linear approximation accuracy question

**satis** · October 14th 2010, 03:36 PM

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.

**zzzoak** · October 14th 2010, 04:08 PM

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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