The given function was
$$f(x)=ln(\frac{2}{x})$$
and I had to compute the linear approximation at x = 2. I obtained the answer of
$$L(x)=-\frac{1}{2}(x-2)$$
I am then supposed to use that approximation to estimate $ln(1.9)$.
At this point I don't understand what to do, because my class only covered examples where the x had the same modifier, i.e. the function was $f(x)=\sqrt{x}$ and the point we had to estimate was $\sqrt2$, and a search on the web only returns the same types of examples.
I assume I have to use a number that will be nicely divisible by 2 and close to 1.9, so I would choose 2. Therefore, $\frac{2}{2}=1$, and when I plug that in to $L(x)$ I would obtain .5, which is a slight underestimate, and
$$f''(1) = \frac{1}{1^2} = 1 $$
so the concavity confirms that it is an underestimate.
I think this is the solution, but I just had to make sure. Is my methodology correct? Should I have used a different number for the estimate, should I have changed the number at all? Any help will be great. Thank you!