:24,770 --> 00:29:28,120
So it's going down.
509
00:29:28,120 --> 00:29:32,100
In this way, it's decaying very
fast. It almost touches
510
00:29:32,100 --> 00:29:34,480
that line, but never quite.
511
00:29:34,480 --> 00:29:35,730
This way, it's climbing.
512
00:29:35,730 --> 00:29:38,860
513
00:29:38,860 --> 00:29:41,600
It's growing, growing really--
514
00:29:41,600 --> 00:29:43,243
well, it's growing
exponentially.
515
00:29:43,243 --> 00:29:46,650
516
00:29:46,650 --> 00:29:49,710
And that's what this
graph looks like.
517
00:29:49,710 --> 00:29:55,690
And now I would like to connect
back, at the end of
518
00:29:55,690 --> 00:30:01,650
this lecture, to the insurance
business--
519
00:30:01,650 --> 00:30:07,680
sorry, the interest business,
the bank compounding interest.
520
00:30:07,680 --> 00:30:15,940
Can I take your time with that
important example of the
521
00:30:15,940 --> 00:30:17,130
exponential function?
522
00:30:17,130 --> 00:30:20,430
And we'll see a new
way to reach e.
523
00:30:20,430 --> 00:30:22,300
I like this way.
524
00:30:22,300 --> 00:30:26,320
I like the way we did it with
the infinite series.
525
00:30:26,320 --> 00:30:29,280
But here's another way.
526
00:30:29,280 --> 00:30:37,540
So suppose you're getting 100%
interest. Generous bank.
527
00:30:37,540 --> 00:30:38,870
OK.
528
00:30:38,870 --> 00:30:43,870
And you start with
$1 at 100% now.
529
00:30:43,870 --> 00:30:46,696
It's 100%.
530
00:30:46,696 --> 00:30:49,450
And the bank gives you
interest at the
531
00:30:49,450 --> 00:30:51,770
end of every year.
532
00:30:51,770 --> 00:30:57,020
So at the end of the first year,
you had $1 dollar in the
533
00:30:57,020 --> 00:31:00,890
bank, it adds in 100%.
534
00:31:00,890 --> 00:31:02,780
It adds in another dollar.
535
00:31:02,780 --> 00:31:08,360
So now you've got $2 in the
bank after the first year.
536
00:31:08,360 --> 00:31:13,390
At the end of the second year,
it gives you 100% of what
537
00:31:13,390 --> 00:31:14,360
you've got in the bank.
538
00:31:14,360 --> 00:31:15,500
So it gives you 2 more.
539
00:31:15,500 --> 00:31:18,030
It give you 4.
540
00:31:18,030 --> 00:31:22,050
At the end of the third year, it
gives you an additional 4.
541
00:31:22,050 --> 00:31:24,220
You're up 50 to 8.
542
00:31:24,220 --> 00:31:26,190
And you see what's happening.
543
00:31:26,190 --> 00:31:29,130
It's the powers of two.
544
00:31:29,130 --> 00:31:33,770
Well, that's pretty
good growth.
545
00:31:33,770 --> 00:31:37,140
But it's not calculus.
546
00:31:37,140 --> 00:31:41,660
Calculus doesn't do things
in steps of a year.
547
00:31:41,660 --> 00:31:44,160
Calculus says cut
that step down.
548
00:31:44,160 --> 00:31:48,120
You would want to ask your bank,
couldn't you just, like,
549
00:31:48,120 --> 00:31:51,430
figure the interest a little
more often and put it in
550
00:31:51,430 --> 00:31:53,890
there-- like, figure
it every month?
551
00:31:53,890 --> 00:31:55,550
So what would happen
if you figured the
552
00:31:55,550 --> 00:31:58,100
interest every month?
553
00:31:58,100 --> 00:32:01,590
Of course, you wouldn't get
100% interest in a month.
554
00:32:01,590 --> 00:32:06,710
You'd get 100% divided by 12,
because we're only talking
555
00:32:06,710 --> 00:32:08,540
about one month.
556
00:32:08,540 --> 00:32:12,900
So if it was months,
you start with 1.
557
00:32:12,900 --> 00:32:18,350
You have 1 plus 1/12.
558
00:32:18,350 --> 00:32:21,880
That's what you'd have
after a month.
559
00:32:21,880 --> 00:32:24,970
Now, what would you have after
2 months and what would you
560
00:32:24,970 --> 00:32:29,640
have after 12 months?
561
00:32:29,640 --> 00:32:31,935
Well, we're going to
follow the rule.
562
00:32:31,935 --> 00:32:35,670
563
00:32:35,670 --> 00:32:39,030
They gave you the 1/12 at
the end of January.
564
00:32:39,030 --> 00:32:41,730
So through all of February,
you've got 1
565
00:32:41,730 --> 00:32:43,680
plus 1/12 in there.
566
00:32:43,680 --> 00:32:51,420
At the end of February, they
take 1/12 of that, add it in.
567
00:32:51,420 --> 00:33:00,400
What you get the next time
is 1 plus 1/12 squared.
568
00:33:00,400 --> 00:33:04,860
That's what you have.
569
00:33:04,860 --> 00:33:08,310
Essentially every time, they're
going to multiply what
570
00:33:08,310 --> 00:33:10,860
you've got by this number
1 plus 1/12.
571
00:33:10,860 --> 00:33:13,140
1 to give you--
572
00:33:13,140 --> 00:33:14,320
leave the money in.
573
00:33:14,320 --> 00:33:15,560
You have to leave your money.
574
00:33:15,560 --> 00:33:17,790
I'm sorry.
575
00:33:17,790 --> 00:33:23,060
Plus 1/12 of it for the
interest. And then twice, and
576
00:33:23,060 --> 00:33:27,060
after 1 year, it's done this.
577
00:33:27,060 --> 00:33:30,360
You see what happens
after 1 year, it's
578
00:33:30,360 --> 00:33:34,540
multiplied 12 times.
579
00:33:34,540 --> 00:33:38,010
1 plus 1/12 to the 12th power.
580
00:33:38,010 --> 00:33:40,720
And that's better
than 2, right?
581
00:33:40,720 --> 00:33:45,000
You've got the 2 only when they
put the interest in just
582
00:33:45,000 --> 00:33:45,630
once a year.
583
00:33:45,630 --> 00:33:51,380
Now we're speeding up the bank
and getting more out of it.
584
00:33:51,380 --> 00:33:55,170
So I don't know exactly what 1
plus 1/12 to the 12th power
585
00:33:55,170 --> 00:33:57,700
is, but I know it's
more than 2.
586
00:33:57,700 --> 00:34:01,080
And actually, I'm sure
it's not more than 3.
587
00:34:01,080 --> 00:34:04,160
588
00:34:04,160 --> 00:34:09,420
In fact, yeah, I'm claiming
that it's not
589
00:34:09,420 --> 00:34:12,699
as much as e, 2.7.
590
00:34:12,699 --> 00:34:18,610
But it was worth doing, to get
them to compound every month.
591
00:34:18,610 --> 00:34:21,790
But, of course, you think, okay,
I'm on to a good thing.
592
00:34:21,790 --> 00:34:22,600
Every day.
593
00:34:22,600 --> 00:34:24,120
Why not?
594
00:34:24,120 --> 00:34:26,770
So what would every day be?
595
00:34:26,770 --> 00:34:31,830
1 plus 1/365.
596
00:34:31,830 --> 00:34:34,620
That's the interest you would
get for just that day.
597
00:34:34,620 --> 00:34:39,810
But then they would compound
it 365 times.
598
00:34:39,810 --> 00:34:44,090
So that would be a little more
than this because they're
599
00:34:44,090 --> 00:34:47,600
adding the interest in
more frequently.
600
00:34:47,600 --> 00:34:51,880
And, in general, I'm
going to divide the
601
00:34:51,880 --> 00:34:54,540
year up into n pieces.
602
00:34:54,540 --> 00:34:59,730
In every piece, they multiply my
wealth by 1 plus 1 over n.
603
00:34:59,730 --> 00:35:01,685
And they do it n times
in a year.
604
00:35:01,685 --> 00:35:05,020
605
00:35:05,020 --> 00:35:11,110
And the beautiful thing is that
as n goes to infinity,
606
00:35:11,110 --> 00:35:15,150
and calculus comes in, because
we're asking them to compound
607
00:35:15,150 --> 00:35:20,450
interest continuously, not just
every month, not every
608
00:35:20,450 --> 00:35:26,100
day, every second even,
but all the time.
609
00:35:26,100 --> 00:35:28,850
You don't get an infinite
amount out of this.
610
00:35:28,850 --> 00:35:31,430
You get e.
611
00:35:31,430 --> 00:35:35,140
As n gets bigger, that
approaches this number e.
612
00:35:35,140 --> 00:35:39,330
That's another way to construct
e, as the limit--
613
00:35:39,330 --> 00:35:45,000
you see, as n gets bigger, it's
like 1 to the infinity,
614
00:35:45,000 --> 00:35:48,400
which is kind of meaningless.
615
00:35:48,400 --> 00:35:50,460
I don't want to say
that 1 to the--
616
00:35:50,460 --> 00:35:53,990
I had an email the other day
that said, well, 1 to the
617
00:35:53,990 --> 00:35:56,350
infinity is e.
618
00:35:56,350 --> 00:35:57,140
What's happening?
619
00:35:57,140 --> 00:35:58,890
That's not true.
620
00:35:58,890 --> 00:36:04,190
It's this thing that's going to
1, this thing that's going
621
00:36:04,190 --> 00:36:05,610
to infinity.
622
00:36:05,610 --> 00:36:08,780
Then the combination
goes to e.
623
00:36:08,780 --> 00:36:09,610
OK.
624
00:36:09,610 --> 00:36:16,030
So that's the application
that shows the
625
00:36:16,030 --> 00:36:18,850
number e appearing again.
626
00:36:18,850 --> 00:36:20,100
OK.
627
00:36:20,100 --> 00:36:23,140
628
00:36:23,140 --> 00:36:25,240
You've got the essence
of e to the x.
629
00:36:25,240 --> 00:36:28,280
630
00:36:28,280 --> 00:36:31,780
I just would like to say one
thing, coming back to the very
631
00:36:31,780 --> 00:36:33,310
beginning here.
632
00:36:33,310 --> 00:36:37,730
The great differential equation,
dy dx equal y.
633
00:36:37,730 --> 00:36:39,480
That was beautiful.
634
00:36:39,480 --> 00:36:41,620
Which we've now solved.
635
00:36:41,620 --> 00:36:46,560
Now I want to ask, what if the
differential equation was dy
636
00:36:46,560 --> 00:36:51,570
dx is some multiple of y?
637
00:36:51,570 --> 00:36:53,620
How would that come up?
638
00:36:53,620 --> 00:36:57,360
Well, up to now, c was 1.
639
00:36:57,360 --> 00:37:02,840
We were getting 100%
interest per year.
640
00:37:02,840 --> 00:37:06,260
But now, if c is sort of the
interest rate, the growth
641
00:37:06,260 --> 00:37:10,580
rate, or the decay rate of c is
negative, we may be losing
642
00:37:10,580 --> 00:37:15,890
money in this bank.
643
00:37:15,890 --> 00:37:22,090
So can I just tell you what
is the solution to this
644
00:37:22,090 --> 00:37:24,750
differential equation?
645
00:37:24,750 --> 00:37:27,960
When I tell you, and we
learned about taking
646
00:37:27,960 --> 00:37:30,650
derivatives, you'll see, of
course, that's all it is.
647
00:37:30,650 --> 00:37:33,550
It's just the solution
to this one.
648
00:37:33,550 --> 00:37:35,800
I'll also start at one.
649
00:37:35,800 --> 00:37:40,010
The solution to that one
is y of x is e--
650
00:37:40,010 --> 00:37:41,710
e is coming in again--
651
00:37:41,710 --> 00:37:42,960
to the cx.
652
00:37:42,960 --> 00:37:45,430
653
00:37:45,430 --> 00:37:49,790
What I'm doing is like changing
the rate at--
654
00:37:49,790 --> 00:37:52,490
I've made the rate
of chance c.
655
00:37:52,490 --> 00:37:56,850
And then that c is going to
come up there and in the
656
00:37:56,850 --> 00:38:03,030
derivative, the slope of this
guy, that c will come down.
657
00:38:03,030 --> 00:38:07,530
The slope of this will be c e to
the cx, which is cy, which
658
00:38:07,530 --> 00:38:09,680
is what that second
differential
659
00:38:09,680 --> 00:38:11,540
equation tells us.
660
00:38:11,540 --> 00:38:16,460
So that's just a comment looking
ahead, that we've
661
00:38:16,460 --> 00:38:19,290
solved not only the most
important differential
662
00:38:19,290 --> 00:38:21,380
equation with the most important
function that
663
00:38:21,380 --> 00:38:26,680
calculus creates but a whole
collection of related
664
00:38:26,680 --> 00:38:34,390
equations in which the rate can
be any fixed number, c.
665
00:38:34,390 --> 00:38:35,050
OK.
666
00:38:35,050 --> 00:38:35,780
Thank you.
667
00:38:35,780 --> 00:38:38,010
FEMALE SPEAKER: This has been
a production of MIT
668
00:38:38,010 --> 00:38:40,390
OpenCourseWare and
Gilbert Strang.
669
00:38:40,390 --> 00:38:42,670
Funding for this video was
provided by the Lord
670
00:38:42,670 --> 00:38:43,890
Foundation.
671
00:38:43,890 --> 00:38:47,020
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672
00:38:47,020 --> 00:38:50,090
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673
00:38:50,090 --> 00:38:51,650
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674
00:38:51,650 --> 00:38:53,738