U1S2V04 Intuition for Tangent Lines.txt

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# Captions for MITx 18.01.1x module [Bon3jzROwqU]
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Welcome back.
You just finished answering some questions about what a tangent
line is not, and now we're going to focus
on what a tangent line is.
And to help us with this, we're going
to be using the Tangent Approximation Mathlet.
This is a visualization tool that
is designed to help us build our intuition about what a tangent
line is.
So what you'll notice down here is
that I have the function set to a cubic function,
and I want to zoom in onto this point.
And I can use this slider to specify how zoomed in or out I
am.
And so as I zoom in, I move closer and closer
and closer and closer.
And what you notice is that this function
begins to look more and more like a straight line
until I'm all the way zoomed in.
At that point, the width of this box
is two one hundredth of a unit, so pretty small.
And the function definitely looks like a straight line.
Well, this line is my tangent line.
What do I mean by this?
I mean that my tangent line must point in the same direction
as this function.
So now this kind of makes sense.
When I'm zoomed in far enough, my function looks like a line,
the tangent line is a line, and we
want the tangent line and the function
to have the same slope at this point,
to point in the same direction.
Now, I can use this tool to actually draw
the tangent line by pressing the Tangent button down here.
Then the tangent line appears and I can zoom out
so that you can see the function and the tangent line together.
So notice that the function is curvy
and the tangent line is a line.
But the tangent line is only close to the function
in this very small neighborhood here that we're interested in.
And that makes sense.
The tangent line is only a good approximation
for our function in this small zoomed in neighborhood.
Other places far away from this point, the tangent line
and the function don't agree at all.
So now I'd like you to go ahead and play with this Tangent
Approximation Mathlet yourself.
You can go ahead and change the function using this drop
down menu down here.
I want you to go ahead and zoom in, draw the tangent line,
zoom back out, and get some intuition for what this tangent
line is, how it behaves, and how it is that you draw this.
Then we're going to have to answer
our original problem, which is what
is the formula for this tangent line.