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Thread: Find Epsilion (What did I do wrong?)

  1. #1
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    Find Epsilion (What did I do wrong?)

    Find Epsilion (What did I do wrong?)-capture.png

    My work:

    abs(f(x)-l) = abs(x^2-64) = abs(x-8)abs(x+8)

    the max bound for x+8 is 17 so

    abs(x-8)abs(x+8) < abs(x-8) * 17 < 17S

    so S = E/17

    S = 0.003 / 17 = 0.000176

    I got the answer wrong? The answer is actually 0.000187...

    What went wrong?
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  2. #2
    MHF Contributor
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    Re: Find Epsilion (What did I do wrong?)

    $$\begin{array}{r r c c c l c r c c c l} \left| x^2 - 64 \right| < \frac{3}{1000} \implies & -\frac{3}{1000} & < & x^2 - 64 & < & \frac3{1000} \\ & 64 - \frac3{1000} & < & x^2 & < & 64 + \frac3{1000} \\ & \frac{63997}{1000} & < & x^2 & < & \frac{64003}{1000} \\ & -\sqrt{\frac{64003}{1000}} & < & x & < & -\sqrt{\frac{63997}{1000}} & \text{or} & \sqrt{\frac{63997}{1000}} & < & x & < & \sqrt{\frac{64003}{1000}} \end{array}$$
    Since we need $x \approx 8$ we take
    $$\begin{array}{r c c c l l} \sqrt{\frac{63997}{1000}} & < & x & < & \sqrt{\frac{64003}{1000}} \\ \sqrt{\frac{63997}{1000}} - 8 & < & x - 8 & < & \sqrt{\frac{64003}{1000}} - 8 \\ -0.0001875 & < & x-8 & < & 0.0001875 & \implies |x-8| < 0.000187 \end{array}$$
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