Question

Find the taylor polynomial t1(x) for the function f(x)=ln(x) based at b=1.

Answer 1

Assuming $T_1(x)$ refers to the first degree Taylor polynomial, you need only find the value of $f$ and its derivative at $x=1$. You have $f(1)=\ln1=0$ and $f'(1)=\dfrac11=1$.

The first degree Taylor approximation to $f$ centered at $b=1$ is then $T_1(x)=f(1)+f'(1)(x-1)=x-1$.