Let f(x) = ln(1 + x). Determine the third Taylor polynomial of f(x) at x = 0.
A) x + $\frac { 1 } { 2 }$ $x ^ { 2 }$ + $x ^ { 3 }$
B) x + $\frac { 1 } { 2 }$ $x ^ { 2 }$ + $\frac { 1 } { 3 }$ $x ^ { 3 }$
C) x - $\frac { 1 } { 2 }$ $x ^ { 2 }$ + $\frac { 1 } { 3 }$ $x ^ { 3 }$
D) x + $x ^ { 2 }$ + $x ^ { 3 }$

Question

Quizplus · Accepted Answer

The third Taylor polynomial of $f(x) = \ln(1 + x)$ at $x = 0$ is found by calculating the derivatives of $f(x)$ at $x = 0$ and using the formula for the Taylor series. The derivatives of $f(x)$ at $x = 0$ are: $f'(x) = \frac{1}{1+x}$, $f''(x) = -\frac{1}{(1+x)^2}$, and $f'''(x) = \frac{2}{(1+x)^3}$. Evaluating these at $x = 0$ gives $f'(0) = 1$, $f''(0) = -1$, and $f'''(0) = 2$. The third Taylor polynomial is thus $f(x) \approx f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 = 0 + x - \frac{1}{2}x^2 + \frac{1}{3}x^3$.

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

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