To approximate $\sin(x)$ over the interval $[0, 1]$ using polynomial interpolation with $n + 1$ equispaced points, we can analyze the error associated with polynomial interpolation using the formula for the error bound:

$E_n(x) = \frac{M}{(n+1)!} |x - x_0||x - x_1| \cdots |x - x_n|$

where $M$ is the maximum value of the $(n+1)$-th derivative of $\sin(x)$ over the interval $[0, 1]$, and $x_0, x_1, \ldots, x_n$ are the interpolation nodes.

Step 1: Determine $M$

The $n$-th derivative of $\sin(x)$ oscillates between -1 and 1, so the maximum $M$ is:

$M = 1$

Step 2: Error Bound Calculation

The maximum value of $|x - x_i|$ for $x \in [0, 1]$ is achieved at the endpoints. For $n + 1$ points, the largest distance between any two nodes is at most $h = \frac{1}{n}$. Therefore, we can express the error as:

$|E_n(x)| \leq \frac{1}{(n+1)!} \cdot \frac{1}{(n+1)^{n+1}} \cdot (1) = \frac{1}{(n+1)!} \cdot \left(\frac{1}{n}\right)^{n}$

Step 3: Set Up the Inequality

We want this error bound to be less than $1 \times 10^{-6}$:

$\frac{1}{(n+1)!} \cdot \left(\frac{1}{n}\right)^{n} < 1 \times 10^{-6}$

Step 4: Simplifying and Estimating $n$

To find the smallest $n$ satisfying this inequality, we can calculate $(n+1)!$ for various $n$:

1. For small values of $n$, compute $\frac{1}{(n+1)!}$ and $\left(\frac{1}{n}\right)^{n}$ until the product is less than $1 \times 10^{-6}$.

Let’s compute this for increasing values of $n$:

• $n = 5$: $\frac{1}{6!} = \frac{1}{720}, \quad \left(\frac{1}{5}\right)^{5} = \frac{1}{3125}$ $\frac{1}{720} \cdot \frac{1}{3125} \approx 4.55 \times 10^{-7} \text{ (valid)}$

• $n = 4$: $\frac{1}{5!} = \frac{1}{120}, \quad \left(\frac{1}{4}\right)^{4} = \frac{1}{256}$ $\frac{1}{120} \cdot \frac{1}{256} \approx 3.91 \times 10^{-6} \text{ (not valid)}$

Conclusion

Thus, to achieve an error less than $1 \times 10^{-6}$, at least $n = 5$ points (which corresponds to $n + 1 = 6$ equispaced points) are required.

Follow-Up Questions

1. How does the choice of equispaced points affect the accuracy of polynomial interpolation?
2. What are the advantages and disadvantages of using polynomial interpolation compared to other methods like spline interpolation?
3. Can you explain the concept of the Runge phenomenon in polynomial interpolation?
4. How would the error bound change if we used Chebyshev nodes instead of equispaced nodes?
5. What are some applications of polynomial interpolation in numerical analysis?

Tip: Always consider the characteristics of the function you're approximating when choosing the interpolation method, as this can significantly impact the error.