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3 questions, need help

  1. Sep 30, 2007 #1
    FIRST
    What is the HIGHEST root? If there is a highest root, are there available any LOWEST root? Say, what is the highest and lowest root of [tex]f(x) = x^2 - 2x + 1[/tex]?

    SECOND
    What happen if I sum the EVEN and ODD function? I don’t think I am a good on geometry.

    THIRD
    What is the integral of [tex]\frac {1}{x^2}[/tex] or [tex]\int\frac{1}{x^2}[/tex]?

    Is it [tex]\frac{x^{-2 + 1}}{-2+1} + C[/tex]?

    Can I make it in natural logarithm [tex]ln[/tex] form?
     
    Last edited: Sep 30, 2007
  2. jcsd
  3. Sep 30, 2007 #2
    Three you got right. As for One: (X-1)^2, what do you mean highest or lowest?
     
  4. Oct 1, 2007 #3
    Sorry for not being careful. Now I understand what is meant by HIGHEST and LOWEST root. Say, for the following equation:

    [tex]f(x) = x^2 - x - 6[/tex]

    The highest root is [tex]x = 3[/tex] and lowest is [tex]x = -2[/tex].

    In case of [tex]f(x) = x^2 - 2x + 1[/tex] or [tex]f(x) = (x - 1)^2[/tex] there is no highest and lowest root because the only available root is [tex]x = 1[/tex].

    Correct me if I am wrong.

    Thanks
     
  5. Oct 1, 2007 #4

    matt grime

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    Largest and smallest, not highest and lowest.
     
  6. Oct 1, 2007 #5
    Why? Is it about interval?
     
  7. Oct 1, 2007 #6

    matt grime

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    No, it just a matter of the meaning of the words in English. If you talk about the highest root no one will know for sure what you mean. The largest root is better.
     
  8. Oct 1, 2007 #7

    HallsofIvy

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    Of a specific equation? If by "highest" you mean largest and by "lowest" you mean smallest, and the equation has a finite number of roots, then yes, it must have a largest and a smallest root. It happens that the polynomial you give x2- 2x+1= (x-1)2 has only a single root, x= 1, so that is both the largest and smallest root is 1.
    On the other hand, the equation x2+ 1= 0 has only imaginary roots and so does not have a "largest" and "smallest" root- the field of complex numbers cannot be ordered.
    Also, the equation sin(x)= 0 has an infinite number of roots, any multiple of [itex]\pi[/itex] and so does not have either a "largest" or "smallest" root.

    (By the way, strictly speaking, an equation has "roots". A function has "zeroes": the roots of the equation f(x)= 0. But people are seldom that strict!)

    What do you mean by "the" even and "the" odd functions? If you mean add an arbitrary odd and an arbitrary even function, you get a function that is neither even nor odd. Functions that are neither even nor odd are far more common than even or odd functions.

    Have you considered the possibility that a function may be BOTH even and odd? It is possible!

    You don't need natural logarithm: your first formula is correct. you have x-2 so you use "int of xn is [itex]\frac{x^{n+1}}{n+1}[/itex] as long as n+1 is not 0". The only time you need natural logarithm is when n+1= 0- in other words, when n= -1.
     
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