Determine the values of x for which the linear approximation is accurate to within 0.1.

Question

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?

Answer 1

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.