Part 5: Now, let's see how we did. Since we cannot find an antiderivative for ∫₀₁ cos(x²)dx, we cannot know the exact "true value" of the integral; however, we can use the error bounds formula to determine the "worst-case" amount that our answers might differ from this hypothetical true value. To use it, you will need not only the values for a, b, and n, but also the constant K that describes the maximum possible curvature of the graph of f(x) = cos(x²) (the extent to which it might not match our Simpson's Rule estimations). For Simpson's Rule, this value depends on the maximum (or minimum) possible value of the fourth derivative f⁽⁴⁾(x), so you will need to find this fourth derivative as well as its maximum (and minimum) possible values over the interval [-1, 0]. What are the values of f⁽⁴⁾(x) at each endpoint? These would be the most extreme possibilities, so since we need to find a value of K for which |f⁽⁴⁾(x)| ≤ K, which of these endpoint values of f⁽⁴⁾(x) is farthest from 0? Choose K to be this "worst" of the two endpoint options. (You might re-examine the textbook authors' similar work with this same function, over a different interval, in Example 7 of Section 7.7 for inspiration.) Now, equipped with this value for K, what is the maximum possible value for |E_s|, according to the error-bounds formula? Remember, you will have to calculate separate error estimates for each value of n.
Can you see evidence of the error trends we discussed in Part 3? Specifically, can you determine whether the maximum possible error is being reduced to (roughly) one-sixteenth of its previous level each time n is doubled?