Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.
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ˇnd ˇnding ˇrst algebraic Appendix arcsin arctan calculus Cauchy Cauchy sequence Chapter choose complex numbers compute Consequently consider continuous function converges uniformly convex cosx deˇned deˇnition denoted derivative difˇcult easy equation example exist expression f and g f is continuous f is differentiable f(x+ fact Find fn(x formula function deˇned function f geometric graph of f Hint inˇnite induction inequality interval irrational least upper bound lemma lim f(x lim x→a limit local maximum logx mathematical maximum Mean Value Theorem minimum point natural numbers notation Notice obtain one-one pairs partition polar coordinates polynomial function power series Problem properties Prove that lim radius radius of convergence rational numbers real numbers root satisˇes satisfying sequence simple sin2 sinx sufˇciently Suppose that f tangent line Taylor polynomial Taylor's Theorem true