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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 84

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 84. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 26.1 Approximating Sums Ln Rn Tn and Mi 811 1 1.5 2 2.5 3 L 8 1.828. 3.5 4 4.5 5 1.428. 1.628. 2 2 1 8 9 2 2 2 21 8 9 10 We know that f5 1 1.428 . Rs M8 1.599 . -dx T 1.628 . L8 1.828 . J1 x f x 1 is decreasing and concave up on 1 5 . Therefore we know that for any n and 51 Mn I - dx Tn. 1x If we are interested in more decimal places we can simply choose larger values of n. 50 Subdivisions Suppose n 50 we chop 1 5 into 50 equal pieces each of length kx i-1 0.08. We don t actually want to sum up 50 terms by hand. Work like this is painful to do by hand but it s child s play for a programmable calculator or computer. Get out your programmed calculator or computer and check the figures given below.5 R50 1.577 . M50 1.60918 . T50 1.60994 . L50 1.641 . 400 Subdivisions Suppose n 400 we chop 1 5 into 400 equal pieces each of length kx 0.01. We obtain R400 1.60544 . M400 1.60943 . T400 1.60944 . L400 1.61344 . Using T400 as an upper bound and M400 as a lower bound we ve nailed down the value of this integral to 4 decimal places. 5 You ll have to enter the following information the function often as Y1 the endpoints of integration and the number of pieces into which you d like to partition the interval. And if you re using a calculator without Ln Rn Tn and Mn programmed you ll have to enter the program. 812 CHAPTER 26 Numerical Methods of Approximating Definite Integrals REMARK M50 and T50 give better approximations of 15 1 dx than do R400 and L400. Summary of the Underlying Principles If f is increasing on a then Ln fa6 f t dt Rn. If f is decreasing on a then Rn f f t dt Ln. If f is concave up on a then Mn f f t dt Tn. If f is concave down on a then Tn f f t dt Mn. Tn M For any given n generally the trapezoidal and midpoint sums are much closer to the actual value of the deflnite integral than are the left- and right-hand sums. EXAMPLE 26.2 Approximate 02 e 2x2 dx by flnding upper and lower bounds differing by no more than 0.001.6 SOLUTION This is a great problem on which