MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
A function g is continuous on [0,10]. Its graph is described verbally as follows.
• From x=0 to x=4, the graph rises steadily, and the slope at every point is positive.
• From x=4 to x=7, the graph descends steadily, and the slope at every point is negative.
• From x=7 to x=10, the graph again rises steadily, with positive slope throughout.
A uniform partition of [0,10] into five subintervals is used. Let L5 and R5 denote the left– and right–endpoint Riemann sums, respectively, for ∫010g(x)dx.
(a) For each of the three subinterval clusters [0,4], [4,7], and [7,10], decide whether the contribution of L5 on that cluster is an underestimate or an overestimate of the corresponding portion of the integral. Explain your reasoning with specific reference to the verbal description of the graph.
(b) Combine your conclusions from part (a) to determine whether L5 is an overall underestimate, an overall overestimate, or whether no definitive conclusion can be drawn. Provide a logical justification.
(c) Repeat parts (a) and (b) for R5.