Let .The first two nonzero terms of the Taylor expansion for about are

Question

Let  .The first two nonzero terms of the Taylor expansion for  about  are ​

Quizplus · Accepted Answer

For the Taylor expansion, we need to find the first few derivatives at the given point and plug them into the formula:
$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$
In this case, the given function is $\ln(1+x)$ and we need to find the first two derivatives at $x=0.01$:
$$f(x) = \ln(1+x) \quad \longrightarrow \quad f(0.01) = \ln(1+0.01)$$
$$f'(x) = \frac{1}{1+x} \quad \longrightarrow \quad f'(0.01) = \frac{1}{1+0.01} = \frac{100}{101}$$
$$f''(x) = -\frac{1}{(1+x)^2} \quad \longrightarrow \quad f''(0.01) = -\frac{1}{(1+0.01)^2} = -\frac{10000}{10201}$$
Plugging these into the formula, we get:
$$f(x) = \ln(1+0.01) + \frac{\frac{100}{101}}{1!}(x-0.01) + \frac{-\frac{10000}{10201}}{2!}(x-0.01)^2 + ...$$
Simplifying, the first two nonzero terms are:
$$f(x) = \ln(1+0.01) + \frac{\frac{100}{101}}{1!}(x-0.01) + \mathcal{O}((x-0.01)^2)$$
The correct choice is (C), which matches the first two terms above.

Let The First Two Nonzero Terms of the Taylor Expansion for Expansion