Uploaded by UCBerkeley on 17.11.2009

Transcript:

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.

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.