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## Secant lines

# Slope of a line secant to a curve

## Video transcript

So let's review
the idea of slope, which you might remember
from your algebra classes. The slope is just the
rate of change of a line. Or the rate of change
of y, with respect to x, as we go along a line. And you could also
view it as a measure of the inclination of a line. So the more incline the line
is, the more positive of a slope it would have. So this right over here,
this has a positive slope. It's increasing as x increases. And if this had an even
higher inclination like this, if it increased even
more as x increased, then it would even
have a higher slope. So this right over
here is some line. So that's some line. And just as a reminder,
we can figure out the slope between two points. Two points define a line. And between those
two points, we can find the rate of change
of y, with respect to x. So let's put two points on here. So let's say that this point
right over here, this x value, is x sub-- well, this
is pronounced x naught, or x sub 0 is just x naught--
and when x is x0 for this line, y is y0. So this is a point x0, comma y0. And let's say we have another
point all the way over here. And let's say that this x value,
this x value right over here is x sub 1. And the y value over
here is y sub 1. So this is the point
x sub 1, y sub 1. So just as a review,
the slope of this line, and a line by definition,
has a constant slope between any two
points that you pick. The slope of this line, which is
often denoted by the letter m, is your rate of change
of y with respect to x. Or another way of
thinking about it, for a given change in x,
how much are you changing y? Or a change in y divided
by a change in x. Just as a reminder,
this triangle, that's the Greek letter delta. It's shorthand for
change in whatever. So change in y over change in x. So let's think about what this
is going to be for this example right over here. Well, let's think about
change in x, first. So we are moving from x0 to x1. So our change in x. So this is our change
in x, right over here. We're starting at
x0 and going to x1. That is our change in x. I'll put that in
that pink color. That is our change in x. And what is it equal to? Well if we're ending
here and we started here, let's just do ending point
minus starting point. So it is x1 minus x0. And that way, doing
it this way, I would have made sure that
I have a positive value. I'm just assuming that
x1 is larger than x0. And what is my change in y? Well, once again,
ending point, ending y value minus starting y value. y1 minus y0. Now, you might be
saying hey, could I have done y0 minus y1
over x0 zero minus x1? Absolutely. You could have done that. Then you would have just
gotten the negative of each of these values in the
numerator and denominator, but they would
have canceled out. The important thing is
that you're consistent. If you're subtracting you're
starting value from your ending value in the numerator, you have
to subtract your starting value from your ending value in
the denominator as well. So this right here you probably
remember from algebra class. The definition of
slope is the rate of change of y
with respect to x. Or it's the rate of change
of our vertical axis, I should say, with respect
to our horizontal axis. Or change in y, or change
in our vertical axis over change in a
horizontal axis. Now I'm going to introduce
a little bit of a conundrum. So let me draw another
axis right over here. Scroll over a little
bit just so we have some space to work with. So that was for a line. And a line, by definition,
has a constant slope. If you calculate this between
any two points on the line, it's going to be
constant for that line. But what happens when we
start dealing with curves? ' When we start dealing with
non-line or non-linear curves. So let's imagine a curve that
looks something like this. So what is the
rate of change of y with respect to x of this curve? Well, let's look at it
at different points. And we could at least
try to approximate what it might be in any moment. So let's say that this
is one point on a curve. Let's call that x1,
and then this is y1. And let's say that this is
another point on a curve right over here, x2. And let's call this y2. So this is a point x1, y1,
this is a point x2, y2. So we don't have the tools yet. And this is what's
exciting about calculus, we will soon have the
tools to figure out, what is the rate of
change of y with respect to x at exactly this point? But we don't have that tool yet. But using just the
tools from algebra, we could at least
start to think about, what is the average
rate of change over the interval from x1 to x2? Well, what's the
average rate of change? Well, that's just how
much did my y change-- so that's my change in
y-- for this change in x. And so we would calculate
it the same way. y2 minus y1 over x2 minus x1. So our change in y
over this interval is equal to y2 minus
y1, and our change in x is going to be
equal to x2 minus x1. So just like that we
were able to figure out the rate of change
between these two points. Or another way of
thinking about it is, this is the average rate of
change for the curve between x equals x1, and x is equal to x2. This is the average
rate of change of y with respect to
x over this interval. But what if we have we
also figured out here? Well, we figured out
the slope of the line that connects these two points. And what we call a
line that intersects a curve in exactly two places? Well, we figured out, we
call that a secant line. So this right over
here is a secant line. So the big idea here is we're
extending the idea of slope. We said, OK, we already knew
how to find the slope of a line. A curve, we don't
have the tools yet, but calculus is about
to give it to us. But let's just use
our algebraic tools. We can at least figure
out the average rate of change of a curve, or a
function, over an interval. That is the same exact thing
as the slope of a secant line. Now just as a little
bit of foreshadowing, where is this all going? How will we eventually
get the tools, so that we can figure out the
instantaneous rate of change, not just the average? Well just imagine what happens
if this point right over here got closer and
closer to this point. Then the secant line is
going to better and better and better approximate
the instantaneous rate of change right over here. Or you could even think of it as
the slope of the tangent line.