First order approximation of $y=2^x$

Question

I wanted to find a first order estimation of $y$ in the form $y\approx y_0\pm \delta y$ where $y=2^x$ and $x=1\pm 0.1$

Thank you for telling me how to proceed. It seems like we need to write $f(x)\approx f(a)+f'(a)(x-a) $ where $f(x)=2^x$ and for each $x$ close to $a$, but i don't see how.

Answer 1

You can use the Taylor polynomial of degree 1 around the point x=1, that’s reasonably good. f(1+x)≈f(1) + x f'(1), and you change that to the form you need. (Joe’s comment).

You will then see that the error is 0 at x=1 and larger at x=0.9 and x = 1.1. Assume that the error is roughly eps in these points, then subtracting eps/2 gives an overall better error.

You get the best maximum error if you get two points with an error eps and a point with error -eps between them (Chebyshev polynomial).