Let be a differentiable function that is strictly increasing on the interval
MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change
Problem
Let g be a differentiable function that is strictly increasing on the interval [2,6]. For a positive integer n, let Pn denote the uniform partition of [2,6] into n subintervals of equal length, and let Ln and Rn denote, respectively, the sum obtained by using the left‐hand and right‐hand sample points of each subinterval of Pn.
Answer the following questions about the relationship among Ln, Rn, and ∫26g(x)dx.
(a) Write an inequality that relates Ln, ∫26g(x)dx, and Rn for every positive integer n. Clearly indicate the ordering of the three quantities.
(b) Provide a convincing explanation for the ordering you stated in part (a). Your explanation should cite relevant properties of g and of the way the sums are constructed.
(c) Describe how the numerical difference Rn−Ln is affected when n is replaced by 2n. Give a reasoned argument, without performing explicit calculations, to support your claim.
(d) For each n, let Tn be the approximation obtained by averaging Ln and Rn. Explain whether Tn is guaranteed to underestimate, overestimate, or neither, and justify your conclusion.