Main content

## Differential Calculus

### Course: Differential Calculus > Unit 2

Lesson 2: 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

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Secant line with arbitrary difference

Sal finds the slope of the secant line on the graph of ln(x) between the points (2,ln2) and (2+h,ln(2+h)).

## Want to join the conversation?

- Clarification of terms requested. At4:05, Sal refers to slope of the secant line as delta y / delta x (rise / run). The line is highlighted in blue. Is not rise over run the same as opp / adj and isn't this the tangent, not the secant? And the blue highlighted line, isn't that the hypotenuse, not the secant. I thought the secant is the inverse of the cosine, or hyp / adj. Thanks for the answers.(7 votes)
- While there is a historical etymological similarity for the trig function terms and the terms as used here in the context of a graph of a function, they are not quite the same thing. You can get an idea of the derivation of the terms here:

https://en.wikipedia.org/wiki/Trigonometric_functions#Sine.2C_cosine_and_tangent

For the purposes of the terms with respect to this aspect of differential calculus:

A**secant line**is a straight line joining two points on a function. It is also equivalent to the average rate of change, or simply the slope between two points.

A**tangent line**is a straight line that touches a function at only one point. The tangent line represents the instantaneous rate of change of the function at that one point. The slope of the tangent line at a point on the function is equal to the derivative of the function at the same point .

http://mathworld.wolfram.com/TangentLine.html

http://mathworld.wolfram.com/SecantLine.html

and

http://mathworld.wolfram.com/Circle-LineIntersection.html(13 votes)

- What is arbitrary difference?(3 votes)
- "Arbitrary" means "chosen in no particular way". So if we have an arbitrary number, x, between 0 and 1, that means that x can be any number between 0 and 1.

Arbitrariness is important in math, especially with proofs, because if we prove that (for instance) some property holds with an arbitrary distance, then we know that the property holds for any distance at all.(10 votes)

- Dears, I am confused a little. Are these two "guys" the same:

- "secant line"

and

- "slope of secant line"

or not?(3 votes)- The secant line is a geometric construction. It's your normal concept of a line, drawn between two points.

The slope of the secant line is a real number. It describes how tilted the secant line is.(7 votes)

- What does arbitrary difference or arbitrary point mean? It is in the title but not in video.(3 votes)
- Arbitrary in this case is used to mean "as close as you want it to be," which is extremely important when we're discussing tangents to the curve and a derivative of the function at a point.(6 votes)

- Doesn't "h" have to be specified to be a number greater than -2?(3 votes)
- Strictly speaking I would say you are correct – in the context of derivatives
**h**is usually thought of as being "arbitrarily" small (i.e. close to zero), which is why Sal probably didn't think (or bother) to mention that ...(2 votes)

- what is the difference between ln(x) and log(x)?(1 vote)
- Ln is a logarithm taken to the base e (also called the natural log)

e=2.718...

More on e here.

https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/e-and-the-natural-logarithm/v/e-through-compound-interest

Log is by convention, logarithm taken to the base 10.

If you have confusion on what logarithms and bases are, I suggest seeing the logarithms playlist on KA.

https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/introduction-to-logarithms/v/logarithms(5 votes)

- Couldn't you further simplify to ln(h) / h? from ln(2) + ln(h) - ln(2) / h(1 vote)
- Couldn't you simplify the answer down further for it to become (ln(h)-ln(2))/h since ln = log base e?(0 votes)
- There might be way to simplify even further but there is no point in doing it, because it is not goal of these examples.(2 votes)

- a normal formula could solve all these kinds of question(0 votes)
- Many things could be solved differently. But important in mathematics and these lessons is to show us the connection between all of these thing instead just blindly teach us how to use formulas.(5 votes)

## Video transcript

- [Voiceover] A secant
line intersects the curve y equals the natural log of x at two points, with x-coordinates two and two plus h. What is the slope of the secant line? Well, they're giving us
two points on this line. It might not be immediately
obvious, but they're giving us the points when x is equal to two, when x is equal to two, what is y? Well y, they tell us, y is
equal to the natural log of x, so in this case it is going
to be the natural log of two, and when x is equal to two plus h, what is y? Well, y is always going
to be the natural log of whatever x is. So it's going to be the
natural log of two plus h. And so these are two points
that sit on the secant line. This happens to be where
the secant line intersects our curve, but these are
two points on the line, and if you know two points on a line, you will then be able to figure
out the slope of that line. Now we can just remind ourselves that a slope is just change
in y over change in x, and so what is this going to be? Well if we view the second
one as our endpoint, our change in y going from ln
of two to ln of two plus h, so our change in y is
going to be our endpoint. So, natural log of two plus
h minus our starting point, or our end y-value minus
our starting y-value. Natural log of two and then our change in x, our change in x is going
to be our ending x-value, two plus h, minus our starting
x-value, minus two, and of course these twos cancel
out, and if we look here it looks like we have a
choice that directly matches what we just wrote. This right over here, natural log of two plus h minus natural log of two over h. Now, if you wanna visualize
this a little bit more, we could draw a little bit,
I'm gonna clear this out so I have space to draw the graph, just so you can really visualize
that this is a secant line. So let me draw my y-axis, and let me draw my x-axis, and y equals the natural
log of x is going to look, so let me underline that, that is going to look something like this. I'm obviously hand drawing it, so not a great drawing, right over here. And so when we have the point two comma natural log of two, which would be, lets say it's over, so if this is two, then this right over here is
the natural log of two, so that's the points two comma
natural log of two, and then we have some other, we just
noted the abstract two plus h, so it's two plus something. So let's say that is two plus h. And so this is going to be the point where we sit on the graph. That's going to be two plus
h comma the natural log of two plus h, and the exercise
we just did is finding the slope of the line
that connects these two. So the line will look
something like that, and and the way that we did
this is we figured out, okay what is our change in y? So our change in, so let's
see, we are going from y equals natural log of two to y equals natural log of two plus h. So our change in y, our change in y is our natural log of two plus h minus natural log of two. Minus natural log of
two, and our change in x? Well we're going from two to two plus h, we're going from two to two
plus h, so our change in x, we just increased by h. We're going from two to two plus h, so our change in x is equal to h. So the slope of the secant line, the slope of the secant line, a secant that intersects
our graph in two points is going to be change
in y over change in x, which is once again exactly
what we have over there.