The given problem involves using the Remainder Estimation Theorem to find an interval containing $x = 0$ over which the function $f(x) = \sin(x)$ can be approximated by the polynomial $p(x) = x - \frac{x^3}{3!}$ with a three-decimal-place accuracy.

Let's go step by step:

Step 1: Taylor Series Approximation

The Taylor series for $\sin(x)$ around $x = 0$ is: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$ The polynomial $p(x) = x - \frac{x^3}{6}$ is the Taylor polynomial of degree 3 for $\sin(x)$.

Step 2: Remainder Estimation Theorem

The remainder after the third term in the Taylor series for $\sin(x)$ is given by: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}$ For $p(x)$, we are using the third-degree polynomial, so the remainder $R_3(x)$ is the next term in the Taylor series: $R_3(x) = \frac{x^5}{5!} = \frac{x^5}{120}$ We need to find the maximum value of $x$ such that this remainder is less than or equal to $0.0005$ (since three decimal places means the error must be less than $0.0005$).

Step 3: Solve for $x$

We want: $\left| \frac{x^5}{120} \right| \leq 0.0005$ Solving for $x$: $\frac{x^5}{120} \leq 0.0005$ $x^5 \leq 0.0005 \times 120 = 0.06$ $x \leq (0.06)^{1/5}$ Now, I'll compute the fifth root of $0.06$.The value of $x$ such that the remainder is less than or equal to $0.0005$ is approximately $x \leq 0.57$.

Thus, the interval over which $f(x) = \sin(x)$ can be approximated by $p(x) = x - \frac{x^3}{3!}$ with three-decimal accuracy is $|x| \leq 0.57$.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How does the Remainder Estimation Theorem apply to higher-degree polynomials?
2. What happens to the interval if we include more terms in the Taylor expansion?
3. How does the error bound change if we require four decimal places of accuracy?
4. What are other common functions approximated using Taylor series?
5. How do we handle the error in approximations for functions that are not smooth?

Tip: When approximating functions using Taylor series, increasing the degree of the polynomial generally increases the accuracy over a larger interval.