Mathematics - Multivariable Calculus - Lecture 2

Welcome back to our little math theater.

So you guys doing all right?
Yeah.
So last time we started to talk about parameter curves.
And today we will continue with this topic and we will learn
various important things about parameter curves, such as
tangent lines, arc length and various areas of surfaces,
which are obtained from parameter curves.

So the first thing we will discuss today is tangent line.
And this is actually very easy to introduce.
This topic is very easy to introduce, because this is
something familiar from single variable calculus.
In single variable calculus we study graphs of
functions, right?
And so graph of a function is something which we
draw on a plane with two coordinates, x and y.
And it looks something like this.
We usually write y equals f of x.

Where f is a function in one variable.
This is a graph.
This is a graph of the function f of x.
So now, oftentimes in mathematics or in science and
engineering you want to know various approximate results
about your object.
Your object may be too complicated and you want to
get sort of the first order approximation as they say.
Or it might be that you want to understand it qualitatively.
For example, let's say if this is a graph of the temperature
in Berkeley, California where x is a time.
So let's say over the last week, you know.
Actually it would be more like this I suppose.
It was very hot last weekend, right?
Then maybe you don't want to know exactly what the graph
looks like, but you want to know for example, the trends.
Does the temperature increase?
Does it decrease?
Things like that.
And for this, it's very useful to find some approximate tools,
which would, you know, you'll be able to say a lot of things
about your object without really getting 100% of the
information about it.
And tangent lines is the first thing that you
can use for that.
And the reason is very simple.
The reason is that out of all the curves that you can draw,
out of all the curves-- so this is an example of a curve, but
of course you can draw many more, very complicated
examples.
I mean, circle and so on goes without saying, but even
kind of really wacky curves you can draw.
Anything you want to draw like a Picasso you know.
It's also curve oftentimes.
He oftentimes just drew it in one stroke.
So out of all of these curves, there's a whole variety of
those curves that are very complicated, but there
is a very simple class.
There's a class of the simplest ones and those are the lines.
The lines are the simple curves.
Lines are the simplest curves.

Equation of a line, lines can be graphs of functions and
those functions usually look like this.
It would be something like k times x minus x 0 plus y 0.

Where k is called the slope. k is called the slope and it's a
line with a slope k and also a line which passes through
the point x 0, y 0.
Let's draw it actually.

So let's say this is x 0, y 0 on the plane and the
line is going to look something like this.
So to draw a line what you need to know is a point through
which it passes and also the slope.
What do I mean by the slope?
The slope is the tangent of this angle.
So in other words, you need to know the angle between
this line and the x-axis.
That's this angle, theta.
Ant the tangent of this angle is called the slope and that's
what we call k in that formula, the slope of this line.
So this is the simplest graph and the simplest curve, really.
I mean, what could be simpler than this?
See the point is on the right-hand side you have a
function of x and the simplest function of x that you can
write is a constant function and then the next simplest
function is a constant function plus a linear function,
which is what we've written.
But the constant function could also be thought of
as a special case of this.
Namely, if k is equal to 0, slope 0, this will just
disappear and then you would have this.
So in other words the case of the constant function is
included because you're allowed to have arbitrary k.
So k equal 0 would give you the constant function.
So that would be just a special case of this
when y just equals y 0.
It's also a line, it's a horizontal line.
It has slope 0.
It's just a special case of this.
In some sense, there's no point of distinguishing this case
from this more general case of lines.
So that's why I'm saying that the simplest curve that you
can draw are the lines.
Because dependence on x, dependence of the function, f
of x, is the simplest possible.
It only contains a constant term and a linear term in x.
In other words, degree one.
It doesn't have x squared, it doesn't have x cubed, not to
mention, you know, logarithms, cosines and all those other
complicated functions.
That's the first fact which you have to remember.
That out of all the curves, there are the simplest ones
and these are the lines.
And we know the equations of the lines.
They're given by this formula.
And so the next thing, the next idea of calculus and really the
most important idea maybe of all calculus is that for many,
many functions, namely the so called smooth functions or
differentiable functions you can approximate your function
very well on a small scale by a line. or more precisely
approximate the graph of a function by a line.
Two ways to think about it.
A smooth function can be approximated by a linear
function like this.
Which geometrically means that the graph can be
very nicely and usefully approximated by a line.
And in practice, the way it works is that if we pick a
point on this curve -- and let's call it again, x 0 y 0.

Then we can think of a whole variety of lines, which
pass through this point.
There are many of them.
Infinitely many, in fact.
Infinitely many lines.
But out of all of those lines there will be one line which
will be the closest to this graph, and that's
the tangent line.

So I forgot to bring my colored chalk again, but
I hope you get the idea.
So this is a tangent line.
And what's so special about this is that it is the closest
one the graph of the function.
In the following sense, that if you move it just slightly it
will intersect the graph at two different points.
If we blow this up it's going to look like this.
And if I just blow up a very small neighborhood of this
point it's going to look like this.
If I change it just slightly like this it already intersects
at two different points.
So you adjust your line just so that it touches the
graph at one point.
That's intuitively what the tangent line is.

The closest one, the closest of all lines to this graph
at this particular point.
If you change your reference point of course you're going
to get a different one.
In other words, here I'm talking about the tangent to
this particular point, which is that point, x 0, y
0 on the big picture.
But if you go to a different place-- here, for example.
Of course, you will get an entirely different line.
So when you talk about tangent line you have to
say tangent at which point.
Otherwise it doesn't make any sense.
Each point on the graph has its own tangent line, and a
priori they're all different.
In almost all cases they will be different.
This is not to say that the line can replace the graph
of the funtion because you see they diverge.
If you go sufficiently far from this point
they become different.
You know, this expression going off on a tangent?
Sometimes you'll catch me doing that I suppose.
Anyone is capable of doing that.
And what it means precisely Is that if you go off on a
tangent, soon enough you'll be far away from the
object itself.
But the good news is that as long as you are in the very
small neighborhood of this point, the difference
is almost negligible.
This is a key idea of calculus really and of calculus of
single variable now because we're talking now about
functions in one variable.
But you will see that the same idea will be applied
very usefully also for multi-variable functions.
For example, if you have function two variables you'll
be approximately graphs by planes instead on lines and so
So I hope I convinced you of the importance of this because,
what does it give you?
For example, you see that the function is, you function is
increasing at this point if the slope is positive like this.
And the function is decreasing if the slope is negative.
So already you can learn a lot of things about your function
by studying the tangent line.
Now the next question is how to find the equation
of the tangent line.
We know that it's going to look like this because
all lines look like this.
More precisely all line, which are graphs of functions.
They all look like this.
So which one is it?
In other words, we have to find the coefficient k, and we have
to find this number y 0, and we have to find the number x 0.
That's what determines the line as we discussed.
We need x 0, y 0 and the slope.
But we already know x 0 and y 0, because that's
the reference point.
That's the point on the curve to which we look
for the tangent line.
So we already know x 0 and y 0.
And the question is, how to find the slope of
the tangent line?
k of the tangent line.

And the answer was given before in single variable calculus
and the answer is very nice.
Or one might say beautiful.
The answer is k is equal to the derivative of this function
at that point x 0.
In other words, you don't need to draw anything, you
don't need to make any complicated calculations.
All you need to know is the derivative of your function and
usually your function is described in a very explicit
way, like you know, say polynomial function, or cosine
or sine, or exponential function for which you know
what the derivatives look like because you've learned
them from you know, by calculating them once.
And then you make a list and you remember them.
So taking derivative with something, which we
know quite well.
All you need to know to find the slope is to
find the derivative.
Is to know how to find the derivative.
Once you know the derivative you know the slope and so the
you can put these things together and you get the
equation of the tangent line.
So the final result is the equation of the tangent line.
Is just obtained by combining all this information
into this formula.
So instead of k you will put f prime over x 0, which is what
I wrote here and then you have x minus x 0 plus y 0.
That's the equation of this tangent line.

No matter how complicated your function is the equation of the
tangent line is going to always look like this.
Of course, I am cheating a little bit because all of this
is applicable to functions for which the derivative exists in
the first place and not every function has a derivative.
This has a so called differentiable functions.
But the functions that we are going to study in this course
there are going to be differentiable and so
this method will apply.
You have to realize that this is something which works, in
some sense, for the nicest possible functions.
Namely differentiable functions or smooth
functions if you will.
The ones for which the graph is sort of a smooth curve
as opposed to a curve which has angles.
Which has sharp angles.
Because if you have a sharp angle it's not clear how
to make a tangent line.
It's not going to touch the graph because there
is a sharp corner.
So this case we don't consider.
We only consider the smooth ones, but the class of smooth
functions is very large.
And we are focusing in this course on the smooth functions,
so we are fine, our method is applicable here.
And once it is smooth it has a derivative and so you can wrute
easily the equation of this line.
That's what we've learned in single variable calculus.
Now in single variable calculus you study graphs of functions
on one variable, which are curves on the plane.
And last week we talked about more general curves.
We said OK, there are many curves that you can get as
graphs of functions, but not all.
There are more general curves, which are not graphs of
functions, and there are two ways to represent them.
One is by an equation, like x squared plus y squared equals
1, like a circle with radius 1.
Or in the parametric form.
That means that we have now a larger class of curves, which
includes but is not equal, is bigger than the
class of graphs.
Now we'd like to ask the same question about
this parameter curves.
In other words we want to learn how to compute the tangent the
line to such a curve at a given point.
So that's the question that we're going to ask.
In other words, now we have a parameter curve and the
parameter curve is given by a pair of functions, f of
t and g of t, as we discussed last time.
Where t is an auxillary variable.
The parameter on the curve.
We want to learn the equation, we want to find out what is the
tangent line to this curve at a given point, x 0, y 0.
Where of course, for this point x 0, y 0 to belong to this
curve both of those values z xero, and y 0 have to be values
of f and g at the given parameter values.
So that would have to be f of t 0 and that would have to be-- y
0 would have to g of t 0 for some t 0 value.

We'll look at examples in a bit, so you'll see
what I'm talking about.
So what is a tangent line to this curve?
In other words, we'd would like to extrapolate this formula.
We would like to generalize this formula.
Which by the way, is the way mathematics is done.
You know, you don't immediately get the answer in all cases.
You work out the simplest cast and then you try to generalize.
So if you look at this formula the answer may not be so
obvious because here this answer involves this function
f, which you know, because we're talking about the
graph of function, which is really the special case.
Special case of the general parameter curves in which the
current position is x equal t and y is equal to f of t.
This shows you right away how special this case is.
How special?
Well it's just that x is t and y is some function.
Because if you have this parameterization, than
the first equation tells you that x is just t.
So you can just substitute x instead of t and you get y
equals f of x, you get the graph of the function.
So in other words, in this special case, the function f, f
small is just the function t and the function g small is
another function, F capital.
Now we have a more general case, where both f small and g
small are some complicated functions, which are not
necessarily equal to t or anything given on the
[UNINTELLIGIBLE].
So we need to generalize this formula and it's not obvious
immediately what the answer should be because it really
appeals to this particular case, to this very special.
But it's actually not so hard to guess the answer.
To guess the answer we have to remember how we derived this
formula in the single variable calculus in the first place.
And actually for that I will use this picture.
So you see, what is the slope?
The slope is the ratio of delta y delta x.
The increment in y over increment in x.
That's because that's how-- let's look at
the graph of the line.
Let's just recall the definition of a tangent.
Because remember k, I said k is a tangent of the angle.
So what is tangent?
Tangent when you draw this triangle is the ratio of the
change in y in this triangle over the change in x.
So k is equal to delta y over delta x on this line.
So for this tangent line we have the same thing.
There's a delta y and a delta x.
So this is delta y over delta x on the tangent line.
But the point is that this is approximately, that this
increment in y, on a very small scale is almost equal to the
increment in y on the graph itself.
And the change in x of course is the same for both.
So you should look at this picture and see that even
though the tangent line that it goes off on a tangent.
They diverge a little bit, but not so much.
And the closer you are to the point, actually the
difference is less and less.
So in fact this ratio is almost the same as delta y over delta
x, but now on the graph itself.
So the slope can be computed as the ratio of the increment
of the function, delta y.
This one.
To the increment in x.
So in other words this is delta y on the tangent and this is
delta y on the graph and they're almost the same.
So the slope, if we're computing the slope we mine as
well take delta y of the graph and divide it by delta x.
And when this becomes very small, when delta has become
smaller and smaller so you are getting closer and closer to
the point, this becomes what we call the derivative, dy
over dx, which is f prime.
So that's the reason why you actually get the derivative
because what you get is dy over dx, and y is f of x, so dy
dx is F prime at your reference point x 0.
That's how you derive this formula.
In other words, this calculation is what
gives you this.
So now we can use the same formula because we have worked
out now the formula for the slope and we see that the
slope, k, is again dy over dx.
But remember y and x are given by this formula.
In other words, y is g of t and x is f of t, so we can use that
to write dx is f prime of t dt and dy is g prime of t dt.
What you get here is g prime of t dt divided
by f prime of t dt.
And now it's tempting to just cancel out these 2 dt's,
and actually you can.
You are allowed to do this, under some mild restrictions.

We're not going to get too much into details, I'm just giving
you an intuitive derivation of the formula.
But what I'm saying now can actually be made
rigorous and precise.
And it took centuries to really work out all the details and to
really explain what dt really means.
We'll talk more about this when we talk about differentials of
functions on two and three variables and you will see why
this kind of calculation is legitimate.
Because the way it is now dt is kind of a mysterious object
and people don't explain.
In the book it's not really explained what dt is and I'm
not going to explain it now.
I'll just explain it later because for now we just have to
take it for granted, the fact that we are allowed
to cancel them out.
The only condition which needs to be satisfied is
that f prime is not 0.
If f primes is not 0 for a good reason.
Because if f prime is 0 in this formula you are dividing by it
and you're not allowed to divide by 0.
So if f prime is not 0.
This formula makes sense as long as f prime is not 0.
And the formula again, reads just like this.

More precisely we have to say at which value of t, but
remember that's why I was careful here when we talked
about the question.
I said, what is the tangent line to this curve at the
given point, x 0, y 0.
And x 0, y 0 was the point which corresponded to a
particular value of the parameter, which I call t 0.
So to be absolutely precise, this is a formula for the slope
of that tangent line where both of the derivatives are
evaluated at t 0.
Any questions?
[UNINTELLIGIBLE PHRASE]
Why is it at t 0?
Because we are calculating the slope at the given point.
And the point on a parameter curve, the point is determined
by the value of the parameter.

I'm sort of writing all over the place, but here, this is
the answer to the question, which is written on
the opposite board.
Where t 0 is introduced.
So you have to have to look at both of these.

So let's see.
What do we want to do with this?
OK, let's do an example.
Here's an example.
So find the tangent line to the parameter curve given by these
equations at the point t equal 1.
So this means that t 0 is 1.
This infamous t 0 which appears in this formula
in this exercise is 1.
Because well, here'e written t equal 1.
My point is I'm trying to use notation in the following way.
That when I say t I kind of view it as an
independent variable.
It can take any values.
When I talk about a specific point then I want to say that
t is equal to some specific value.
In a general formula I don't want to say which value it is.
1,2,3, I don't want to say it so that's why I use
the notation t 0.
So t 0 just means a particular value of t as opposed
to a variable itself.
It's a subtle difference, but if it's lost on you don't
worry about it too much.

Find the equation of the tangent line. the equation of
the tangent line in the case of a graph of a function
is written here.
In the case of curves, parameter curves, the equation
is going to be y equals-- I take this-- the equation
is always like this.
And now k is equal to this.
So I just substitute this k into this formula.
So I get g prime of t 0 of f prime of t 0. x minus x 0 plus
y 0 where x 0 and y 0 are the values of the function to begin
with, so that's going to be f of t 0 and that's
going to be g of t 0.
So it's a very straightforward exercise because all you need
to do is just to calculate each of these numbers,
which show up here.
Let's first calculate x 0 and y 0.

x 0 is going to be the value of x when t is eqal to 1.
This is f.
This is g.
So what you need to do is you need to calculate the value of
this function first at t equal 1.
That's logarithm at 1 and logarithm at 1 is 0.
So now y 0 is you calculate 1 times e to the 1.
So e to the 1 is e.
And if you don't remember what e is, you should go
back and read Math 1B.
It's a particular constant, which is defined by the
property that the derivative of this function, e to the t at t
equals 0, is equal to 1.
It's the base of the natural logarithm.
So it's a particular number like 2.7-- it's like pi,
it's a very important universal constant.
So I on purpose chose this example because you know, in
the homework you will have to do deal with this.
Constants like e and things like that, so you have to
remember them and get used to them.
OK, so e is a particular number.
It's not a variable.
It's a particular number, which is equal to-- I don't remember
exactly, but I think something like 781, or whatever.
Maybe I shouldn't write it because if it's not correct
then I'll make a fool of myself.
OK, so we found x 0 and y 0.
That's great and next we need to find the slope.
So for that we need to find the derivatives of this.
So f prime of t is 1/t and g prime of t-- so again, if you
don't remember how to differentiate logarithm you
have to remember because this is something from single
variable calculus and we are going to use these
results freely.
We are using everything we've learned so far, which means
single variable calculus.
In particular, we have to know derivatives of
all these functions.
And all the rules of how to calculate
derivatives like this.
So g prime, you use the derivative of the product,
so that's going to be e to the t plus t e to the t.
I'm sorry, plus.
And now, if I want the value at at t 0, which is 1 I'm going
to get 1 for this one.
And I'm going to get 2.
And now I need to calculate the ratio between them.
So let me just write above.
So I have to take the ratio of g prime over f prime and g
prime is 2e and this is 1.
That's going to be 2 times 2e and then I have x minus x
0, which we found is 0.
So actually it's going to be 2e times x plus y 0
and y 0 we found to be e.
That's the answer.
That's the equation of the tangent line.
So next-- what else can we learn about this?
Amongst all the lines there are special ones.
There are ones which are vertical and the ones
which are horizontal.
So how to find out when it's vertical or horizontal.
You just look at the slope.
So the slope again, is a tangent.
So the slope is 0 if and only if the tangent
line is horizontal.

If the tangent line s horizontal.
This we learned in single variable calculus.
But we never talked about when the tangent line is vertical.
There's good reason for this because the tangent line for
a graph of a function is never going to be vertical.
You kind of have to think about it a little bit and then you'll
see that it's not possible.
So the tangent line can only be horizontal, but not vertical in
the case of a graph of a function, which is our
special case like this.
But in the most general case it surely can be
vertical or horizontal.
For one thing, you could switch x and y.
And when you switch x and y what I mean is switching
f of t and g of t.
We're allowed to do that because x and y are now on
completely equal footing.
And so, if you switch x and y a vertical becomes
horizontal and vice versa.
So clearly vertical tangent lines will be something
which will show up as well.
So the way to see that then is it's better to look at this
formula, at the more general formula, which we have
just found for the slope.
And so we see that if g prime of t 0 is 0 it means that
the tangent is horizontal.
Right?
I would like to say that, but the problem is I have to make
sure that this formula is valid and the formula is valid
if f prime is not 0.
So you have to have two conditions satisfied.
g prime is 0, but f prime of t 0 is not 0.
By this thing I mean the end, that both conditions
are satisfied.
Once again, g prime is 0 but f prime is not 0
than it's horizontal.

If you want to understand when it's vertical you
just switch x and y.
So when you switch x and y f starts playing the role of g
and g starts playing the role of f, so just without thinking,
just switch them and you will get the condition for the
vertical one, for the vertical tangent lines. f prime is
0, but g prime is not 0.
That's vertical.

So let's see some example.
When tangent lines are vertical or horizontal.
How to find out.
See the problem is if both of them are 0 you're kind
of dividing 0 over 0.
And that's really not well defined.
So it really depends on the situation.
It could be anything, so it really depends.
You have to study in more detail.
But if just one of them is 0, but the other one is not 0 then
you can say for sure that it's vertical or horizontal
depending on which one is 0 and which one is not 0.
So example two.

x is t times t squared minus 3 and y is 3t squared minus 3.
I'm sorry, 3 times t squared minus 3.
So let's compute.
So this is f, right.
Again, this is f of t and this is g of t.
So what is f prime?
Well we can write this as t cubed minus 3t.
So that's going to be 3t squared minus 3. g prime, this
is now 3t squared minus 9, so that's going to be what? 6t.
And to find out when it's vertical, when it's horizontal
we have to find the values of t for which one of these
two functions is 0.
So f prime of t equals z means that 3t squared minus 3 is 0.
Or in other words, 3t squared is equal to 3, which is the
same as t squared is equal to 1.
So there are two solutions: t equal 1 or t equal negative 1.
OK, so for those values f prime is 0, but to be able to say
conclusively whether the tangent is vertical we also
have to check the values of the other function or more
precisely, the derivative of the other function.
So we have to substitute these two values into
the other derivative.
So we get g prime of 1 is 6.
So no 0, good.
g prime of negative 1 is negative 6, not 0.
Good, well good in the sense that we caught a point or two
points in this case at which the tangent is vertical.
So what are these points?
Points where tangent line-- sometimes I'll just write the
word tangent to make it short, but it means tangent line.
It's the same thing.
The tangent line is vertical r.
The points -- it's pointing to the value t equal 1, which is
the point -- if we substitute t equal 1, we get what? 1 minus
3, which is negative 2.
So negative 2 and y, if you substitute 1 we get 1 minus
3 is negative 2, times 3 is negative 6.
And the second one when 2 is equal to negative 1.
So if it's negative 1 we get negative 1.
Negative 1 minus 3 times negative 1, which means plus 3
so that's 2 and here doesn't matter because we square
so it's going to be the same, 2 and negative 6.
So that's how you find.
And horizontal, for the horizontal one you have to do
the same, but with g prime.
Say g prime equals z, which means t equals 0.

And then you substitute here and you see that f prime is
negative 3, which is not 0.
So that means great.
At this point we also get horizontal tangent line.
And then you find that point in the same way.
I'm not going to do it.
I think it's clear.
Simply substitute in x and y, substitute the value t equal 0.
Any questions?
Yes.
[UNINTELLIGIBLE PHRASE]
That's a good question actually.
Let me give you one example kind of to show you that
it could be anything.
A very simple example.
Yes, I have repeat and also for our worldwide audience,
I hope we're being filmed.
So the question is give an example when both f prime
and g prime are equal to 0.
What does it mean geometrically?

So let's do this.
So I want to find two function, which at a particular value
of a point, a value of the parameter have 0 derivatives.

The simplest function which could have derivative
0 is t squared.
Not t because the derivative of t is 1.
So it's a constant.
It can not be 0. t squared already has derivative
equal to 2 times t and if t is 0 that's 0.
So let's say this is t squared.
And let's say that this one is also t squared.
So in other words, what I mean to say is that x is t squared
and y is equal to t squared.

So what does it look like?
Actually in this case x is equal to y.
So it looks like a line, right?
Which is kind of diagonal.
In other words, the slope, the angle here is 45 degrees.
The slope is 1.
Because it's a funny things.
You say this is t squared and this is t squared, which
means that x is equal to y.
It's almost like we are eliminating the parameter.
But there is a catch and the catch is we have to be careful.
What are the ranges of the variables?
Something which I mentioned at the end of last lecture.
Because the point is that for both positive and negative
values of t this is going to be positive or more
precisely nonnegative.
Could be 0 or positive.
And likewise here.
So that's why on purpose I didn't draw the entire
line, but only half of it.
So the image which represents this parameter curve
is this half a line.
And actually then you have to be careful.
What are the ranges?
I didn't say anything about the ranges.
If you take just the positive ranges, positive values
of t is going to be this.
And if you take negative values of t it's going
to be the same thing.
So actually, if you don't say anything about the range of t
and kind of implicitly say that it's from negative infinity to
positive infinity then it's going to be this curve twice.
You come from here and then you come back.
And if you say for example, t from 0 to infinity then you're
going to get just this half a line once.

So we see very clearly, graphically we see very clearly
what the object is, which makes it much more easy to analyze.
Now let's compute the derivatives.
So f prime of t is 2t and g prime of t is 2t.
So at t equals 0, so there is a point equal 0, which is here.
This a point equal 0.
So when t is equal to 0 both are 0.
And so the slope, as I said, you can not use this formula
for the slope because you're dividing 0 by 0.
But then of course the question, what is a
slope in this case?
Well the slope is 1.
It's sort of half a line so you may feel a little bit uneasy
because there's no other end, but you can still think about
the slope of this, right?
Of a tangent line.
In this case the tangent is going to be parallel, it's
going to coincide with the curve itself.
At least on this part and so the slope is going to be
just the slope of this curve, which is 1.
So you have sort of 0 by 0, but what happens is that
the derivative is 0 just at this point.
But outside of this point it's not 0, so you can approximate
the ratio of the derivatives by the ratio these two functions
and then take t to 0.
Because you see, what you get you see is 2t
over 2t and that's 1.
So in this particular case, even though you can not
apply the formula you know the answer's 1.
But to show you that you are sort of really on a slippery
slope-- no pun intended-- let's suppose you put
something like 5.
OK, 2.
Let's put 2.
So then you have this and so instead is going to be a more
sharper line, a steeper line.
So then y is going to be 2 times x.
This is y equal x and this is y equal 2 times x.
So for this one the slope is 2.
But the derivatives now are going to be 2t and 4t.
So you see both again are 0 and you can not use a
formula because 0 over 0 is undetermined.
And the point is that even though it's again, 0 by 0,
but now the answer is not 1.
But the answer is 2.
So that sort of illustrates that 0 over 0 could
actually be anything.
So in these two examples it's 1 or 2.
[INAUDIBLE]
That's right.

So to repeat, he's saying that it looks like we're just
applying the L'hopital's rule.
Which I hope you remember what L'hopital's rule is.
Which is to say that we're actually looking at
these two derivatives.
So now it's going to be 2t over 4t and we don't substitute t
equal 0 immediately in the numerator and the denominator,
but we look at this function for t very close to 0.
And we see, what is it?
Well if t is not 0 this makes perfect sense.
It's going to be 2/4, which is 1/2.
Except I'm taking the ratio in the opposite order.
Sorry.
So we have to do g prime over f prime.
Si it's 4/2.
And so 4/2, and that's 2.
The ratio itself is well-defined even though if we
substitute too quickly, too soon then it will be 0 over 0.
I don't want to go too off on a tangent here.
Sorry, I couldn't resist.
But I will let you play with other examples.
For example, try to do, say this square right here, but
here put t cubed or something like this, and see
what will happen.
It's really a very nice example to consider.
Let's go back to our curve.
So now we know about tangent lines, when they're vertical,
when they're horizontal.
What else do we need to know?
Well in the single variable calculus we also talked
about second derivatives.
So you see the point is that the first derivative--
It's good music.

But we're not here to listen to music so, should avoid this.
At least not during the lecture.
The first derivative is the slope.
It's dy over dx.
And it tells you the general direction of the function.
If the slope is positive it means that y is increasing
as x is increasing.
If the slope is negative it means that y is decreasing
as x is increasing.
And oftentimes we call this the first order approximation.
First order, now you can see why it's first order because
it's the first derivative.
It's the first order approximation.
And oftentimes it's also good to look at the second
order approximation.
In other words, to look at the second derivative.
And the second derivative is d squared y over dx squared.
Or if you will, d dx of dy dx.

And that's something which tells us -- if you think of
this as the velocity, this is acceleration.
It tells you the trend.
In other words, whether -- say this function is increasing.
But is it increasing faster as time goes by or is it
increasing more slowly?
And the way this can be seen geometrically is from the
concavity of the function.
If the function is like this it means that the second
derivative is positive.
We will call this concave upward.
That's what it's called in the book.
Actually I'm not sure.
Maybe I would call it concave downward.
But that's the terminology so we'll stick to it.
So this means concave upward.

And it means that this is greater than 0.
And if it's negative it's concave downward.
So it looks like this.
So it gives you more qualitative features
of the curve.
Even if you can not draw the curve right away, it tells
you by calculating these derivatives you will know the
ranges of parameters for which the curve looks like this
approximately or like this.
And there is a simple formula for this in terms of f ang g,
but I will let you read about it in the book.
It's very straightforward, so I don't want to
waste time on this.
So that's the other thing you need to know in addition to the
first derivative, which gives you the equation of
the tangent line.
You can also have the second derivative tell you about the
quality of behavior of the graph.
Kind of a second order approximation.
So what are we going to do next.
What we're going to do next is we're going to talk about other
features of parameter curves.
So far, we talked about differentiation.
So this information about the tangent lines has to
do with the derivatives.
And now we'll talk about integrals.
And integrals are not about the local behavior of the function
like the derivatives, which tell us about the behavior
of the graph on a very small scale.
But integrals are about the global behavior.
About averaging of the function.
In other words, about areas, various areas, which are
related to the graphs.
So here again, we use as a sort of a guiding principle the
material that we learned in singel variable calculus and
then we generalize it to the more general parameter curves.
Namely, we have to remember the formula about the area under
the graph of a function.
So again, I go back to the single variable situation.
I'm drawing graph of a function, f of x.
Suppose on the x line, on the x-axis I mark two points, a
and b and I look at the graph above it.
Let's assume that the graph in this range from a to b, that
the graph is entirely above the axis like shown.
And not below or not like going from upper half of the plane
to the lower half plane.
Let's assume for simplicity it's like this.
Then we can ask, what is the area, which is enclosed between
the x-axis, the graph and the vertical lines, which are
x equal a and x equal b.
And one of the triumphs, so to speak, of single variable
calculus was a formula for this, which I actually kind of
alluded to in my first lecture.
Which is that, this is what's called the integral.
So I'm going to say area under the graph is given
by the integral f of x dx.
This we can write as g of x from b to a.
Now let's just say g of b minus g of a, where g is
the anti-derivative of x.

Anti-derivative of f.
So in other words, finding areas involves integration.
And this formula shows that integration is really the
procedure which is inverse or opposite to differentiation.
Because to find the integral you have to not differentiate
this function.
But rather find a function whose derivative f is.
That's what we call anti-derivative.
So in other words, find g such that g prime is f.
Find the anti-derivative.
So that was a story.
And now we would like to generalize it because again,
we view now this graph as a special parameter curve.
How special the curve given by that parameterization.
Now we want to generalize it to the case of a parameter curve
for which the functions, f and g are arbitrary functions.
I mean, the small f and g.
Not the big ones, which I use here.
So the question becomes, suppose we are given the
parameter curve, so the same for a parameter curve.
Suppose you have a parameter curve now.
Well bad drawing because it looks exactly the
same as that one.
So let me make it-- that's the only one I can draw-- it's
just a natural impulse.
OK, and again, we pick some interval here for a to b.
And we again ask, what is this area?
So in other words, now again, f is f of t y is g of t and let's
say t is between some alpha and beta.
So that a-- this is x and this is y, so I called them a
and b, so what are they?
If t is from alpha to beta it means that a is f of
alpha and b is g of beta.
Somehow, a and b are not so important now.
What is important is the values of the parameter, t goes
from alpha to beta.
So now again, when this kind of question is asked you have to
be careful to make sure that the graph is indeed
above the x-axis.
And after I write down the formula we'll discuss briefly
about what happens if it's not the case.
Now the question is to generalize now this formula.
And it's actually very easy because another way to look at
this formula is to say that it's given by the formula--
there's a way to rewrite this formula, we just have to
remember that f of x is actually y.
So another way to think about this formula is to just say the
integral of y dx from a to b.
So now this makes sense even for parameter curves because
x and y still make sense.
So the area between let's just say, the area of this figure is
going to be equal to the integral again, of y dx.
But the problem is that x and y are given as functions
of variable t.
So at first it looks a little bit like a nuisance, but then
you have to remember that actually something we've
learned before when we studied the integrals we oftentimes saw
that it was beneficial to substitute a different
variable instead of x.
Oftentimes in single variable calculus to actually
technically evaluate the integral to find the
anti-derivative it was oftentimes useful to make
substitution and say that x
is some function of t of some other variable.
So
let's just call that other variable t and say that
x is equal to f of t.
And then we had a formula for this.
For calculating the integral in terms of this new variable t.
And what was this formula?
Actually it's very easy to write it if we just remember
how to compute the differential, dx.
So the main formula here is dx is equal to f prime of t dt.
That's what dx is.
[UNINTELLIGIBLE PHRASE]
Yes, you're right.
Thank.
You Very good.
I should develop some system of prizes for people who find--
extra points-- I'll think about this.
But you should be on the lookout because
I can make mistakes.
Sometimes on purpose and sometimes-- because no
one's perfect, you know.
In this case I just made a mistake, so thank
you for correcting me.
This is a formula we're going to use.
So dx is just f prime of t dt and y is g of t.
So substituting the variable t here simply means using the old
substitution formula, which gives us the integral
from alpha to beta.
g of t, f prime of t dt.

That's already a very nice formula, which now does not
involve x and y, but only these two functions, f and g.
So it becomes a very formula because just as soon as you
know what the g of t is and f of t is you can
calculate the area.
So it's the same formula, but just generalized to this more
general context of parameter curves.
Now let's talk about the subtle point, which is what to do if
in fact the picture's not like this, but it also involves
some part of that lower half plane, below the x-axis.
Because this happens.
So this is something we learned even in single
variable calculus.

And the point here is the point is that when you write this
formula if you really think of y dx, if it lies above the
x-axis it means that y is always positive.
So you actually get a positive answer, which is the area.
But if y is negative than you're going to get
a negative answer.
So this already suggests to you that for example, if you were
to consider this cased where actually it is below the axis,
what you're going to get is not this area, but this area
with negative sign.
So minus negative the area will be equal to this integral.
And I want to write it like this because it includes both
this case where you can just write y equals f of x and also
includes this case where you simply substitute this.
So in other words, area is equal to negative of this.
The integrals is negative and area is always positive.
So to extract the area out of the integral you have to
put an extra negative sign.
So the integral is going to be negative to begin with, you put
another negative sign, the answer will be positive.
So if the graph is entirely below the x-axis, you
just put a negative sign.
So then of course, there's a mixed case where it
could go like this.
So in this case let's call this area, a1 and let's
call this area, a2.
So what you are going to get is the difference between the two.
In other words, the part which lies above the axis is going
to contribute with a positive sign.
And the part which lies below the x-axis will
contribute negative sign.
So a1 minus a2 will be the integral of y dx.
So whenever you fall below the x-axis you're going
to get negative things.
So you don't get the area, you get negative
area minus the area.
And that's why you get this formula.
So that's the most general formula.
And the same thing will be true here.
It's not going to be the area, but rather a1 minus a2 in
general, where a1 and a2 mean the same thing as here. a1 is
the part which is above the axis and a2 is part
below the axis.
Is this clear.
Yes?
[UNINTELLIGIBLE PHRASE]
The graph doubles on itself, OK
[UNINTELLIGIBLE PHRASE]
OK, that's like a doomsday scenario.
So the question is, and this is not the worst
case, you're right.
There is a worse situation when it goes like this.
And see, before when we studied graphs of functions
this could not happen.
Could not possibly have happened because for each
value of x we would have just one value of y.
But now because we're doing parameter curves
anything is possible.

So in this case what's going to happen is-- here
there's a different issue.
In my formula, which I wrote here, I said t
goes from alpha to beta.
So I put the limits like this and I'm assuming that
alpha is less than beta.
So smaller value of t corresponds to smaller
value f x and bigger value of t corresponds
to bigger value of x.
So then you get this formula.
What could happen is that in this picture the curve is
traverse like this, but if the curve were traversing like this
you would get the integral in which the lower limit will be
bigger than the higher limit and we make sense of that by
saying that you switch the limits, but you
put a minus sign.
So that's another way by which you could introduce a negative
sign in this integral.
Namely, in the case when the left end point cprresponds
to a larger value of t.
So if alpha is less and alpha is less than beta, so alpha
goes to a, which is less than b.
Beta goes to b.
But the other possibilities could be that you could have
exactly the same picture, but now I could change
the parameterization.
For example, I could just substitute.
Instead of t I put negative t.
So that what was before, say a would corresponds to value of
alpha, which is bigger than value of beta.

So this is x, this is t.
So then you will end up with an integral where-- this formula
would still be correct, will be integral from alpha to beta,
but when you start calculating it you will have integral let's
say not from 0 to 1, but from one to 0.
And here you will have g of t and then f prime of t.
So the rule is that this is the same as the integral from 0
to 1, but with negative sign.
So you have to be careful that first of all, the thing is
above the axis or below the axis.
This kind of stuff, which I talked about before.
The second subtlety is that you have to be careful as to what
direction the curve is going with respect to the
parameterization.
When t is increasing are you going from left the right or
you going from right to left?
Now to fo back to this, the interesting thing that
happens here is let's say it goes like this.
So then on this segment it will go from left to right, but then
from this segment will go from right to left.
Then again, we'll go from left to right.
The fact that actually there are three different parts
is not so important.
Because the point is that in setting up the integral we will
be using not a and b for general parameter curves,
but alpha and beta.
In other words, we will have to specify from the beginning
which branch, out of this three branches we are talking about.
Are we talking about the area under this one or the area
under this one or the area under this one?
Because the formula really explicitly involves the end
points, alpha and beta with respect to t.
Not with respect to a and b.
So usually we will just pick a particular branch and we'll
just say-- in other words, we will be saying that this
is t equal alpha and this t equal beta.
And this segment will correspond to some other alpha
beta and this will corresponds to some other alpha beta.
Of course, in principle you could say what is the meaning
of the integral when t will go from this value to this value?
And this actually is very easy to figure out.
But at this point, we are kind of losing the
geometric meaning of this.
So we have a clean geometric meaning when we're talking
about the branch, which doesn't double on itself like you said.
But you have a single branch over the segment in the x line.
In principle you could also give interpretations to the
more general integrals.
But you kind of lose the interpretation.
So will not do that.
We will consider the ones which have a more clear
meaning like this.
Does that answer your question?
OK.
Any other questions?
So that's the integrals.
And the last thing I want to talk about is arc lengths.
So the other thing which is very interesting is to find
a length of a segment of a parameter curve.
So less than 10 minutes and then you'll be free to go.
The finish line and then we'll be done.
All of this stuff is really not so difficult.
If you see, in each case there is a formula.
So what I'm trying to do here today is to kind of give you
intuitive understanding of the formula.
Kind of introduce the formula for you and
explain why it's true.
But once you know the formula you basically
just have to substitute.
If you look at the homework exercise most of them is
just about substituting.
There are a few subtle points.
And one of the subtle points is the way, when you do the
integral, what exactly are you calculating?
So there are subtle points about the signs like the ones,
which I mentioned here.
But other than that it's fairly straightforward.
And likewise, what is also straightforward
is the arc length.

So the question really is, what is the length?
Say you have a parameter curve, what is the length of the part
of this curve between these two points?
Now of course you could say, what do I mean by the length?
And for this you can just think of the same analogy which I
explained last time, which is think of your curve
as this wide.
So the curve is curvy when you look at on the plane.
In other words, when I make it like this it's curvy.
But at the end of the day I can just take it and stretch it out
and measure it, so that's the length.
And of course, the point is when I stretch it out it
should not be stretchable.
You know, it should be sturdy because otherwise I can stretch
it as far as I want, right?
When I say I stretch it I mean, just kind of making it into
straight line segment.
But I should assume that I'm not displacing anything.

It's length is something which is well-defined, it doesn't
depend on me the way I hold it.
So that's exactly this.
Just kind of take it and put like this and then of course
it's clear what its length is.
You could measure it.
That's what we mean by arc length.
So in other words, even though it is curvy there's
a notion of a length.
Even for curved objects.
And of course a great example of this is that
arc length of a circle.
A circle is curvy, but we know what the length is, what it it?
There seems to be disagreement on that.
2 pi.
2 pi, that's right.
So the length of the circle is 2 pi and that's how we-- sorry?
[UNINTELLIGIBLE]
Circumference. 2 pi r, that's right.
I'm thinking of a circle of radius 1.
But if it has a radius-- OK.
I think now we are in agreement.
Circle of radius r, the lengths will be 2 pi r.
And in fact there's a very important constant showing
up, which is called pi.
We'll talk about it.
Have you seen the movie Pi, by the way?
You should see it.
It's cool.
It's one of those movies with a crazy mathematician, you know.
But it's a cool movie.

So, settle down.
Now pi is a universal constant which actually is defined by
this property that 2 pi r is the circumference of a
circle of radius r.
So this is just a good example to show you that even
though the circle is curvy, there is some length.
It has a length.
And of course the way you measure this length is you can
think of a circle as being formed by a rope, which you
kind of open up and stretch and OK you can measure it and you
can see that it's 2 pi r.
But even for more general parameter curves we have a
notion of the length and we would like to calculate it by
using these functions f and g.
And there is actually a very nice formula which involves
integration for this.
The point is that the formula is also very easy to derive.
And the reason why it's easy to derive is because we model
everything on straight lines.
So today when I started the lectures I said, the simplest
curves are lines and to really understand various
characteristics of complicated curves we have to understand
them first for lines and then we kind of extrapolate--
generalize to the more general curves.
To the most general parameter curves.
So if our curve were in fact a line or a line segment we would
be able to find the length very easily by using the
Pythagoras Theorem.

So this would be delta x and this would be delta y.
The length would be the square root of delta x squared
plus delta y squared.
So now we have a more complicated curve here, but as
we discussed earlier, no matter how complicated a curve is, if
it is smooth or differentiable it can actually be approximated
on a very small scale.
It can always be approximated by a line.
So small scale.
So what that means is that we should-- OK, if we just try to
approximate the whole thing by a line it's not a very
good approximation.
But we could break it into segments like this and now each
of those segments-- each of the small segments actually does
look like an interval, a segment of a line.
And so for each of them you can have delta x and delta y.

So it will look something like.
Or maybe on this side it will look like this.
So this is just one of the segments, which I blew up.
I zoom on it.
So I have delta x and delta y.
So then I have the lengths of this line segment, which is
now going to be very close to the actual length.
So I'm going to approximate the entire length by the sum of the
lengths of those line segments.
Just kind of like a snake like picture.
On each of the small segments I have an interval.
So at the end of the day what I'm going to get is a sum, is
the integral, which looks like this.
So it's going to be kind of like delta x squared again,
delta y squared, but on the small-- in the limit when those
little pieces become smaller and smaller it's going to be
dx squared plus dy squared.
And then what I'm going to do I'm going to put this.
And this will be from alpha to beta.
Then we can take this under the integral.
What the result is going to be is I'm going to have d squared,
plus dy dt squared, dt.

That's the answer.
That's the formula for the arc length of the curve between
the points alpha and beta. t equal alpha and t equal beta.
And I have now given you a kind of very informal intuitive
understanding of why this formula holds.
It essentially comes from the Pythagoras Theorem.
[UNINTELLIGIBLE]
So this was just a trick.
I wanted to rewrite this in a nicer way.
So I just put this and I put this other.
So we're out of time.
So see you on Thursday.