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Introduction to average rate of change

What's the average rate of change of a function over an interval?

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  • leafers ultimate style avatar for user pascal5
    This is probably a silly question, but why do you need differential calculus to find the instantaneous slope of the line? Why couldn't you just look at it like:
    y = mx+b
    y = t^2+1 which is the same as y = t * t + 1
    t = m = slope of the line at time = t?
    (21 votes)
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    • blobby green style avatar for user Pavelsu
      It's impossible to determine the instantaneous rate of change without calculus. You can approach it, but you can't just pick the average value between two points no matter how close they are to the point of interest. If you zoom in you'd see that the curve before the point of interest is different from the curve after the point of interest.
      That being said, a close approximation may provide a good enough answer for your work.
      (12 votes)
  • marcimus pink style avatar for user Nitya
    While finding average of numbers,etc., we usually add up all those and divide by their count,but in here to find the average speed, we are actually taking up the slope formula.Would anyone please explain . Or am I thinking it in a wrong way?
    (6 votes)
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    • leafers ultimate style avatar for user Alex
      On a position-time graph, the slope at any particular point is the velocity at that point. This is because velocity is the rate of change of position, or change in position over time. Here, the average velocity is given as the total change in position over the time taken (in a given interval).

      Using your idea of an average, to find the average velocity we'd want to measure the velocity at a bunch of (evenly spaced) points in that interval, and find the average of those. The question you might ask then would be: how many points should we take?

      If we just took 2 points (the start and the end), we might get some idea of the average but this would likely be a bad representation of the true average. If the car started off stationary and ended stationary, its velocity is zero at those two points, which would suggest it's average velocity was zero - that can't be right! By taking just two points, we lost all the information about what happened between those points.

      So we have to take some more points, and the more points we take, the more information we take into account, and so the closer our estimation should get to the actual answer. In fact, it seems like if we were able to take an infinite number of points we'd get the most accurate value possible. But since infinity is hard to do, let's just use a "large" number instead. So now we have two ways of finding an average velocity, Sal's way and your way. So you now may ask, what's the difference, what makes his way right and my way wrong? In fact, there is no difference, the two ways will give exactly the same answer!

      An exact proof of this requires calculus or limits, but you could play around with this idea on paper or on a computer or even run some experiments to test this for yourself.
      (20 votes)
  • mr pants teal style avatar for user Ira B.
    I don't get this at all! Can anyone help?
    (6 votes)
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  • leaf green style avatar for user sst
    Why that line is called secant line?
    (4 votes)
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  • aqualine ultimate style avatar for user John He
    Is the average rate of change really means"average"value of the slope?How can people just call it "average" rate of change?
    (2 votes)
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    • leaf blue style avatar for user Stefen
      Here is my answer, I hope I have understood your question.
      Slope = Rate of Change
      For a straight line, the slope is the exact rate of change.
      We are using the, by now familiar, concept of the slope of a function whose output is a straight line to introduce how we can think about the rate of change of a function that is not a straight line.

      Using the Δy/Δx idea, we choose two points on our non-linear graph of some function f, and draw a straight line (a secant line) to calculate the slope of this straight line. Now this is not the exact value of the slope of the curved line, but it is a reasonable average of the rate of change of f between the two values we selected.

      In time, you will learn how to calculate the instantaneous rate of change of a curvy graph of some function - that is, the exact slope (via a tangent line) at a point on the graph. We can do this by finding the derivative of f (this is calculus), and then plugging in the x value for which we we want to know the slope, and out pops the instantaneous rate of change of f at x.
      Here is a sneak peek:
      https://www.khanacademy.org/math/differential-calculus/derivative-intro-dc/derivative-as-tangent-slope-dc/v/derivative-as-slope-of-tangent-line

      Example:
      Let f(x)=x², the derivative of f is f'(x)=2x, so the slope of the graph, when x=3, for our example is f'(3)=(2)(3) = 6. This is the instantaneous rate of change of f at x=3.

      Don't worry about all this differentiation stuff right now, but do study algebra to be able to take a pre-calculus course to get into the calculus.
      (6 votes)
  • leaf orange style avatar for user YanSu
    What relationship does a tangent line in graphs have with the tangent of a circle?How about secant lines?
    (4 votes)
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  • blobby green style avatar for user monicabrettler
    This video has a mistake at the end. The d(x) for 3 is 10, not 9, and that makes the drawing more logical.
    (4 votes)
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  • stelly orange style avatar for user Sachika
    Hi! I was wondering what the ∆ symbol means and where it can be used. Thank you!
    (1 vote)
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  • piceratops ultimate style avatar for user Anish Madireddy
    At , Sal talks about slope-intercept form. Can anyone give me an explanation of what that is and how it can be used to find average rate of change.
    (4 votes)
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  • eggleston blue style avatar for user dena escot
    is the average rate of change between two points on a curve represent the two points on the a curve as two points on straight line, I mean make a segment on a curve which i want to calculate the average of change between two points on this segment on a curve , when i take the average for this segment, that mean this segment is converted to a line, straight line which i can take the slope for it?
    (3 votes)
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    • female robot grace style avatar for user sa.ma
      but that's actually what we do we turn the curve ( not the whole curve we part the curve which its points near each other and easy to be turned to a straight line) to a straight line then take the slope by two points on it. so,yes the segment is line . in lines, you get the exact slope. When you apply it to 2 points on a curved line, you get the average slope between those 2 points.

      DON'T forget that we just approach it. it is not the exact value the exact value as sal said solved by calculus
      (1 vote)

Video transcript

- So we have different definitions for d of t on the left and the right and let's say that d is distance and t is time, so this is giving us our distance as a function of time, on the left, it's equal to 3t plus one and you can see the graph of how distance is changing as a function of time here is a line and just as a review from algebra, the rate of change of a line, we refer to as the slope of a line and we can figure it out, we can figure out, well, for any change in time, what is our change in distance? And so in this situation, if we're going from time equal one to time equal two, our change in time, delta t is equal to one and what is our change in distance? We go from distance is equal to four meters, at time equals one, to distance in seven meters at time equal two and so our change in distance here is equal to three and if we wanna put our units, it's three meters for every one second in time and so our slope would be our change in our vertical divided by our change in our horizontal, which would be change in d, delta d over delta t, which is equal to three over one or we could just write that as three meters per second and you might recognize this as a rate, if you're thinking about your change in distance over change in time, this rate right over here is going to be your speed. This is all a review of what you've seen before and what's interesting about a line, or if we're talking about a linear function, is that your rate does not change at any point, the slope of this line between any two points is always going to be three, but what's interesting about this function on the right is that is not true, our rate of change is constantly changing and we're going to study that in a lot more depth, when we get to differential calculus and really this video's a little bit of a foundational primer for that future state, where we learn about differential calculus and the thing to appreciate here is think about the instantaneous rate of change someplace, so let's say right over there, if you ever think about the slope of a line, that just barely touches this graph, it might look something like that, the slope of a tangent line and then right over here, it looks like it's a little bit steeper and then over here, it looks like it's a little bit steeper, so it looks like your rate of change is increasing as t increases. As I mentioned, we will build the tools to later think about instantaneous rate of change, but what we can start to think about is an average rate of change, average rate of change, and the way that we think about our average rate of change is we use the same tools, that we first learned in algebra, we think about slopes of secant lines, what is a secant line? Well, we talk about this in geometry, that a secant is something that intersects a curve in two points, so let's say that there's a line, that intersects at t equals zero and t equals one and so let me draw that line, I'll draw it in orange, so this right over here is a secant line and you could do the slope of the secant line as the average rate of change from t equals zero to t equals one, well, what is that average rate of change going to be? Well, the slope of our secant line is going to be our change in distance divided by our change in time, which is going to be equal to, well, our change in time is one second, one, I'll put the units here, one second and what is our change in distance? At t equals zero or d of zero is one and d of one is two, so our distance has increased by one meter, so we've gone one meter in one second or we could say that our average rate of change over that first second from t equals zero, t equals one is one meter per second, but let's think about what it is, if we're going from t equals two to t equals three. Well, once again, we can look at this secant line and we can figure out its slope, so the slope here, which you could also use the average rate of change from t equals two to t equals three, as I already mentioned, the rate of change seems to be constantly changing, but we can think about the average rate of change and so that's going to be our change in distance over our change in time, which is going to be equal to when t is equal to two, our distance is equal to five, so one, two, three, four, five, so that's five right over there and when t is equal to three, our distance is equal to 10, six, seven, eight, nine, 10, so that is 10 right over there, so our change in time, that's pretty straightforward, we've just gone forward one second, so that's one second and then our change in distance right over here, we go from five meters to 10 meters is five meters, so this is equal to five meters per second and so this makes it very clear, that our average rate of change has changed from t equals zero, t equals one to t equals two to t equals three, our average rate of change is higher on this second interval, than on this first one and as you can imagine, something very interesting to think about is what if you were to take the slope of the secant line of closer and closer points? Well, then you would get closer and closer to approximating that slope of the tangent line and that's actually what we will do when we get to calculus.