U1S2V09 Limits of Secant Lines.txt

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I hope you had a good time playing
with the secant approximation mathlet.
I want to remind you what we're looking for.
Remember that we want to find the slope of the tangent line,
but we don't know how to do that yet.
So we introduced the secant line,
and we were able to find the slope of a secant line.
So we're going to use what we know about the secant line
to understand the slope of the tangent line better.
Let's see how.
The first thing I'm going to do is to pick a value,
x equals a, using this horizontal slider.
So I can choose any point that I want here.
The value of this point is not special in any way.
Any point that you choose will be fine.
Then by clicking the Secant button down here,
you notice that the secant line appears,
and it goes through the function above our point a
and through our function above a point
b, a distance delta x away from a.
We can change the value of delta x
by moving this horizontal slider.
Observe what happens.
As the point b moves closer and closer to the point a,
the secant line appears to become tangent
to our function at the point a.
Let's go ahead and hit the Tangent button down here.
The tangent line appears, and we can go ahead and verify
that, in fact, as the point b moves closer and closer
to the point a, the secant line is, in fact, becoming
the tangent line.

Let's go ahead and recap what we've just seen.
We saw that in the limit as b approaches a,
the secant line approaches the tangent line.
So in particular, the slope of the secant line as b
approaches a becomes the slope of the tangent line.
Now we've already computed a formula for the slope
of the secant line.
We saw that this was delta f over delta x, which
is f of b minus f of a all over b minus a,
and we're interested in what happens as b approaches a.
Now I hope this looks a little bit familiar to you
because in the limit, as b approaches a,
this is our definition for the derivative f prime of a.
So this is a formula for the slope of the tangent line.
The slope of the tangent line is the derivative
of the function at a.
And we found an interpretation for the derivative
as the instantaneous rate of change of our function at a.
So just as the slope of the secant line
is measuring the average rate of change of our function,
as we take the limit of the secant line as b approaches a,
we're getting the tangent line slope,
which is measuring the instantaneous rate of change
of our function.

So what we see on this left-hand side
are three interpretations for the same object.
We have a geometric interpretation
as the slope of the secant line, we
have a symbolic interpretation which
is a formula for that slope, and we
have a physical interpretation in terms
of the average rate of change.
And then when we take the limit as b approaches a,
we approach the object on the right-hand side.
It's pretty amazing how all these objects are related.
So I want you to do a quick concept check,
and then we'll be back to look at a worked example
where we approximate derivatives by estimating
slopes of tangent lines.