To determine how large we need to choose $n$ so that the approximations $T_n$ (trapezoidal rule) and $M_n$ (midpoint rule) are accurate to within $0.0001$, we need to use the error bounds for each approximation method.

1. Error for the Trapezoidal Rule: The error bound for the trapezoidal rule is given by: $E_T \leq \frac{K(b-a)^3}{12n^2}$ where $K$ is an upper bound on the absolute value of the second derivative of the integrand $f(x)$, $[a, b]$ is the interval of integration, and $n$ is the number of subintervals.

2. Error for the Midpoint Rule: The error bound for the midpoint rule is: $E_M \leq \frac{K(b-a)^3}{24n^2}$ where $K$ is again an upper bound on the absolute value of the second derivative of the integrand $f(x)$.

To ensure that the errors are within $0.0001$, we set up the following inequalities:

Trapezoidal Rule:

$\frac{K(b-a)^3}{12n^2} \leq 0.0001$

Solving for $n$: $n \geq \sqrt{\frac{K(b-a)^3}{12 \times 0.0001}}$

Midpoint Rule:

$\frac{K(b-a)^3}{24n^2} \leq 0.0001$

Solving for $n$: $n \geq \sqrt{\frac{K(b-a)^3}{24 \times 0.0001}}$

Procedure:

1. Step 1: Use the value of $K$ from part (b) to calculate both bounds for $n$.
2. Step 2: Solve for $n$ in both cases (for $T_n$ and $M_n$).
3. Step 3: Choose the larger of the two $n$ values to ensure the error is within bounds for both approximations.

Please provide the value of $K$ from part (b) and the limits of integration $a$ and $b$, and I can help you complete the calculations.

Would you like to proceed with that, or do you need more clarification on any steps?

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Related Questions:

1. What is the difference between the trapezoidal rule and the midpoint rule in numerical integration?
2. How does the second derivative $K$ influence the error in numerical integration?
3. Can the midpoint rule sometimes yield more accurate results than the trapezoidal rule?
4. How would increasing $n$ improve the accuracy of both $T_n$ and $M_n$?
5. Are there other numerical methods that can provide more accurate approximations than the trapezoidal and midpoint rules?

Tip: Higher values of $n$ reduce the error but increase computation. Choose $n$ wisely based on required precision and available computational resources.