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Antiderivative graphing question - help

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data
    http://img64.imageshack.us/img64/5430/80433637.jpg [Broken]


    2. Relevant equations



    3. The attempt at a solution
    for a, would it be local.max: x=7.7 and local.min: x=4.8 ?
    and for b, abs.max: x=11 and abs.min: x=2 ?

    im not sure if i did it right but looks like this isnt what the question wants me to do..
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jan 17, 2010 #2

    Mark44

    Staff: Mentor

    The graph you show is f(x) = xsinx. The function you're investigating is [tex]g(x)~=~\int_0^x t sin(t)dt[/tex]

    How would you ordinarily go about find the local max or min of a function?

    Have you learned about the Fundamental Theorem of Calculus recently? (That's a hint.)
     
  4. Jan 17, 2010 #3
    [tex]
    g(x)~=~\int_0^{pi/4} x sin(x)dx
    [/tex]

    i would look at the graph and see which point looks like local min/max then plug it into the antiderivative of the integral. (which i dont really know how to get cuz its xsinx...)
    so whatever i get for G(x) i will plug that number i got for the local min/max and that will be the value for local min/max.. am i making any sense? lol

    and yes i have learned about the Fundamental Theorem of Calculus but how's that helping me?
     
  5. Jan 17, 2010 #4

    Char. Limit

    User Avatar
    Gold Member

    Well, one of the most common applications of the first derivative is in finding the minimum or maximum of a function...

    Think about the first derivative as a slope. If [tex]g(x)[/tex] is at a min or max, what is the slope?
     
  6. Jan 18, 2010 #5

    Mark44

    Staff: Mentor

    This integral doesn't have anything to do with your problem. This is not a function of x - it's a constant.
    This is the integral you're interested in
    [tex]g(x)~=~\int_0^x t sin(t)dt[/tex]

    What I've been getting at and what Char.Limit made more explicit is for you to look at the derivative of this function. And no, you don't have to evaluate the integral directly. Use the FTC to find g'(x).
     
  7. Jan 18, 2010 #6

    Char. Limit

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    Gold Member

    Sorry, Mark, but it didn't seem to me as if he fully caught what you were saying...
     
  8. Jan 18, 2010 #7

    Mark44

    Staff: Mentor

    If you're apologizing to me for jumping in, I don't mind. The more the merrier.
     
  9. Jan 18, 2010 #8
    Char. Limit-
    "If g(x) is at a min or max, what is the slope? "
    its 0.

    Mark44-
    the FTC just tells me that g'(x)=f(x) (for my case)
    which means that g'(x)=xsinx
    now g(x) = antiderivative of g'(x) which is <i dont know how to get it, tried everything already>
    but how does this help me with any of the question a,b,c or d?
     
  10. Jan 18, 2010 #9

    Char. Limit

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    Gold Member

    Ok... Now what does the first derivative tell you about the min/max points, if you know that the slope of those points must be zero?
     
  11. Jan 18, 2010 #10
    im not sure... i think that it tells me that that min/max point will be positive or negative for g(x)'s graph..
    i really have no idea. im not really good with calc and i feel dumb right now :(
     
  12. Jan 19, 2010 #11

    Mark44

    Staff: Mentor

    What I've been trying to steer you to is that you don't need to evaluate g(x) directly; all you need is g'(x), which you have already said is equal to x*sinx. If you want to find the max and min values of a function, say g, look at where its derivative g' is zero.

    g'(x) = ?
    g'(x) = 0 for what x?
     
  13. Jan 19, 2010 #12
    g'(x) = xsinx
    g'(x) = 0 for x=0,pi, 2pi, 3pi, 4pi
     
  14. Jan 19, 2010 #13

    Char. Limit

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    Gold Member

    So now you know where the min and max are, now you need to figure out which ones are which.

    See a way to do that?
     
  15. Jan 19, 2010 #14
    i would know if i had the graph of g(x)..
    but i kinda remember from highschool that from g'(x) graph, if it goes from + to - then its a local max, and if from - to + its a local min.. but when is it global max/min?
     
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