Calculus Volume 2 · 免费预览
数值积分
梯形公式和辛普森公式根据采样值近似计算定积分，其误差界在网格加密时分别减半和降为四分之一，适用于原函数未知或不存在的被积函数。
1. The trapezoid rule
The trapezoid rule
2. Simpson's rule
Simpson's rule
3. Choosing rules and checking
Choosing rules and checking
4. Error bound computation
Error bound computation
5. Trapezoid weights
In the trapezoid rule T_n, how are the sample values weighted?
6. Simpson requirement
Simpson's rule requires which condition on the strip count n?
7. Error order comparison
How do the trapezoid and Simpson error orders compare?
8. Error bound inputs
The trapezoid error bound needs M bounding which derivative of f?
9. Gaussian example accuracy
For the integral from 0 to 1 of e^(-x^2) dx, Simpson's rule with n = 4 gives about 0.74683. The true value is about 0.74682. Roughly how many digits agree?
10. Trapezoid with two strips
For the integral from 0 to 2 of f dx with n = 2, the samples are f(0), f(1), f(2), and Delta x = 1. The trapezoid rule gives (1/2)(f(0) + 2 f(1) + f(2)). For f(x) = x^2 this equals what number? Answer with a number.
11. Simpson weight pattern
In Simpson's rule the interior weights alternate starting with 4. As a number, what weight follows the first interior 4? Answer with a number.
12. Simpson error order
Simpson's error behaves like 1 over n to which power? Answer with a number.
13. Singularity impact
Why do both trapezoid and Simpson rules struggle near an integrand's singularity?
14. Refinement agreement
When no antiderivative is known, agreement between S_n and S_2n to the target digits certifies the value. As a single word, what kind of check is this? Answer in lowercase.