WEBVTT

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PROFESSOR: Hi.

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Well, this is sort of a summary
lecture for the big

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group about differential
calculus.

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And it's got a fancy title, Six
Functions, that we know.

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Well, five of them
that we know.

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And a new one-- of course,
there has to

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be something new--

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Six Rules, and Six Theorems.
So I haven't emphasized

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theorems, but it seemed like
this was an occasion where we

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could see the main points of the
math behind the functions

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and the rules.

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OK.

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So, here are my first five
functions, all familiar.

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And, what I'm happy about is
that, if we understand those

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five and the rules to create
more out of them, we get

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practically everything,
everything we frequently use.

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OK.

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So, I wrote down function
one, power of x, and its

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derivative, function
two and its

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derivative, function three.

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Function four has,
a little bit,

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something that is important.

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If it's e to the x,
then we know the

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derivative is e to the x.

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But, if it's e to the c, x
of factor c, comes down.

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Important case, you could
say the chain rule.

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The derivative is that times
the derivative of what's

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inside, which is the c.

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And, finally, the natural
logarithm with the great

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derivative of 1/x.

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And now, oh, I left space
to go from function one

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backwards, to remember the
function that came before it.

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So, what function has
this derivative?

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I'm looking here at the other
generation, the older

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generation.

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Well, the function with that
derivative is we need the

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power to be one higher, right?

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And then, the derivative of
that, we need to divide by n

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plus 1 so that, when we take the
derivative, the n plus 1

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comes down, cancels this, and
gives us x to the n-th.

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The function that comes before
sine x will be--

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oh, there was cos x
in that direction.

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In this direction, we need
minus cos x because the

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derivative of minus cos
x is plus sine x.

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But, for this guy, cosine x,
that came from sine x.

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And, what about this one?

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What function has
this derivative?

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Well, with exponentials, we
expect to see that exponential

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always, e to the c, x again,
but, since this would bring

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down a c and here we don't
want it, we'd better

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divide by that c.

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So then, if I take that, that's
e to the c, x divided

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by c, so the c will come
down, cancel the c,

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just the way here.

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And, oh, we've never
figured out log x.

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That'll be something novel to
do for integral calculus.

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But, I think, if I write down
the answer, I think it's x

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times log x minus x.

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I believe that works.

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I would use the product
rule on that.

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x times the derivative of that
would be a 1 minus that.

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And the derivative of that
would be a 1, so two ones

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would cancel, and the product
rule would leave me with log x

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times the derivative of that.

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It works.

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It works.

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And notice the one beautiful
thing in this list, that the

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case here is great unless
I'm dividing by 0.

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If n is minus 1,
I'm in trouble.

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If n is minus 1, I don't have
here something whose--

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if n is minus 1, I can't
divide by 0.

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I don't get x to the minus
1 out of x to the 0.

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That rule fails at
n equal minus 1.

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But look, here, is exactly
fills in that whole.

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Wonderful.

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Here is the minus 1 power, and
here is where it comes from.

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So that log just filled in the
one hole that was left there.

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OK.

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Otherwise, you know
these guys.

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But here's a new one:
a step function.

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A step function, it's 0 and it
jumps up to 1 at x equals 0.

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So, here's x.

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The function is 0 until
it gets to that point.

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So it's level, then it takes
a step up, a jump up, to 1.

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And let's say it's 1 at that
point, so it takes that jump.

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All right.

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OK.

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That's a function that's
actually quite important.

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And it's sort of like a two-part
function, it's got a

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part to the left and a
part to the right.

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And they don't meet, it's a
non-continuous function.

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Can I figure out what is it
that-- so here will be the old

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generation, what graph do
I put there so that the

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derivative is 0 and then 1?

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Well, that's not too hard.

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If I put 0's here, the
derivative will be 0.

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And now, over here, I want the
derivative to be a constant 1.

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And we know that the derivative
of x is what I

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need, so this is 0 and then
x, two parts again.

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And the derivatives of those
parts are 0 and then 1.

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And I often call that a ramp
function because it looks a

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little like a ramp.

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OK.

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What about going this way?

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Ah, that's a little more
interesting because what's the

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derivative of a step function?

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What's the slope of
a step function?

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Well, the slope here is
certainly 0, and the slope

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along here is certainly 0,
so, is the answer 0?

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Well, of course not.

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All the action is
at this jump.

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And what's the derivative
there?

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Now, a careful person
would say there is

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no derivative there.

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The limit of delta f/delta x,
you don't get a correct answer

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there because delta f jumps
by one, and delta x

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could be very small.

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And, as delta x goes to 0, we
have 1/0, we have infinite.

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Well, I say, what?

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Let's go for infinite.

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So my derivative is
0 and 0, and, at

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this point, it's infinite.

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It's a spike, or sometimes
called a delta function.

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It's 0, and then infinite at
one point, and then 0.

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And the oddball thing is that
the area under that one-point

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tower, spike, is supposed
to be 1.

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Because, do you remember--

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and we'll do more areas if we
get to integral calculus--

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but, the area under
this function is

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supposed to be this one.

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The area under the cosine
function is sine x.

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The area under this function
should be this one, so the

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area is 0 here.

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Run along here.

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No area under it.

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Then, I have a one-point
spike, and the area is

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supposed to jump to 1
under that spike,

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at that single point.

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That spike is infinitely tall,
and it actually has a little

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area under it.

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Ah, well, your teacher may say
get that function out of here.

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That's not a function.

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And I'm afraid that's
a true fact that

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it's not a real function.

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So you could say I don't
want to see this

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thing, clear it out.

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But, actually, that's
very useful.

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It's a model for something that
happens very quickly: an

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instant, an impulse, so
I'll leave it there.

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I'll leave it there,
but I'll go on.

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So, if you don't like it, you
don't have to look at it.

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OK.

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So those were the
six functions,

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now for the six rules.

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Nothing too fancy here.

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I don't think I really
emphasized the most important

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and simplest rule that, if you
have as a combination, like

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you add two functions, then
the derivatives add.

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Or, if you multiply that
function by 2 and that

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function by 3 before you add,
then you multiply the

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derivatives by 2 and
3 before you add.

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It's that fact that
allowed us--

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I mean, you've used
it all the time.

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If you integrated x plus x
squared, you used the sum rule

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to integrate--

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ah, sorry--

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took the derivative.

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If you want the slope of x plus
x squared, you would say

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oh, no problem: 1 plus 2x.

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1 coming from the first
function, 2x from the x

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squared function.

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So the slope of a sum is just
the sum of the slopes.

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You constantly use that to build
many more functions out

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of the simple, anything, x
squared plus x cubed plus x 4,

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if you know its derivative and
you're using this rule.

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Now, the product rule,
we worked through.

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You've practiced that.

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The quotient rule is a little
messier with this minus sign

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and the division by g squared.

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It's a fraction.

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And then, a little more
complicated, was

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this inverse function.

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Do you remember that if you
start from y equals f of x--

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which is what we always
have been doing--

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and then you say all right,
switch it so that x isn't the

00:11:39.170 --> 00:11:44.010
input anymore, it's now
the output, and the

00:11:44.010 --> 00:11:45.280
input is the y.

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So you're reversing
the function.

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You're flipping the graph.

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We did this to get between
e to the x and log x.

00:11:55.920 --> 00:12:00.350
That was the most important
case of doing this flip

00:12:00.350 --> 00:12:06.820
between y equals e to the
x and x equals log of y.

00:12:06.820 --> 00:12:13.120
And the chain rule tells us that
the derivative of this

00:12:13.120 --> 00:12:17.810
inverse function is 1 over the
derivative of the original.

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Nice rule.

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And here's the full-scale
chain rule.

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Oh, that deserves to be put
inside a box or something

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because this is a really
great way to create new

00:12:35.180 --> 00:12:38.590
functions as a chain.

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You start with x.

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You do g of x, and then
that's the input to f.

00:12:44.370 --> 00:12:46.830
You will know that chain rule.

00:12:46.830 --> 00:12:51.450
And you remember that that
produces a product, the

00:12:51.450 --> 00:12:56.100
derivative of f times the
derivative of g, but there was

00:12:56.100 --> 00:12:57.350
this little trick, right?

00:13:00.340 --> 00:13:03.180
This g of x was the y.

00:13:03.180 --> 00:13:07.210
I'll just remind you that this
g of x is the y, and you have

00:13:07.210 --> 00:13:11.420
to get y out of the answer.

00:13:11.420 --> 00:13:15.590
Use this to get an answer
in terms of x.

00:13:15.590 --> 00:13:20.220
Wherever you see y, you
have to put in g of x.

00:13:20.220 --> 00:13:21.560
So, that's the chain rule.

00:13:21.560 --> 00:13:27.010
And then the final rule that
I want to mention is this

00:13:27.010 --> 00:13:30.550
L'hopital rule about--

00:13:30.550 --> 00:13:34.620
well, a lot of calculus is about
a ratio of f of x to g

00:13:34.620 --> 00:13:39.110
of x when it's going to 0/0.

00:13:39.110 --> 00:13:41.900
What do you do about 0/0?

00:13:41.900 --> 00:13:48.320
Well, as we're going to some
point, like x equals a, if

00:13:48.320 --> 00:13:56.230
this is going to 0/0, then
you're allowed to look.

00:13:56.230 --> 00:14:01.720
The slopes will tell you how
quickly each one is going to

00:14:01.720 --> 00:14:08.830
0, and the ratio becomes a
ratio of the two slopes.

00:14:08.830 --> 00:14:16.670
So, normally then, this answer
would be the derivative at a

00:14:16.670 --> 00:14:19.000
divided by the derivative
at a.

00:14:21.940 --> 00:14:28.130
If we're lucky, this 0/0 thing,
when we look at the

00:14:28.130 --> 00:14:31.300
slopes, isn't 0/0 any more.

00:14:31.300 --> 00:14:33.950
It's good numbers,
and L'hopital

00:14:33.950 --> 00:14:36.750
gets the answer right.

00:14:36.750 --> 00:14:37.330
OK.

00:14:37.330 --> 00:14:44.040
That's a review of L'hopital's
rule, just really remembering

00:14:44.040 --> 00:14:47.440
that that's an important rule
that came directly from the

00:14:47.440 --> 00:14:50.030
idea of the derivative.

00:14:50.030 --> 00:14:56.440
We're using the important part
of the function because the

00:14:56.440 --> 00:14:58.660
constant term in that
function is 0.

00:14:58.660 --> 00:15:00.390
Good.

00:15:00.390 --> 00:15:05.060
OK, are you ready for
six theorems?

00:15:05.060 --> 00:15:12.880
That is a handful, but
let's just tackle it.

00:15:12.880 --> 00:15:13.740
Why not?

00:15:13.740 --> 00:15:14.290
Why not?

00:15:14.290 --> 00:15:14.750
OK.

00:15:14.750 --> 00:15:18.310
So, six functions were easy.

00:15:18.310 --> 00:15:22.510
Well, we start with the big
theorem, the big theorem, the

00:15:22.510 --> 00:15:24.440
fundamental theorem
of calculus.

00:15:24.440 --> 00:15:28.320
The fundamental theorem
of calculus, OK,

00:15:28.320 --> 00:15:32.540
that ought to be important.

00:15:32.540 --> 00:15:34.720
And what does it say?

00:15:34.720 --> 00:15:38.630
It says that the two operations
of going from

00:15:38.630 --> 00:15:44.030
function one to two by taking
the derivative, the slope, the

00:15:44.030 --> 00:15:48.740
speed, is the reverse of
going the other way,

00:15:48.740 --> 00:15:50.170
from two back to one.

00:15:50.170 --> 00:15:58.295
It's really saying that, if
I start with a function--

00:15:58.295 --> 00:15:59.740
Here, this would be one way.

00:15:59.740 --> 00:16:05.540
If I start with a function, f,
I take the derivative to get

00:16:05.540 --> 00:16:08.490
function two, the speed,
the slope.

00:16:08.490 --> 00:16:10.130
Then, if I go backwards--

00:16:10.130 --> 00:16:16.560
which is this integrating that
integration symbol that's the

00:16:16.560 --> 00:16:19.220
core in integral calculus--

00:16:19.220 --> 00:16:22.000
if I take the derivative
and then take the

00:16:22.000 --> 00:16:26.080
integral, I'm back to f.

00:16:26.080 --> 00:16:30.990
And what you actually get
in this number is f at--

00:16:30.990 --> 00:16:31.940
it depends.

00:16:31.940 --> 00:16:36.000
It's like a delta f, really.

00:16:36.000 --> 00:16:44.190
It's the f at the end minus
the f at the start.

00:16:44.190 --> 00:16:45.770
Maybe you'll remember that.

00:16:45.770 --> 00:16:54.750
When we talked about it, there
was one lecture on big picture

00:16:54.750 --> 00:16:59.940
of the integral, and there may
be more coming, but that was

00:16:59.940 --> 00:17:03.490
the one where we had
this kind of thing.

00:17:03.490 --> 00:17:08.569
And, in the other direction, if
I start with function two,

00:17:08.569 --> 00:17:11.869
do its integral to get function
one, take the

00:17:11.869 --> 00:17:16.040
derivative of that, then I'm
back to function two.

00:17:16.040 --> 00:17:22.240
Actually, you're going to say
I knew that: function one to

00:17:22.240 --> 00:17:24.780
two, back to one.

00:17:24.780 --> 00:17:28.280
Or start with two, go to one,
then back to two, that's the

00:17:28.280 --> 00:17:29.800
fundamental theorem.

00:17:29.800 --> 00:17:34.030
That those two operations,
of taking the

00:17:34.030 --> 00:17:36.160
derivative, that limit--

00:17:36.160 --> 00:17:40.860
You remember what's tricky about
all that is that this

00:17:40.860 --> 00:17:45.940
d,f, d,x, involves a limit
as delta x goes to 0.

00:17:45.940 --> 00:17:49.010
And this integral will also
involve a limit as

00:17:49.010 --> 00:17:51.230
delta x goes to 0.

00:17:51.230 --> 00:17:54.690
So that's the point at which it
became calculus instead of

00:17:54.690 --> 00:17:56.870
just algebra.

00:17:56.870 --> 00:17:58.340
Well, important.

00:18:02.060 --> 00:18:07.160
I should say, let's assume here,
that these functions are

00:18:07.160 --> 00:18:09.240
all continuous functions.

00:18:09.240 --> 00:18:14.150
And I'm going to assume that
these theorems will apply to

00:18:14.150 --> 00:18:15.990
continuous functions.

00:18:15.990 --> 00:18:19.140
And do you remember
what that meant?

00:18:19.140 --> 00:18:22.880
Basically, it meant that that
jump function is not

00:18:22.880 --> 00:18:24.630
continuous.

00:18:24.630 --> 00:18:26.420
And that delta function is--

00:18:26.420 --> 00:18:29.280
well, that's not even
a function.

00:18:29.280 --> 00:18:33.650
The ramp function is continuous
but, of course, the

00:18:33.650 --> 00:18:35.240
derivative isn't.

00:18:35.240 --> 00:18:37.780
OK.

00:18:37.780 --> 00:18:38.750
All right.

00:18:38.750 --> 00:18:42.950
So, we've got functions that
we can draw without raising

00:18:42.950 --> 00:18:45.290
our pen, without lifting
the chalk.

00:18:45.290 --> 00:18:49.890
And here's the fact about
them, that if I have a

00:18:49.890 --> 00:18:53.340
continuous function on an
interval-- so, here is some

00:18:53.340 --> 00:18:56.450
point, a, and here is some
point, b, and my

00:18:56.450 --> 00:18:57.830
function goes like that.

00:18:57.830 --> 00:18:59.290
Oh, it doesn't do that.

00:18:59.290 --> 00:19:02.330
It goes like that.

00:19:02.330 --> 00:19:09.690
Then this thing says that this
maximum is actually reached,

00:19:09.690 --> 00:19:12.430
and this minimum is
actually reached.

00:19:12.430 --> 00:19:15.620
And any value in-between,
anywhere between this height

00:19:15.620 --> 00:19:20.590
and this height, there are
points where the function

00:19:20.590 --> 00:19:22.470
equals that.

00:19:22.470 --> 00:19:28.320
The continuous function hits its
maximum, hits its minimum,

00:19:28.320 --> 00:19:29.920
hits every point in-between.

00:19:29.920 --> 00:19:33.150
Where, if it wasn't continuous,
you see it could

00:19:33.150 --> 00:19:38.330
go up, and then, suddenly,
never reach that point,

00:19:38.330 --> 00:19:39.580
suddenly drop to there.

00:19:42.340 --> 00:19:45.910
There's a function not
continuous, of course, because

00:19:45.910 --> 00:19:48.410
it fell down there.

00:19:48.410 --> 00:19:53.560
And it never reached m because
it was this close, as close as

00:19:53.560 --> 00:19:54.120
it could be.

00:19:54.120 --> 00:19:57.290
But it never got there because,
at the last minute,

00:19:57.290 --> 00:19:58.780
it jumped down.

00:19:58.780 --> 00:20:00.740
OK.

00:20:00.740 --> 00:20:08.130
So, that's sort of a good
theoretical bit about

00:20:08.130 --> 00:20:09.550
continuous functions.

00:20:09.550 --> 00:20:10.800
OK.

00:20:20.510 --> 00:20:22.450
So, that's new.

00:20:22.450 --> 00:20:25.665
That was not mentioned before.

00:20:25.665 --> 00:20:34.500
But you can see it by just
drawing a picture where it

00:20:34.500 --> 00:20:38.610
hits the max, hits the min, hits
all values in-between.

00:20:38.610 --> 00:20:42.720
And then, you see the point,
y, continuous was needed

00:20:42.720 --> 00:20:47.250
because, if you let it jump,
the result doesn't work.

00:20:47.250 --> 00:20:47.620
OK.

00:20:47.620 --> 00:20:48.800
Here's another thing.

00:20:48.800 --> 00:20:51.470
This is now called the
mean value theorem.

00:20:54.040 --> 00:20:56.030
That's a neat theorem.

00:20:56.030 --> 00:20:57.100
OK.

00:20:57.100 --> 00:20:57.280
Oh.

00:20:57.280 --> 00:21:01.780
Now, here, our function is going
to have a derivative

00:21:01.780 --> 00:21:03.660
over some region.

00:21:03.660 --> 00:21:05.900
That function probably
had a derivative.

00:21:05.900 --> 00:21:06.820
OK.

00:21:06.820 --> 00:21:07.820
OK.

00:21:07.820 --> 00:21:16.110
So, that function, or this
function, f of x, here's the

00:21:16.110 --> 00:21:21.310
idea of the mean
value theorem.

00:21:21.310 --> 00:21:27.870
This is like delta f/delta
x for the whole

00:21:27.870 --> 00:21:29.810
interval from a to b.

00:21:29.810 --> 00:21:34.100
Delta x is b minus a,
the whole jump.

00:21:34.100 --> 00:21:38.270
Delta f is f at the end
minus f at this end.

00:21:38.270 --> 00:21:42.380
So that delta f/delta x is like
your average speed over

00:21:42.380 --> 00:21:44.630
the whole trip.

00:21:44.630 --> 00:21:47.800
Like you went on the
MassPike, right?

00:21:47.800 --> 00:21:50.560
And you entered at 1:00 o'clock
and came out at 4:00

00:21:50.560 --> 00:21:53.980
o'clock, so you were on the
pike for three hours.

00:21:53.980 --> 00:21:58.930
And your trip meter
shows 200 miles.

00:21:58.930 --> 00:22:07.640
So your average speed, average
speed, was 200 divided by 3,

00:22:07.640 --> 00:22:09.750
that number of miles per hour.

00:22:09.750 --> 00:22:13.480
Yeah, about 66 miles-- well,
probably illegal.

00:22:13.480 --> 00:22:15.670
OK.

00:22:15.670 --> 00:22:20.160
A little over 66 miles an
hour: 200/3, so you're

00:22:20.160 --> 00:22:21.710
slightly over the speed limit.

00:22:21.710 --> 00:22:28.900
Well, the mean value theorem
catches you because you could

00:22:28.900 --> 00:22:37.490
say well, but when did
I pass the limit?

00:22:37.490 --> 00:22:41.210
When was I going more than 65?

00:22:41.210 --> 00:22:45.190
And the mean value theorem says
there was a time, there

00:22:45.190 --> 00:22:52.080
was a moment when your speed,
when the speedometer, itself,

00:22:52.080 --> 00:22:53.600
was exactly.

00:22:53.600 --> 00:22:59.950
This instant speed equaled
the average speed.

00:22:59.950 --> 00:23:01.750
Shall I say that again?

00:23:01.750 --> 00:23:10.180
If you travel with a smooth
changes of speed, no jumps in

00:23:10.180 --> 00:23:18.770
speed, then, if I look at the
average speed over a delta t,

00:23:18.770 --> 00:23:23.400
there is some point inside that
one where the average

00:23:23.400 --> 00:23:29.530
speed agrees with the
instant speed.

00:23:29.530 --> 00:23:35.890
Or you could say, if
you prefer slope--

00:23:35.890 --> 00:23:41.530
Suppose the average slope, the
up over a cross, is 10, So in

00:23:41.530 --> 00:23:45.870
the time at cross, you
eventually got up 10.

00:23:45.870 --> 00:23:48.910
Then there will be some
point when your

00:23:48.910 --> 00:23:51.300
climbing rate was 10.

00:23:51.300 --> 00:23:56.650
There'd be some point when that
instant slope is also 10.

00:23:56.650 --> 00:23:59.340
OK.

00:23:59.340 --> 00:24:02.360
That's the mean value theorem.

00:24:02.360 --> 00:24:04.250
This is called the mean value.

00:24:04.250 --> 00:24:07.880
Mean value is another
word for average.

00:24:07.880 --> 00:24:12.200
So the mean value equals the
instant value at some point.

00:24:12.200 --> 00:24:16.900
But we don't know, that point
could be anywhere.

00:24:16.900 --> 00:24:17.430
OK.

00:24:17.430 --> 00:24:24.840
Now, I'm ready for the last two
theorems. And the first

00:24:24.840 --> 00:24:29.300
one is called Taylor Series,
the Taylor's theorem.

00:24:29.300 --> 00:24:33.800
And we have touched on that.

00:24:33.800 --> 00:24:38.110
And what is Taylor
Series about?

00:24:38.110 --> 00:24:44.840
Taylor Series is when you know
what's going on at some point

00:24:44.840 --> 00:24:50.060
x equal a, and you want to know
what the function is at

00:24:50.060 --> 00:24:53.990
some point x near a.

00:24:53.990 --> 00:24:57.230
So x is near a.

00:24:57.230 --> 00:25:03.780
And, to a very low
approximation, f of x is

00:25:03.780 --> 00:25:05.580
pretty close to f of a.

00:25:05.580 --> 00:25:08.970
This is the constant term.

00:25:08.970 --> 00:25:10.890
That's where the trip started.

00:25:10.890 --> 00:25:15.380
So this is like a trip meter
for a very short trip.

00:25:15.380 --> 00:25:19.620
The first thing would be to know
what was the trip meter

00:25:19.620 --> 00:25:22.650
reading at the start.

00:25:22.650 --> 00:25:30.900
But then the correction term, so
this is the calculus term,

00:25:30.900 --> 00:25:35.985
it's the speed at the start
times the time of the trip.

00:25:39.760 --> 00:25:43.910
If you only keep this, the
trip meter isn't moving.

00:25:43.910 --> 00:25:48.280
When you add on this, you're
like following a tangent line.

00:25:48.280 --> 00:25:55.050
If I try to describe it, you're
pretending the speed

00:25:55.050 --> 00:25:56.380
didn't change.

00:25:56.380 --> 00:25:59.030
Here, you're pretending the
trip meter didn't change.

00:25:59.030 --> 00:26:00.190
Nothing happened.

00:26:00.190 --> 00:26:01.550
Here is the next term.

00:26:01.550 --> 00:26:08.490
But now, of course, this speed
normally changes too.

00:26:08.490 --> 00:26:12.630
So calculus says there is
a term from the second

00:26:12.630 --> 00:26:15.570
derivative, there's
a bending term.

00:26:15.570 --> 00:26:19.720
This, we would be correct to
stop right there on a straight

00:26:19.720 --> 00:26:23.170
line: constant speed.

00:26:23.170 --> 00:26:27.730
But now, if the speed is
increasing, your trip meter

00:26:27.730 --> 00:26:34.300
graph is bending upwards, you'd
better have a correction

00:26:34.300 --> 00:26:36.750
from the second derivative.

00:26:36.750 --> 00:26:41.880
That's the slope of the slope,
the rate of change

00:26:41.880 --> 00:26:45.310
of the rate of change.

00:26:45.310 --> 00:26:47.710
It's the acceleration.

00:26:47.710 --> 00:26:53.980
So, if I had constant
acceleration, like I drop this

00:26:53.980 --> 00:26:56.120
chalk, it accelerates.

00:26:56.120 --> 00:27:02.650
So, from where I drop it, that
gives me its original height.

00:27:02.650 --> 00:27:06.280
Its original speed might be
0, if I hold onto it.

00:27:06.280 --> 00:27:11.350
But then, this term would
account for the second

00:27:11.350 --> 00:27:12.890
derivative, the acceleration.

00:27:12.890 --> 00:27:20.660
And that would give me the right
answer, the right answer

00:27:20.660 --> 00:27:25.030
to the next term, but now I've
drawn the famous three dots.

00:27:25.030 --> 00:27:30.730
So three dots is the way to
say there are more terms

00:27:30.730 --> 00:27:34.660
because the acceleration
might not be constant.

00:27:34.660 --> 00:27:36.380
What's the next term?

00:27:36.380 --> 00:27:38.340
If you know the next
term, then you

00:27:38.340 --> 00:27:40.820
and Taylor are square.

00:27:40.820 --> 00:27:46.470
The next term will be
1/3 factorial, 1/6.

00:27:46.470 --> 00:27:51.470
It'll be a third derivative of
f at the known point times

00:27:51.470 --> 00:27:53.750
this x minus a cubed.

00:27:53.750 --> 00:27:57.680
You see that these terms
are getting,

00:27:57.680 --> 00:28:00.340
typically, for a nice function--

00:28:00.340 --> 00:28:03.150
and we saw this for
e to the x.

00:28:03.150 --> 00:28:06.370
We saw the Taylor Series
for e to the x.

00:28:06.370 --> 00:28:12.490
Can I remind you of the Taylor
Series for e to the x around

00:28:12.490 --> 00:28:18.610
the point 0 because e to the x
is the greatest function I've

00:28:18.610 --> 00:28:21.070
spoken about, at all?

00:28:21.070 --> 00:28:27.540
So, if this was e to the x, it
would start out at e to the 0,

00:28:27.540 --> 00:28:29.260
which is 1.

00:28:29.260 --> 00:28:36.530
Its slope is 1, so this
is 1 times x.

00:28:36.530 --> 00:28:40.790
Its second derivative
is, again, 1.

00:28:40.790 --> 00:28:44.640
And a is 0 here, so
this would be 1/2,

00:28:44.640 --> 00:28:47.670
1/2 factorial x squared.

00:28:47.670 --> 00:28:50.160
And then that next three-dot
term would be

00:28:50.160 --> 00:28:54.100
1/3 factorial x cube.

00:28:54.100 --> 00:28:56.660
And you remember what
it looks like.

00:28:56.660 --> 00:29:00.880
So the Taylor Series just
looks messy because I'm

00:29:00.880 --> 00:29:03.450
writing any old f.

00:29:03.450 --> 00:29:08.520
I'm allowing it to be the start
point, to be a, and not

00:29:08.520 --> 00:29:09.800
necessarily 0.

00:29:09.800 --> 00:29:12.390
But, typically, it's 0.

00:29:12.390 --> 00:29:16.930
And the e to the x series
is the best example.

00:29:16.930 --> 00:29:20.300
But I want to show you
one more example.

00:29:20.300 --> 00:29:23.090
That'll be my last theorem.

00:29:23.090 --> 00:29:26.510
I just mention it here because
it's just like

00:29:26.510 --> 00:29:29.240
the mean value theorem.

00:29:29.240 --> 00:29:37.220
If I do stop, suppose I stop
here and I don't include the x

00:29:37.220 --> 00:29:40.890
cube term, the third derivative
term, then I've

00:29:40.890 --> 00:29:42.570
made an error.

00:29:42.570 --> 00:29:45.700
And, of course, that error
depends on what the third

00:29:45.700 --> 00:29:50.970
derivative is, the one I
skipped, the x minus a cube,

00:29:50.970 --> 00:29:54.445
the thing I skipped, and
the 1/3 factorial.

00:29:57.810 --> 00:30:03.600
And this third derivative
is, at some point,

00:30:03.600 --> 00:30:04.850
between a and x.

00:30:07.465 --> 00:30:13.570
That's a lot to put in, but the
mean value theorem said

00:30:13.570 --> 00:30:16.870
you could take the derivative at
some point in-between, some

00:30:16.870 --> 00:30:19.360
point along the MassPike.

00:30:19.360 --> 00:30:22.990
And this is just the same thing,
but I'm keeping more

00:30:22.990 --> 00:30:28.060
terms. I'm quitting at any
point, and then I would take

00:30:28.060 --> 00:30:31.530
the next derivative at somewhere
along the MassPike.

00:30:34.410 --> 00:30:37.520
What should you learn
out of that?

00:30:37.520 --> 00:30:42.920
I think the idea is
Taylor Series.

00:30:42.920 --> 00:30:49.320
And, of course, we have
two possibilities.

00:30:49.320 --> 00:30:58.860
Either we cut the series off and
we make some error, but we

00:30:58.860 --> 00:31:04.440
get a pretty good answer, or we
let the series go forever.

00:31:04.440 --> 00:31:06.770
And then comes the question.

00:31:06.770 --> 00:31:09.750
Then we have an infinite number
of terms, and then the

00:31:09.750 --> 00:31:13.510
question is does that series add
up to a finite thing like

00:31:13.510 --> 00:31:15.200
e to the x?

00:31:15.200 --> 00:31:19.120
Or does it add up to a delta
function or something

00:31:19.120 --> 00:31:21.580
impossible?

00:31:21.580 --> 00:31:25.320
So that leads to the question
of learning

00:31:25.320 --> 00:31:29.340
about infinite series.

00:31:29.340 --> 00:31:32.710
In calculus, Taylor
Series is where

00:31:32.710 --> 00:31:35.280
infinite series come from.

00:31:35.280 --> 00:31:39.920
And, if we want to go all the
way with them, then we have to

00:31:39.920 --> 00:31:43.050
begin to think about what does
it mean for that infinite

00:31:43.050 --> 00:31:46.180
series to add up to a
number, or maybe it

00:31:46.180 --> 00:31:49.930
just goes off to infinity.

00:31:49.930 --> 00:31:53.890
Does it converge, or
does it diverge?

00:31:53.890 --> 00:31:58.410
Ah, that would be another
lecture or two.

00:31:58.410 --> 00:32:06.740
Let me complete today with one
more theorem, a famous one,

00:32:06.740 --> 00:32:07.970
the binomial theorem.

00:32:07.970 --> 00:32:10.750
So, what's the binomial
theorem about?

00:32:10.750 --> 00:32:14.870
The binomial theorem is about
powers of 1 plus x.

00:32:14.870 --> 00:32:20.690
1 plus x is a typical binomial:
two things, 1 and x.

00:32:20.690 --> 00:32:22.900
And we have various powers.

00:32:22.900 --> 00:32:26.960
Well, if the powers are the
first power, the second power,

00:32:26.960 --> 00:32:32.020
the third power, we can write
out, we can square 1 plus x,

00:32:32.020 --> 00:32:34.850
and we can get 1 plus x cubed.

00:32:34.850 --> 00:32:36.830
And, out of it, we get this.

00:32:36.830 --> 00:32:40.200
And there would be 1 plus
x to the 0-th power.

00:32:40.200 --> 00:32:43.110
And do you see that there's a
whole lot of ones in the neat

00:32:43.110 --> 00:32:44.440
pattern there?

00:32:44.440 --> 00:32:46.900
And then there's a
2, and a 3, 3.

00:32:46.900 --> 00:32:53.500
And if you'd like to know this
one, it would be 1, 4, 6, 4, 1

00:32:53.500 --> 00:32:56.430
would be the next
row of Pascal.

00:32:56.430 --> 00:33:03.630
Pascal really had a sense of
beauty or art in this triangle

00:33:03.630 --> 00:33:06.570
of numbers.

00:33:06.570 --> 00:33:11.550
And that's the triangle you
get, Pascal's triangle, if

00:33:11.550 --> 00:33:12.770
you're taking--

00:33:12.770 --> 00:33:17.790
A whole number, a power is 1
plus x to the third power,

00:33:17.790 --> 00:33:23.170
fourth power, fifth power,
sixth power, but what if

00:33:23.170 --> 00:33:27.390
you're taking to some other
power, any power, p?

00:33:27.390 --> 00:33:32.200
So now I'm interested in this
guy to a power of p that,

00:33:32.200 --> 00:33:34.250
maybe, is not two, three,
four, five.

00:33:34.250 --> 00:33:42.260
It could be 1/2, 1 plus x square
root to the 1/2 power,

00:33:42.260 --> 00:33:46.440
or 1 plus x to the
minus 1 power.

00:33:46.440 --> 00:33:54.480
All other powers are possible
and, for those,

00:33:54.480 --> 00:33:56.240
the Taylor's theorem.

00:33:56.240 --> 00:33:57.490
And here's my function.

00:34:01.540 --> 00:34:06.690
And I could apply Taylor's
theorem to find the--

00:34:06.690 --> 00:34:11.190
and I'll do it at x equals 0,
that's the place Taylor liked

00:34:11.190 --> 00:34:16.670
the best. So the
constant term--

00:34:16.670 --> 00:34:20.210
think of this Taylor expansion
that we just did--

00:34:20.210 --> 00:34:22.880
at x equals 0, this
thing is 1.

00:34:22.880 --> 00:34:27.500
So, the big theory starts
out with a 1 for

00:34:27.500 --> 00:34:29.080
the constant term.

00:34:29.080 --> 00:34:33.730
Then what I do for the next
term of the Taylor Series?

00:34:33.730 --> 00:34:37.840
I take the derivative and
I put x equals 0.

00:34:37.840 --> 00:34:39.719
And what do I get then?

00:34:39.719 --> 00:34:42.420
I get p times x.

00:34:42.420 --> 00:34:46.610
So this is the constant
term: f of 0.

00:34:46.610 --> 00:34:50.270
This is the derivative:
times x minus a

00:34:50.270 --> 00:34:52.270
divided by 1 factorial.

00:34:52.270 --> 00:34:55.389
Well, you didn't see all those
things because one factorial I

00:34:55.389 --> 00:34:56.530
didn't write.

00:34:56.530 --> 00:34:59.740
And then the next term would be
the next derivative, of p

00:34:59.740 --> 00:35:03.510
minus 1 will come down, so
you'll have p, p minus 1.

00:35:03.510 --> 00:35:05.600
You're supposed to divide
by 2 factorial.

00:35:05.600 --> 00:35:08.690
That multiplies x squared.

00:35:08.690 --> 00:35:17.070
Well, my point is just that
this binomial formula is

00:35:17.070 --> 00:35:19.090
Taylor's formula.

00:35:19.090 --> 00:35:24.260
The binomial theorem, with
these, this is called a

00:35:24.260 --> 00:35:26.100
binomial coefficient.

00:35:26.100 --> 00:35:31.480
Gamblers know all about
that, you know?

00:35:31.480 --> 00:35:34.650
If you've got p things and you
want to take two, how many

00:35:34.650 --> 00:35:36.140
ways to do it?

00:35:36.140 --> 00:35:39.000
You know, how many ways to get
two aces out of a deck, all

00:35:39.000 --> 00:35:45.230
these things are hidden in those
numbers, which gamblers

00:35:45.230 --> 00:35:47.820
learn or lose.

00:35:47.820 --> 00:35:49.190
OK.

00:35:49.190 --> 00:35:52.460
So, I'll make one last point
about the binomial theorem.

00:35:57.780 --> 00:36:00.050
Those were Taylor Series.

00:36:00.050 --> 00:36:01.490
This is a Taylor series.

00:36:01.490 --> 00:36:03.220
What's the difference?

00:36:03.220 --> 00:36:07.990
The difference is these
series stop.

00:36:07.990 --> 00:36:10.220
This is a series: 1
plus x squared.

00:36:10.220 --> 00:36:12.770
That's the Taylor Series,
but the third

00:36:12.770 --> 00:36:14.790
derivative is 0, right?

00:36:14.790 --> 00:36:18.480
The third derivative of that
function, because that

00:36:18.480 --> 00:36:21.700
function's only going up to x
squared, the third derivative

00:36:21.700 --> 00:36:25.170
is 0, so the rest of Taylor
Series has died.

00:36:25.170 --> 00:36:26.310
It's not there.

00:36:26.310 --> 00:36:27.830
So that's all there is.

00:36:30.810 --> 00:36:33.870
The derivative of any of those
powers, one, two, three, four,

00:36:33.870 --> 00:36:39.040
five powers, after I take enough
derivatives, gone.

00:36:39.040 --> 00:36:44.270
But, if I take a power like
minus 1, or 1/2, or pi, or

00:36:44.270 --> 00:36:48.910
anything, then I can take
derivatives forever

00:36:48.910 --> 00:36:54.080
without hitting 0.

00:36:54.080 --> 00:36:56.570
In other words, this series
goes on, and on, and on.

00:36:56.570 --> 00:36:57.650
Those three dots--

00:36:57.650 --> 00:37:01.160
let me move that eraser so you
see those three dots--

00:37:01.160 --> 00:37:07.040
that signals an infinite series
and the question of

00:37:07.040 --> 00:37:10.560
does it add up to a finite
number, what's going on with

00:37:10.560 --> 00:37:11.950
infinite series?

00:37:11.950 --> 00:37:19.440
But, for the moment, my point
is just this is what

00:37:19.440 --> 00:37:22.390
calculus can do.

00:37:22.390 --> 00:37:26.320
If you not only take that slope,
but the slope of the

00:37:26.320 --> 00:37:28.890
slope, and the third derivative,
and all higher

00:37:28.890 --> 00:37:33.340
derivatives, that's what Taylor
Series tells you.

00:37:33.340 --> 00:37:34.250
OK.

00:37:34.250 --> 00:37:40.590
So that's the, in some way, high
point of the highlights

00:37:40.590 --> 00:37:45.680
of calculus, and I sure hope
they're helpful to you.

00:37:45.680 --> 00:37:47.440
Thank you.

00:37:47.440 --> 00:37:49.650
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