To determine the value of $n$ that guarantees the Simpson's Rule approximation to the integral

$I = \int_0^1 \frac{1}{17} e^{x^2} \, dx$

is accurate within $0.00001$, we can use the error bound for Simpson's Rule.

The error bound for Simpson's Rule is given by the formula:

$E_S = \frac{M(b - a)^5}{180n^4}$

where:

• $E_S$ is the error in the approximation,
• $M$ is the maximum value of the fourth derivative of the integrand over the interval $[a, b]$,
• $n$ is the number of subintervals (which must be even),
• $a$ and $b$ are the bounds of the integration.

Step-by-Step Solution:

1. Identify the integrand: The function is $f(x) = \frac{1}{17} e^{x^2}$, so the integrand is $f(x) = e^{x^2} / 17$.

2. Find the fourth derivative: We need to compute the fourth derivative of $f(x) = e^{x^2} / 17$. Let's first find the derivatives of $e^{x^2}$.

   • $f'(x) = 2x e^{x^2}$,
   • $f''(x) = 2 e^{x^2} + 4x^2 e^{x^2}$,
   • $f^{(3)}(x) = 8x e^{x^2} + 8x^3 e^{x^2}$,
   • $f^{(4)}(x) = 8 e^{x^2} + 32x^2 e^{x^2} + 8x^4 e^{x^2}$.

3. Determine the maximum value of the fourth derivative: To ensure a safe bound for the error, we can estimate the maximum value of $|f^{(4)}(x)|$ over the interval $[0, 1]$. For simplicity, we can approximate that $f^{(4)}(x)$ attains its maximum at $x = 1$, so we can evaluate $f^{(4)}(1)$.

   $f^{(4)}(1) = 8 e^{1^2} + 32 \cdot 1^2 e^{1^2} + 8 \cdot 1^4 e^{1^2} = 48 e$ Since $e \approx 2.718$, we get: $f^{(4)}(1) \approx 48 \times 2.718 = 130.464$ Therefore, we use $M \approx 130.464$.

4. Use the error formula: The error bound formula is:

   $E_S \leq \frac{M(b - a)^5}{180n^4}$

   Substituting the known values ($b = 1$, $a = 0$, $M \approx 130.464$, and $E_S \leq 0.00001$):

   $0.00001 \geq \frac{130.464 \times (1 - 0)^5}{180n^4}$

   Simplifying:

   $0.00001 \geq \frac{130.464}{180n^4}$

   Solving for $n^4$:

   $n^4 \geq \frac{130.464}{180 \times 0.00001} = \frac{130.464}{0.0018} \approx 72536.67$

   Taking the fourth root:

   $n \geq \sqrt[4]{72536.67} \approx 18.5$

5. Conclusion: Since $n$ must be even, the smallest integer $n$ is $n = 20$.

Therefore, the value of $n$ should be at least 20 to guarantee that the Simpson's Rule approximation is accurate to within 0.00001.

Would you like any further details or explanations? Here are some related questions for exploration:

1. How do you find the error bound for the Trapezoidal Rule?
2. What are the practical applications of Simpson's Rule in numerical integration?
3. How does increasing $n$ improve the accuracy of the approximation?
4. How do we compute the exact value of the integral $\int_0^1 e^{x^2} \, dx$?
5. What is the relationship between the degree of the polynomial and the accuracy of Simpson's Rule?

Tip: To ensure more accurate numerical approximations, always verify that $n$ is sufficiently large and that the integrand is smooth over the interval of integration.