To solve Part 4, which asks for the error in the trapezoid rule approximation with $2n$ subintervals, we'll follow these steps:

1. Exact Value of the Integral: First, compute the exact value of the integral $\int_0^1 e^{7x} \, dx$.

   $\int_0^1 e^{7x} \, dx = \left[ \frac{1}{7} e^{7x} \right]_0^1 = \frac{1}{7} (e^7 - e^0) = \frac{e^7 - 1}{7}$

2. Evaluate $e^7$: Calculate $e^7$ using a calculator or software to get a more precise value.

   $e^7 \approx 1096.633158$

   So, the exact value of the integral is:

   $\text{Exact Value} = \frac{1096.633158 - 1}{7} \approx 156.661880$

3. Trapezoid Rule Approximation with $2n$ Subintervals: Given from the problem, the trapezoid rule approximation with $2n$ subintervals (i.e., $n = 40$) is $T(40) = 156.918268$.

4. Calculate the Error: The error in the trapezoid rule is the difference between the exact value and the trapezoid rule approximation:

   $\text{Error} = T(40) - \text{Exact Value} = 156.918268 - 156.661880$

   Calculate the error:

   $\text{Error} \approx 0.256388$

   Convert the error into scientific notation and round to three decimal places:

   $\text{Error} \approx 2.564 \times 10^{-1}$

So, the error in the trapezoid rule approximation with $2n$ subintervals is approximately $2.564 \times 10^{-1}$.

Would you like more details on this, or have any other questions?

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Here are 5 questions related to the topic:

1. What are the general formulas for the trapezoid rule and Simpson's rule?
2. How does the error behave as you increase the number of subintervals in the trapezoid rule?
3. Why is Simpson's rule generally more accurate than the trapezoid rule?
4. How would you modify the trapezoid rule if the function has more complex behavior over the interval?
5. What are the conditions under which Simpson's rule is exact?

Tip: For functions that are not polynomial, Simpson's rule provides better accuracy as it approximates the integrand with a quadratic polynomial.