It's another day of calculus and I'm having trouble with linear approximations; perhaps you guys can help. I am unsure of how to calculate the accuracy of these approximations; let me give you an example.
Verify the given linear approximation at $a = 0$. Then determine the values of $x$ for which the linear approximation is accurate to within $0.1$.
$$1/(1+2x)^4 \approx 1 – 8x$$
I can verify the linear approximation easily enough, but how do I determine its accuracy? Thanks!
Best Answer
You question is a little ambiguous but I assume you mean find the values of $x$ for which the computed function values differ by less than 0.1.
If you are allowed to use a graphing calculator or something similar, just graph the functions
$$f1(x) = \left| {\frac{1}{{{{(1 + 2x)}^4}}} - (1 - 8x)} \right|$$
and
$$f2(x) = 0.1$$
and see where they intersect. I get the result
$$- 0.04536 \leqslant x \leqslant 0.05539$$
to five decimal places.
Another way is to use the second derivative but that seems too advanced for Precalculus.