Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007

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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
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S… Speaker 3 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so I'd like to begin the second lecture by reminding
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you what we did last time.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So last time.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Last time we defined
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative as a
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
slope of a tangent
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
line.
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S… Speaker 3 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that was our geometric point of view.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And we also did a couple of computations.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We worked out that the derivative of
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
1 over x was minus
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
1 over x squared.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And we also computed the derivative of
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
x to the nth power.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
for n equals 1,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
2, etc.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that turned out to be x,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
sorry,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
n x to the n -1.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's what we did last time.
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S… Speaker 3 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
today I want to finish up with other
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
points of view.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
on what a derivative is.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is extremely important.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It's almost the most important thing I'll be saying in the class.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
But you'll have to think about it again when you start over and start using calculus in
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the real world.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So again,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we're talking about what is a derivative.
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S… Speaker 3 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this is just a continuation of last time.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So as I said last time,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we talked about geometric interpretations.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
What we're going to talk about is rate of change as
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
an interpretation of the
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
derivative.
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S… Speaker 3 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
remember,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we drew graphs of functions,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
y equals f ,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and we kept track of the
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
change in x,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and here the change in y,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
let's say.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
From this new point of view,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
a rate of change,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
keeping track of the rate of change of x and the rate of change of y,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it's the relative rate of change we're interested in.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's delta y over delta x.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that has another interpretation.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
This is the average change.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Usually we would think of that if x
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
were measuring time.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so the average,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and that's when this becomes a rate.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And the average is over the time interval delta x.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then the limiting value is
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
denoted dy dx.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that is the,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so this one is the average rate of change.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And this one is the instantaneous rate.
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S… Speaker 3 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's the point of view that I'd like to discuss now and give you just a couple of examples.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's see.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe some examples from physics here.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So q is usually the name for a
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
charge.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then dq
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
dt is what's known as current.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's one physical example.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
A second example,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is probably the most tangible one,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is we could denote the letter s
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
by distance.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The rate of change is what we call speed.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Two typical examples.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I want to illustrate the second example
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
in a little bit more detail.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because I think it's important to have some visceral sense of this
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
notion of instantaneous speed.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I get to use the example of
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this very building to do that.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Probably you know,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
or maybe you don't,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that on Halloween
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
there's an event that takes place in this building,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
or really from the top of this building,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is called the pumpkin drop.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So let's
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
illustrate this idea of rate of change with
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the pumpkin drop.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So what happens is,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this building,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
well let's see,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
here's the building.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And here's the dot,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's the beautiful grass out on this side of the building.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then there's some people up
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
here and very small objects.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
they're not that small when you're close to them.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
They get dumped over the side
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
there.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And they fall down.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
You know,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
everything at MIT,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
or a lot of things at MIT,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
are physics experiments.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the pumpkin drop.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So roughly speaking,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
actually the building's about 300 feet high.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We're down here on the first
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
usable floor.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so we're going to use,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
instead of 300 feet,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
just for convenience purposes,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we'll use 80 meters.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because that makes the numbers come out simply.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So we have the height,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which starts out at 80 meters at time 0.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then the acceleration due to gravity gives
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you this formula for h.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
This is the height.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So at time t equals 0,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we're up at the top,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
h is 80 meters.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So the units here are meters.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
At time t equals 4,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you notice 5 times 4 squared is 80.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So I picked these numbers conveniently so that we're down at
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the bottom.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So this notion of
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
average change here,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so the average change,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
or
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the average speed here,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
maybe we'll call it the average speed,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
since that's
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
over this time that it takes for the pumpkin to drop,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is going to be the change in h divided
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
by the change in t.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Which is,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it starts out at,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
what does it start out as?
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It starts out as 80,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
right?
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And it ends at 0.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So actually we have to do it backwards.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We have to take 0 minus 80.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because the first value is the final position,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and the second value is the initial position.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's divided by 4 minus 0,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
times 4 seconds minus times 0 seconds.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so that,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
of course, is minus 20 meters per second.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is 20 meters a second.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So why did I pick
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this example?
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because, of course,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the average,
9:03
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
although interesting,
9:04
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is not really what anybody cares about who actually goes to the event.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
All we really care about is the instantaneous speed when it
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
hits the pavement.
9:14
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so that can be
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
calculated.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
at the bottom.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So what's the instantaneous speed?
9:25
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
That's the derivative.
9:25
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Or maybe I should,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
to be consistent with the notation I've been using so far,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's d by dt .
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
d by dt .
9:38
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Now remember,
9:39
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we have formulas for these things.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We can differentiate this function now.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We did that yesterday.
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We're going to take the rate of change,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and if you take a look at it,
9:51
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it's just the rate of change of 80 is 0,
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S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
minus the rate of change for this minus 5t squared.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So that's using the fact that d by dt of 80
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is equal to 0.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And d by dt of t squared is
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
equal to 2t.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The special case,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
well, I'm cheating here,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
but there's a special case that's obvious.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I didn't throw it in over here.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The case n equals 2 is that second case there.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
But the case n equals 0 also works.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because that's constants,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the derivative of a constant is 0,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and then the factor n there is 0.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's consistent.
10:36
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And actually,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
if you look at the formula above it,
10:38
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you'll see that it's the case n equals minus 1.
10:42
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So we'll get a larger pattern soon enough
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
with the powers.
10:48
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
anyway, back over here,
10:51
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we have our rate of change.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is what it is.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And at the bottom,
10:56
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
at the point of impact,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
We have t is equal to 4.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so h',
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is the derivative,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is equal to minus 40 meters
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
per second.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So twice as fast
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
as the average speed here.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And if you need to convert that,
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's about 90 miles an hour.
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S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Which is why the police are there.
11:27
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
at midnight on Halloween to make sure you're all safe.
11:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And also why when you come you have to be prepared to clean
11:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
up afterwards.
11:36
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So anyway,
11:37
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's what happens.
11:39
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It's 90 miles an hour.
11:40
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It's actually, the building's a little taller,
11:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
but there's air resistance.
11:43
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'm sure you can do a much more thorough study
11:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
of this example.
11:50
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
11:50
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so now I want to give you a couple of more examples because time
11:55
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and these kinds of parameters and variables are not the only
11:59
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
ones that are important for calculus.
12:02
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
If it were only this kind of physics that was involved,
12:05
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
then this would be a much more specialized subject than it is.
12:08
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so I want to give you a couple of examples that
12:13
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
don't involve time as a variable.
12:16
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So the third example I'll give here is the
12:21
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
letter t often denotes temperature.
12:23
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then
12:28
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
dt dx would be what is
12:32
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
known as the temperature gradient.
12:35
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Which
12:39
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we really care about a lot when
12:43
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we're predicting the weather.
12:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because it's that temperature difference that causes air flows and causes
12:48
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
things to change.
12:51
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then there's another theme which is throughout
12:56
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the sciences and engineering,
13:00
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which I'm going to talk about under the heading of sensitivity of
13:05
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
measurements.
13:09
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
13:15
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
let me explain this.
13:17
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I don't want to belabor it because I just am doing
13:21
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this in order to introduce you to the ideas on your problem set,
13:26
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which are the first.
13:28
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So on problem set one,
13:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you have an example which is based on a
13:35
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
simplified model of GPS.
13:36
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Sort of the flat Earth model.
13:38
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And in that situation,
13:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
well if the Earth is flat,
13:42
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it's just a horizontal line like this.
13:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then you have a satellite,
13:46
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is over here.
13:49
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Preferably
13:53
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
above the Earth.
13:57
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The satellite knows exactly,
14:01
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
or the system knows exactly where the point directly below
14:05
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the satellite is.
14:06
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So this point is treated as known.
14:08
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And
14:12
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm sitting here with
14:17
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
my little GPS device.
14:20
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I want to know where
14:25
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I am.
14:25
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And the way I locate where I am is I communicate
14:29
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
with this satellite.
14:32
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
by radio signals.
14:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I can measure this distance here,
14:36
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is called h.
14:37
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then the system will compute this
14:42
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
horizontal distance,
14:43
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is l.
14:46
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So in other words,
14:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
what's measured,
14:48
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so
14:53
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
h measured by radios.
15:00
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
radio waves and a clock,
15:01
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
or various clocks.
15:03
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then L is deduced
15:08
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
from H.
15:12
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And what's critical in all of these systems is
15:17
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that you don't know H exactly.
15:19
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
There's an error in H,
15:22
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which will denote delta H.
15:26
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
There's some degree of uncertainty.
15:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The main uncertainty in GPS is from the ionosphere.
15:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
But there are lots of corrections that are made of
15:39
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
all kinds.
15:40
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And also,
15:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
if you're inside a building,
15:42
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it's a problem to measure it.
15:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
But it's an extremely important issue,
15:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
as I'll explain in a second.
15:49
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So the idea is we then get to the,
15:53
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
get at delta L is estimated by
16:01
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
considering this ratio,
16:02
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
delta L over delta H,
16:04
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is going to be approximately the same as the
16:09
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
derivative of L with respect to H.
16:11
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the thing that's easy because,
16:16
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
of course, it's calculus.
16:17
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Calculus is the easy part.
16:21
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that allows us to deduce something about the real world that's
16:25
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
close by,
16:26
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
over here.
16:28
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So the reason why you should care about this quite a bit is that it's used all
16:32
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the time to land airplanes.
16:33
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so you really do care that they actually know to within
16:37
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
a few feet or even closer where your plane is
16:42
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and how high up it is and so forth.
16:43
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
16:48
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's it for the general introduction to what a derivative is.
16:51
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm sure you'll be getting used to this in a lot of different contexts throughout the course.
16:59
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
to some rigorous details.
17:03
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
OK?
17:04
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Is everybody happy with what we've got so far?
17:07
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Yeah?
17:09
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Ah,
17:13
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
good question.
17:14
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The question was,
17:15
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
how did I get this equation for height?
17:16
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I just made it up because
17:20
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it's the formula from physics that you
17:25
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
will learn when you take eta 1.
17:29
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And it has to do with the fact that,
17:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
in fact,
17:32
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it has to do with the fact that if you,
17:33
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this is speed,
17:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
if you differentiate another time,
17:36
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you get acceleration.
17:37
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And this is the acceleration due to gravity is 10 meters per second,
17:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which happens to be the second derivative of this.
17:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
But anyway,
17:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I just pulled it out of a hat from your physics class.
17:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So you can just say C801.
17:51
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
All right,
17:55
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
other questions?
18:02
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so let's go on now.
18:04
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Now I have to be a little bit more systematic about limits.
18:07
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And so let's do that now.
18:11
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
18:20
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
now what I'd like to talk about is limits and continuity.
18:24
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And
18:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this is a warm -up for deriving all the rest
18:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
of the formulas.
18:35
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
All the rest of the formulas that I'm going to need to differentiate
18:39
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
every function you know.
18:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Remember, that's our goal,
18:42
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and we only have about a week left.
18:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So we'd better get started.
18:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So first of all,
18:48
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
there's what I will call easy limits.
18:52
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So
18:59
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
what's an easy limit?
19:00
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
An easy limit is something like the limit as x
19:04
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
goes to 4 of x plus 3 over
19:08
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
x squared plus 1.
19:10
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And with this kind of limit,
19:13
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
all I have to do to evaluate it is to plug in x
19:17
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
equals 4.
19:19
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because, so what I get here is 4 plus 3 divided
19:23
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
by 4 squared plus 1.
19:25
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's just 7
19:29
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
divided by 17.
19:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And that's the end of it.
19:32
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So those are the easy limits.
19:34
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The second
19:38
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
kind of limit,
19:39
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
well,
19:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so this isn't the only second kind of limit,
19:43
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
but I just want to point this out,
19:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
it's very important,
19:45
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is that derivatives are always
19:52
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
harder than
19:57
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this.
20:00
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
with nothing here.
20:02
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So why is that?
20:04
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Well,
20:05
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
when you take a derivative,
20:06
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
you're taking the limit as x goes to x0 of
20:11
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
f of x,
20:12
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
well,
20:13
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
f of x,
20:15
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
let's try,
20:16
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
we'll write it all out in all its glory.
20:21
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Here's the formula
20:27
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
for a derivative.
20:28
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
20:29
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
notice that if you plug in x equals x0,
20:35
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
always gives 0 over 0.
20:38
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So it just basically never works.
20:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So we always are going to need some cancellation to
20:45
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
make
20:51
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
sense out of the limit.
20:53
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Now,
21:06
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
in order to make
21:10
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
things a little easier for myself to explain what's
21:14
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
going on with limits,
21:15
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I need to introduce just one more piece of notation.
21:18
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
What I'm going to introduce here is what's known as a left hand and a right hand
21:22
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
limit.
21:23
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
If I take the limit as x tends to x0 with a plus
21:27
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
sign here of some function,
21:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this is what's known as the right -hand limit.
21:43
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I can display it visually.
21:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So what does this mean?
21:45
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It means practically the same thing as x tends to x0,
21:48
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
except there's one more restriction which has to do with this plus sign,
21:52
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is we're going from the plus side of x0.
21:55
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
That means x is bigger than x0.
21:58
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I say right hand,
21:59
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so there should be a hyphen here,
22:01
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
right hand limit.
22:03
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Because on the number line,
22:05
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
if x0 is over here,
22:08
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the x is to
22:13
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
the right.
22:14
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
All right, so that's a right -hand limit.
22:16
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then,
22:17
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
this being the left side of the board,
22:18
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I'll put on the right side of the board the left limit,
22:22
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
just to make things confusing.
22:23
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So that one has the minus sign here.
22:27
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I'm just a little
22:31
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
dyslexic,
22:32
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and I hope you're not.
22:33
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So I may have gotten that wrong.
22:37
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So this is the left -hand limit.
22:40
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And I'll draw,
22:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so of course that just means x goes to x0,
22:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
but x is to the left of x0.
22:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And again on the number line,
22:49
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
here's the x0,
22:51
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
and the x is on the other side of it.
22:56
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
OK, so those two notations are going to help us to clarify a
23:01
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
bunch of things.
23:01
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It's much more convenient to have this extra bit of description
23:06
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
of limits than to just consider limits from both sides.
23:11
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
OK,
23:16
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so I want to give an example of this.
23:19
S… Speaker 2 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And
23:26
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
also an example of how you're going to think about these
23:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
sorts of problems.
23:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which has two different definitions.
23:37
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Say it's x plus 1 when x is bigger
23:41
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
than 0, and minus x plus 2 when x
23:45
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
is less than 0.
23:46
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So maybe put commas there.
23:49
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So when x is greater than
23:53
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
0, it's x plus 1.
23:54
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Now I
23:58
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
can draw a picture of this.
24:00
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
It's going to be kind of a little small,
24:02
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
because I'm going to try to fit it down in here.
24:06
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
axis down below.
24:07
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So at height 1,
24:10
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I have,
24:11
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
to the right,
24:12
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I have something of slope 1.
24:14
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So it goes up like this.
24:15
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then to the left
24:20
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
of 0,
24:21
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
I have something which has slope negative 1.
24:24
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
But it hits the axis at 2,
24:28
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so it's up here.
24:30
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
So I have this sort of strange antenna figure here,
24:34
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
which is my graph.
24:35
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Maybe I should draw these in another color to depict
24:40
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that.
24:40
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And then
24:44
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
if I calculate these two limits here,
24:47
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
what I see is that the
24:51
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
limit as x goes to
24:56
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
0 from above of f ,
24:58
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
that's
25:00
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
The same is the limit as x goes to 0 of the
25:04
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
formula here.
25:05
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
x plus 1.
25:06
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
Which turns out to be 1.
25:09
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
And if I take the limit from the,
25:13
S… Speaker 1 (Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007)
so that's the left -hand limit,

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