Elementary Calculus: An Infinitesimal ApproachThis first-year calculus book is centered around the use of infinitesimals, an approach largely neglected until recently for reasons of mathematical rigor. It exposes students to the intuition that originally led to the calculus, simplifying their grasp of the central concepts of derivatives and integrals. The author also teaches the traditional approach, giving students the benefits of both methods. Chapters 1 through 4 employ infinitesimals to quickly develop the basic concepts of derivatives, continuity, and integrals. Chapter 5 introduces the traditional limit concept, using approximation problems as the motivation. Later chapters develop transcendental functions, series, vectors, partial derivatives, and multiple integrals. The theory differs from traditional courses, but the notation and methods for solving practical problems are the same. The text suggests a variety of applications to both natural and social sciences. |
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angle approximation base calculus called Chapter circle compute constant continuous converges coordinates cost curve cylinder decr defined density depends derivative differential equation direction distance diverges dy/dx equal equation error Evaluate EXAMPLE exists Find finite force formula function function f(x give given graph hyperreal numbers incr infinite infinitely close infinitesimal initial integral interval length limit mass maximum means method moves negative object origin partial plane positive power series Problems PROBLEMS FOR SECTION PROOF Prove radius radius of convergence real number rectangle region rotating Rule shown in Figure shows side sketch slope ſº solid solve standard Step Suppose surface tangent Test Theorem unit variable vector vertical x-axis y-axis zero