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3370Midpoint Rule Misses a Symmetric BowlUsing a single-panel midpoint rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3371Simpson on a CubicUsing one Simpson panel, approximate \int 0 2 x 3\,dx.数学中等derivation未尝试面试订阅3376Halving h in the Trapezoid RuleA smooth integrand is approximated with the trapezoid rule. If the current error at step size h is about 0.08, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3378Halving h in Simpson's RuleA smooth integrand is approximated with the Simpson rule. If the current error at step size h is about 0.016, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3381Degree of Exactness of Two-Point GaussFor two-point Gauss-Legendre quadrature on [-1,1], what is the highest polynomial degree integrated exactly?数学中等derivation未尝试面试订阅3382Two-Point Gauss on x^2+1Using two-point Gauss-Legendre on [-1,1], approximate \int -1 1 (x 2+1)\,dx.数学中等derivation未尝试面试订阅3383Two-Point Gauss on x^4Using two-point Gauss-Legendre on [-1,1], what approximation do you get for \int -1 1 x 4\,dx?数学中等derivation未尝试面试订阅3386Why Trapezoid Overestimates a Convex FunctionWhy does the trapezoid rule typically overestimate the integral of a convex function on one panel?数学中等essay未尝试面试订阅3387Why Midpoint Often Beats Trapezoid on Smooth ProblemsWhy can the midpoint rule outperform the trapezoid rule on smooth integrands even though both are second-order?数学中等essay未尝试面试订阅3388When Simpson's Rule Is Especially AttractiveWhen is Simpson's rule especially attractive compared with midpoint or trapezoid?数学中等essay未尝试面试订阅3389Why Oscillatory Integrals Need CareWhy can standard coarse-grid quadrature be unreliable on highly oscillatory integrals?数学中等essay未尝试面试订阅3390Why Adaptive Refinement Helps Near KinksWhy is adaptive refinement often better than a uniform fine grid when the integrand has a kink or localized sharp feature?数学中等essay未尝试面试订阅3391Central-Difference Richardson Upgrade 1A second-order central-difference estimate of f'(x) is 1.28 at h=0.2 and 1.22 at h=0.1. Using an O(h 2) Richardson extrapolation, what improved estimate do you get?数学简单derivation未尝试面试订阅3392Second-Derivative Richardson Upgrade 2A second-order estimate of f''(x) is 5.8 at h=0.4 and 5.2 at h=0.2. Assuming an O(h 2) truncation error, what Richardson-extrapolated estimate do you get?数学简单derivation未尝试面试订阅3393First-Order Extrapolation From Two Forward MeshesA first-order one-sided derivative estimate is 2.6 at h=0.1 and 2.45 at h=0.05. Assuming an O(h) bias, what extrapolated estimate cancels the leading error?数学简单derivation未尝试面试订阅3394Recover the Leading O(h^2) Error CoefficientSuppose D h = L + c h 2 + O(h 4). If D 0.3 = 4.18 and D 0.15 = 4.045, what is c?数学简单derivation未尝试面试订阅3395Recover the Limit of a First-Order SchemeA first-order approximation obeys D h = L + c h + O(h 2). If D 0.4 = 1.9 and D 0.2 = 1.6, what is the extrapolated estimate of L?数学简单derivation未尝试面试订阅3396Max Mesh Under a Central-Difference Error BudgetA central-difference first-derivative formula has truncation error bounded by M h 2 / 6, and a third-derivative bound gives M=12. How large can h be if you want the truncation error below 0.002?数学中等derivation未尝试面试订阅3397Forward-Difference Mesh Under a Linear Error BoundA forward-difference derivative formula has truncation error bounded by M h / 2 with M=8. How large can h be if the truncation budget is 0.01?数学中等derivation未尝试面试订阅3398Second-Derivative Mesh From a Fourth-Derivative BoundA centered second-derivative formula has error bounded by M h 2 / 12 with M=24. What h keeps the truncation error at or below 0.005?数学中等derivation未尝试面试订阅