We are given that:

• $|f''(x)| \leq 44$ and $|f^{(4)}(x)| \leq 44$ for all $x \in [0, \pi]$,
• We want to determine how large we need to choose $n$ so that the approximations $T_n$, $M_n$, and $S_n$ (the Trapezoidal, Midpoint, and Simpson's rules) to the integral of $f(x)$ from 0 to $\pi$ are accurate to within 0.00001.

Step 1: Error bounds for numerical integration methods

We will use the error bounds for the Trapezoidal, Midpoint, and Simpson's rules to determine the required number of subintervals, $n$, for each method. The error bounds for these methods are given by the following formulas:

1. Trapezoidal Rule: $E_T \leq \frac{(b-a)^3}{12n^2} \max_{x \in [a,b]} |f''(x)|$

2. Midpoint Rule: $E_M \leq \frac{(b-a)^3}{24n^2} \max_{x \in [a,b]} |f''(x)|$

3. Simpson's Rule: $E_S \leq \frac{(b-a)^5}{180n^4} \max_{x \in [a,b]} |f^{(4)}(x)|$

Where:

• $[a, b] = [0, \pi]$, so $b-a = \pi$,
• The error should be less than $0.00001$,
• $\max |f''(x)| \leq 44$ and $\max |f^{(4)}(x)| \leq 44$.

Step 2: Solve for $n$

For the Trapezoidal Rule:

The error bound is: $E_T \leq \frac{\pi^3}{12n^2} \cdot 44$ We want $E_T \leq 0.00001$, so: $\frac{\pi^3}{12n^2} \cdot 44 \leq 0.00001$ Solving for $n$: $n \geq \sqrt{\frac{\pi^3 \cdot 44}{12 \cdot 0.00001}}$

For the Midpoint Rule:

The error bound is: $E_M \leq \frac{\pi^3}{24n^2} \cdot 44$ We want $E_M \leq 0.00001$, so: $\frac{\pi^3}{24n^2} \cdot 44 \leq 0.00001$ Solving for $n$: $n \geq \sqrt{\frac{\pi^3 \cdot 44}{24 \cdot 0.00001}}$

For Simpson's Rule:

The error bound is: $E_S \leq \frac{\pi^5}{180n^4} \cdot 44$ We want $E_S \leq 0.00001$, so: $\frac{\pi^5}{180n^4} \cdot 44 \leq 0.00001$ Solving for $n$: $n \geq \left( \frac{\pi^5 \cdot 44}{180 \cdot 0.00001} \right)^{\frac{1}{4}}$

Step 3: Compute the values for $n$

I'll calculate the values of $n$ for each method.The required values of $n$ for each method to achieve the desired accuracy of 0.00001 are:

1. Trapezoidal Rule: $n \geq 3372$
2. Midpoint Rule: $n \geq 2385$
3. Simpson's Rule: $n \geq 53$

These values indicate how many subintervals you need to use for each method to ensure the approximation is accurate within the given tolerance.

Would you like more details on how these were derived, or do you have any questions?

5 Related Questions:

1. How do these error bounds change if the bounds on $f''(x)$ and $f^{(4)}(x)$ are different?
2. What happens to the accuracy if we reduce $n$ by a small amount from the calculated values?
3. Can these methods be used for functions with discontinuities or rapid changes in derivatives?
4. How does the choice of integration method affect computational efficiency?
5. How would the error change if we use adaptive quadrature instead of fixed intervals?

Tip:

Simpson's Rule often provides higher accuracy with fewer intervals, especially when the function is well-behaved, making it more efficient for smoother functions.