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### Course: Calculus, all content (2017 edition) > Unit 2

Lesson 4: Secant lines- Slope of a line secant to a curve
- Secant line with arbitrary difference
- Secant line with arbitrary point
- Secant lines & average rate of change with arbitrary points
- Secant line with arbitrary difference (with simplification)
- Secant line with arbitrary point (with simplification)
- Secant lines & average rate of change with arbitrary points (with simplification)
- Secant lines: challenging problem 1
- Secant lines: challenging problem 2

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# Slope of a line secant to a curve

What is the slope of a line between two points on a curve? This is called a "secant" line. Created by Sal Khan.

## Want to join the conversation?

- What are the main differences between a tangent line and a secant line in a curve?(20 votes)
- A tangent line touches the curve at one point and has the same slope as the curve does at that point. A secant line intersects at 2 or more points and has a slope equal to the average rate of change between those points.(34 votes)

- I'm having difficulty understanding the concept of a secant line as it pertains to a sine graph. Specifically, there is a problem in the "Slope of secant lines" exercise, where there are four questions. In each you are asked to evaluate the rates of change between secant lines for four different points.

The second question asks whether [Sin 5/2 Pi - Sin 1/2 Pi / 5/2 Pi - 1/2 Pi] is greater than, less than or equal to [Sin 2/3 Pi - Sin 1/3 Pi / 2/3 Pi - 1/3 Pi]. Evaluating the first part, I see that 5/2 Pi is the same as 1/2 Pi, and therefore the change in Y is zero. However, when evaluating the second part, it seems to me that Delta Y/Delta X is Sin 1/3 Pi / 1/3 Pi.

But the lesson tells me that the secant line is horizontal, which I do not understand. Can anyone help me with understanding secant lines on a sine graph? I did all of the trig lessons and most of the videos already, but if anyone can clarify this concept or point me in the direction of some valuable review materials, that would be legit. ( :(17 votes)- You're correct that sin(5π/2) - sin(π/2) = 0, so the slope of the first line is 0 making it horizontal.

Let's look at the second slope. In the numerator it has sin(2π/3) - sin(π/3).

Can we simplify that?

Remember that sin x = sin(π - x), so

sin(2π/3) = sin(π - 2π/3) = sin(π/3).

The numerator then becomes

sin(2π/3) - sin(π/3) = sin(π/3) - sin(π/3) = 0.

So, both slopes are horizontal and thus equal.

Maybe the mistake you made was that you thought that

sin(2π/3) - sin(π-3) = sin(2π/3 - π/3)?

The two aren't equal.(22 votes)

- Does a secant line always only touch 2 points on the graph? (is this not doable for some periodic functions where it would hit it 3 times + for example?)(8 votes)
- As the term is typically used in calculus, a secant line intersects the curve in two places
**locally**-- it may or may not intersect the curve somewhere else. So the requirement of just two intersections applies just to the small region of interest and is not a strict requirement for regions you are not concerned with at the moment.

Note: in some fields of geometry the requirement of exactly two points of intersection is much more strict than what we usually have in calculus.(18 votes)

- what is the difference between a chord and a secant?(2 votes)
- A chord is a line segment, a secant is a line.(11 votes)

- Why does he use x not and sub 1 in the first graph but sub 1 and sub 2 in the second graph? Is there any special situation that those are reserved for?(4 votes)
- No, they are just placeholders. Nothing is stopping you from writing x205, but for the sake of clarity he just uses smaller values.

X(anything) just means a non determined value of x.(4 votes)

- Can't we calculate slope in the same way as we do for lines?We have coordinates in the curve.(3 votes)
- We can, and that's the we did in the second part of the video. However the slope that our algebraic tools (for geometry) gives us is the average slope between two points.

However we can't calculate the slope exactly at a point.(1 vote)

- At5:05Sal mentions interval. Can someone explain to me what this 'interval' means? I keep on hearing this term more and more often.(0 votes)
- In this context, "interval" just refers to the section of points (along the x-axis) between x1 and x2. If it were an interval of time, for example, it could be the segment of time between two events. Hope this helps!(7 votes)

- Can't we call that secant line is a chord? Or is it both? Or one in specific?(1 vote)
- At the high school level, the word 'chord' is usually reserved for a line segment across a circle. But in the broader mathematical community, 'chord' and 'secant line' are the same thing.(3 votes)

- In the end, he ends up with finding the slope of a line with points (X0, Y0), (X1, Y1). This way, when I know two points on the line, I can find out the equation of that line. Now, that line, suppose line m, was secant to the curve, suppose L. So Line m is secant to the curve L. Now, let's imagine a line n, which is parallel to line m and tangent to the curve L. So i know the slope of the line m is equal to the slope of the line n, because they are parallel. Hence, can I find out the point at which the tangent n intersects the curve L using just algebra? I think I can, using some concepts in co-ordinate geometry. Suppose that point is (X3, Y3). So without actually using Differential calculus, we have the value of the instantaneous point. So if this was the example of a car running on a road and a graph or it's speed on y and time on x axis, then i guess that at X3, dy/dx = Y3....i.e. instantaneous velocity at X3 = Y3. So we solved it without using calculus. Someone please tell me if i'm correct or wrong!(2 votes)
- The problem is that the slope of the tangent line, while it may be very close, is not exactly the same as the secant line. In the interval between the two point where the secant line and curve are the same, there are an infinite number of tangent lines each with a slightly different slope. You can use algebra to estimate a tangent line, but to actually find the tangent line, you need to find the limit of the slope of the secant lines which is calculus.(2 votes)

- A secant gives the average rate of change of a function. However, this is not the accurate value of the average change. How to find out the average rate of change of an arbitrary function?(1 vote)
- You can use differential calculus to determine the rate of change of functions at given points. So for instance, if I want to know what is the slope of a line tangent to a part of sinx, I need to be able to calculate its derivative.(4 votes)

## 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.