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Derivatives in relation to curve sketching

  1. Apr 4, 2005 #1
    Hi Again!

    Right now I'm taking derivatives in relation to curve sketching, and I just wanted to make sure I am doing these right.

    The first question is to determine using the first derivative where the graph of y = x is rising.
    (x^2)+1

    This is what I have done so far:

    I tried to find the derivative using the quotient rule.....

    ((x^2)+1)(d/dx)(x)-(x)(d/dx)((x^2)+1)
    ((x^2)+1)^2


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

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

    That is the value I found for the first derivative, although I am unsure whether I have done it right or not. Then to find where the curve was rising I said that since the derivative must be greater than 0, the value I found for the derivative must be greater than zero. Then I tried solving that and I ended up with -x^2 + 1 > x^4 + 2x^2 + 2 and although I realize that that is not quite finished yet, it just doesn't seem right to me. I am obviously going wrong somewhere, but where I do not know.


    The second question asks you to determine where f(x) = x^2(1-x) is concave upward using the second derivative. This is what I did:

    (x^2)(d/dx)(1-x) + (1-x)(d/dx)(x^2) =


    (x^2)(-1) + (1-x)(2x) =

    (-3x^2) + 2x This is the first derivative I found using the product rule and the steps shown above.

    Then I found the second derivative to be -6x + 2

    Have I found the derivatives correctly for these questions?????? It just seems like something is very wrong, mostly with the first question. Any help you guys can give me I would really appreciate. Thanks in advance.
     
  2. jcsd
  3. Apr 4, 2005 #2

    dextercioby

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    a) The first derivative is correct.However,you solved this inequation

    [tex]\frac{-x^{2}+1}{\left(x^{2}+1\right)^{2}} >0 [/tex]

    incorrectly...

    Do it again HINT:The denominator is always positive
     
  4. Apr 4, 2005 #3

    dextercioby

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    For the second,everything is correct so far.How do you interpret the result...?

    Daniel.
     
  5. Apr 4, 2005 #4
    Thank you so much for the help, I really appreciate it!

    Ok for the concave one ( second question) I think it is concave upward from (-infinity, (1/3))

    I solved it like this:

    -6x+2>0
    -6x>-2
    x < (1/3)

    I'm redoing the last part of the first question right now, I post again in a minute.
     
  6. Apr 4, 2005 #5

    dextercioby

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    Okay.It's okay,so far.

    Daniel.
     
  7. Apr 4, 2005 #6
    OK now for the first one,

    I went back and did this:

    (-x^2)+1 >0
    ((x^2)+1)^2

    =

    x^2 < 1

    x<1
    x>-1

    therefore -1 < x < 1


    OK now an I on the right track? Thanks again for the help!!!!
     
  8. Apr 4, 2005 #7

    dextercioby

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    It's perfect,u can plot it to get a graphical confirmation,but it's everything okay now.

    Daniel.
     
  9. Apr 4, 2005 #8
    Alright!!!!!!!!!!!!! Thank you so much!!!!
     
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