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### Course: AP®︎ Calculus AB (2017 edition) > Unit 2

Lesson 2: Estimating derivatives with secant lines and average rate of change# Secant lines & average rate of change

Learn how to calculate the average rate of change for a function and its connection to the slope of a secant line. Grasp the concept of instantaneous rate of change and its significance in calculus, leading to the idea of the derivative.

## Want to join the conversation?

- At3:15, if the secant line isn't the exact instantaneous rate of change, what's its purpose? It seems like it's just a crude version of the tangent line. If the tangent line is what's actually important in calculating derivatives, etc., why do we bother with the secant? I understand that the concept of the tangent line was probably derived from the secant line, but why not just use the tangent line and not bother with the secant?(21 votes)
- Calculating average change using secant lines is actually an important intermediate step to finding instantaneous change via tangent lines (and thus also derivatives). As you say, the process of finding the slope of the tangent line descends from the process of finding the slope of a secant line (the only difference is that a certain limit is taken of the
*difference quotient*which is the expression for the slope of a general secant). As for "not bothering with the secant", there is no way to explain the process of finding instantaneous change without explaining how to find average change for the reason you just identified: the principles involved in finding average rate of change are part of finding instantaneous rate of change.

Also, average rates of change have advantages in their own right. In particular, in physics, there are a lot of phenomena that occur that have to deal with average rates of change instead of instantaneous rates of change. Furthermore, if you are looking at discrete data (as is the case in every real world observation), there is no way to get an instantaneous rate of change from that data because it is not continuous.

Lastly, "not having a purpose" (which is not the case with secant lines and average rates of change) is a poor argument for neglecting to study anything – especially in mathematics. There are plenty of things in mathematics that have no real world application, though they are studied nonetheless.

Comment if you have questions!(62 votes)

- What's the best way to memorize the difference between a tangent and a secant line?(9 votes)
- Secant is intersection at 2 points of a curve. Tangent is at 1 ponit.(34 votes)

- because the secant line is basically the hypotenuse of a triangle, could a^2 + b^2 = c^2 work? I tried it myself and got the answer 8.24.. which does not match sal's slope of 4. Am I missing something?(3 votes)
- The Pythagorean theorem involves the
*length*of the hypotenuse, not the slope. We're approximating the slope of the function, so we don't care about the length of the secant line.(23 votes)

- Can anyone explain how the terms 'tangent' and 'secant' lines used here are connected to those which I learnt in trigonometry?

[In short; Why are they called tangent and secant lines?](8 votes)- Secant line is a line that touches a curve at two points, pretty much the average rate of change because it is the rate of change between two points on a curve (x1,y1), (x2,y2) the average rate of change is = (y2-y1)/(x2-x1) which is the slope of the secant line between the two points on the curve.

The tangent line is a line that touches a curve at one point, this line's slope at a point is the derivative in a sense the limit as the change in x between two points of a secant line approach 0. its slope is the derivative of the curve at the point.

Hope that helps(11 votes)

- What is the difference between a secant line and tangent line?(5 votes)
- A secant line represents the rate of change over a 'longer' interval.(The average rate of change) A tangent line represents the rate of change over an infinitesimally 'short' interval.(The instantaneous rate of change)(15 votes)

- can you explain what is the difference between scant line and slope??(4 votes)
- Say you have a curve, and this curve has two points on it. The line between those two points is called a secant line.

The slope is the m in the equation for any line, y=mx+b. The slope describes whether the line is going down or up on the graph, and how quickly it is doing so.

While a secant line has a slope, the two are otherwise rather unrelated.(5 votes)

- what is the difference between the slope of secant line and the slope of tangent line ?(4 votes)
- Basically, both are slopes, except secant and tangent lines are totally different. However, the method of finding the slop for each line remains the same. Secant line is one that intersects two points in a line, whereas tangent line intersects exactly one point on the curve.(3 votes)

- The slope of the secant line was the same as the derivative at 2. Are there other functions that have this trait where the average rate of change is the same as the rate of change in the center of the interval?(2 votes)
- For
*any*function continuous on [a, b] and differentiable on (a, b), there is some point inside the interval where the derivative is equal to the slope of the secant line between a and b.

This is the Mean Value Theorem, which is discussed elsewhere on Khan Academy. This point only occurs at the midpoint of the interval in certain specific cases, and this is largely coincidental.(4 votes)

- Find the slope of the secant line between x=−2 and x=3 on the graph of the function f(x)=−2x2−5x+1.

how do you do problems like this?(3 votes)- The slope of the secant line is the average rate of change. So in general, the slope of the secant line between x=a and x=b on the graph of f(x) is [f(b)-f(a)] / (b-a).

So in your example, calculate {[-2(3)^2-5(3)+1] - [-2(-2)^2-5(-2)+1]} / [3-(-2)].

Have a blessed, wonderful day!(2 votes)

- But what is an anti derivative?(3 votes)
- For a function
`f(x)`

, its anti-derivative is some new function`F(x)`

such that the derivative of`F`

is equal to the original function - that is,`F'(x) = f(x)`

.

With this in mind, anti-differentiating a function is done by basically**reversing**differentiation techniques.

With a simple example: We know that the derivative function for`x^2`

is`2x`

, right?

- The derivative of`x^2`

is`2x`

.

- Then, the antiderivative of`2x`

is`x^2 + C`

.`C`

is a constant, and`+ C`

is added because the derivative of a constant is`0`

and this introduces ambiguity. For instance, the derivatives of`x^2`

,`x^2 + 4`

and`x^2 - 1`

are all`2x`

.

- There is a reverse power rule that allows us to find the antiderivative of a term - for`f(x) = x^n`

, the antiderivative`F(x) = x^(n+1)/(n+1) + C`

.

Antidifferentiation is important later on in calc and goes into how**integration**works.

So yeah, hope it helped :)(2 votes)

## Video transcript

- [Instructor] So right over here we have the graph of y
is equal to x squared, or at least part of the graph
of y is equal to x squared. And the first thing I'd like to tackle is think about the average rate of change of y with respect to x over the interval from x
equaling 1 to x equaling 3. So let me write that down. We want to know the average rate of change of y with respect to x over the interval from
x going from 1 to 3. And that's a closed interval, where x could be 1, and
x could be equal to 3. Well we could do this even
without looking at the graph. If I were to just make a table here, where, if this is x, and this
is y is equal to x squared, when x is equal to 1, y
is equal to 1 squared, which is just 1. You see that right over there. And when x is equal to 3, y is equal to 3 squared,
which is equal to 9. And so you can see when x is equal to 3, y is equal to 9. And to figure out the
average rate of change of y with respect to x, you say, "Okay, well
what's my change in x?" Well, we could see very clearly that our change in x over this interval is equal to positive 2. Well, what's our change in
y over the same interval? Our change in y is equal to ... When x increased by 2 from 1 to 3, y increases by 8, so it's
going to be a positive 8. So what is our average rate of change? Well, it's going to be our change in y, or our change in x, which is equal to 8 over
2, which is equal to 4. So that would be our
average rate of change. Over that interval, on average, every time x increases by
1, y is increasing by 4. And how did we calculate that? We looked at our change in x, let me draw that here ... We looked at our change in x, and we looked at our change in y, which would be this right over here, and we calculated change
in y over change of x for average rate of change. Now this might be looking
fairly familiar to you, because you're used to thinking about change in y over change in x as the slope of a line
connecting two points. And that's indeed what we did calculate. If you were to draw a secant
line between these two points, we essentially just calculated the slope of that secant line. And so the average rate of
change between two points, that is the same thing as
the slope of the secant line. And by looking at the secant line, in comparison to the
curve over that interval, it hopefully gives you a visual intuition for what even average
rate of change means. Because in the beginning
part of the interval, you see that the secant line is actually increasing at a faster rate, but then as we get closer to 3, it looks like our yellow curve is increasing at a faster
rate than the secant line, and then they eventually catch up. And so that's why the
slope of the secant line is the average rate of change. Is it the exact rate of
change at every point? Absolutely not. The curve's rate of change
is constantly changing. It's at a slower rate of change in the beginning part of this interval, and then it's actually
increasing at a higher rate as we get closer and closer to three. So over the interval, the change in y over the change
in x is exactly the same. Now one question you might be wondering is why are you learning this
is in a calculus class? Couldn't you have learned
this in an algebra class? The answer is yes. But what's going to be interesting, and is really one of the
foundational ideas of calculus is well what happens as these points get closer and closer together? We found the average rate
of change between 1 and 3, or the slope of the secant
line from (1, 1) to (3, 9). But what instead if you found
the slope of the secant line between (2, 4) and (3, 9)? So what if you found this slope? But what if you wanted to get even closer? Let's say you wanted to find
the slope of the secant line between the point (2.5, 6.25) and (3, 9)? And what if you just kept getting closer and closer and closer? Well then, the slopes
of these secant lines are going to get closer and closer to the slope of the
tangent line at x equals 3. And if we can figure out the
slope of the tangent line, well then we're in business. Because then we're not talking
about average rate of change, we're going to be talking about instantaneous rate of change, which is one of the central ideas, that is the derivative, and we're going to get there soon. But it's really important to appreciate that average rate of
change between two points is the same thing as the
slope of the secant line. And as those points get
closer and closer together, and as the secant line is connecting two closer and closer points together, that distance between the points, between the x values of
the points approach 0, very interesting things
are going to happen.