A continuous function is defined on

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A continuous function hh is defined on [a,b][a,b]. Two sequences of partitions are constructed: • PnP_n divides [a,b][a,b] into nn subintervals of equal length. • QnQ_n divides [a,b][a,b] into n\sqrt{n} equal subintervals and then further subdivides only the leftmost of these subintervals into n\sqrt{n} equal pieces. Which statement about the convergence of the Riemann sums of hh corresponding to PnP_n and QnQ_n is true? (a) Only the sequence using PnP_n is guaranteed to converge because QnQ_n leaves most subintervals unchanged at each stage. (b) Only the sequence using QnQ_n is guaranteed (and does so more quickly) since it already contains subintervals of length (b−a)/n(b-a)/n. (c) Both sequences converge to the same integral because max⁡{Δx}→0\max\{\Delta x\}\to 0 for each partition as n→∞n\to\infty. (d) Neither sequence is certain to converge; QnQ_n introduces only 2n−12\sqrt n-1 subintervals, which is too few compared with nn.

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