CalculusSpivak'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. |
Contents
Basic Properties of Numbers | 3 |
PART II | 25 |
Derivatives and Integrals | 145 |
10 | 152 |
Significance of the Derivative | 185 |
Inverse Functions | 227 |
Integrals | 250 |
The Fundamental Theorem of Calculus | 282 |
Fields | 571 |
29 | 578 |
Uniqueness of the Real Numbers | 591 |
Common terms and phrases
ˇnd ˇrst algebraic arcsin arctan calculus Cauchy Cauchy sequence Chapter complex numbers compute consider continuous function converges uniformly convex deˇned deˇnition defined definition denoted derivative domain equation example exists expression f is continuous f is differentiable fact Find fn(x formula function f ƒ and g ƒ is integrable graph of f h→0 lim Hint inequality infinite interval irrational least upper bound lemma Let f lim f lim f(x lim h→0 limit local maximum mathematical maximum Mean Value Theorem minimum point natural numbers notation nth root obtain one-one ordered field pairs partition polynomial function power series Problem properties Prove that f radius radius of convergence rational numbers real numbers root satisfying sequence simple sin2 Suppose that f tangent line Taylor polynomial true