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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 a, is less than or equal to, x, is less than or equal to, b is given by this expression:
start fraction, f, left parenthesis, b, right parenthesis, minus, f, left parenthesis, a, right parenthesis, divided by, b, minus, a, end fraction
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 0, is less than or equal to, x, is less than or equal to, 9:
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, left parenthesis, 0, right parenthesis, equals, minus, 7 and f, left parenthesis, 9, right parenthesis, equals, 3.
Average rate of change=f(9)f(0)90=3(7)9=109\begin{aligned} \text{Average rate of change}&=\dfrac{f(9)-f(0)}{9-0} \\\\ &=\dfrac{3-(-7)}{9} \\\\ &=\dfrac{10}{9} \end{aligned}

Example 2: Average rate of change from equation

Let's find the rate of change of g, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 9, x over the interval 1, is less than or equal to, x, is less than or equal to, 6.
g, left parenthesis, 1, right parenthesis, equals, 1, cubed, minus, 9, dot, 1, equals, minus, 8
g, left parenthesis, 6, right parenthesis, equals, 6, cubed, minus, 9, dot, 6, equals, 162
Average rate of change=g(6)g(1)61=162(8)5=34\begin{aligned} \text{Average rate of change}&=\dfrac{g(6)-g(1)}{6-1} \\\\ &=\dfrac{162-(-8)}{5} \\\\ &=34 \end{aligned}
Problem 1
What is the average rate of change of g over the interval minus, 8, is less than or equal to, x, is less than or equal to, minus, 2?
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 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.
    (11 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
    (4 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.
      (5 votes)
  • 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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  • winston baby style avatar for user cr
    can there be no solution to this type of problem?
    (1 vote)
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  • blobby green style avatar for user EL GATO Y PTO
    why does it not tell me exactly where it hits the line while i find the average I got a question and it was impossible to tell where it hit the lone exactly I and all the possible answers where less then 2 numbers close together
    (2 votes)
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  • 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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  • piceratops seed style avatar for user Jatavian Allen
    f(x)=x
    3
    −9xf, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 9, x
    What is the average rate of change of fff over the interval [1,6][1,6]open bracket, 1, comma, 6, close bracket?
    (2 votes)
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  • blobby green style avatar for user Chandan
    f(x)=x
    2
    −x−1f, left parenthesis, x, right parenthesis, equals, x, squared, minus, x, minus, 1
    Over which interval does fff have an average rate of change of zero?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Your function creates a parabola when graphed. So have an average rate of change = 0, your interval would need 2 points on direct opposite sides of the parabola. A line thru those 2 points would be a horizontal line and have a slope of 0.
      (2 votes)
  • blobby green style avatar for user Foxen
    How do you find rate of change from a equation such as y=3.75+1.5(x-1)?
    (1 vote)
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    • duskpin sapling style avatar for user proxima
      The rate of change would be the coefficient of x. To find that, you would use the distributive property to simplify 1.5(x-1). Once you do, the new equation is y = 3.75 + 1.5x -1.5. Subtract 1.5 from 3.75 next to get: y = 1.5x + 2.25. Since 1.5 is the coefficient of x, 1.5 would be the rate of change.
      Hope that helps! If you have any more questions, you can ask me :)
      (2 votes)