To solve this problem systematically, we’ll address the following:

1. Determine the remainder term $R_3(x)$ using the given formula.
2. Find the interval for $x$ where the approximation error is within $0.001$.

1. Remainder Term $R_3(x)$:

The remainder for the Taylor polynomial is given by:

$R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!} (x-c)^{n+1},$ where $c$ is the center (here $c = 0$) and $z$ lies between $x$ and $0$. For $T_3(x)$, we calculate $R_3(x)$, the remainder after the 3rd degree term.

Derivatives of $f(x) = \ln(1+x)$:

• $f(x) = \ln(1+x)$
• $f'(x) = \frac{1}{1+x}$
• $f''(x) = -\frac{1}{(1+x)^2}$
• $f'''(x) = \frac{2}{(1+x)^3}$
• $f^{(4)}(x) = -\frac{6}{(1+x)^4}$

Thus, the 4th derivative $f^{(4)}(z) = -\frac{6}{(1+z)^4}$.

Remainder $R_3(x)$ for $n = 3$:

Substitute into the remainder formula:

$R_3(x) = \frac{f^{(4)}(z)}{4!} x^4$

Since $f^{(4)}(z) = -\frac{6}{(1+z)^4}$ and $4! = 24$, we get:

$R_3(x) = \frac{-6}{24 (1+z)^4} x^4 = -\frac{x^4}{4 (1+z)^4},$ where $z$ lies between $0$ and $x$.

2. Find $x$ for Approximation Error $|R_3(x)| < 0.001$:

We need the absolute value of the remainder:

$\left| R_3(x) \right| = \frac{x^4}{4 (1+z)^4}.$

• Since $z$ is between $0$ and $x$, the maximum value of $(1+z)$ occurs at $z = 0$, giving $(1+z)^4 \geq 1$.
• Thus, to bound the remainder, we approximate:

$\left| R_3(x) \right| \leq \frac{x^4}{4}.$

We require $\left| R_3(x) \right| < 0.001$, so:

$\frac{x^4}{4} < 0.001.$

Multiply through by 4:

$x^4 < 0.004.$

Take the 4th root:

$x < \sqrt[4]{0.004}.$

Calculate $\sqrt[4]{0.004}$:

$\sqrt[4]{0.004} = \sqrt{\sqrt{0.004}} = \sqrt{0.0632} \approx 0.251.$

Final Answer:

The Taylor polynomial approximation is accurate within $0.001$ for:

$0 < x \leq 0.251.$

Summary:

1. Remainder Term: $R_3(x) = -\frac{x^4}{4(1+z)^4}$, where $z$ is between $0$ and $x$.
2. Approximation Interval: $0 < x \leq 0.251$.

Would you like a detailed explanation of any step or further clarifications?

Related Questions:

1. How do you derive the remainder formula for Taylor polynomials?
2. Why does the error decrease as the polynomial degree increases?
3. How do higher-order derivatives influence the Taylor remainder?
4. What is the geometric significance of the remainder term?
5. How can we improve the accuracy of a Taylor polynomial approximation?

Tip:

When approximating with Taylor polynomials, always use the remainder term to ensure the desired level of accuracy!