## 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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a₁ antiderivative approximation arcsin arctan Ax˛ b₁ calculus Chain Rule compute constant continuous converges coordinates cos˛ curve cylinder decr defined definite integral density differential equation diverges dx dx dx dy dy dx Evaluate EXAMPLE 1 Find Find the area Find the volume finite formula function f(x given graph hyperreal numbers improper integral incr Infinite Sum Theorem infinitely close infinitesimal interval inverse function length lim f(x limit line integral maximum parabola parametric equations particular solution plane polynomial position vector positive infinite positive infinitesimal power series PROBLEMS FOR SECTION PROOF Prove radius radius of convergence real function real number rectangle region Riemann sum rotating sec˛ shown in Figure shows sin˛ sketch slope Step surface Transfer Principle velocity vertical x-axis y-axis zero дх ду