U1S2V07 Slopes of Secant Lines.txt

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Let's get back to thinking about the secant line,
and what we want to do is identify
the slope of this line in terms of our function f.
So let's start with the rise, delta y.
The value of delta y is given by the value
of our function at the point b f of b
minus the value of this function at the point a
giving us f of b minus f of a.
This is the difference or the change in f.
So a good notation for this might be delta f.

Keep in mind that delta f and delta y
mean exactly the same thing.
Both of these notations are used.
So you're going to have to get used to it
and understand that delta y is a quantity which
is equal to the quantity delta f,
and they may be used interchangeably.
OK.
Now, enough about delta f.
Let's think about the run, delta x.
What is this?
Well, that's the change in this horizontal distance.
That's just b minus a.
So what is this slope?
It is given by the ratio f of b minus f of a all
over b minus a.
Now, hopefully, this expression looks
a little bit familiar to you.
This is the expression that computes
the average rate of change of our function
f with respect to x.
So what have we just done?
We've seen that the slope of the secant line
is measuring the average rate of change of this function f.
Pretty cool, huh?
Now, I want you to take a second,
and you're going to play with the secant approximation
mathlet to see what happens to the secant line
if you move the point b closer and closer to the point a.
Good luck, and we'll see you shortly.