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Homework Help: Find interval

  1. Nov 30, 2005 #1
    Hello
    How do I do the following
    Find the interval on which the curve
    [tex]y = \int_x^0 \frac{1}{1 + t + t^2} dt[/tex]
    is concave upward.

    any help would be great.

    P

    Just in case Latex does not show up
    x
    Large S 1/1(1 + t + t^2) dt
    0
     
  2. jcsd
  3. Nov 30, 2005 #2

    Galileo

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    When, in general, is some function f concave upwards?
     
  4. Nov 30, 2005 #3
    I belive a function f should be concave upward when f''(c) > 0. Am I correct. Don't have my text book on me.
     
  5. Nov 30, 2005 #4

    BobG

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    Correct.

    Now, if the function f (t) is [tex]y = \int_x^0 \frac{1}{1 + t + t^2} dt[/tex], then what's f'(t). (This is very easy).

    Finding f''(t) is a little harder. Once found set it up as an inequality and solve.
     
  6. Nov 30, 2005 #5
    f' is (x)[tex]y = \int_x^0 \frac{-1-2t}{(1 + t + t^2)^2}[/tex]

    f'' would then be
    [tex]y = \int_x^0 \frac{-1 + 2t + 2t^2}{(1 + t + t^2)^4} dt[/tex]

    is this correct??

    How do I solve an inequality that is this complex(it is complex to me)?? I am really not sure about this.
     
  7. Nov 30, 2005 #6

    BobG

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    The integral is also called the anti-derivative, so the derivative of [tex]y = \int_x^0 \frac{1}{1 + t + t^2} dt[/tex]
    is just:
    [tex]f'(t)= \frac{1}{1 + t + t^2} [/tex]

    So your second derivative is just:
    [tex]f''(t) =\frac{-1-2t}{(1 + t + t^2)^2}[/tex]
     
  8. Nov 30, 2005 #7

    Zurtex

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    Not quite BobG, think about where the x is and what your actually differentiating.
     
  9. Nov 30, 2005 #8
    anybody with some guidence?
     
  10. Nov 30, 2005 #9

    Zurtex

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    Erm, you've been given loads and loads, what have you done with what you have been given?
     
  11. Nov 30, 2005 #10

    HallsofIvy

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    "Fundamental Theorem of Calculus"!!!


    What is the derivative of [tex]\int_a^x f(t)dt[/tex] according to the Fundamenta Theorem of Calculus? (This is what Bobg was doing.)

    Knowing that, what is the derivative of [tex]\int_x^a f(t)dt[/tex]? (This is what Bobg should have done!)


    What is the derivative of [tex]\int_x^a f(t)dt[/tex]
     
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