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Thread: Average rate of change

  1. #1
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    Average rate of change

    I am unsure as to which forum to put this in. I took a shot and put it here, apologies if wrong.

    Okay, to the question I am having trouble with.

    "Find the average rate of change of $\displaystyle f(x) = \sqrt{x + 11}$ with respect to x from $\displaystyle x = 5$ to $\displaystyle x = 5 +h$. [4 marks]"

    My attempt to answer:
    $\displaystyle f(5) = \sqrt{5 + 11}$
    $\displaystyle f(5) = \sqrt{16}$
    $\displaystyle f(5) = 4 $

    $\displaystyle f(5 + h) = \sqrt{5 + h + 11}$
    $\displaystyle f(5 + h) = \sqrt{16 + h}$
    $\displaystyle f(5 + h) = 4 + \sqrt{h}$
    *I am unsure if I am correct in square rooting the 16, so having 4 + the square root of h.*

    Then I use the $\displaystyle \frac{y2 - y1}{x2 - x1}$ formula to get the slope of the secant *rate of change*:

    $\displaystyle \frac{4 + \sqrt{h} - 4}{5 - 5 + h}$
    $\displaystyle = \frac{\sqrt{h}}{h}$
    That's my final solution. My problem with this solution is when I sub a number in for h, I get a different answer from running the numbers through, and simply going *square root of the number divided by the number*.
    Example:
    Let's pretend h is 20.
    So I have $\displaystyle f(5) = 4$.
    I can sub in 20 for h:
    $\displaystyle f(5 + 20) = \sqrt{5 + 20 + 11} $
    $\displaystyle f(25) = \sqrt{36}$
    $\displaystyle f(25) = 6 $

    Run that through the slope formula:
    $\displaystyle \frac{6 - 4}{20 - 5}$
    $\displaystyle = \frac{2}{5} $
    $\displaystyle = 0.4$

    Then, using my average rate of change thing: $\displaystyle \frac{\sqrt{h}}{h}$ I get:
    $\displaystyle \frac{\sqrt{20}}{20}$
    $\displaystyle = 0.223606797$
    which is a completely different answer. So I must be doing something wrong here.

    Okay, I tried using 25 for h in the $\displaystyle \frac{\sqrt{h}}{h}$ and it gives the same ansewr as running the math through. So it is not technically h, but it is h + 5, so is this actually right??? I see that as: $\displaystyle \frac{\sqrt{h+5}}{h+5}$ which I think is different, please correct me, I am so confused.

    EDIT:There, I have successfully turned this into something readable using latex
    Last edited by Kakariki; Sep 15th 2009 at 01:04 AM.
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Kakariki View Post
    I am unsure as to which forum to put this in. I took a shot and put it here, apologies if wrong.

    Okay, to the question I am having trouble with.

    "Find the average rate of change of $\displaystyle f(x) = \sqrt{x + 11}$ with respect to x from $\displaystyle x = 5$ to $\displaystyle x = 5 +h$. [4 marks]"

    My attempt to answer:
    $\displaystyle f(5) = \sqrt{5 + 11}$
    $\displaystyle f(5) = \sqrt{16}$
    $\displaystyle f(5) = 4 $

    $\displaystyle f(5 + h) = \sqrt{5 + h + 11}$
    $\displaystyle f(5 + h) = \sqrt{16 + h}$
    $\displaystyle f(5 + h) = 4 + \sqrt{h}$
    *I am unsure if I am correct in square rooting the 16, so having 4 + the square root of h.*

    Then I use the $\displaystyle \frac{y2 - y1}{x2 - x1}$ formula to get the slope of the secant *rate of change*:

    $\displaystyle \frac{4 + \sqrt{h} - 4}{5 - 5 + h}$
    $\displaystyle = \frac{\sqrt{h}}{h}$
    That's my final solution. My problem with this solution is when I sub a number in for h, I get a different answer from running the numbers through, and simply going *square root of the number divided by the number*.
    Example:
    Let's pretend h is 20.
    So I have $\displaystyle f(5) = 4$.
    I can sub in 20 for h:
    $\displaystyle f(5 + 20) = \sqrt{5 + 20 + 11} $
    $\displaystyle f(25) = \sqrt{36}$
    $\displaystyle f(25) = 6 $

    Run that through the slope formula:
    $\displaystyle \frac{6 - 4}{20 - 5}$
    $\displaystyle = \frac{2}{5} $
    $\displaystyle = 0.4$

    Then, using my average rate of change thing: $\displaystyle \frac{\sqrt[h]][h]]$ I get:
    $\displaystyle \frac{\sqrt{20}}{20}$
    $\displaystyle = 0.223606797$
    which is a completely different answer. So I must be doing something wrong here.

    Okay, I tried using 25 for h in the $\displaystyle \frac{\sqrt{h}}{h}$ and it gives the same ansewr as running the math through. So it is not technically h, but it is h + 5, so is this actually right???

    the wrong thing as what you said you can't separate 16 from the square root

    $\displaystyle f(5+x)=\sqrt{16+h}$ you can't simplify it more than that


    $\displaystyle s=\frac{\sqrt{16+h} - 4 }{h}$\

    EDIT:There, I have successfully turned this into something readable using latex
    CONGRATULATIONS LATEX is simple and useful
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