TITLE: Determine the values of x for which the linear approximation is accurate to within 0.1.
QUESTION [2 upvotes]: So I've got a function:
$$\frac{1}{(1+2x)^4}$$
with its linear approximation:
$$1-8x$$
For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the approximation from the function and set up an inequality.
$$-0.1 \leq \frac{1}{(1+2x)^4} - (1-8x) \leq 0.1$$
I can't work out how to progress from here though.
Multiplying out the brackets seems like a nightmare, and I don't think it will help? Any ideas?

REPLY [1 votes]: Presumably your approximation is around $x=0$.  If you plot the error you see it is roughly parabolic and the error is never less than zero.  So we can do well with the quadratic approximation.  Near $x=0,\ \frac 1{(1+2x)^4} \approx 1-8x+40x^2-160x^3+\dots$  The error is then about $40x^2$ and solving $40x^2=0.1$ gives $x=\pm 0.05$.  By comparison $160x^3$ evaluated at $0.05$ is $\pm 0.02$ so the approximation is not too bad.
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