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Math Help - difference quotient help

  1. #1
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    difference quotient help

    I have 2 graded problems that are due monday, september 22.
    The directions are to simplify the difference quotients of the following functions:
    #1:
    f(x) = (√x)
    and
    #2:
    f(x) = 1/x
    The difference quotient is: m = (f(x+h) - f(x)) / h
    All work/steps must be shown.

    I attempted both of the problems but did not get very far. I put the functions into the difference quotient equation but I don't know how to simplify the expressions properly.

    #1:
    f(x) = (√x)
    (√(x+h) - √(x)) / h

    #2:
    f(x) = 1/x
    (((1/(x+h)) - (1/x)) / h

    Please respond quickly; all responses are greatly appreciated.
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  2. #2
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    for #1 ... rationalize the numerator and simplify to determine the limit.

    for #2 ... get a common denominator for the two fractions 1/x and 1/(x+h), combine them, then simplify. might be easier if you view the difference quotient like this ...

    \frac{1}{h}\left(\frac{1}{x+h} - \frac{1}{x}\right)
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  3. #3
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    1. You can't really "simplify" this expression but you can change it in order to calculate the limit as h tends to zero. We begin with:

    \frac{\sqrt{x+h} - \sqrt{x}}{h}

    Multiply top and bottom by \sqrt{x+h} + \sqrt{x}:

    \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})}

    Distributively multiply out the top:

    \frac{x + h - x}{h(\sqrt{x+h} + \sqrt{x})}

    \frac{h}{h(\sqrt{x+h} + \sqrt{x})}

    Cancel the factors of h in the top and bottom:

    \frac{1}{\sqrt{x+h} + \sqrt{x}}
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  4. #4
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    Quote Originally Posted by skeeter View Post
    for #2 ... get a common denominator for the two fractions 1/x and 1/(x+h), combine them, then simplify. might be easier if you view the difference quotient like this ...
    \frac{1}{h}\left(\frac{1}{x+h} - \frac{1}{x}\right)
    I did this and it all simplified down to just -h. Can you try it and verify that this is the correct answer?
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  5. #5
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    no, it does not simplify to just -h.

    here's something to get you going in the right direction ...

    \frac{1}{h} \left(\frac{1}{x+h} - \frac{1}{x}\right) = \frac{1}{h} \left(\frac{x - (x+h)}{x(x+h)}\right)
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  6. #6
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    Yeah I got that part right. After that I simplify it to:
    (\frac{-h}{x^2+xh}) (\frac{1}{h})
    then
    (\frac{-h^2}{h(x^2+xh})(\frac{x^2+xh}{h(x^2+xh)})
    then
    \frac{-h^2(x^2+xh)}{h(x^2+xh)}
    then
    \frac{-h^2}{h}
    then
    -h

    What am I doing wrong?
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  7. #7
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    you don't need a common denominator to multiply two fractions ... the h's cancel.

    \frac{1}{h} \left( \frac{-h}{x(x+h)}\right) = \frac{-1}{x(x+h)}
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