PROFESSOR: Hi.
Well, this is sort of a summary
lecture for the big
group about differential
calculus.
And it's got a fancy title, Six
Functions, that we know.
Well, five of them
that we know.
And a new one-- of course,
there has to
be something new--
Six Rules, and Six Theorems.
So I haven't emphasized
theorems, but it seemed like
this was an occasion where we
could see the main points of the
math behind the functions
and the rules.
OK.
So, here are my first five
functions, all familiar.
And, what I'm happy about is
that, if we understand those
five and the rules to create
more out of them, we get
practically everything,
everything we frequently use.
OK.
So, I wrote down function
one, power of x, and its
derivative, function
two and its
derivative, function three.
Function four has,
a little bit,
something that is important.
If it's e to the x,
then we know the
derivative is e to the x.
But, if it's e to the c, x
of factor c, comes down.
Important case, you could
say the chain rule.
The derivative is that times
the derivative of what's
inside, which is the c.
And, finally, the natural
logarithm with the great
derivative of 1/x.
And now, oh, I left space
to go from function one
backwards, to remember the
function that came before it.
So, what function has
this derivative?
I'm looking here at the other
generation, the older
generation.
Well, the function with that
derivative is we need the
power to be one higher, right?
And then, the derivative of
that, we need to divide by n
plus 1 so that, when we take the
derivative, the n plus 1
comes down, cancels this, and
gives us x to the n-th.
The function that comes before
sine x will be--
oh, there was cos x
in that direction.
In this direction, we need
minus cos x because the
derivative of minus cos
x is plus sine x.
But, for this guy, cosine x,
that came from sine x.
And, what about this one?
What function has
this derivative?
Well, with exponentials, we
expect to see that exponential
always, e to the c, x again,
but, since this would bring
down a c and here we don't
want it, we'd better
divide by that c.
So then, if I take that, that's
e to the c, x divided
by c, so the c will come
down, cancel the c,
just the way here.
And, oh, we've never
figured out log x.
That'll be something novel to
do for integral calculus.
But, I think, if I write down
the answer, I think it's x
times log x minus x.
I believe that works.
I would use the product
rule on that.
x times the derivative of that
would be a 1 minus that.
And the derivative of that
would be a 1, so two ones
would cancel, and the product
rule would leave me with log x
times the derivative of that.
It works.
It works.
And notice the one beautiful
thing in this list, that the
case here is great unless
I'm dividing by 0.
If n is minus 1,
I'm in trouble.
If n is minus 1, I don't have
here something whose--
if n is minus 1, I can't
divide by 0.
I don't get x to the minus
1 out of x to the 0.
That rule fails at
n equal minus 1.
But look, here, is exactly
fills in that whole.
Wonderful.
Here is the minus 1 power, and
here is where it comes from.
So that log just filled in the
one hole that was left there.
OK.
Otherwise, you know
these guys.
But here's a new one:
a step function.
A step function, it's 0 and it
jumps up to 1 at x equals 0.
So, here's x.
The function is 0 until
it gets to that point.
So it's level, then it takes
a step up, a jump up, to 1.
And let's say it's 1 at that
point, so it takes that jump.
All right.
OK.
That's a function that's
actually quite important.
And it's sort of like a two-part
function, it's got a
part to the left and a
part to the right.
And they don't meet, it's a
non-continuous function.
Can I figure out what is it
that-- so here will be the old
generation, what graph do
I put there so that the
derivative is 0 and then 1?
Well, that's not too hard.
If I put 0's here, the
derivative will be 0.
And now, over here, I want the
derivative to be a constant 1.
And we know that the derivative
of x is what I
need, so this is 0 and then
x, two parts again.
And the derivatives of those
parts are 0 and then 1.
And I often call that a ramp
function because it looks a
little like a ramp.
OK.
What about going this way?
Ah, that's a little more
interesting because what's the
derivative of a step function?
What's the slope of
a step function?
Well, the slope here is
certainly 0, and the slope
along here is certainly 0,
so, is the answer 0?
Well, of course not.
All the action is
at this jump.
And what's the derivative
there?
Now, a careful person
would say there is
no derivative there.
The limit of delta f/delta x,
you don't get a correct answer
there because delta f jumps
by one, and delta x
could be very small.
And, as delta x goes to 0, we
have 1/0, we have infinite.
Well, I say, what?
Let's go for infinite.
So my derivative is
0 and 0, and, at
this point, it's infinite.
It's a spike, or sometimes
called a delta function.
It's 0, and then infinite at
one point, and then 0.
And the oddball thing is that
the area under that one-point
tower, spike, is supposed
to be 1.
Because, do you remember--
and we'll do more areas if we
get to integral calculus--
but, the area under
this function is
supposed to be this one.
The area under the cosine
function is sine x.
The area under this function
should be this one, so the
area is 0 here.
Run along here.
No area under it.
Then, I have a one-point
spike, and the area is
supposed to jump to 1
under that spike,
at that single point.
That spike is infinitely tall,
and it actually has a little
area under it.
Ah, well, your teacher may say
get that function out of here.
That's not a function.
And I'm afraid that's
a true fact that
it's not a real function.
So you could say I don't
want to see this
thing, clear it out.
But, actually, that's
very useful.
It's a model for something that
happens very quickly: an
instant, an impulse, so
I'll leave it there.
I'll leave it there,
but I'll go on.
So, if you don't like it, you
don't have to look at it.
OK.
So those were the
six functions,
now for the six rules.
Nothing too fancy here.
I don't think I really
emphasized the most important
and simplest rule that, if you
have as a combination, like
you add two functions, then
the derivatives add.
Or, if you multiply that
function by 2 and that
function by 3 before you add,
then you multiply the
derivatives by 2 and
3 before you add.
It's that fact that
allowed us--
I mean, you've used
it all the time.
If you integrated x plus x
squared, you used the sum rule
to integrate--
ah, sorry--
took the derivative.
If you want the slope of x plus
x squared, you would say
oh, no problem: 1 plus 2x.
1 coming from the first
function, 2x from the x
squared function.
So the slope of a sum is just
the sum of the slopes.
You constantly use that to build
many more functions out
of the simple, anything, x
squared plus x cubed plus x 4,
if you know its derivative and
you're using this rule.
Now, the product rule,
we worked through.
You've practiced that.
The quotient rule is a little
messier with this minus sign
and the division by g squared.
It's a fraction.
And then, a little more
complicated, was
this inverse function.
Do you remember that if you
start from y equals f of x--
which is what we always
have been doing--
and then you say all right,
switch it so that x isn't the
input anymore, it's now
the output, and the
input is the y.
So you're reversing
the function.
You're flipping the graph.
We did this to get between
e to the x and log x.
That was the most important
case of doing this flip
between y equals e to the
x and x equals log of y.
And the chain rule tells us that
the derivative of this
inverse function is 1 over the
derivative of the original.
Nice rule.
And here's the full-scale
chain rule.
Oh, that deserves to be put
inside a box or something
because this is a really
great way to create new
functions as a chain.
You start with x.
You do g of x, and then
that's the input to f.
You will know that chain rule.
And you remember that that
produces a product, the
derivative of f times the
derivative of g, but there was
this little trick, right?
This g of x was the y.
I'll just remind you that this
g of x is the y, and you have
to get y out of the answer.
Use this to get an answer
in terms of x.
Wherever you see y, you
have to put in g of x.
So, that's the chain rule.
And then the final rule that
I want to mention is this
L'hopital rule about--
well, a lot of calculus is about
a ratio of f of x to g
of x when it's going to 0/0.
What do you do about 0/0?
Well, as we're going to some
point, like x equals a, if
this is going to 0/0, then
you're allowed to look.
The slopes will tell you how
quickly each one is going to
0, and the ratio becomes a
ratio of the two slopes.
So, normally then, this answer
would be the derivative at a
divided by the derivative
at a.
If we're lucky, this 0/0 thing,
when we look at the
slopes, isn't 0/0 any more.
It's good numbers,
and L'hopital
gets the answer right.
OK.
That's a review of L'hopital's
rule, just really remembering
that that's an important rule
that came directly from the
idea of the derivative.
We're using the important part
of the function because the
constant term in that
function is 0.
Good.
OK, are you ready for
six theorems?
That is a handful, but
let's just tackle it.
Why not?
Why not?
OK.
So, six functions were easy.
Well, we start with the big
theorem, the big theorem, the
fundamental theorem
of calculus.
The fundamental theorem
of calculus, OK,
that ought to be important.
And what does it say?
It says that the two operations
of going from
function one to two by taking
the derivative, the slope, the
speed, is the reverse of
going the other way,
from two back to one.
It's really saying that, if
I start with a function--
Here, this would be one way.
If I start with a function, f,
I take the derivative to get
function two, the speed,
the slope.
Then, if I go backwards--
which is this integrating that
integration symbol that's the
core in integral calculus--
if I take the derivative
and then take the
integral, I'm back to f.
And what you actually get
in this number is f at--
it depends.
It's like a delta f, really.
It's the f at the end minus
the f at the start.
Maybe you'll remember that.
When we talked about it, there
was one lecture on big picture
of the integral, and there may
be more coming, but that was
the one where we had
this kind of thing.
And, in the other direction, if
I start with function two,
do its integral to get function
one, take the
derivative of that, then I'm
back to function two.
Actually, you're going to say
I knew that: function one to
two, back to one.
Or start with two, go to one,
then back to two, that's the
fundamental theorem.
That those two operations,
of taking the
derivative, that limit--
You remember what's tricky about
all that is that this
d,f, d,x, involves a limit
as delta x goes to 0.
And this integral will also
involve a limit as
delta x goes to 0.
So that's the point at which it
became calculus instead of
just algebra.
Well, important.
I should say, let's assume here,
that these functions are
all continuous functions.
And I'm going to assume that
these theorems will apply to
continuous functions.
And do you remember
what that meant?
Basically, it meant that that
jump function is not
continuous.
And that delta function is--
well, that's not even
a function.
The ramp function is continuous
but, of course, the
derivative isn't.
OK.
All right.
So, we've got functions that
we can draw without raising
our pen, without lifting
the chalk.
And here's the fact about
them, that if I have a
continuous function on an
interval-- so, here is some
point, a, and here is some
point, b, and my
function goes like that.
Oh, it doesn't do that.
It goes like that.
Then this thing says that this
maximum is actually reached,
and this minimum is
actually reached.
And any value in-between,
anywhere between this height
and this height, there are
points where the function
equals that.
The continuous function hits its
maximum, hits its minimum,
hits every point in-between.
Where, if it wasn't continuous,
you see it could
go up, and then, suddenly,
never reach that point,
suddenly drop to there.
There's a function not
continuous, of course, because
it fell down there.
And it never reached m because
it was this close, as close as
it could be.
But it never got there because,
at the last minute,
it jumped down.
OK.
So, that's sort of a good
theoretical bit about
continuous functions.
OK.
So, that's new.
That was not mentioned before.
But you can see it by just
drawing a picture where it
hits the max, hits the min, hits
all values in-between.
And then, you see the point,
y, continuous was needed
because, if you let it jump,
the result doesn't work.
OK.
Here's another thing.
This is now called the
mean value theorem.
That's a neat theorem.
OK.
Oh.
Now, here, our function is going
to have a derivative
over some region.
That function probably
had a derivative.
OK.
OK.
So, that function, or this
function, f of x, here's the
idea of the mean
value theorem.
This is like delta f/delta
x for the whole
interval from a to b.
Delta x is b minus a,
the whole jump.
Delta f is f at the end
minus f at this end.
So that delta f/delta x is like
your average speed over
the whole trip.
Like you went on the
MassPike, right?
And you entered at 1:00 o'clock
and came out at 4:00
o'clock, so you were on the
pike for three hours.
And your trip meter
shows 200 miles.
So your average speed, average
speed, was 200 divided by 3,
that number of miles per hour.
Yeah, about 66 miles-- well,
probably illegal.
OK.
A little over 66 miles an
hour: 200/3, so you're
slightly over the speed limit.
Well, the mean value theorem
catches you because you could
say well, but when did
I pass the limit?
When was I going more than 65?
And the mean value theorem says
there was a time, there
was a moment when your speed,
when the speedometer, itself,
was exactly.
This instant speed equaled
the average speed.
Shall I say that again?
If you travel with a smooth
changes of speed, no jumps in
speed, then, if I look at the
average speed over a delta t,
there is some point inside that
one where the average
speed agrees with the
instant speed.
Or you could say, if
you prefer slope--
Suppose the average slope, the
up over a cross, is 10, So in
the time at cross, you
eventually got up 10.
Then there will be some
point when your
climbing rate was 10.
There'd be some point when that
instant slope is also 10.
OK.
That's the mean value theorem.
This is called the mean value.
Mean value is another
word for average.
So the mean value equals the
instant value at some point.
But we don't know, that point
could be anywhere.
OK.
Now, I'm ready for the last two
theorems. And the first
one is called Taylor Series,
the Taylor's theorem.
And we have touched on that.
And what is Taylor
Series about?
Taylor Series is when you know
what's going on at some point
x equal a, and you want to know
what the function is at
some point x near a.
So x is near a.
And, to a very low
approximation, f of x is
pretty close to f of a.
This is the constant term.
That's where the trip started.
So this is like a trip meter
for a very short trip.
The first thing would be to know
what was the trip meter
reading at the start.
But then the correction term, so
this is the calculus term,
it's the speed at the start
times the time of the trip.
If you only keep this, the
trip meter isn't moving.
When you add on this, you're
like following a tangent line.
If I try to describe it, you're
pretending the speed
didn't change.
Here, you're pretending the
trip meter didn't change.
Nothing happened.
Here is the next term.
But now, of course, this speed
normally changes too.
So calculus says there is
a term from the second
derivative, there's
a bending term.
This, we would be correct to
stop right there on a straight
line: constant speed.
But now, if the speed is
increasing, your trip meter
graph is bending upwards, you'd
better have a correction
from the second derivative.
That's the slope of the slope,
the rate of change
of the rate of change.
It's the acceleration.
So, if I had constant
acceleration, like I drop this
chalk, it accelerates.
So, from where I drop it, that
gives me its original height.
Its original speed might be
0, if I hold onto it.
But then, this term would
account for the second
derivative, the acceleration.
And that would give me the right
answer, the right answer
to the next term, but now I've
drawn the famous three dots.
So three dots is the way to
say there are more terms
because the acceleration
might not be constant.
What's the next term?
If you know the next
term, then you
and Taylor are square.
The next term will be
1/3 factorial, 1/6.
It'll be a third derivative of
f at the known point times
this x minus a cubed.
You see that these terms
are getting,
typically, for a nice function--
and we saw this for
e to the x.
We saw the Taylor Series
for e to the x.
Can I remind you of the Taylor
Series for e to the x around
the point 0 because e to the x
is the greatest function I've
spoken about, at all?
So, if this was e to the x, it
would start out at e to the 0,
which is 1.
Its slope is 1, so this
is 1 times x.
Its second derivative
is, again, 1.
And a is 0 here, so
this would be 1/2,
1/2 factorial x squared.
And then that next three-dot
term would be
1/3 factorial x cube.
And you remember what
it looks like.
So the Taylor Series just
looks messy because I'm
writing any old f.
I'm allowing it to be the start
point, to be a, and not
necessarily 0.
But, typically, it's 0.
And the e to the x series
is the best example.
But I want to show you
one more example.
That'll be my last theorem.
I just mention it here because
it's just like
the mean value theorem.
If I do stop, suppose I stop
here and I don't include the x
cube term, the third derivative
term, then I've
made an error.
And, of course, that error
depends on what the third
derivative is, the one I
skipped, the x minus a cube,
the thing I skipped, and
the 1/3 factorial.
And this third derivative
is, at some point,
between a and x.
That's a lot to put in, but the
mean value theorem said
you could take the derivative at
some point in-between, some
point along the MassPike.
And this is just the same thing,
but I'm keeping more
terms. I'm quitting at any
point, and then I would take
the next derivative at somewhere
along the MassPike.
What should you learn
out of that?
I think the idea is
Taylor Series.
And, of course, we have
two possibilities.
Either we cut the series off and
we make some error, but we
get a pretty good answer, or we
let the series go forever.
And then comes the question.
Then we have an infinite number
of terms, and then the
question is does that series add
up to a finite thing like
e to the x?
Or does it add up to a delta
function or something
impossible?
So that leads to the question
of learning
about infinite series.
In calculus, Taylor
Series is where
infinite series come from.
And, if we want to go all the
way with them, then we have to
begin to think about what does
it mean for that infinite
series to add up to a
number, or maybe it
just goes off to infinity.
Does it converge, or
does it diverge?
Ah, that would be another
lecture or two.
Let me complete today with one
more theorem, a famous one,
the binomial theorem.
So, what's the binomial
theorem about?
The binomial theorem is about
powers of 1 plus x.
1 plus x is a typical binomial:
two things, 1 and x.
And we have various powers.
Well, if the powers are the
first power, the second power,
the third power, we can write
out, we can square 1 plus x,
and we can get 1 plus x cubed.
And, out of it, we get this.
And there would be 1 plus
x to the 0-th power.
And do you see that there's a
whole lot of ones in the neat
pattern there?
And then there's a
2, and a 3, 3.
And if you'd like to know this
one, it would be 1, 4, 6, 4, 1
would be the next
row of Pascal.
Pascal really had a sense of
beauty or art in this triangle
of numbers.
And that's the triangle you
get, Pascal's triangle, if
you're taking--
A whole number, a power is 1
plus x to the third power,
fourth power, fifth power,
sixth power, but what if
you're taking to some other
power, any power, p?
So now I'm interested in this
guy to a power of p that,
maybe, is not two, three,
four, five.
It could be 1/2, 1 plus x square
root to the 1/2 power,
or 1 plus x to the
minus 1 power.
All other powers are possible
and, for those,
the Taylor's theorem.
And here's my function.
And I could apply Taylor's
theorem to find the--
and I'll do it at x equals 0,
that's the place Taylor liked
the best. So the
constant term--
think of this Taylor expansion
that we just did--
at x equals 0, this
thing is 1.
So, the big theory starts
out with a 1 for
the constant term.
Then what I do for the next
term of the Taylor Series?
I take the derivative and
I put x equals 0.
And what do I get then?
I get p times x.
So this is the constant
term: f of 0.
This is the derivative:
times x minus a
divided by 1 factorial.
Well, you didn't see all those
things because one factorial I
didn't write.
And then the next term would be
the next derivative, of p
minus 1 will come down, so
you'll have p, p minus 1.
You're supposed to divide
by 2 factorial.
That multiplies x squared.
Well, my point is just that
this binomial formula is
Taylor's formula.
The binomial theorem, with
these, this is called a
binomial coefficient.
Gamblers know all about
that, you know?
If you've got p things and you
want to take two, how many
ways to do it?
You know, how many ways to get
two aces out of a deck, all
these things are hidden in those
numbers, which gamblers
learn or lose.
OK.
So, I'll make one last point
about the binomial theorem.
Those were Taylor Series.
This is a Taylor series.
What's the difference?
The difference is these
series stop.
This is a series: 1
plus x squared.
That's the Taylor Series,
but the third
derivative is 0, right?
The third derivative of that
function, because that
function's only going up to x
squared, the third derivative
is 0, so the rest of Taylor
Series has died.
It's not there.
So that's all there is.
The derivative of any of those
powers, one, two, three, four,
five powers, after I take enough
derivatives, gone.
But, if I take a power like
minus 1, or 1/2, or pi, or
anything, then I can take
derivatives forever
without hitting 0.
In other words, this series
goes on, and on, and on.
Those three dots--
let me move that eraser so you
see those three dots--
that signals an infinite series
and the question of
does it add up to a finite
number, what's going on with
infinite series?
But, for the moment, my point
is just this is what
calculus can do.
If you not only take that slope,
but the slope of the
slope, and the third derivative,
and all higher
derivatives, that's what Taylor
Series tells you.
OK.
So that's the, in some way, high
point of the highlights
of calculus, and I sure hope
they're helpful to you.
Thank you.
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