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Next I'm going to take a few minutes to define what a secant line is and what a tangent line
is, and then in the next video we'll look at the relationship between secant lines and
tangent lines. Now, if you were paying close attention in the previous videos, all of this
that I'm about to talk about should seem pretty familiar to you. It's a lot like the relationship
between average velocity and instantaneous velocity. So, let me start by just writing
a definition of a secant line. A secant line is just a line joining two points on a curve
-- its pretty simple, really. So, a line joining two points on a curve is a secant line. So, to relate this back to the
videos you just watched, the average velocity is the slope of a secant line from a position
versus time graph. So, in those previous videos when Jim was calculating the average velocity,
what he was really doing was calculating the slope of a secant line from the position versus
time graph. But, let me show you what secant line looks like on a graph. So, here's a graph
of a simple parabola, y equals one half x squared. So, like I said, a secant line is
just a line joining two points on a curve. So, let's start by drawing a secant line.
I'm going to actually draw several secant lines, so I'm going to use different colors
to make them a little easier to see. So, the first secant line I'm going to draw is going
to go between x equals one here and x equals three. Now, of course the line would continue
on in either direction here, but those are kind of the reference points I'm using. So,
let's take a look at the secant line here and let's find its slope. I'm going to create
a chart out here to the side. I'm going to actually end up drawing several secant lines
and we're going to calculate the slope of those different secant lines. So, this first
one I'm doing -- the interval I'm looking at is between one and four. Sorry -- I said
three before, but its actually between x equals one and x equals four. So, how do I find the
slope of the secant line? Well, I just find that slope like I would find the slope of
any other line -- change in y over change in x. What that's going to be in this case
is f of four minus f of one -- so the function value at four minus the function value at
one -- over four minus one. So, that's just change in y over change in x. Now, f of four,
if I look at my equation there, that's going to be one half times sixteen, so that's eight.
F of one is one half times one, so that's one half, or 0.5, and that's over three. And
so if you work that out, you should get 2.5. So, this is a lot like the formula for average
velocity, right? Because, like I said, the average velocity is just the slope of a secant
line. So, that's the slope of our red secant line. Let's do a few more secant lines. Let
me draw one between x equals one and x equals three. And I'll make it a blue line this time.
And I want to go between x equals one and x equals three. Let's calculate this slope.
So, now I'm looking between one and three. Once again, the slope of the secant line is
just the normal slope formula. So, that's going to be f of 3 minus f of one over three
minus one. F of three is one half times three squared, so that's one half times nine, or
four point five. F of one we've already calculated, right? That's .5. And this is over two. So,
that's four over two, or two. So, the slope of the blue secant line is two. Alright, let's
do just one more secant line. We'll do green this time and this time I'm going to go between
one and two. Hopefully you can see that green line there. Once again, let's go ahead and
calculate this slope as well. Now I'm looking at the secant line between one and two. So
I'm going to do f of two minus f of one over two minus one. F of two is going to give me
one half times four, so that's two. Minus f of one which we found to be .5. Two minus
one in the denominator is one. So, that's 1.5. So, what do we see when we look at these
slopes? So, I've got the slopes of three different secant lines, and I can see that actually
these slopes are decreasing, right? I've got 2.5, 2, and then 1.5. And if I look at the
graph that makes sense, right? Because the secant lines are getting a little flatter,
right, they're getting less steep as I go down. And we'll go into a little more detail
about that in a few minutes. But now, let me switch gears and tell you what a tangent
line is. So, the easiest way for me to tell you what a tangent line is is just going to
be to draw it on this graph. So, let me go ahead and do that. To make it clear, let me
use a different color. Maybe I'll use this purple color here and a dotted line. Let me
draw this in and then I'll explain what's going on. Hopefully you can see that line
there. This is the tangent line at x equals one. Remember, a secant line connects two points on a curve. Now, what a tangent
line does, is it just skims one point on the curve. So, my tangent line just touches at
x equals one, but that's the only point it touches on the curve. So, that's what a tangent
line is. In just a second we're going to go into detail about the relationship between
secant lines and tangent lines.