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Math Help - Help please!!

  1. #1
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    Help please!!

    The problem is differentiate x^2+1/x
    Find the following:
    1.)Intercepts

    2.)Asymptotes
    3.)Symmetry
    4.)Domain
    5.)Increasing/decreasing behavior
    6.)Relative extrema
    7.)Concavity
    8.)Inflection Points


    What i have so far
    no intercepts
    V.A(vertical asymptote) at x=0
    H.A none
    S.A y=x
    symmetry at origin
    increasing (-infinity,-1)
    relative max (-1,-2)
    relative min (1,2)
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  2. #2
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    Look at its graph and see if you did it right.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Tskate View Post
    The problem is differentiate x^2+1/x
    Find the following:
    1.)Intercepts

    2.)Asymptotes
    3.)Symmetry
    4.)Domain
    5.)Increasing/decreasing behavior
    6.)Relative extrema
    7.)Concavity
    8.)Inflection Points


    What i have so far
    no intercepts
    V.A(vertical asymptote) at x=0
    H.A none
    S.A y=x
    symmetry at origin
    increasing (-infinity,-1)
    relative max (-1,-2)
    relative min (1,2)
    In the future you might try a larger font. This is difficult to read.

    -Dan
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  4. #4
    Super Member

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    Hello, Tskate!

    y \:=\:\frac{x^2+1}{x} . ← Is this what you meant?

    Find the following:
    1) Intercepts
    x-intercepts: let y = 0
    . . \frac{x^2+1}{x} \:=\:0\quad\Rightarrow\quad x^2+1\:=\:0\quad\Rightarrow\quad x^2 \:=\:-1 . . . no x-intercepts

    y-intercepts: let x = 0
    . . But \frac{0^2+1}{0} is undefined . . . no y-intercept



    2) Asymptotes
    Vertical asymptote: . y \,=\,0 (y-axis)

    Horizontal asymptote: . None

    Slant asymptote: . \lim_{x\to\infty}\frac{x^2+1}{x} \:=\:x\quad\Rightarrow\quad y \:=\:x



    3) Symmetry
    Symmetry to origin (only).


    4) Domain
    All real x \neq 0


    5) Increasing/decreasing behavior
    y' \:=\:\frac{x\cdot2x - (x^2+1)\cdot1}{x^2} \:=\:\frac{x^2-1}{x^2}

    Increasing: . x^2-1\:>\:0\quad\Rightarrow\quad x^2\:>\:1\quad\Rightarrow\quad|x| > 1\quad\Rightarrow\quad(-\infty,\,-1) \cup (1,\infty)

    Decreasing: . |x| < 1,\;x \neq 0\quad\Rightarrow\quad (-1,0) \cup (0,1)



    6) Relative extrema
    Solve y' = 0\!:\;\;\frac{x^2-1}{x^2}\:=\:0\quad\Rightarrow\quad x \:=\:\pm1\quad\Rightarrow\quad y \:=\:\pm2

    Second derivative: . y'' \:=\:\frac{x^2\cdot2x - (x^2-1)\cdot2x}{x^4} \;=\;\frac{2}{x^3}

    At x = 1\!:\;y'' = +2 concave up . . . minimum at (1,2)

    At x = -1\!:\;y'' = -2 concave down . . . maximum at (-1,-2)



    7) Concavity
    Concave up where y'' > 0\!:\;\frac{2}{x^3} > 0 \quad\Rightarrow\quad x > 0

    Concave down where y'' < 0\!:\;\;\frac{2}{x^3} < 0 \quad\Rightarrow\quad x < 0



    8) Inflection Points

    Inflection point occur where y'' = 0

    . . But: . \frac{2}{x^3} \:=\:0 has no solutions . . . There are no inflection points.


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