Give the 3rd order Taylor Polynomial for \( f(x)=\ln x \) at \( a=1 \) and then use it to approximate \( \ln 2 \)

\( f(x) = \ln x \) , \( f(1) = 0 \)

\( f^\prime (x) = \frac{1}{x} \) , \( f^\prime(1) = 1 \)

\( f^{\prime \prime} (x) = \frac{-1}{x^2} \) , \( f^{\prime \prime}(1) = -1 \)
\( f^{\prime \prime \prime} (x) = \frac{2}{x^3} \) , \( f^{\prime \prime}(1) = 2 \)

\( P_3(x) = \frac{0}{0!}(x-1)^0 + \frac{1}{1!}(x-1)^1 +\frac{-1}{2!}(x-1)^2+\frac{2}{3!}(x-1)^3= (x-1)-\frac{1}{2}(x-1)^2 +\frac{1}{3}(x-1)^3\)

\( \ln 2 \approx P_3(2)=1-\frac{1}{2}+\frac{1}{3} =\frac{5}{6} \)