How to use linear approximation to estimate $2\ln(1.1)+3$? Can anyone please share how to solve this?
The question is to use the linear approximation to estimate $2\ln(1.1)+3$
The hint given to me was to find f(x) first and find a and f'(a) but I can't follow the hint and don't know how to find those.
Thanks.
I have not tried anything because I am lost at how to find f(x)first or a.
We have only been doing answers where the original f(x) is given then I find the linear approximation and estimate certain values afterwards. This is the first time the f(x) is not provided to me.

First, we have the following formula for the linear approximation of f(x) at a point $x=a$:
$f(x) ≈f(a) + f'(a)(x - a)$
The nearest value to $1.1$ that is easy to compute is $x=1$ as we know $\ln(1)=0$
If $f(x) = \ln (x)$ and $a = 1$, we get:
$\ln(x) = \ln (1) + x - 1=x-1$
If we set $x=1.1$ we can approximate $\ln(1.1)$ by $1.1-1=0.1$, we can multiply it by $2$ now and get $0.2$, then we add $3$ to it as we need $2\ln(1.1)+3$ and get $3.2$ as our approximation.
Actual value is $2\ln(1.1)+3=3.1906...$ so we are pretty close.
