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Math Help - linear approximation accuracy question

  1. #1
    Junior Member
    Joined
    Sep 2009
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    74

    linear approximation accuracy question

    I have some problems that ask you go verify a linear approximation (easy enough), and then asks you to find the range of values for x which would make it accurate within 0.1. The latter part has me somewhat stumped.

    The problem is:
    (1-x)^\frac{1}{3} \approx 1-\frac{1}{3}x, a=0

    f'(x) = \frac{1}{3}(1-x)^\frac{-2}{3}(-1) which is f'(x) = \frac{1}{3(1-x)^\frac{2}{3}}

    f'(0) = -\frac{1}{3} and f(0) = 1
    so
    L(x) =1-\frac{1}{3}x... so that's verified. To get a tolerance of +/- 0.1 you set up the inequality

    (1-x)^\frac{1}{3} - 0.1 < 1-\frac{1}{3}x < (1-x)^\frac{1}{3} + 0.1

    Here I'm stuck. Can someone point me in the right direction? At this point I'm probably supposed to be doing algebra, but I'm hitting a wall.
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  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    I think that the accuracy of the linear approximation is
    <br />
| \; (1-x)^\frac{1}{3} -( 1-\frac{1}{3}x) \; |= \; the \; third \; term \; of \; Teylor \; series<br />
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