Let f(x) = ln x. Find the second Taylor polynomial of f(x) at x = 2.
A) 1 + $\frac { 1 } { 2 }$ (x - 2) + $\frac { 1 } { 8 }$ $( x - 2 ) ^ { 2 }$
B) ln 2 - $\frac { 1 } { 2 }$ (x - 2) - $\frac { 1 } { 8 }$ $( x - 2 ) ^ { 2 }$
C) 1 + $\frac { 1 } { 2 }$ (x - 2) - $\frac { 1 } { 8 }$ $( x - 2 ) ^ { 2 }$
D) ln 2 + $\frac { 1 } { 2 }$ (x - 2) - $\frac { 1 } { 8 }$ $( x - 2 ) ^ { 2 }$

Question

Quizplus · Accepted Answer

The second Taylor polynomial of $f(x) = \ln x$ at $x = 2$ includes the function evaluated at 2, the first derivative evaluated at 2 times $(x-2)$, and the second derivative evaluated at 2 times $\frac{(x-2)^2}{2!}$. The first derivative of $\ln x$ is $\frac{1}{x}$, and the second derivative is $-\frac{1}{x^2}$. Evaluating these at $x = 2$ gives $f(2) = \ln 2$, $f'(2) = \frac{1}{2}$, and $f''(2) = -\frac{1}{4}$. Thus, the second Taylor polynomial is $f(2) + f'(2)(x-2) + \frac{f''(2)}{2}(x-2)^2 = \ln 2 + \frac{1}{2}(x-2) - \frac{1}{8}(x-2)^2$.

Let F(x) = Ln X 12\frac { 1 } { 2 }21​

Let F(x) = Ln X $\frac { 1 } { 2 }$