MATHEMATICS: A COURSE IN CALCULUS & REAL ANALYSIS

By CHARLOTTE DAVIS

CONTENTS NUMBERS AND FUNCTIONS PROPERTIES OF REAL NUMBERS INEQUALITIES FUNCTIONS AND THEIR GEOMETRIC PROPERTIES EXERCISES SEQUENCES CONVERGENCE OF SEQUENCES SUBSEQUENCES AND CAUCHY SEQUENCES CONTINUITY AND LIMITS CONTINUITY OF FUNCTIONS BASIC PROPERTIES OF CONTINUOUS FUNCTIONS LIMITS OF FUNCTIONS OF A REAL VARIABLE DI?ERENTIATION THE DERIVATIVE AND ITS BASIC PROPERTIES THE MEAN VALUE AND TAYLOR THEOREMS MONOTONICITY, CONVEXITY, AND CONCAVITY L’HˆOPITAL’S RULE APPLICATIONS OF DI?ERENTIATION ABSOLUTE MINIMUM AND MAXIMUM LOCAL EXTREMA AND POINTS OF IN?ECTION LINEAR AND QUADRATIC APPROXIMATIONS THE PICARD AND NEWTON METHODS INTEGRATION THE RIEMANN INTEGRAL INTEGRABLE FUNCTIONS THE FUNDAMENTAL THEOREM OF CALCULUS RIEMANN SUMS ELEMENTARY TRANSCENDENTAL FUNCTIONS LOGARITHMIC AND EXPONENTIAL FUNCTIONS TRIGONOMETRIC FUNCTIONS SINE OF THE RECIPROCAL POLAR COORDINATES TRANSCENDENCE REVISION EXERCISES APPLICATIONS AND APPROXIMATIONS OF RIEMANN INTEGRALS AREA OF A REGION BETWEEN CURVES VOLUME OF A SOLID ARC LENGTH OF A CURVE AREA OF A SURFACE OF REVOLUTION CENTROIDS QUADRATURE RULES IN?NITE SERIES AND IMPROPER INTEGRALS CONVERGENCE OF SERIES CONVERGENCE TESTS FOR SERIES POWER SERIES CONVERGENCE OF IMPROPER INTEGRALS CONVERGENCE TESTS FOR IMPROPER INTEGRALS RELATED INTEGRALS REFERENCES NUMBERS AND FUNCTIONS LET US BEGIN AT THE BEGINNING WHEN WE LEARN THE SCRIPT OF A LANGUAGE, SUCH AS THE ENGLISH LANGUAGE, WE BEGIN WITH THE LETTERS OF THE ALPHABET A, B, C, ; WHEN WE LEARN THE SOUNDS OF MUSIC, SUCH AS THOSE OF WESTERN CLASSICAL MUSIC, WE BEGIN WITH THE NOTES DO, RE, MI, LIKEWISE, IN MATHEMATICS, ONE BEGINS WITH 1, 2, 3, ; THESE ARE THE POSITIVE INTEGERS OR THE NATURAL NUMBERS WE SHALL DENOTE THE SET OF POSITIVE INTEGERS BY N THUS, N = {1, 2, 3, } THESE NUMBERS HAVE BEEN KNOWN SINCE ANTIQUITY OVER THE YEARS, THE NUMBER WAS CONCEIVED1 AND SUBSEQUENTLY, THE NEGATIVE INTEGERS TOGETHER, THESE FORM THE SET Z OF INTEGERS 2 THUS, Z = { , ?3, ?2, ?1, 0, 1, 2, 3, } QUOTIENTS OF INTEGERS ARE CALLED RATIONAL NUMBERS WE SHALL DENOTE THE SET OF ALL RATIONAL NUMBERS BY Q THUS, Q = M N : M, N ?Z, N ?= GEOMETRICALLY, THE INTEGERS CAN BE REPRESENTED BY POINTS ON A LINE BY ?XING A BASE POINT (SIGNIFYING THE NUMBER 0) AND A UNIT DISTANCE SUCH A LINE IS CALLED THE NUMBER LINE AND IT MAY BE DRAWN AS IN FIGURE 1 1 BY SUITABLY SUBDI- VIDING THE SEGMENT BETWEEN 0 AND 1, WE CAN ALSO REPRESENT RATIONAL NUMBERS SUCH AS 1/N, WHERE N ?N, AND THIS CAN, IN TURN, BE USED TO REPRESENT ANY 1 THE INVENTION OF ‘ZERO’, WHICH ALSO PAVES THE WAY FOR THE PLACE VALUE SYSTEM OF ENUMERATION, IS WIDELY CREDITED TO THE INDIANS GREAT PSYCHOLOGICAL BARRIERS HAD TO BE OVERCOME WHEN ‘ZERO’ WAS BEING GIVEN THE STATUS OF A LEGITIMATE NUMBER FOR MORE ON THIS, SEE THE BOOKS OF KAPLAN [39] AND KLINE [41] 2 THE NOTATION Z FOR THE SET OF INTEGERS IS INSPIRED BY THE GERMAN WORD ZAHLEN FOR NUMBERS