Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007

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S… Speaker 3 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 3 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 3 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 3 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So again,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
welcome to 1801.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We're getting started today with what
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we're calling
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Unit 1,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
highly imaginative title.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And it's differentiation.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let me first tell you briefly what's in store in the next couple
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
of weeks.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The main topic today is what
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is a derivative.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we're going to look at
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this from several different points of view.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And the first one is the geometric interpretation.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that's what we'll spend most of today on.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then we'll
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
also talk about a physical interpretation of what
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
a derivative is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
then there's going
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
to be something else,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which I guess is maybe the reason why calculus is so fundamental
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
and why we always start with it.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in most science and engineering schools.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Which is the importance of derivatives
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
of this to all measurements.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that means pretty much every place.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That means in science,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in engineering,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in economics,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in political science,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
et cetera.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Polling,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
lots of commercial applications.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Just about everything.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Now, so that's what we'll be getting started with.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then there's another thing that we're going to do in
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this unit, which is we're going to explain
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
how to differentiate anything.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
how to differentiate any function you know.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that's kind of a tall order.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But let me just give you an example.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
If you want to take the derivative,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this we'll see today as the notation for the derivative of something,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
of some messy function like e to the x arc
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
tan of x,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we'll work this out by
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the end of this unit.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So anything you can think of,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
anything you can write down,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we can differentiate it.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's what we're going to do.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We're going to spend most of our time on this geometric interpretation.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's begin with
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that. So
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
here we go with the geometric interpretation
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
of derivatives.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And what we're
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
going to do is just ask the geometric
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
problem of finding the tangent line to
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
some graph of some function.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
at some point.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the problem that we're addressing here.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I guess I should probably turn this
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
off. All right.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So here's our problem.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And now let me show you the solution.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
well,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let's graph the function.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let's say here's its graph.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And here's some point.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe I should draw it just a bit lower
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so that I don't.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So here's a point P.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe it's above the point x0.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
x0,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
by the way,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this was supposed to be an x0.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That was some fixed place on the x -axis.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
now,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in order to perform this mighty feat,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I will
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Use another color of chalk.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
How about red?
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So here it is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
There's the tangent line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, not quite straight.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Close enough.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I did it.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the end.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the geometric problem.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I achieved what I wanted to do.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And it's kind of an interesting question,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which unfortunately I can't
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
solve for you in this class,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is, how did I do that?
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That is,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
how physically did I manage to know what to do to draw this tangent
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
line? But that's what geometric problems are like.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We visualize it.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We can figure it out.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Somewhere in our brains it happens.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And the task that we have now is to figure out
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
how to do it analytically,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
to do it in a way that
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
A machine could do just as well as I did in drawing this tangent line.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
what did we learn in high school about what a tangent line
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is?
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, a tangent line has an equation.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Any line through point has the equation y -y0 is
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
equal to m the slope times x -x0.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So here's the equation
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
for that line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And now there are two pieces of information that we're going
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
to need to work out what the line is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The first one is the point.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's that point P there.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And to specify P,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
given x,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we need to know the level of y,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is of course just f .
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Now, that's not a calculus problem.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But anyway, that's a very important part of the process.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the first thing we need to know.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And the second thing we need to know is
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the slope.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's this number
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
m.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And in calculus,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we have another name for it.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We call it f' ,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
namely the derivative.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
of f.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the calculus part,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that's the tricky part,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
and that's the part that we have to discuss now.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So just to make that explicit
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
here, I'm going to make a definition,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is that f' ,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is known as the derivative of f
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
at x0,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is the
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
slope of the
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
tangent line to
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
y equals f at the point,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let's just call it p.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that's what it is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But still I haven't made any progress in figuring out
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
any better how I drew that line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So I have to say something that's more concrete,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
because I want to be able to cook up what these numbers are.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I have to figure out what this number m is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And one way of thinking about that,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let me just try it.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So I certainly am taking for granted that in sort of non -calculus part,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that I know what a line through a point is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So I know this equation.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But another possibility might be,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
you know,
9:26
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this line here.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
How do I know,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
well, unfortunately I didn't draw it quite straight,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
but there it is.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
How do I know that this orange line is not a
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
tangent line,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
but this other line is a tangent line?
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
9:45
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
it's
9:49
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
actually not so obvious.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But I'm going to describe it a little bit.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
It's not really the fact,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this thing crosses at some other
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Which is this point q.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But it's not really the fact that the thing crosses it to place.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Because the line could be wiggly,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the curve could be wiggly.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And it could cross back and forth a number of times.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's not what distinguishes the tangent line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So I'm going to have to somehow grasp
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this. And I first do it in language.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And it's the following idea.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this orange line,
10:30
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is called a secant line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And you think of the q,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the point q is getting closer and closer to p.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Then the slope of that line will get closer and closer to
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the slope of the red line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And if we draw it close
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
enough, then
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's going to be the correct line.
10:54
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's really what I did sort of in my brain when I drew that first line.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And so that's the way I'm going to articulate it first now.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So the tangent line is
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
equal to the limit of
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
what's so -called secant lines,
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
pq,
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
as q tends to p.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And here we're thinking of p as being fixed.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And q as varying.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Again,
11:35
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this is still a geometric discussion.
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But now we're going to be able to put symbols and formulas
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
to this computation.
11:43
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And we'll be able to work out formulas
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S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in any example.
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S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
11:54
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let's do that.
11:57
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
11:59
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to write out these points p
12:03
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
and q again.
12:04
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe we'll put p here and q here.
12:11
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm thinking of this line through them.
12:12
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I guess it was orange,
12:13
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so we'll leave it as orange.
12:16
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And now
12:20
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to compute its slope.
12:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And so this will gradually,
12:25
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we'll do this in two steps.
12:26
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And these steps will introduce us to the basic notations which are used throughout calculus,
12:31
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
including multivariable calculus across the board.
12:39
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Here's the x -axis underneath.
12:41
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And here's the x0,
12:43
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the location directly below the point p.
12:46
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And we're traveling here a horizontal distance,
12:50
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is denoted by delta x.
12:53
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's delta x,
12:56
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so -called.
12:57
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And we could also call it the change in
13:01
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
x.
13:02
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
13:07
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that's one thing we want to measure in order to get the slope of this
13:11
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
line pq.
13:11
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And the other thing is this height.
13:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's this distance here,
13:15
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which we denote delta f,
13:18
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is the change in f.
13:19
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then the
13:23
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
slope is just the ratio,
13:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
delta f over delta x.
13:28
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the slope of the
13:32
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
secant.
13:34
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And
13:39
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the process I just described over here with this limit applies
13:44
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
not just to the whole line itself,
13:46
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
but also in particular to its slope.
13:48
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And the way we write that is the limit.
13:51
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x goes to 0.
13:53
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's going to be our slope.
13:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the slope of the tangent line.
13:58
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
OK.
14:11
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
14:12
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this
14:16
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is still a little general.
14:19
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And I want to work out a more usable
14:23
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
form here.
14:24
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I want to work out a better formula for this.
14:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And in order to do that,
14:29
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to write delta f,
14:32
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the numerator,
14:33
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
more explicitly.
14:35
S… Speaker 3 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
here.
14:35
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The change in f.
14:37
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So remember that the point P is
14:41
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the point x0,
14:42
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
f .
14:43
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
14:47
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that's what we got from our formula for the point.
14:49
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And in order to compute
14:53
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
these distances,
14:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
and in particular the vertical distance here,
14:57
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to have to get a formula for q as well.
15:00
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So if this horizontal distance is delta x,
15:04
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
then this location is x0 plus delta x.
15:09
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And so the point above that point has
15:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
a formula, which is x0
15:18
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
plus delta x f of,
15:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
and this is a mouthful,
15:23
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
x0 plus delta x.
15:26
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
15:32
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
there's the formula for the point q.
15:33
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Here's the formula for the point p.
15:36
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And now I can write a different formula for
15:40
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative,
15:44
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is the following.
15:48
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this f' ,
15:51
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is the same as m,
15:53
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is going to be the limit,
15:56
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x goes to 0,
15:59
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
of the change in f.
16:01
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Well, the change in f is the value of f at
16:06
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the upper point here,
16:08
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is x0 plus delta x.
16:12
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And minus its value at the lower point,
16:15
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
p, which is f ,
16:16
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
divided by delta
16:21
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
x.
16:21
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the formula.
16:24
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm going to put this in a little box,
16:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
because this is by far the most important formula
16:31
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
today, which we use to derive pretty much everything else.
16:34
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is the way that we're going to be able to compute
16:38
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
these numbers.
16:39
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
16:46
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let's do an example.
17:06
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
This example,
17:07
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so we'll call this example 1.
17:09
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We'll
17:13
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
take the function f ,
17:15
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is 1 over x.
17:17
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's sufficiently complicated to have
17:21
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
an interesting answer.
17:23
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And sufficiently straightforward that we can compute
17:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative fairly quickly.
17:29
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So what is it that we're
17:33
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
going to do here?
17:34
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
All we're going to do is
17:38
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we're going to
17:40
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Plug in this formula here for that function.
17:44
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's all we're going to do.
17:45
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And visually,
17:47
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
what we're accomplishing is somehow to take the
17:51
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
hyperbola and take a point on the hyperbola and figure out
17:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
some tangent
18:00
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
line.
18:00
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's what we're accomplishing when we do that.
18:02
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So we're accomplishing this geometrically,
18:04
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
but we'll be doing it algebraically.
18:08
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
we consider this difference delta f over
18:13
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
delta x.
18:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And write out its formula.
18:16
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So I have to have a place,
18:18
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so I'm going to make it again above this point x0,
18:21
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is a general point.
18:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
We'll make the general calculation.
18:24
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So the value of f at the top,
18:28
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
when we move to the right by f ,
18:30
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so I just read off from here the
18:35
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
formula.
18:38
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The first thing I get here is 1 over x0 plus delta x.
18:42
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the left -hand term.
18:45
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Minus 1 over x0.
18:47
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the right -hand term.
18:49
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I have to divide that by delta x.
18:52
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So here's our expression.
18:57
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And by the way,
18:58
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this has a name.
19:00
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
This thing is called a difference quotient.
19:09
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
It's pretty complicated because there's always a difference in the numerator.
19:13
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And in disguise,
19:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the denominator is a difference because it's the difference between the value on the
19:18
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
right side and the value on the left side here.
19:22
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
19:26
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so now we're going to simplify
19:30
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
it by some algebra.
19:32
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's just take a look.
19:35
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is equal to,
19:37
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let's continue on the next.
19:39
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
level here.
19:40
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
This is equal to 1 over delta x times,
19:44
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
now all I'm going to do is put it over a common denominator.
19:48
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So the common denominator is x0
19:52
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
plus delta x times x0.
19:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And so in the numerator for the first expression,
19:58
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I have x0.
20:00
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And for the second expression I have x0 plus delta x.
20:03
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the same thing as I had in the numerator
20:08
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
before, factoring out this denominator.
20:10
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And here I put that numerator into this
20:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
more amenable form.
20:16
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And now there are two basic cancellations.
20:20
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The first one is that x0 and x0 cancel.
20:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So we have this.
20:36
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Two expressions cancel.
20:38
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The numerator and denominator,
20:40
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
now we have a cancellation that we can make use of.
20:43
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So we'll write that under here.
20:46
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is equals minus
20:50
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
1 over x0 plus delta x times
20:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
x0.
20:57
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then the very last step is to take the limit
21:01
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
as delta x tends to 0.
21:08
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Now we can do it.
21:09
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Before we couldn't do it.
21:10
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Why?
21:11
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Because the numerator and the dominator gave us 0 over 0.
21:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But now that I've made this cancellation,
21:17
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I can pass to the limit.
21:18
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And all that happens is I set this delta x equal to 0,
21:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
and I get minus 1 over x0 squared.
21:24
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's the answer.
21:26
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
21:32
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
in other words, what I've shown,
21:33
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let me put it up here,
21:35
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is that f prime of x0 is minus 1 over
21:39
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
x0 squared.
21:53
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Now, let's look at the graph just a little bit to check this for
21:57
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
plausibility.
21:58
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
What's happening
22:02
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
here is, first of all,
22:03
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
it's negative.
22:04
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
It's less than 0,
22:06
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is a good thing.
22:08
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
You see that slope there is negative.
22:10
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's
22:16
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the simplest check that you could make.
22:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that I would just like to point out is that as x goes
22:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
to infinity,
22:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
that is, as we go farther to the right,
22:29
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
it gets less and less steep.
22:31
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So less and,
22:35
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
whoops,
22:36
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
as x0 goes to infinity,
22:38
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
not 0.
22:39
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
As x0 goes to infinity,
22:41
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
less and less steep.
22:45
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's also consistent here.
22:47
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
When x0 is very large,
22:49
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this is a smaller and smaller number in magnitude.
22:52
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Although it's always negative.
22:53
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
It's always sloping down.
22:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
23:02
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I've managed to fill the boards,
23:03
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
so maybe I should stop for a question or two.
23:05
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Yes?
23:06
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
23:15
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the question is to explain again this limiting
23:20
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
process.
23:21
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So the formula here is we have basically two numbers.
23:26
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So in other words,
23:27
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
why is it that this expression,
23:28
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
when delta x tends to 0,
23:30
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
is equal to minus 1 over x0 squared?
23:33
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Let me illustrate it by sticking in a number for x0
23:37
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
to make it more explicit.
23:39
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So for instance,
23:41
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
let me stick in here for x0 the number 3.
23:45
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Then it's minus 1 over 3 plus delta x
23:49
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
times 3.
23:52
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the situation that we've got.
23:54
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And now the question is,
23:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
what happens as this number gets smaller and smaller and smaller and gets to
23:59
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
be practically 0?
24:01
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
24:02
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
literally what we can do is just plug in 0 there.
24:04
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Then you get 3 plus 0 times 3 in the denominator,
24:07
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
minus 1 in the numerator.
24:08
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So this tends to minus
24:12
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
1 over 9,
24:13
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
over 3 squared.
24:14
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's what I'm saying in general with this extra
24:18
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
number here.
24:21
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Other questions?
24:25
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Yes.
24:26
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So
24:37
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
the question is,
24:38
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
what happened between this step and this step,
24:42
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
right? Explain this step here.
24:45
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So there were two parts to that.
24:47
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
The first is,
24:49
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
this delta x,
24:51
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which was sitting in the denominator,
24:52
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I factored all the way out front.
24:55
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And so what's in the parentheses is supposed to be the same as
25:00
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
What's in the numerator of this other expression.
25:02
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then,
25:03
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
at the same time as doing that,
25:05
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I put that expression,
25:07
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
which is the difference of two fractions,
25:09
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
I expressed it with a common denominator.
25:11
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
So in the denominator here,
25:13
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
you see the product of the denominators of the two fractions.
25:16
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And then I just figured out what the numerator had to be.
25:19
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
Without really,
25:20
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
yeah,
25:22
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
other questions.
25:34
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That on the whole,
25:36
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
calculus gets a bad rap.
25:40
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
That it's actually easier than most
25:45
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
things.
25:45
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
But there's a perception that it's
25:50
S… Speaker 1 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
harder.
25:51
S… Speaker 2 (Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007)
And so I really have a duty to give

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