Suppose that you use the interval [−5, 0] with 5 subintervals instead. Explain why your answers would be different for the above questions.

Breakdown of the Problem

1. We are asked to explain why the answers would differ when using the interval $[-5, 0]$ with 5 subintervals, as opposed to potentially other intervals or numbers of subintervals.
2. Identifying key factors that affect the results of calculations based on intervals and subintervals.

Relevant Concepts

3. The concepts involved in this question include partitioning intervals into subintervals and the effects of different subdivision methods on results, particularly in the context of numerical integration or approximation methods.

Analysis and Detail

4. When using the interval $[-5, 0]$ and dividing it into 5 subintervals, each subinterval will have a width of: $\Delta x = \frac{b-a}{n} = \frac{0 - (-5)}{5} = \frac{5}{5} = 1$ The endpoints of the subintervals would be: $-5, -4, -3, -2, -1, 0$.
5. The choice of subintervals can significantly affect the outcomes of various numerical methods (e.g., Riemann sums, trapezoidal rule) applied to the function over that interval.
6. If sought answers depend on the function evaluations at the bounds or midpoints of these subintervals, different selections can yield different aggregate properties (like areas or averages).

Verify and Summarize

7. Different intervals or number of subintervals can lead to different approximations or exact evaluations because of the method of sampling used (left endpoint, right endpoint, midpoints, etc.).
8. The difference in answers arises due to the sensitivity of numerical methods to the width of the intervals and the choice of points at which the function is evaluated.

Final Answer

In summary, using the interval $[-5, 0]$ with 5 subintervals results in different answers because the chosen partitioning and width influence the sampling points for numerical calculations, which can alter the convergence or accuracy of an approximation method.