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PROF.
 

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JERISON: So, we're ready to
begin Lecture 10, and what

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I'm going to begin with
is by finishing up some

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things from last time.
 

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We'll talk about
approximations, and I want to

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fill in a number of comments
and get you a little bit more

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oriented in the point of view
that I'm trying to express

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about approximations.
 

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So, first of all, I want to
remind you of the actual

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applied example that I
wrote down last time.

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So that was this business here.
 

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There was something from
special relativity.

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And the approximation that
we used was the linear

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approximation, with a -
1/2 power that comes

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out to be t( 1
 

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1/2 v^2 / C^2).
 

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I want to reiterate why
this is a useful way

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of thinking of things.
 

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And why this is that this
comes up in real life.

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Why this is maybe more
important than everything

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that I've taught you
about technically so far.

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So, first of all, what this is
telling us is the change in t /

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t, if you do the arithmetic
here and subtract t that's

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using the change in
t is t' - t here.

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If you work that out, this
is approximately the

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same as 1/2 (v^2 / C^2).
 

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So what is this saying?
 

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This is saying that if you have
this satellite, which is going

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at speed v, and little c is the
speed of light, then the change

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in the watch down here on
earth, relative to the time on

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the satellite, is going to
be proportional to

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this ratio here.
 

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So, physically,
this makes sense.

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This is time divided by time.
 

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And this is velocity squared
divided by velocity squared.

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So, in each case, the
units divide out.

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So this is a
dimensionless quantity.

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And this is a
dimensionless quantity.

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And the only point here that
we're trying to make is just

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this notion of proportionality.
 

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So I want to write this down.
 

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Just, in summary.
 

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So the error fraction, if you
like, which is sort of the

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number of significant digits
that we have in our

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measurement, is proportional,
in this case, to this quantity.

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It happens to be proportional
to this quantity here.

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And the factor is,
happens to be, 1/2.

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So these proportionality
factors are what

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we're looking for.
 

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Their rates of change.
 

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Their rates of change of
something with respect

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to something else.
 

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Now, on your homework, you
have something rather

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similar to this.
 

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So in Problem, on Part 2b, Part
II, Problem 1, there's the

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speed of a pitch, right?
 

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And the speed of the pitch
is changing depending on

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how high the mound is.
 

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And the point here is that
that's approximately

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proportional to the change
in the height of the mound.

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In that problem, we had this
delta h, that was the x

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variable in that problem.
 

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And what you're trying to
figure out is what the constant

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of proportionality is.
 

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That's what you're aiming
for in this problem.

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So there's a linear
relationship, approximately,

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to all intents and purposes
this is an equality.

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Because the lower order
terms are unimportant

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for the problem.
 

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Just as over here, this
function is a little

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bit complicated.
 

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This function is a
little more simple.

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For the purposes of this
problem, they are the same.

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Because the errors are
negligible for the particular

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problem that we're working on.
 

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So we might as well work with
the simpler relationship.

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And similarly, over here, so
you could do this with, in

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this case with square roots,
it's not so hard here with

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reciprocals of square roots.
 

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It's also not terribly hard
to do it numerically.

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And the reason why we're not
doing it numerically is

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that, as I say, this is
something that happens

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all across engineering.
 

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People are looking for these
linear relationships between

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the change in some input and
the change in the output.

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And if you don't make these
simplifications, then when you

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get, say, a dozen of them
together, you can't figure

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out what's going on.
 

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In this case the design of the
satellite, it's very important.

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The speed actually
isn't just one speed.

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Because it's the relative
speed of u to the satellite.

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And you might be, it depends on
your angle of sight with the

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satellite what the speed is.
 

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So it varies quite a bit.
 

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So you really need
this rule of thumb.

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Then there are all kinds
of other considerations

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in this question.
 

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Like, for example, there's the
fact that we're sitting on

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Earth and so we're rotating
around on what's called

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a non-inertial frame.
 

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So there's the question
of that acceleration.

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There's the question that the
gravity that I experience here

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on Earth is not the same
as up at the satellite.

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And that also creates a
difference in time, as a

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result of general relativity.
 

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So all of these considerations
come down to formulas which

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are this complicated or
maybe a tiny bit more.

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Not really that much.
 

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And then people simplify them
enormously to these very

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simple-minded rules.
 

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And they don't keep track
of what's going on.

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So in order to design the
system, you must make these

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simplifications, otherwise
you can't even think

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about what's going on.
 

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This comes up in everything.
 

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In weather forecasting,
economic forecasting.

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Figuring out whether there's
going to be an asteroid that's

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going to hit the Earth.
 

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Every single one of these
things involves dozens of

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these considerations.
 

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OK, there was a question
that I saw, here.

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Yes.
 

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STUDENT: [INAUDIBLE]
 

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PROF.
 

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JERISON: Yeah.
 

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Basically, any problem where
you have a derivative, the rate

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of change also depends upon
what the base point is.

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That's the question.
 

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You're saying, doesn't this
delta v also depend, I had a

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base point in that problem.
 

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I happened to decide that
pitchers pitch on average

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about 90 miles an hour.
 

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Whereas, in fact, some pitchers
pitch at 100 miles an hour,

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some pitch at 80 miles an hour,
and of course they vary

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the speed of the pitch.
 

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And so, this varies
a little bit.

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In fact, that's sort of
a second order effect.

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It does change the constant
of proportionality.

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It's a rate of change at
a different base point.

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Which we're considering fixed.
 

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In fact, that's sort of
a second order effect.

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When you actually do the
computations, what you discover

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is that it doesn't make
that much difference.

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To the a.
 

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And that's something that
you get from experience.

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That it turns out, which things
matter and which things don't.

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And yet again, that's exactly
the same sort of consideration

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but at the next order of what
I'm talking about here is.

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You have to have enough
experience with numbers to know

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that if you take, if you vary
something a little bit it's not

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going to change the answer that
you're looking for very much.

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And that's exactly the
point that I'm making.

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So I can't make them all
at once, all such points.

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So that's my pitch for
understanding things from

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this point of view.
 

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Now, we're going to go on, now,
to quadratic approximations,

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which are a little
more complicated.

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So, we talked a little
bit about this last time

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but I didn't finish.
 

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So I want to finish this up.
 

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And the first thing that
I should say is that you

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use the when the linear
approximation is not enough.

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OK, so, that's something that
you really need to get a

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little experience with.
 

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In economics, I told you
they use logarithms.

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So sometimes they use
log linear functions.

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Sometimes they use log
quadratic functions when the

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log linear ones don't work.
 

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So most modeling in
economics is with log

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quadratic functions.
 

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And if you've made it any
more complicated than

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that, it's useless.
 

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And it's a mess.
 

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And people don't do it.
 

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So they stick with the
quadratic ones, typically.

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So the basic formula here, and
I'm going to take the base

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point to be 0, is that f ( x
) is approximately f ( 0 )

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f' ( 0 )x.
 

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That's the linear part.
 

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Plus this extra term.
 

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Which is f'' ( 0 ) / 2x^2.
 

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And this is supposed
to work for x near 0.

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So it shows in the base point
as simply as possible.

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So here's more or less where
we left off last time.

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And one thing that I said I was
going to explain, which I will

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now, is why it's
(1/2) f'' ( 0 ).

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So we need to know that.
 

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So let's work that out
here first of all.

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So I'm just going to
do it by example.

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So if you like, the answer
is just, well, what happens

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00:10:00 --> 00:10:03
when you have a parabola?
 

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A parabola's a quadratic.
 

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00:10:06 --> 00:10:09
It had better, its quadratic
approximation had

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better be itself.
 

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It's got to be the best one.
 

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So it's got to be itself.
 

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So this formula, if it's
going to work, has

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00:10:15 --> 00:10:19
to work on the nose.
 

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00:10:19 --> 00:10:21
For quadratic functions.
 

209
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So, let's take a look.
 

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00:10:23 --> 00:10:26
If I differentiate, I get b
 

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2cx.
 

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If I differentiate a
second time, I get 2c.

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And now let's plug it in.
 

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Well, we can recover, what is
it that we want to recover?

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00:10:39 --> 00:10:42
We want to recover these
numbers a, b and c using the

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00:10:42 --> 00:10:45
derivatives evaluated at 0.
 

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00:10:45 --> 00:10:49
So let's see.
 

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It's pretty easy,
actually. f ( 0 ) = a.

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00:10:52 --> 00:10:53
That's on the nose.
 

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If you plug in x = 0
here, these terms drop

221
00:10:56 --> 00:10:58
out and you get a.
 

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00:10:58 --> 00:11:02
And now, f' ( 0 ),
whoops that was wrong.

223
00:11:02 --> 00:11:05
So I wrote f' but
what I meant was f.

224
00:11:05 --> 00:11:07
So f ( 0 ) is a.
 

225
00:11:07 --> 00:11:13
Let's back up. f ( 0 ) is a, so
if I plug in x = 0 I get a.

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00:11:13 --> 00:11:18
Now, f' ( 0 ), that's this next
formula here, f' ( 0 ), I

227
00:11:18 --> 00:11:21
plug in 0 here, and I get b.
 

228
00:11:21 --> 00:11:22
That's also good.
 

229
00:11:22 --> 00:11:24
And that's exactly what
the linear approximation

230
00:11:24 --> 00:11:25
is supposed to be.
 

231
00:11:25 --> 00:11:28
But now you notice, f'' is 2c.
 

232
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So to recover c, I
better take half of it.

233
00:11:34 --> 00:11:35
And that's it.
 

234
00:11:35 --> 00:11:38
That's the reason.
 

235
00:11:38 --> 00:11:40
There's no chance that any
other formula could work.

236
00:11:40 --> 00:11:45
And this one does.
 

237
00:11:45 --> 00:11:50
So that's the explanation
for the formula.

238
00:11:50 --> 00:11:54
And now I remind you that I had
a collection of basic formulas

239
00:11:54 --> 00:11:55
written on the board.
 

240
00:11:55 --> 00:12:00
And I want to just make sure
we know all of them again.

241
00:12:00 --> 00:12:07
So, first of all, there was
sine x is approximately x.

242
00:12:07 --> 00:12:10
Cosine x is approximately
1 - 1/2 x^2.

243
00:12:12 --> 00:12:15
And e ^ x is approximately 1
 

244
00:12:15 --> 00:12:17
x
 

245
00:12:17 --> 00:12:17
1/2 x^2.
 

246
00:12:19 --> 00:12:22
So those were three that
I mentioned last time.

247
00:12:22 --> 00:12:28
And, again, this is
all for x near 0.

248
00:12:28 --> 00:12:31
All for x near 0 only.
 

249
00:12:31 --> 00:12:35
These are wildly wrong, Far
away, but near 0 they're nice,

250
00:12:35 --> 00:12:37
good, quadratic approximations.
 

251
00:12:37 --> 00:12:39
Now, the other two
approximations that I want to

252
00:12:39 --> 00:12:46
mention are the logarithm and
we use the base point shifted.

253
00:12:46 --> 00:12:50
So we can put it at x near 0.
 

254
00:12:50 --> 00:12:52
And this one - sorry,
this is an approximately

255
00:12:52 --> 00:12:55
equals sign there.
 

256
00:12:55 --> 00:12:57
Turns out to be x - 1/2 x^2.
 

257
00:12:59 --> 00:13:02
And the last one is one (1
 

258
00:13:02 --> 00:13:07
x) ^ r, which turns
out to be 1

259
00:13:07 --> 00:13:08
rx
 

260
00:13:09 --> 00:13:12
r ( r - 1) / 2x^2.
 

261
00:13:14 --> 00:13:20
Now, eventually, your mind will
converge on all of these and

262
00:13:20 --> 00:13:23
you'll find them relatively
easy to memorize.

263
00:13:23 --> 00:13:25
But it'll take some
getting used to.

264
00:13:25 --> 00:13:30
And I'm not claiming that you
should recognize them and

265
00:13:30 --> 00:13:32
understand them all now.
 

266
00:13:32 --> 00:13:35
But I'm going to put a
giant box around this.

267
00:13:35 --> 00:13:42
STUDENT: [INAUDIBLE]
 

268
00:13:42 --> 00:13:42
PROF.
 

269
00:13:42 --> 00:13:42
JERISON: Yes.
 

270
00:13:42 --> 00:13:45
So the question was, you get
all of these if you use

271
00:13:45 --> 00:13:45
that equation there.
 

272
00:13:45 --> 00:13:48
That's exactly what are you
going to do. so I already

273
00:13:48 --> 00:13:52
did it actually for
these three, last time.

274
00:13:52 --> 00:13:55
But I didn't do it
yet for these two.

275
00:13:55 --> 00:13:58
But I will do it in
about two minutes.

276
00:13:58 --> 00:14:00
Well, maybe five minutes.
 

277
00:14:00 --> 00:14:10
But first I want to explain
just a few things about these.

278
00:14:10 --> 00:14:12
They all follow from
the basic formula.

279
00:14:12 --> 00:14:16
In fact, that one deserves a
pink box too, doesn't it.

280
00:14:16 --> 00:14:18
That one's pretty important.
 

281
00:14:18 --> 00:14:19
Alright.
 

282
00:14:19 --> 00:14:23
Yeah.
 

283
00:14:23 --> 00:14:25
Maybe even some
little sparkles.

284
00:14:25 --> 00:14:32
Alright.
 

285
00:14:32 --> 00:14:33
OK.
 

286
00:14:33 --> 00:14:36
So that's pretty important.
 

287
00:14:36 --> 00:14:39
Almost as important as the
more basic one without

288
00:14:39 --> 00:14:41
this term here.
 

289
00:14:41 --> 00:14:47
So now, let me just tell you
a little bit more about

290
00:14:47 --> 00:14:54
the significance.
 

291
00:14:54 --> 00:14:56
Again, this is just to
reinforce something that

292
00:14:56 --> 00:14:57
we've already done.
 

293
00:14:57 --> 00:14:59
But it's closely related
to what you're doing

294
00:14:59 --> 00:15:00
on your problem set.
 

295
00:15:00 --> 00:15:05
So it's worth your
while to recall this.

296
00:15:05 --> 00:15:10
So, there's this expression
that we were dealing with.

297
00:15:10 --> 00:15:13
And we talked about
it in lecture.

298
00:15:13 --> 00:15:19
And we showed that this tends
to e as k goes to infinity.

299
00:15:19 --> 00:15:21
So that's what we
showed in lecture.

300
00:15:21 --> 00:15:25
And the way that we did that
was, we took the logarithm

301
00:15:25 --> 00:15:29
and we wrote it as k
times, sorry, the ln of 1

302
00:15:29 --> 00:15:32
(1 / k).
 

303
00:15:32 --> 00:15:35
And then we evaluated
the limit of this.

304
00:15:35 --> 00:15:38
And I want to do this
limit again, using

305
00:15:38 --> 00:15:40
linear approximation.
 

306
00:15:40 --> 00:15:41
To show you how easy it is.
 

307
00:15:41 --> 00:15:44
If you just remember the
linear approximation.

308
00:15:44 --> 00:15:46
And then we'll explain
where the quadratic

309
00:15:46 --> 00:15:48
approximation comes in.
 

310
00:15:48 --> 00:15:52
So I claim that this is
approximately equal

311
00:15:52 --> 00:15:59
to k ( 1 / k).
 

312
00:15:59 --> 00:16:00
Now, why is that?
 

313
00:16:00 --> 00:16:04
Well, that's just this
linear approximation.

314
00:16:04 --> 00:16:05
So what did I use here?
 

315
00:16:05 --> 00:16:07
I used ln of 1
 

316
00:16:07 --> 00:16:10
x is approximately x.
 

317
00:16:10 --> 00:16:14
For x = 1 / k.
 

318
00:16:14 --> 00:16:18
That's what I used in
this approximation here.

319
00:16:18 --> 00:16:20
And that's the linear
approximation to the

320
00:16:20 --> 00:16:24
natural logarithm.
 

321
00:16:24 --> 00:16:27
And this number is relatively
easy to evaluate.

322
00:16:27 --> 00:16:28
I know how to do it.
 

323
00:16:28 --> 00:16:31
It's equal to 1.
 

324
00:16:31 --> 00:16:34
That's the same, well, so
where does this work?

325
00:16:34 --> 00:16:37
This works where this
thing is near 0.

326
00:16:37 --> 00:16:41
Which is when k is
going to infinity.

327
00:16:41 --> 00:16:44
This thing is working only
when k is going to infinity.

328
00:16:44 --> 00:16:46
So what it's really saying,
this approximation formula,

329
00:16:46 --> 00:16:50
it's really saying that as
we go to infinity, in k,

330
00:16:50 --> 00:16:54
this thing is going to 1.
 

331
00:16:54 --> 00:16:58
As k goes to infinity.
 

332
00:16:58 --> 00:17:00
So that's what it's saying.
 

333
00:17:00 --> 00:17:01
That's the substance there.
 

334
00:17:01 --> 00:17:05
And that's how we want to
use it, in many instances.

335
00:17:05 --> 00:17:06
Just to evaluate limits.
 

336
00:17:06 --> 00:17:09
We also want to realize
that it's nearby when k

337
00:17:09 --> 00:17:12
is pretty large, like 100
or something like that.

338
00:17:12 --> 00:17:16
Now, so that's the idea of
the linear approximation.

339
00:17:16 --> 00:17:21
Now, if you want to get the
rate of convergence here,

340
00:17:21 --> 00:17:27
so the rate of what's
called convergence.

341
00:17:27 --> 00:17:34
So convergence means how fast
this is going towards that.

342
00:17:34 --> 00:17:36
What I have to do is
take the difference.

343
00:17:36 --> 00:17:40
I have to take ln ak, and I
have to subtract 1 from it.

344
00:17:40 --> 00:17:43
And I know that this is going
to 0, and the question

345
00:17:43 --> 00:17:47
is how big is this.
 

346
00:17:47 --> 00:17:51
We want it to be very small.
 

347
00:17:51 --> 00:17:56
And the answer we're going to
get, so the answer just uses

348
00:17:56 --> 00:18:03
the quadratic approximation.
 

349
00:18:03 --> 00:18:06
So if I just have a little
bit more detail, then this

350
00:18:06 --> 00:18:10
expression here, in other
words, I have the next

351
00:18:10 --> 00:18:11
higher order term.
 

352
00:18:11 --> 00:18:13
This is like 1 / k,
this is like 1 / k^2.

353
00:18:15 --> 00:18:21
Then I can understand how big
the difference is between the

354
00:18:21 --> 00:18:24
expression that I've
got and its limit.

355
00:18:24 --> 00:18:26
And so that's what's
on your homework.

356
00:18:26 --> 00:18:31
This is on your problem set.
 

357
00:18:31 --> 00:18:35
OK, so that is more or less an
explanation for one of the

358
00:18:35 --> 00:18:38
things that quadratic
approximations are good for.

359
00:18:38 --> 00:18:42
And I'm going to give you
one more illustration.

360
00:18:42 --> 00:18:45
One more illustration.
 

361
00:18:45 --> 00:18:47
And then we'll actually
check these formulas.

362
00:18:47 --> 00:18:48
Yeah, another question.
 

363
00:18:48 --> 00:18:55
STUDENT: [INAUDIBLE]
 

364
00:18:55 --> 00:18:56
PROF.
 

365
00:18:56 --> 00:18:58
JERISON: That's a very
good question here.

366
00:18:58 --> 00:19:02
When they, which in this case
means maybe, me, when I give

367
00:19:02 --> 00:19:10
you a question, does one
specify whether you want

368
00:19:10 --> 00:19:14
to use a linear or a
quadratic approximation.

369
00:19:14 --> 00:19:18
The answer is, in real life
when you're faced with a

370
00:19:18 --> 00:19:23
problem like this, where some
satellite is orbiting and you

371
00:19:23 --> 00:19:25
want to know the effects of
gravity or something like

372
00:19:25 --> 00:19:28
that, nobody is going
to tell you anything.

373
00:19:28 --> 00:19:30
They're not even going to
tell you whether a linear

374
00:19:30 --> 00:19:33
approximation is relevant,
or a quadratic or anything.

375
00:19:33 --> 00:19:36
So you're on your own.
 

376
00:19:36 --> 00:19:40
When I give you a question,
at least for right now, I'm

377
00:19:40 --> 00:19:42
always going to tell you.
 

378
00:19:42 --> 00:19:47
But as time goes on I'd like
you to get used to when it's

379
00:19:47 --> 00:19:49
enough to get away with
a linear approximation.

380
00:19:49 --> 00:19:53
And you should only use a
quadratic approximation if

381
00:19:53 --> 00:19:55
somebody forces you to.
 

382
00:19:55 --> 00:19:57
You should always start
trying with a linear one.

383
00:19:57 --> 00:20:00
Because the quadratic ones are
much more complicated as you'll

384
00:20:00 --> 00:20:03
see in this next example.
 

385
00:20:03 --> 00:20:06
OK, so the example that I want
to use is, you're going to be

386
00:20:06 --> 00:20:09
stuck with it because I'm
asking for the quadratic.

387
00:20:09 --> 00:20:16
So we're going to find the
quadratic approximation

388
00:20:16 --> 00:20:23
near, for x near 0.
 

389
00:20:23 --> 00:20:24
To what?
 

390
00:20:24 --> 00:20:29
Well, this is the same
function that we used

391
00:20:29 --> 00:20:30
in the last lecture.
 

392
00:20:30 --> 00:20:34
I think this was
it. e ^ - 3x (1

393
00:20:34 --> 00:20:37
x) ^ - 1/2.
 

394
00:20:37 --> 00:20:40
OK.
 

395
00:20:40 --> 00:20:45
So, unfortunately, I stuck
it in the wrong place

396
00:20:45 --> 00:20:47
to be able to fit this
very long formula here.

397
00:20:47 --> 00:20:51
So I'm going to switch it.
 

398
00:20:51 --> 00:20:57
I'm just going to
write it here.

399
00:20:57 --> 00:21:00
And we're going to just
do the approximation.

400
00:21:00 --> 00:21:03
So we're going to say
quadratic, in parentheses.

401
00:21:03 --> 00:21:08
And we'll say x near 0.
 

402
00:21:08 --> 00:21:12
So that's what I want.
 

403
00:21:12 --> 00:21:15
So now, here's what
I have to do.

404
00:21:15 --> 00:21:18
Well, I have to write in the
quadratic approximation for e

405
00:21:18 --> 00:21:25
^ - 3x, and I'm going to use
this formula right here.

406
00:21:25 --> 00:21:27
And so that's (1
 

407
00:21:27 --> 00:21:29
(- 3x)
 

408
00:21:30 --> 00:21:33
(- 3x)^2 / 2).
 

409
00:21:33 --> 00:21:39
And the other factor, I'm going
to have to use this formula

410
00:21:39 --> 00:21:43
down here. because r is - 1/2.
 

411
00:21:43 --> 00:21:48
And so that's (1 - 1/2 x
 

412
00:21:48 --> 00:21:53
1/2 ( - 1/2)( - 3/2)x^2).
 

413
00:21:59 --> 00:22:09
So this is the r term, and
this is the r - 1 term.

414
00:22:09 --> 00:22:12
And now I'm going to do
something which is the

415
00:22:12 --> 00:22:15
only good thing about
quadratic approximations.

416
00:22:15 --> 00:22:18
They're messy, they're long,
there's nothing particularly

417
00:22:18 --> 00:22:19
good about them.
 

418
00:22:19 --> 00:22:21
But there is one good
thing about them.

419
00:22:21 --> 00:22:25
Which is that you always get to
ignore the higher order terms.

420
00:22:25 --> 00:22:29
So even though this looks like
a very ugly multiplication,

421
00:22:29 --> 00:22:31
I can do it in my head.
 

422
00:22:31 --> 00:22:33
Just watching it.
 

423
00:22:33 --> 00:22:38
Because I get a 1 * 1, I'm
forced with that term here.

424
00:22:38 --> 00:22:41
And then I get the cross
terms which are linear,

425
00:22:41 --> 00:22:44
which is - 3x - 1/2 x.
 

426
00:22:44 --> 00:22:46
We already did that when
we calculated the linear

427
00:22:46 --> 00:22:50
approximation, so that's
this times the 1 and

428
00:22:50 --> 00:22:51
this times that 1.
 

429
00:22:51 --> 00:22:54
And now I have three
cross-terms which

430
00:22:54 --> 00:22:55
are quadratic.
 

431
00:22:55 --> 00:22:58
So one of them is these two
linear terms are multiplying.

432
00:22:58 --> 00:23:00
So that's plus 3/2 x^2.
 

433
00:23:02 --> 00:23:05
That's (- 3)( - 1/2).
 

434
00:23:05 --> 00:23:07
And then there's this
term, multiplying the

435
00:23:07 --> 00:23:10
1, that's plus 9/2 x^2.
 

436
00:23:11 --> 00:23:15
And then there's one last term,
which is this monster here.

437
00:23:15 --> 00:23:23
Multiplying 1, and
that is - 3/8.

438
00:23:23 --> 00:23:31
So the great thing is, we drop
x^3, x ^ 4, etc., terms.

439
00:23:31 --> 00:23:37
Yeah?
 

440
00:23:37 --> 00:23:39
STUDENT: [INAUDIBLE]
 

441
00:23:39 --> 00:23:40
PROF.
 

442
00:23:40 --> 00:23:42
JERISON: OK, well
so copy it down.

443
00:23:42 --> 00:23:45
And you work it out
as I'm doing it now.

444
00:23:45 --> 00:23:47
So what I did is, I
multiplied 1 by 1.

445
00:23:47 --> 00:23:50
I'm using the
distributive law here.

446
00:23:50 --> 00:23:51
That was this one.
 

447
00:23:51 --> 00:23:55
I multiplied this 3x by this
one, that was that term.

448
00:23:55 --> 00:23:58
I multiplied this by
this, that's that term.

449
00:23:58 --> 00:24:01
And then I multiplied
this by this.

450
00:24:01 --> 00:24:03
In other words, 2 x terms
that gave me an x^2

451
00:24:03 --> 00:24:06
and a (- 3)( - 1/2).
 

452
00:24:06 --> 00:24:08
And I'm going to
stop at that point.

453
00:24:08 --> 00:24:10
Because the point is it's
just all the rest of

454
00:24:10 --> 00:24:12
the terms that come up.
 

455
00:24:12 --> 00:24:14
Now, the reason, the only
reason why it's easy is

456
00:24:14 --> 00:24:16
that I only have to go
up to x squared term.

457
00:24:16 --> 00:24:21
I don't have to do
the higher ones.

458
00:24:21 --> 00:24:22
Another question,
way back here.

459
00:24:22 --> 00:24:23
Yeah, right there.
 

460
00:24:23 --> 00:24:29
STUDENT: [INAUDIBLE]
 

461
00:24:29 --> 00:24:29
PROF.
 

462
00:24:29 --> 00:24:29
JERISON: OK.
 

463
00:24:29 --> 00:24:38
So somebody can check
my arithmetic, too.

464
00:24:38 --> 00:24:39
Good.
 

465
00:24:39 --> 00:24:40
STUDENT: [INAUDIBLE]
 

466
00:24:40 --> 00:24:40
PROF.
 

467
00:24:40 --> 00:24:43
JERISON: Why do I get to drop
all the higher-order terms.

468
00:24:43 --> 00:24:46
So, that's because the
situation where I'm going to

469
00:24:46 --> 00:24:52
apply this is the situation
in which x is, say, 1/100.

470
00:24:52 --> 00:24:55
So if here's about 1/100.
 

471
00:24:55 --> 00:24:57
Here's something which
is on the order of 100.

472
00:24:57 --> 00:24:58
This is on the
order of 1/100^2.

473
00:25:00 --> 00:25:02
1/100^2, all of these terms.
 

474
00:25:02 --> 00:25:06
Now, these cubic and
quartic terms are of

475
00:25:06 --> 00:25:09
the order of 1/100^3.
 

476
00:25:10 --> 00:25:12
That's 10 ^ - 6.
 

477
00:25:12 --> 00:25:14
And the point is that
I'm not claiming that I

478
00:25:14 --> 00:25:16
have an exact answer.
 

479
00:25:16 --> 00:25:19
And I'm going to drop things
of that order of magnitude.

480
00:25:19 --> 00:25:22
So I'm saving everything
up to 4 decimal places.

481
00:25:22 --> 00:25:29
I'm throwing away things which
are 6 decimal places out.

482
00:25:29 --> 00:25:30
Does that answer your question?
 

483
00:25:30 --> 00:25:35
STUDENT: [INAUDIBLE]
 

484
00:25:35 --> 00:25:35
PROF.
 

485
00:25:35 --> 00:25:36
JERISON: So.
 

486
00:25:36 --> 00:25:39
That's the situation, and now
you can combine the terms.

487
00:25:39 --> 00:25:43
I mean, it's not very
impressive here.

488
00:25:43 --> 00:25:48
This is equal to 1
- 7/2 x, maybe

489
00:25:50 --> 00:25:53
51/8 x^2.
 

490
00:25:54 --> 00:25:57
If I've made that, if those
minus signs hadn't canceled,

491
00:25:57 --> 00:25:59
I would have gotten
the wrong answer here.

492
00:25:59 --> 00:26:00
Anyway.
 

493
00:26:00 --> 00:26:03
So, this is a 2 here, sorry.
 

494
00:26:03 --> 00:26:04
7/2.
 

495
00:26:04 --> 00:26:07
This is the linear
approximation we got last

496
00:26:07 --> 00:26:09
time and here's the extra
information that we got

497
00:26:09 --> 00:26:11
from this calculation.
 

498
00:26:11 --> 00:26:17
Which is this 51/8 term.
 

499
00:26:17 --> 00:26:20
Right, you have to accept that
there's a certain degree of

500
00:26:20 --> 00:26:23
complexity to this problem and
the answer is sufficiently

501
00:26:23 --> 00:26:26
complicated so it can't be less
arithmetic because we get this

502
00:26:26 --> 00:26:29
peculiar 51/8 there, right.
 

503
00:26:29 --> 00:26:33
So one of the things to realize
is that these kinds of

504
00:26:33 --> 00:26:37
problems, because they involve
many, many terms are always

505
00:26:37 --> 00:26:43
going to involve a little bit
of complicated arithmetic.

506
00:26:43 --> 00:26:48
Last little bit, I did promise
you that I was going to derive

507
00:26:48 --> 00:26:51
these two relations, as I said.
 

508
00:26:51 --> 00:26:53
Did the ones in
the left column.

509
00:26:53 --> 00:26:56
So let's carry that out.
 

510
00:26:56 --> 00:27:00
And as someone just pointed
out, it all comes from

511
00:27:00 --> 00:27:01
this formula here.
 

512
00:27:01 --> 00:27:07
So let's just check it.
 

513
00:27:07 --> 00:27:12
So we'll start with
the ln function.

514
00:27:12 --> 00:27:16
This is the function,
f, and then f' is 1/1

515
00:27:17 --> 00:27:18
x.
 

516
00:27:18 --> 00:27:24
And f'', so this is f',
this is f'', is - 1/1

517
00:27:24 --> 00:27:25
x^2.
 

518
00:27:25 --> 00:27:28
 


519
00:27:28 --> 00:27:31
And now I have to
plug in x = 0.

520
00:27:31 --> 00:27:35
So at x = 0 this is
ln 1, which is 0.

521
00:27:35 --> 00:27:37
So this is at x = 0.
 

522
00:27:37 --> 00:27:41
I'm getting 0 here, I
plug in 0 and I get 1.

523
00:27:41 --> 00:27:45
And here, I plug in
0 and I get - 1.

524
00:27:45 --> 00:27:49
So now I go and I look up at
that formula, which is way

525
00:27:49 --> 00:27:50
in that upper corner there.
 

526
00:27:50 --> 00:27:53
And I see that the coefficient
on the constant is 0.

527
00:27:53 --> 00:27:55
The coefficient on x is 1.
 

528
00:27:55 --> 00:27:59
And then the other coefficient,
the very last one, is - 1/2.

529
00:27:59 --> 00:28:01
So this is the - 1 here.
 

530
00:28:01 --> 00:28:05
And then in the formula,
there's a 2 in the denominator.

531
00:28:05 --> 00:28:07
So it's half of whatever
I get for this second

532
00:28:07 --> 00:28:10
derivative, at 0.
 

533
00:28:10 --> 00:28:13
So this is the approximation
formula, which is way up

534
00:28:13 --> 00:28:17
in that corner there.
 

535
00:28:17 --> 00:28:19
Similarly, if I do it for (1
 

536
00:28:19 --> 00:28:23
x) ^ r, I have to differentiate
that I get r( 1

537
00:28:23 --> 00:28:28
x) ^ r - 1, and
then r ( r - 1( x

538
00:28:29 --> 00:28:31
1) ^ r - 2.
 

539
00:28:31 --> 00:28:33
So here are the derivatives.
 

540
00:28:33 --> 00:28:41
And so if I evaluate
them at x = 0, I get 1.

541
00:28:41 --> 00:28:43
That's 1 ^ r = 1.
 

542
00:28:43 --> 00:28:47
And here I get r. (1 ^ r - 1)r.
 

543
00:28:49 --> 00:28:59
And here, I plug in x =
0 and I get r ( r - 1).

544
00:28:59 --> 00:29:02
So again, the pattern is
right above it here.

545
00:29:02 --> 00:29:05
The 1 is there, the r is there.
 

546
00:29:05 --> 00:29:08
And then instead of r ( r
- 1), I have half that.

547
00:29:08 --> 00:29:21
For the coefficient.
 

548
00:29:21 --> 00:29:22
So these are just examples.
 

549
00:29:22 --> 00:29:24
Obviously if we had a more
complicated functional,

550
00:29:24 --> 00:29:26
we might carry this out.
 

551
00:29:26 --> 00:29:29
But as a practical matter, we
try to stick with the ones in

552
00:29:29 --> 00:29:42
the pink box and just use
algebra to get other formulas.

553
00:29:42 --> 00:29:46
So I want to shift gears now
and treat the subject that was

554
00:29:46 --> 00:29:48
supposed to be this lecture.
 

555
00:29:48 --> 00:29:53
And we're not quite caught up,
but we will try to do our best

556
00:29:53 --> 00:29:54
to do as much as we can today.
 

557
00:29:54 --> 00:29:59
So the next topic is
curve sketching.

558
00:29:59 --> 00:30:18
And so let's get
started with that.

559
00:30:18 --> 00:30:23
So now, happily in this
subject, there are more

560
00:30:23 --> 00:30:26
pictures and it's a little
bit more geometric.

561
00:30:26 --> 00:30:30
And there's relatively
little computation.

562
00:30:30 --> 00:30:33
So let's hope we can do this.
 

563
00:30:33 --> 00:30:37
So I want to, so here we
go, we'll start with

564
00:30:37 --> 00:30:44
curve sketching.
 

565
00:30:44 --> 00:30:47
And the goal here
 

566
00:30:47 --> 00:30:56
STUDENT: [INAUDIBLE]
 

567
00:30:56 --> 00:30:56
PROF.
 

568
00:30:56 --> 00:31:00
JERISON: So that's like,
liner, the last time.

569
00:31:00 --> 00:31:09
That's kind of sketchy
spelling, isn't it?

570
00:31:09 --> 00:31:12
Yeah, there are certain kinds
of things which I can't spell.

571
00:31:12 --> 00:31:16
But, alright.
 

572
00:31:16 --> 00:31:18
Sketching.
 

573
00:31:18 --> 00:31:19
Alright.
 

574
00:31:19 --> 00:31:21
So here's our goal.
 

575
00:31:21 --> 00:31:38
Our goal is to draw the graph
of f, using f' and f''.

576
00:31:38 --> 00:31:42
 


577
00:31:42 --> 00:31:47
Whether they're
positive or negative.

578
00:31:47 --> 00:31:48
So that's it.
 

579
00:31:48 --> 00:31:52
This is the goal here.
 

580
00:31:52 --> 00:31:57
However, there is a big warning
that I want to give you.

581
00:31:57 --> 00:32:06
And this is one that
unfortunately I now have to

582
00:32:06 --> 00:32:08
make you unlearn, especially
those that you that have

583
00:32:08 --> 00:32:11
actually had a little bit of
calculus before, I want to make

584
00:32:11 --> 00:32:13
you unlearn some of your
instincts that you developed.

585
00:32:13 --> 00:32:15
So this will be harder for
those of you who have

586
00:32:15 --> 00:32:19
actually done this before.
 

587
00:32:19 --> 00:32:22
But for the rest of you, it
will be relatively easy.

588
00:32:22 --> 00:32:35
Which is, don't abandon
your precalculus skills.

589
00:32:35 --> 00:32:42
And common sense.
 

590
00:32:42 --> 00:32:46
So there's a great deal
of common sense in this.

591
00:32:46 --> 00:32:50
And it actually trumps
some of the calculus.

592
00:32:50 --> 00:32:56
The calculus just fills in what
you didn't quite know yet.

593
00:32:56 --> 00:32:59
So I will try to
illustrate this.

594
00:32:59 --> 00:33:02
And because we're running a bit
late, I won't get to the some

595
00:33:02 --> 00:33:05
of the main punchlines
until next lecture.

596
00:33:05 --> 00:33:07
But I want you to do it.
 

597
00:33:07 --> 00:33:09
So for now, I'm just going
to tell you about the

598
00:33:09 --> 00:33:10
general principles.
 

599
00:33:10 --> 00:33:14
And in the process I'm going
to introduce the terminology.

600
00:33:14 --> 00:33:17
Just, the words that we need
to use to describe what

601
00:33:17 --> 00:33:18
is that we're doing.
 

602
00:33:18 --> 00:33:19
And there's also a certain
amount of carelessness

603
00:33:19 --> 00:33:23
with that in many of the
treatments that you'll see.

604
00:33:23 --> 00:33:24
And a lot of hastiness.
 

605
00:33:24 --> 00:33:29
So just be a little patient
and we will do this.

606
00:33:29 --> 00:33:33
So, the first principle
is the following.

607
00:33:33 --> 00:33:40
If f' is positive,
then f is increasing.

608
00:33:40 --> 00:33:44
That's a straightforward idea,
and it's closely related to

609
00:33:44 --> 00:33:48
this tangent line approximation
or the linear approximation

610
00:33:48 --> 00:33:49
that I just did.
 

611
00:33:49 --> 00:33:50
You can just imagine.
 

612
00:33:50 --> 00:33:53
Here's the tangent line,
here's the function.

613
00:33:53 --> 00:33:56
And if the tangent line is
pointing up, then the function

614
00:33:56 --> 00:33:58
is also going up, too.
 

615
00:33:58 --> 00:34:00
So that's all that's
going on here.

616
00:34:00 --> 00:34:10
Similarly, if f' is negative,
then f is decreasing.

617
00:34:10 --> 00:34:12
And that's the basic idea.
 

618
00:34:12 --> 00:34:17
Now, the second step is also
fairly straightforward.

619
00:34:17 --> 00:34:21
It's just a second order
effect of the same type.

620
00:34:21 --> 00:34:27
If you have f'' as positive,
then that means that

621
00:34:27 --> 00:34:33
f' is increasing.
 

622
00:34:33 --> 00:34:36
That's the same principle
applied one step up.

623
00:34:36 --> 00:34:37
Right?
 

624
00:34:37 --> 00:34:41
Because if f'' is positive,
that means it's the

625
00:34:41 --> 00:34:42
derivative of f'.
 

626
00:34:42 --> 00:34:45
So it's the same
principle just repeated.

627
00:34:45 --> 00:34:49
And now I just want to
draw a picture of this.

628
00:34:49 --> 00:34:52
Here's a picture
of it, I claim.

629
00:34:52 --> 00:34:54
And it looks like
something's going down.

630
00:34:54 --> 00:34:56
And I did that on purpose.
 

631
00:34:56 --> 00:34:58
But there is something
that's increasing here.

632
00:34:58 --> 00:35:02
Which is, the slope is
very steep negative here.

633
00:35:02 --> 00:35:06
And it's less steep
negative over here.

634
00:35:06 --> 00:35:10
So we have the slope which is
some negative number, say, - 4.

635
00:35:10 --> 00:35:13
And here it's - 3.
 

636
00:35:13 --> 00:35:14
So it's increasing.
 

637
00:35:14 --> 00:35:18
It's getting less negative,
and maybe eventually it'll

638
00:35:18 --> 00:35:19
curve up the other way.
 

639
00:35:19 --> 00:35:23
And this is a picture of what
I'm talking about here.

640
00:35:23 --> 00:35:25
That's what it means to say
that f' is increasing.

641
00:35:25 --> 00:35:28
The slope is getting larger.
 

642
00:35:28 --> 00:35:34
And the way to describe a curve
like this is that it's concave.

643
00:35:34 --> 00:35:41
So f is concave up.
 

644
00:35:41 --> 00:35:49
And similarly, f'' negative is
going to be the same thing as f

645
00:35:49 --> 00:35:59
concave, or implies
f concave down.

646
00:35:59 --> 00:36:04
So those are the ways in which
derivatives will help us

647
00:36:04 --> 00:36:08
qualitatively to draw graphs.
 

648
00:36:08 --> 00:36:10
But as I said before, we still
have to use a little bit of

649
00:36:10 --> 00:36:13
common sense when we
draw the graphs.

650
00:36:13 --> 00:36:15
These are just the additional
bits of help that we

651
00:36:15 --> 00:36:17
have from calculus.
 

652
00:36:17 --> 00:36:25
In drawing pictures.
 

653
00:36:25 --> 00:36:34
So I'm going to go through
one example to introduce

654
00:36:34 --> 00:36:36
all the notations.
 

655
00:36:36 --> 00:36:40
And then eventually, so
probably at the beginning of

656
00:36:40 --> 00:36:45
next time, I'll give you a
systematic strategy that's

657
00:36:45 --> 00:36:49
going to work when what I'm
describing now goes wrong,

658
00:36:49 --> 00:36:52
or a little bit wrong.
 

659
00:36:52 --> 00:36:58
So let's begin with a
straightforward example.

660
00:36:58 --> 00:37:00
So, the first example that
I'll give you is the

661
00:37:00 --> 00:37:04
function f (x) = 3x - x^3.
 

662
00:37:04 --> 00:37:07
 


663
00:37:07 --> 00:37:11
Just, as I said, to be able to
introduce all the notations.

664
00:37:11 --> 00:37:16
Now, if you differentiate
it, you get 3 - 3x^2.

665
00:37:17 --> 00:37:20
And I can factor that.
 

666
00:37:20 --> 00:37:23
This is 3 ( 1 - x)( 1
 

667
00:37:24 --> 00:37:26
x).
 

668
00:37:26 --> 00:37:27
OK?
 

669
00:37:27 --> 00:37:33
And so, I can decide
whether the derivative

670
00:37:33 --> 00:37:37
is positive or negative.
 

671
00:37:37 --> 00:37:38
Easily enough.
 

672
00:37:38 --> 00:37:50
Namely, just staring at this, I
can see that when -1 < x < 1,

673
00:37:50 --> 00:37:53
in that range there, both
these numbers, both these

674
00:37:53 --> 00:37:55
factors, are positive.
 

675
00:37:55 --> 00:37:57
1 - x is a positive
number and 1

676
00:37:58 --> 00:37:59
x is a positive number.
 

677
00:37:59 --> 00:38:04
So, in this range, f'
( x ) is positive.

678
00:38:04 --> 00:38:09
So this thing is, so
f is increasing.

679
00:38:09 --> 00:38:14
And similarly, in the other
ranges, if x is very, very

680
00:38:14 --> 00:38:17
large, this becomes, if it
crosses 1, in fact, this

681
00:38:17 --> 00:38:19
becomes, this factor
becomes negative and this

682
00:38:19 --> 00:38:21
one stays positive.
 

683
00:38:21 --> 00:38:29
So when x > 1, we have
that f ' (x) is negative.

684
00:38:29 --> 00:38:35
And so f is decreasing.
 

685
00:38:35 --> 00:38:39
And the same thing goes
for the other side.

686
00:38:39 --> 00:38:42
When it's less than - 1,
that also works this way.

687
00:38:42 --> 00:38:45
Because when it's less
than - 1, this number

688
00:38:45 --> 00:38:46
factors positive.
 

689
00:38:46 --> 00:38:50
But the other one is negative.
 

690
00:38:50 --> 00:38:56
So in both of these cases, we
get that it's decreasing.

691
00:38:56 --> 00:39:07
So now, here's the schematic
picture of this function.

692
00:39:07 --> 00:39:12
So here's - 1, here's 1.
 

693
00:39:12 --> 00:39:19
It's going to go
down, up, down.

694
00:39:19 --> 00:39:21
That's what it's doing.
 

695
00:39:21 --> 00:39:23
Maybe I'll just leave
it alone like this.

696
00:39:23 --> 00:39:28
That's what it looks like.
 

697
00:39:28 --> 00:39:31
So, this is the kind of
information we can get

698
00:39:31 --> 00:39:32
right off the bat.
 

699
00:39:32 --> 00:39:38
And you notice immediately that
it's very important, from the

700
00:39:38 --> 00:39:40
features of the function, the
sort of key features of the

701
00:39:40 --> 00:39:44
function that we see here,
are these two places.

702
00:39:44 --> 00:39:49
Maybe I'll even mark
them in a, like this.

703
00:39:49 --> 00:39:59
And these things are
turning points.

704
00:39:59 --> 00:40:00
So what are they?
 

705
00:40:00 --> 00:40:03
Well, they're just
the points where the

706
00:40:03 --> 00:40:05
derivative changes sign.
 

707
00:40:05 --> 00:40:07
Where it's negative here
and it's positive there,

708
00:40:07 --> 00:40:09
so there it must be 0.
 

709
00:40:09 --> 00:40:12
So we have a definition, and
this is the most important

710
00:40:12 --> 00:40:21
definition in this subject,
which is that is if f' (x0) =

711
00:40:21 --> 00:40:33
0, we call x0 a critical point.
 

712
00:40:33 --> 00:40:36
The word 'turning point' is not
used just because, in fact,

713
00:40:36 --> 00:40:38
it doesn't have to turn
around at those points.

714
00:40:38 --> 00:40:42
But certainly, if it turns
around then this will happen.

715
00:40:42 --> 00:40:46
And we also have another
notation, which is the number

716
00:40:46 --> 00:40:59
y0 which is f ( x0 ) is
called a critical value.

717
00:40:59 --> 00:41:02
So these are the key numbers
that we're going to have to

718
00:41:02 --> 00:41:18
work out in order to understand
what the function looks like.

719
00:41:18 --> 00:41:27
So what I'm going to
do is just plot them.

720
00:41:27 --> 00:41:30
We're going to plot the
critical points and the values.

721
00:41:30 --> 00:41:34
Well, we found the critical
points relatively easily.

722
00:41:34 --> 00:41:37
I didn't write it down here
but it's pretty obvious.

723
00:41:37 --> 00:41:43
If you set f(x) = 0, that
implies that (1 - x)( 1

724
00:41:44 --> 00:41:47
x) = 0, which
implies that x is

725
00:41:47 --> 00:41:50
or - 1.
 

726
00:41:50 --> 00:41:52
So those are known as
the critical points.

727
00:41:52 --> 00:41:56
And now, in order to get the
critical values here, I have to

728
00:41:56 --> 00:42:02
plug in f (1), for instance,
the function is 3x - x^2, so

729
00:42:02 --> 00:42:07
there's this 3 * 1
- 1^3, which is 2.

730
00:42:07 --> 00:42:17
And f ( - 1), which is 3 ( -
1) - (- 1)^3, which is - 2.

731
00:42:17 --> 00:42:21
And so I can plot
the function here.

732
00:42:21 --> 00:42:26
So here's the point - 1 and
here's, up here, is 2.

733
00:42:26 --> 00:42:28
So this is - whoops,
which one is it?

734
00:42:28 --> 00:42:29
Yeah.
 

735
00:42:29 --> 00:42:32
This is - 1, so it's down here.
 

736
00:42:32 --> 00:42:35
So it's (- 1, - 2).
 

737
00:42:35 --> 00:42:41
And then over here, I
have the point (1, 2).

738
00:42:41 --> 00:42:47
Alright, now, what information
do I get from - so I've

739
00:42:47 --> 00:42:50
now plotted two, I claim,
very interesting points.

740
00:42:50 --> 00:42:55
What information do
I get from this?

741
00:42:55 --> 00:42:59
The answer is, I know
something very nearby.

742
00:42:59 --> 00:43:02
Because I've already checked
that the thing is coming down

743
00:43:02 --> 00:43:04
from the left, and
coming back up.

744
00:43:04 --> 00:43:07
And so it must be
shaped like this.

745
00:43:07 --> 00:43:08
Over here.
 

746
00:43:08 --> 00:43:11
The tangent line is 0, it's
going to be level there.

747
00:43:11 --> 00:43:14
And similarly over here,
it's going to do that.

748
00:43:14 --> 00:43:22
So this is what we know so far,
based on what we've computed.

749
00:43:22 --> 00:43:22
Question.
 

750
00:43:22 --> 00:43:37
STUDENT: [INAUDIBLE]
 

751
00:43:37 --> 00:43:37
PROF.
 

752
00:43:37 --> 00:43:38
JERISON: The question is,
what happens if there's

753
00:43:38 --> 00:43:38
a sharp corner.
 

754
00:43:38 --> 00:43:45
The answer is, calculus, it's
not called a critical point.

755
00:43:45 --> 00:43:47
It's a something else.
 

756
00:43:47 --> 00:43:50
And it's a very
important point, too.

757
00:43:50 --> 00:43:52
And we will be discussing
those kinds of points.

758
00:43:52 --> 00:43:54
There are much more dramatic
instances of that.

759
00:43:54 --> 00:43:56
That's part of what
we're going to say.

760
00:43:56 --> 00:43:58
But I just want to
save that, alright.

761
00:43:58 --> 00:44:02
We will be discussing.
 

762
00:44:02 --> 00:44:03
Yeah.
 

763
00:44:03 --> 00:44:03
Question.
 

764
00:44:03 --> 00:44:08
STUDENT: [INAUDIBLE]
 

765
00:44:08 --> 00:44:08
PROF.
 

766
00:44:08 --> 00:44:11
JERISON: The question that was
asked was, how did I know at

767
00:44:11 --> 00:44:14
the critical point that it's
concave down over here and

768
00:44:14 --> 00:44:17
concave up over here.
 

769
00:44:17 --> 00:44:22
The answer is that I actually
did not use the second

770
00:44:22 --> 00:44:24
derivative yet.
 

771
00:44:24 --> 00:44:26
What I used is another
piece of information.

772
00:44:26 --> 00:44:28
I used the information
that I derived over here.

773
00:44:28 --> 00:44:32
That f' is positive,
where f' is positive and

774
00:44:32 --> 00:44:33
where it's negative.
 

775
00:44:33 --> 00:44:36
So what I know is that
the graph is going down

776
00:44:36 --> 00:44:39
to the left of - 1.
 

777
00:44:39 --> 00:44:41
It's going up to
the right, here.

778
00:44:41 --> 00:44:44
It's going up here and
it's going down there.

779
00:44:44 --> 00:44:47
I did not use the
second derivative.

780
00:44:47 --> 00:44:49
I used the first derivative.
 

781
00:44:49 --> 00:44:52
OK, but I didn't just use
the fact that there was

782
00:44:52 --> 00:44:56
a turning point here.
 

783
00:44:56 --> 00:44:57
So, actually, I was using
the fact that it was

784
00:44:57 --> 00:44:58
a turning point.
 

785
00:44:58 --> 00:45:00
I wasn't using the fact
that it had the second

786
00:45:00 --> 00:45:01
derivative, though.
 

787
00:45:01 --> 00:45:01
OK.
 

788
00:45:01 --> 00:45:03
For now.
 

789
00:45:03 --> 00:45:09
You can also see it by the
second derivative as well.

790
00:45:09 --> 00:45:14
So now, the next thing that
I'd like to do, I need to

791
00:45:14 --> 00:45:16
finish off this graph.
 

792
00:45:16 --> 00:45:19
And I just want to do it a
little bit carefully here.

793
00:45:19 --> 00:45:22
In the order that
is reasonable.

794
00:45:22 --> 00:45:27
Now, you might happen to
notice, and there's nothing

795
00:45:27 --> 00:45:35
wrong with this, so let's
even fill in a guess.

796
00:45:35 --> 00:45:37
In order to fill in a guess,
though, and have it be even

797
00:45:37 --> 00:45:39
vaguely right, I do have to
notice that this thing

798
00:45:39 --> 00:45:41
crosses, this function
crosses the origin.

799
00:45:41 --> 00:45:48
The function f(x) = 3x -
x^3 happens also have the

800
00:45:48 --> 00:45:50
property that f ( 0 ) = 0.
 

801
00:45:50 --> 00:45:52
Again, common sense.
 

802
00:45:52 --> 00:45:54
You're allowed to use
your common sense.

803
00:45:54 --> 00:45:56
You're allowed to notice
a value of the function

804
00:45:56 --> 00:45:58
and put it in.
 

805
00:45:58 --> 00:46:00
So there's nothing
wrong with that.

806
00:46:00 --> 00:46:03
If you happen to
have such a value.

807
00:46:03 --> 00:46:06
So, now we can guess what our
function is going to look like.

808
00:46:06 --> 00:46:10
It's going to maybe
come down like this.

809
00:46:10 --> 00:46:11
Come up like this.
 

810
00:46:11 --> 00:46:13
And come down like this.
 

811
00:46:13 --> 00:46:15
That could be what
it looks like.

812
00:46:15 --> 00:46:17
But, you know, another
possibility is it sort

813
00:46:17 --> 00:46:19
of comes along here
and goes out that way.

814
00:46:19 --> 00:46:22
Comes along here and goes
out that way, who knows?

815
00:46:22 --> 00:46:25
It happens, by the way,
that it's an odd function.

816
00:46:25 --> 00:46:25
Right?
 

817
00:46:25 --> 00:46:26
Those are all odd powers.
 

818
00:46:26 --> 00:46:28
So, actually, it's symmetric
on the right half

819
00:46:28 --> 00:46:29
and the left half.
 

820
00:46:29 --> 00:46:31
And crosses at 0.
 

821
00:46:31 --> 00:46:33
So everything that we do on the
right is going to be the same

822
00:46:33 --> 00:46:34
as what happens on the left.
 

823
00:46:34 --> 00:46:36
That's another piece
of common sense.

824
00:46:36 --> 00:46:39
You want to make use of that
as much as possible, whenever

825
00:46:39 --> 00:46:40
you're drawing anything.
 

826
00:46:40 --> 00:46:42
Don't want to throw
out information.

827
00:46:42 --> 00:46:46
So this function
happens to be odd.

828
00:46:46 --> 00:46:48
Odd, and f ( 0 ) = 0.
 

829
00:46:48 --> 00:46:52
I'm considering those to be
kinds of precalculus skills

830
00:46:52 --> 00:47:00
that I want you to use
as much as you can.

831
00:47:00 --> 00:47:03
So now, here's the first
feature which is unfortunately

832
00:47:03 --> 00:47:08
ignored in most
discussions of functions.

833
00:47:08 --> 00:47:11
And it's strange, because
nowadays we have

834
00:47:11 --> 00:47:13
graphing things.
 

835
00:47:13 --> 00:47:19
And it's really the only part
of the exercise that you

836
00:47:19 --> 00:47:26
couldn't do, at least on this
relatively simpleminded level,

837
00:47:26 --> 00:47:28
with a graphing calculator.
 

838
00:47:28 --> 00:47:33
And that is what I would call
the ends of the problem.

839
00:47:33 --> 00:47:37
So what happens off the
screen, is the question.

840
00:47:37 --> 00:47:39
And that basically is the
theoretical part of the problem

841
00:47:39 --> 00:47:41
that you have to address.
 

842
00:47:41 --> 00:47:42
You can program this.
 

843
00:47:42 --> 00:47:45
You can draw all the
pictures that you want.

844
00:47:45 --> 00:47:48
But what you won't see is
what's off the screen.

845
00:47:48 --> 00:47:50
You need to know something
to figure out what's

846
00:47:50 --> 00:47:51
off the screen.
 

847
00:47:51 --> 00:47:54
So, in this case, I'm talking
about what's off the screen

848
00:47:54 --> 00:48:01
going to the right, or
going to the left.

849
00:48:01 --> 00:48:06
So let's check the ends.
 

850
00:48:06 --> 00:48:07
So here, let's
just take a look.

851
00:48:07 --> 00:48:12
We have the function f(x),
which is, sorry, 3x - x^3.

852
00:48:12 --> 00:48:14
Again this is a precalculus
sort of thing.

853
00:48:14 --> 00:48:16
And we're imagining now,
let's just do x goes

854
00:48:16 --> 00:48:18
to plus infinity.
 

855
00:48:18 --> 00:48:19
So what happens here.
 

856
00:48:19 --> 00:48:25
When x is gigantic, this term
is completely negligible.

857
00:48:25 --> 00:48:30
And it just behaves like - x^3,
which goes to minus infinity

858
00:48:30 --> 00:48:32
as x goes to plus infinity.
 

859
00:48:32 --> 00:48:41
And similarly, f (x) goes
to plus infinity if x

860
00:48:41 --> 00:48:46
goes to minus infinity.
 

861
00:48:46 --> 00:48:49
Now let me pull down this
picture again, and show

862
00:48:49 --> 00:48:53
you what piece of the
information we've got.

863
00:48:53 --> 00:48:55
We now know that it is
heading up this way.

864
00:48:55 --> 00:48:58
It doesn't go like this,
it goes up like that.

865
00:48:58 --> 00:49:00
And I'm going to put an
arrow for it, And it's

866
00:49:00 --> 00:49:02
going down like this.
 

867
00:49:02 --> 00:49:06
Heading down to minus infinity
as x goes out farther

868
00:49:06 --> 00:49:07
to the right.
 

869
00:49:07 --> 00:49:15
And going out to plus infinity
as x goes farther to the left.

870
00:49:15 --> 00:49:21
So now there's hardly anything
left of this function

871
00:49:21 --> 00:49:21
to describe.
 

872
00:49:21 --> 00:49:27
There's really nothing left
except maybe decoration.

873
00:49:27 --> 00:49:31
And we kind of like that
decoration, so we will

874
00:49:31 --> 00:49:32
pay attention to it.
 

875
00:49:32 --> 00:49:35
And to do that, we'll have to
check the second derivative.

876
00:49:35 --> 00:49:39
So if we differentiate a second
time, the first derivative

877
00:49:39 --> 00:49:41
was, remember, 3 - 3x^2.
 

878
00:49:42 --> 00:49:53
So the second
derivative is - 6x.

879
00:49:53 --> 00:50:02
So now we notice that f'' (x)
is negative if x is positive.

880
00:50:02 --> 00:50:04
And f'' ( x)
 

881
00:50:04 --> 00:50:08
is positive if x is negative.
 

882
00:50:08 --> 00:50:13
And so in this part
it's concave down.

883
00:50:13 --> 00:50:18
And in this part
it's concave up.

884
00:50:18 --> 00:50:21
And now I'm going to
switch the boards so that

885
00:50:21 --> 00:50:24
you'll, and draw it.
 

886
00:50:24 --> 00:50:30
And you see that it was
begging to be this way.

887
00:50:30 --> 00:50:32
So we'll fill in the
rest of it here.

888
00:50:32 --> 00:50:35
Maybe in a nice color here.
 

889
00:50:35 --> 00:50:38
So this is the whole graph and
this is the correct graph.

890
00:50:38 --> 00:50:41
It comes down in one swoop
down here, and comes up here.

891
00:50:41 --> 00:50:46
And then it changes to concave
down right at the origin.

892
00:50:46 --> 00:50:49
So this point is of interest,
not only because it's the place

893
00:50:49 --> 00:50:52
where it crosses the axis, but
it's also what's called

894
00:50:52 --> 00:51:00
an inflection point.
 

895
00:51:00 --> 00:51:04
Inflection point, that's a
point where because f'' at

896
00:51:04 --> 00:51:07
that place is equal to 0.
 

897
00:51:07 --> 00:51:10
So it's a place where the
second derivative is 0.

898
00:51:10 --> 00:51:15
We also consider those to
be interesting points.

899
00:51:15 --> 00:51:21
Now, so let me just making
one closing remark here.

900
00:51:21 --> 00:51:26
Which is that all of this
information fits together.

901
00:51:26 --> 00:51:29
And we're going to have much,
much harder examples of this

902
00:51:29 --> 00:51:32
where you'll actually have to
think about what's going on.

903
00:51:32 --> 00:51:35
But there's a lot of
stuff protecting you.

904
00:51:35 --> 00:51:39
And functions will behave
themselves and turn

905
00:51:39 --> 00:51:40
around appropriately.
 

906
00:51:40 --> 00:51:43
Anyway, we'll talk
about it next time.

907
00:51:43 --> 00:51:43