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Average rate of change review

Review average rate of change and how to apply it to solve problems.

What is average rate of change?

The average rate of change of function f over the interval axb is given by this expression:
f(b)f(a)ba
It is a measure of how much the function changed per unit, on average, over that interval.
It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.
Want to learn more about average rate of change? Check out this video.

Finding average rate of change

Example 1: Average rate of change from graph

Let's find the average rate of change of f over the interval 0x9:
A coordinate plane. The x- and y-axes each scale by one. The function y equals f of x is a continuous curve that contains the following points: the point negative five, five, the point negative three, zero, the point zero, negative seven, the point two, negative three, the point three, negative three, the point five point five, zero, and the point nine, three. The points zero, negative seven and nine, three are plotted on the function.
We can see from the graph that f(0)=7 and f(9)=3.
Average rate of change=f(9)f(0)90=3(7)9=109

Example 2: Average rate of change from equation

Let's find the rate of change of g(x)=x39x over the interval 1x6.
g(1)=1391=8
g(6)=6396=162
Average rate of change=g(6)g(1)61=162(8)5=34
Problem 1
What is the average rate of change of g over the interval 8x2?
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
A coordinate plane. The x- and y-axes each scale by one. The function y equals g of x is a continuous curve that contains the following points: the point negative eight, negative eight, the point negative five, negative five, the point negative three, zero, the point negative two, three, the point zero, six, the point two, three, the point three, zero, and the point four, negative four. The points negative eight, negative eight and negative two, three are plotted on the function.

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

  • leafers seedling style avatar for user Eloy Frias
    Over which interval does h have a negative average rate of change? Can I ask for a some help please? because I looked at the problems above but it still seems a little confusing to me.
    (13 votes)
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  • leaf green style avatar for user s-723724152
    I need help to solve this and I don't know how to solve this.

    Here is the question and the problem:
    Solve the system of equations.
    −9y+4x−20=0
    -7y+16x-80=0
    (3 votes)
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    • blobby green style avatar for user Alex T.
      First, it will simplify things if we convert everything to standard form (Ax+By=C) such that the terms without a variable are on the other side of the equation.

      In this way, we get:

      4x-9y=20 and 16x-7y=80

      Then, we look for a way to get one of the variables to cancel out with the other equation. Thus, we multiply the entirety of the first equation by 4:

      16x-36y=80 and 16x-7y=80

      Since we have identical coefficients for the x-variable in both equations, we can subtract one equation from the other so that the x-terms cancel out.

      16x-36y=80
      -16x-7y =80
      -----------------
      -43y=0

      We have successfully isolated y. From here, we can divide both sides of the equation by -43 to get the value of y:

      y=0

      From here, we can plug the y-value back into one of the previous equations to determine the x-value:

      4x-9y=20
      4x-9(0)=20
      4x=20

      This yields the solution:

      x=5

      In these system of equations problems, your strategy should be as follows: choose one variable and eliminate it, solve for the other variable, and then plug the value of the solved variable into the original equation to solve for the unsolved variable.
      (6 votes)
  • marcimus red style avatar for user 913498
    Find the rate change between f(3)=10 and f(7)=18
    (2 votes)
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  • blobby green style avatar for user mernellejoy
    What interval should I use if I was given 0<t<10?
    (3 votes)
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  • hopper happy style avatar for user corny@hornydog6969
    can there be no solution to this type of problem?
    (2 votes)
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  • stelly blue style avatar for user Lili
    Why are we doing the rate of change with these equations instead of how we were taught in the videos?
    (2 votes)
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    • boggle purple style avatar for user Logan
      The formula in these equations is more applicable to the work you will be doing in maths, especially when functions get involved. It still means the same thing, with 'f(b) - f(a)/ b - a' being a different way of writing 'the change in y/ the change in x'.

      Why the formula is not written in the videos is unclear; I assume it's because writing the same formula over and over again will make the videos feel drawn out.
      (2 votes)
  • purple pi teal style avatar for user 申良平
    Should the name of "Mean Value Theorem" asked in the practice questions in this unit be specified as "Mean Value Theorem for for derivatives" to distinguish that for integrals?
    (2 votes)
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  • blobby green style avatar for user Teairra Pough
    What is the average rate of change of F over the interval -7≤x≤2?
    (2 votes)
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  • blobby green style avatar for user Michael Pinion
    what if the dots on your graft land on the number line itself. Like (-6) and (-4). Would that mean that the other number is "0" for both?
    (2 votes)
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  • aqualine ultimate style avatar for user eabney24
    How do you neaten the lines on the graphs? Mine were always wiggly and they made me get multiple questions wrong.
    (2 votes)
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