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Worked example: average rate of change from table

Finding the average rate of change of a function over the interval -5<x<-2, given a table of values of the function. Created by Sal Khan.

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  • aqualine ultimate style avatar for user Ashish Kadam
    The question says, -5 < x < -2, wouldn't it mean from x greater than -5 upto x less than -2, which would actually mean from x >= -4 upto x <= -3
    i.e., excluding the -5 and -2?
    (60 votes)
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    • piceratops ultimate style avatar for user alexdragon0701
      So, in the two previous videos on this topic Sal mentioned that: The average rate of change is really the slope of the line that connects the two endpoints. If you plotted the function, you would get a line with two endpoints of (-5,6) and (-2,0). Basically the average rate of change is everything between those two points (on the line). If we used (-4,4) and (-3,2) then we would be talking about the average rate of change of a completely different line segment. I'm sorry if this answer confused you; with a graph it would be much easier to explain. :)
      (28 votes)
  • leaf green style avatar for user michael.farghali
    I don't understand why he picks the points -5, 6 and -2, 0. Since the interval is -5 < x < -2 wouldn't he have to pick a number less than -5 and -2 respectively?? Thanks!
    (26 votes)
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    • leaf green style avatar for user Megamind
      The interval applies to the x variable, saying that x is greater than -5 and less than -2. Since he was finding the slope on that interval of -5 to -2 for x, he used the two endpoints: -5 and -2.
      The y coordinates are not bound by the interval, when x = -5, then y = 6, and when x = -2, y = 0. With the two coordinates, (-5, 6) and (-2, 0), you can use the slope formula to find it.
      (0 - 6) / (-2 - -5) = -2
      (18 votes)
  • piceratops ultimate style avatar for user d.eileen.d
    ok, i'm lost trying to figure out the problems in "Average rate of change". i also can't figure out where the method of solution was explained in the Average Rate of Change Examples 1, 2 or 3.
    for example:
    "over which interval does y(x) have an average rate of change of 5/2?"
    a graph is provided with five seemingly random horizontal segments, spread across the four quadrants. each is four units wide. some have solid dots marking end points, some have hollow dots, some have no end point dot and run off the graph. i don't know what these lines are for. they don't match to the possible answers. (and there's five of them, with only four possible solutions). all possible answers are given in form of A < x < B.

    here's what the hint says:
    "average rate of change over an interval is total change in value of y(x) over change in x."
    "let's look at - 6 < x < -2 :
    y (-6) = -3
    y (-2) = 2
    change in y = 2 - (-3) 5
    --------------- ---------- = --
    change in x - 2 - (-6) 4

    i can see that the range of the possible answer is being dropped in to test it for a solution but that's as far as i get. i don't know why y (-6) = -3, or why y (-2) = 2. all help greatly appreciated! :)
    (12 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Let us first explain what the line segments mean:
      If there is a point with an hollow dot, it means that the line segment begins or ends at that point, but does NOT include that point.

      If there is a point with a filled dot, that means the line segment begins or ends at that point and DOES include that point.

      If a line runs off the page without an ending dot, that means that the line just goes on forever in the fashion indicated by the portion that is being shown.

      To find the rate of change for any segment, which is the same thing as the slope of the segment, just take any two points you know the value for in that segment (it doesn't matter which two points or which order you put them in). Call one of the points (x₁ , y₁ ) and the other point (x₂ , y₂). The rate of change will = (y₁ - y₂) / (x₁ -x₂ )

      The part that looks like - 6 < x < -2 means that it is the line segment that begins at x=-6 and ends at x=-2
      (16 votes)
  • blobby green style avatar for user ishotmisha
    Around what does he mean by "with respect to" ?
    (7 votes)
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  • blobby green style avatar for user Scott Johnsen
    Does 'Average Rate of Change' mean slope??
    I feel like this section on graphing went from line intuition to average rate of change examples without a video explaining what rate of change is. Any help is appreciated.
    (6 votes)
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  • purple pi purple style avatar for user Spartacus!
    If the problem was -5 <= x <= -2, would that have changed the answer? Why or why not?
    (5 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      No. The average rate of change is exactly the same for this set of data, within this interval, whether the interval includes the endpoints or not. You still calculate it by the end points.
      ∆y is still -6 and ∆x is still 3, so the average rate of change is still -2.
      All the little rates of changes between points in the interval are also -2, so this part of the graph is a straight line segment.
      Notice that the rate of change is constant within this interval, but it is different outside this interval. Look at the rate of change between 1 < x < 5
      Zowie--it has turned positive, and is +3
      In fact, it is interesting to graph all the points and see that the function is made up of three different slopes. It makes it more clear why Sal chose the end points for calculating the rate of change.
      (1 vote)
  • purple pi purple style avatar for user Spartacus!
    I still don't get this. If we want to compute the average rate of change, why don't we do that like how we usually average things: add some numbers and then divide the sum of those numbers by how my numbers we added together?
    (3 votes)
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    • piceratops ultimate style avatar for user jimstanley49
      You can if you want to, but it's the long way round. The first change in y(x) is from 6 to 4 (-2), then 4 to 2 (-2), then 2 to 0 (-2). (-2). The corresponding changes in x are from -5 to -4 (1), then from -4 to -3 (1), then from -3 to -2 (1). So, (-2)/1 + (-2)/1 + (-2)/1 = -6/1. Divide this by our three samples and you get -2.
      Because the rate of change function is continuous, even if the rate spikes in the middle somewhere, it has to come down through all the values it went up through and they cancel out in the averaging. So only the end points matter.

      Think about driving a car 60 miles in 1 hour (rate of change in position.) If you start and end at a stop, you weren't going 60 mph the whole time, but your average speed was 60 mph. This highlights a shortcoming of the average rate of change: we know nothing about the rate at an arbitrary point in the middle. Maybe you went 100 mph for the first 30 minutes, then spent 20 minutes stopped by a policeman, then finished the trip in the last ten minutes. You still made the trip in an hour and your average speed was still 60 mph, and you get the same answer as if you somehow accounted for all the speeds traveled in between.
      (4 votes)
  • blobby green style avatar for user Diego Piscoya
    does the order matter when solving for the slope; is it more neat or something like that
    (3 votes)
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  • leaf blue style avatar for user Cameron Christensen
    Why are we defining the interval using < instead of <=? It seems to me that we should be using <= to make sure that we include our endpoints...
    (3 votes)
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  • blobby green style avatar for user ALI RAMAZANI
    Hello everyone! I am Ali from Afghanistan. I love learning math on Khan Academy.

    I think there is a mistake in this video. The question says x is greater than-5 and less than -2. So, the average rate of change should be calculated from -4 to -1.

    I read some of the comments and someone said that there is no mistake because Sal calculated the the slope. But the question is, Sal included -5 and -2 when those points are strictly excluded. If Sal is right, then what's the difference between -5<x<-2 and -5<=x<=-2?
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      First, -1 is not in the interval of -5<x<-2, and using -3 instead of -1 doesn't help either.

      You can't change the points to -4 and -3 because you skip all the values that are between -5 and -4; and all those between -3 and -2. There is an infinite set of numbers between every pair of integers. We use the -5 and -2 to simplify the calculations. If you didn't, You need numbers extremely close to -5 and -2, something like -4.99999999999999999 and -2.000000000000000000000001. And, even then, you could get even closer by picking values even closer to the -5 and -2. We're finding an average - so, we simplify the math and just use the -5 and -2
      (1 vote)

Video transcript

What is the average rate of change of y of x over the interval negative 5 is less than x is less than negative 2? So this is x is equal to negative 5. When x is equal to negative 5, y of x is equal to 6. And when x is equal to negative 2, y of x is equal to 0. So to figure out the average rate of change, so the average rate of change, of y of x, with respect-- and we can assume it's with respect to x-- let me make that a little bit neater-- this is going to be the change in y of x over that interval over the change of x of that interval. And the shorthand for change is this triangle symbol, delta. Delta y-- I'll just write y. I could write delta y of x. It's delta y. Change in y over our change in x. That's going to be our average rate of change over this interval. So how much did y change over this interval? So y went from a 6 to a 0. So let's say that we can kind of view this as our endpoint right over here. So this is our end. This is our start. And we could have done it the other way around. We would get a consistent result. But since this is higher up on the list, let's call this the start. And the x is a lower value. We'll call that our start. This is our end. So we start at 6. We end at 0. So our change in y is going to be negative 6. We went down by 6 in the y direction. It's negative 6. You could say that's 0 minus 6. And our change in x, well, we are at negative 5, and we go up to negative 2. We increased by 3. So when we increased x by 3, we decreased y of x by 6. Or if we want to simplify this right over here, negative 6 over 3 is the same thing as negative 2. So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. Every time, on average, x increased 1, y went down by negative 2.