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Secant lines & average rate of change

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.A (LO)
,
CHA‑2.A.1 (EK)
Understanding average rate of change and its relation to slope of a secant line.

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  • duskpin tree style avatar for user G Y
    At , 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?
    (20 votes)
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    • mr pink red style avatar for user andrewp18
      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!
      (51 votes)
  • primosaur seed style avatar for user Felicia Kohn
    What's the best way to memorize the difference between a tangent and a secant line?
    (9 votes)
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  • aqualine ultimate style avatar for user 007euclidd
    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)
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  • piceratops ultimate style avatar for user Bailan
    What is the difference between a secant line and tangent line?
    (4 votes)
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    • winston default style avatar for user John Mao
      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)
      (13 votes)
  • leaf blue style avatar for user Mohamed Saad
    can you explain what is the difference between scant line and slope??
    (3 votes)
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    • duskpin ultimate style avatar for user SherlockHolmes.42
      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.
      (4 votes)
  • leaf green style avatar for user Akshay Raman
    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?]
    (2 votes)
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    • purple pi purple style avatar for user Yahia Khalafalla
      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
      (2 votes)
  • leaf green style avatar for user Moly
    what is the difference between the slope of secant line and the slope of tangent line ?
    (2 votes)
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    • duskpin seedling style avatar for user cholaqshya
      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.
      (1 vote)
  • aqualine ultimate style avatar for user Aseer
    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?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      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.
      (3 votes)
  • blobby green style avatar for user ZionCorley
    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?
    (2 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      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!
      (1 vote)
  • blobby green style avatar for user krm.shreya
    av.speed is total distance covered/total time taken
    but,(delta dis.)/(delta time)
    is change in dis./t1-t2?
    (1 vote)
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    • purple pi pink style avatar for user Sebastian
      Av. speed is Δd/Δt (Δ means 'change in'), which is:
      (final d. -initial d.)/(final t. -initial t.).

      This is the same as saying (total d. traveled)/(total time passed)
      For example, imagine you are waching a car race. If at some random instant after the race has started you start a timer (t=0) and at t=2s you see a car that's at 500m from the starting line. At t=7s you look again and the car is at 600m from the starting line.
      The distance the car traveled over that interval is 600m-500m (change in distance), or 100m, and the change in time is 7s-2s, or 5s. This means over that interval the car's average speed was 100m/5s = 20m/s

      Hope this helps!
      (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.