Rate of change from 1 to 2 for f(x)=2x^3 + x

  1. Find the average rate of change from 1 to 2 for the function f(x)=2x^3 + x

    so I did this:
    [f(2) – f(1)] – [2x^3 + x] / 2-1

    = 2-1-2x^3 + x / 1

    = 1-2x^3 + x

    = -2x^3 + x

    Right?
     
  2. jcsd
  3. Galileo

    Galileo 2,002
    Science Advisor
    Homework Helper

    The average rate of change of f from a to b is
    [f(b)-f(a)]/(b-a)

    and it's (naturally) just a number. It doesn't depend on x. Check your definition.
     
  4. [2(2)^3 - 2(1)^3] - [2x^3 + x] / 2-1

    16-1-2x^3 + x / 1

    15-2x^3 + x

    I don't understand what you are telling me
     
  5. Galileo

    Galileo 2,002
    Science Advisor
    Homework Helper

    You've got the definition of the average rate of change wrong.
    You wrote something like (f(2)-f(1)-f(x))/(2-1).
    By definition, the average rate of change of f on the interval [a,b] is:
    [tex]\frac{f(b)-f(a)}{b-a}[/tex]

    So in your case, the average rate of change is:
    [tex]\frac{f(2)-f(1)}{2-1}[/tex]
     
  6. Ok, I am not sure what to do with 2x^3 + x . So I subtracted it from the f(b) - f(a).
    If I had 2x^3 by it self, I can see just putting 2(2)^3 - 2(1)^3 / 2-1
    but the "+x" is confusing me
     
  7. I think I got it

    2(2)^3 + x - 2(1)^3 + x / 2-1

    =16-2+x+x / 1

    =14+2x

    =-2x + 14

    x = -7
     
  8. Galileo

    Galileo 2,002
    Science Advisor
    Homework Helper

    So you can solve it if the function is 2x^3, but not if it's 2x^3+x? What's the difference, conceptually?

    f(x)=2x^3+x, so what is f(2)? And what is f(1)?
     
  9. BobG

    BobG 2,346
    Science Advisor
    Homework Helper

    Calculate f(2). Calculate f(1). Subtract the result of f(1) from f(2).

    The solution for f(2) is not 16+x. You have to substitute '2' for x everywhere it appears, so the solution for f(2) is 16+2.

    Also, your algebra is wrong (in addition to being not applicable in this case). If you have:

    [tex](3x^2 + 3x) - (2x^2 + 2x)[/tex]
    then the minus sign means both the 2x^2 and the 2x are negative:
    [tex]3x^2 + 3x - 2x^2 - 2x[/tex]
    [tex](3x^2 - 2x^2) + (3x - 2x)[/tex]
    etc.
     
  10. 2(2)^3 + (2) -1 / 2-1

    16+2-1 / 2-1

    17/1

    17

    I'm sorry if I'm just not getting it
     
  11. Galileo

    Galileo 2,002
    Science Advisor
    Homework Helper

    Alright, let's take some steps back.
    You are given a function f. It's a machine that eats a number and spits out a (usually different) number.
    f(x)=2x^3+x tells you the value of the function at each point, it's an equality that holds for each number x.
    For example:
    f(1)=2(1)^3+1=2+1=3
    f(5)=2(5)^3+5=2(125)+5=255

    So if you want to calculate [f(2)-f(1)]/(2-1) you have to calculate f(2) and f(1). I already did f(1) for you above. Now you do f(2) and calculate [f(2)-f(1)]/(2-1)
     
  12. 2(2)^3 + (2) - 2(1)^3 +1 / 2-1

    =18-3 / 2-1

    =15/1

    =15
     
  13. Galileo

    Galileo 2,002
    Science Advisor
    Homework Helper

    Right that's correct.

    BTW: Mind your brackets: -2(1)^3+1 is not the same as -(2(1)^3+1)
     
  14. Great! Thanks for your help!
    I have another one, maybe I should start a new post
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?