U1S2V05 Secant lines defined.txt

#
# File:   content-mit-18-01-1x-captions/U1S2V05 Secant lines defined.txt
#
# Captions for MITx 18.01.1x module [gUsr2VuO7Ek]
#
# This file has 41 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

Recall that we want to find the slope of the tangent line.
In order to do this, we're going to introduce a new concept--
the secant line.
To get started, we're going to draw
some function y equals f of x.
And then we're going to choose two points.
Here's the point a and here is the point b.
We can find the point on the graph above the point a
and we can find the point on the graph above the point b.
Then we can draw a line through these two points.
And this line is a secant line.
Now a natural question that we can
ask ourselves is what is the slope of this line?
From your other math classes, you
may remember that the slope is defined
as the rise over the run.
So let's go ahead and identify the rise and the run
on our graph.
This vertical distance here is the rise,
which we write with this symbol, delta y.
This triangle here is the Greek letter delta.
It stands for difference because it
is the difference in the height of these two points.
We often read this symbol as the change in y.
Now where is the run?
The run is this horizontal distance here, delta x.
Again, it stands for the different
because it is the difference in the x values of these two
points.
We often read this as the change in x.
This allows us to identify the slope
of the secant line, the rise over the run, which
is delta y over delta x.
Of course, we'd like a formula that's
a little bit more explicit in terms of our function f of x.
But before we do that, you're going
to go ahead and do some problems to recall
the information about average rate of change
that we learned before.