U2S1V08 Concavity and Linear Approximation.txt

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So we've established that making a linear approximation
to a function means using a tangent line
to the graph as an estimate for the actual value
of the function.
So this would be exactly correct if the graph of g
was the line itself.
If it was completely straight.
But of course, most of the time it isn't.
We know that most graphs are going
to have some curviness to them.
And that's going to create the small difference
between the tangent line and the graph.
What allows us to get a handle on this curviness,
well, that's the second derivative.
So when we estimated the square root of 104 for instance,
we had our function being the square root function.
And we started with this point, which was 100, 10.
And the first derivative gave us the slope of this line.
And our estimate then was this point on the line,
it was 10.2, that was our estimate for g of 104.
When you calculated the second derivative of g,
you should have gotten something negative.
So that means that the graph is concave down
and it bends like this.
So the actual value of g of 104, that's going to be around here.
And we can see that it's going to be
a little bit lower than the number our linear approximation
gave us.
So in fact, let's pull out our calculator here.
If we take 104 square root, we're going to get 10.198.
So very close to a 10.2, which was our approximation.
And it's just a little bit lower than our approximation
as we predicted.

While we're on the subject, most of the time
when you ask an computer to do this sort of calculation,
they don't have all the values in memory.
They actually use techniques that are very similar to this.
They get a little bit more complicated,
but linear approximation is really the basis for all of it.

One last thing we should mention,
so we've been saying that the tangent
line is a good approximation to the graph
near the point of tangency.
But how near is near?
Geometrically, we want to know how quickly does the graph of g
bend away from the tangent line.
Does it bend away slowly like this?
Or does it bend away very quickly?
And again, the second derivative is going to help us.
So the larger the second derivative,
we know the bendier the curve.
So if you think about our boat, well we
know our estimate of the position of our boat
would have been exact if the velocity was always constant.
So if the boat just putters along at a constant rate
and is completely predictable.
But of course, the boat doesn't have to do that.
It can accelerate or it can decelerate.
And acceleration, that's our second derivative again.
So that's why the oil tanker's position is easier
to predict then the canoe's position.
So if you took any physics, you know that force
is mass times acceleration.
So the larger the mass of the boat,
the harder it is to accelerate.
And if you have very little acceleration,
like for an oil tanker, than the graph of position
can't be very curvy.
And so our linear approximation is
going to be more accurate for a longer period of time.

So that wraps up linear approximation.
Later in the course, we'll teach you
how to use these second derivatives to actually
quantify how close the linear approximation gets.
And even to improve on the linear approximation.
But linear approximation is really useful just by itself.
And we're going to use it to help
us figure out all sorts of key facts in the rest of this unit.
So hope you enjoyed it.