calculus_notes

# calculus_notes - Notes on Calculus by Dinakar Ramakrishnan...

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Notes on Calculus by Dinakar Ramakrishnan 253-37 Caltech Pasadena, CA 91125 Fall 2001 1

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Contents 0 Logical Background 4 0 .1 S e t s ........................................ 4 0 .2 F un c t ion s ..................................... 5 0 .3 C a rd in a l i ty . ................................... 5 0 .4 Equ iv a l en c eR e la t s............................... 6 1 Real and Complex Numbers 8 1 .1 D e s i r edP r op e r t i e s ................................ 8 1.2 Natural Numbers, Well Ordering, and Induction . . . . . . . . . . . . . . . . 10 1 .3 In t eg e r s ...................................... 1 2 1 .4 Ra t a lNumb e r s................................. 1 3 1 .5 O e r edF i e ld s .................................. 1 5 1 .6 R ea e r s................................... 1 6 1 .7 Ab s o lu t eV a e .................................. 2 0 1 .8 C omp l exNumb e r 2 1 2 Sequences and Series 24 2 .1 C onv e r g c eo fs equ c e s............................. 2 4 2 .2 C au chy sc r i t e r ion. ................................ 2 8 2 on s t ru c t iono fR e r sr ev i s i t ed. .................... 2 9 2.4 In±nite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 .5 T e s t sfo rC e r g c e............................... 3 3 2 .6 A l t e rn a t gs e r i e 3 5 3 Basics of Integration 38 3.1 Open, closed and compact sets in R ....................... 3 8 3 .2 In t r a l so fbound edfun c t s.......................... 4 1 3.3 Integrability of monotone functions . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Computation of b R a x s dx .............................. 4 5 3 .5 Examp l fan - t r ab l e ,bound c t ion . ............... 4 7 3 .6 P r e r t i e fin t r a l s .............................. 4 8 3.7 The integral of x m r i s i t ed ,andpo lyn om ia l s ................. 5 0 4 Continuous functions, Integrability 53 4 .1 L im i t sandC t inu i ............................. 5 3 4 .2 S eth r em soncon t ou sfun c t s..................... 5 7 4.3 Integrability of continuous functions . . . . . . . . . . . . . . . . . . . . . . . 59 4 .4 T r ig e t r i cfun c t s ............................. 6 0 4 .5 F c t sw i thd i s con t i t i e 6 4 1
5 Improper Integrals, Areas, Polar Coordinates, Volumes 66 5 .1 Imp r op e rIn t eg r a l s ................................ 6 6 5 .2 A r ea s........................................ 6 9 5 .3 P o la rcoo rd in a t e s................................. 7 1 5 .4 V o lum e s...................................... 7 3 5.5 The integral test for inFnite series . . . . . . . . . . . . . . . . . . . . . . . . 75 6 Diferentiation, Properties, Tangents, Extrema 78 6 .1 D e r iv a t e s..................................... 7 8 6 .2 Ru l e so fd e r en t ia t ion ,con s equ c e s...................... 8 1 6 r oo f fth eru l e 8 4 6 .4 T an g t 8 6 6 .5 Ex t r emao e r t iab l efun c t s ....................... 8 7 6 .6 Th em eanv a lu eth eo r em. ............................ 8 8 7 The Fundamental Theorems o± Calculus, Methods o± Integration 91 7 .1 Th efund am t a lth r s............................ 9 1 7 .2 Th eind eFn i t ein t r a l .............................. 9 4 7 .3 In t r a t ionbysub s t i tu t ion. ........................... 9 4 7 .4 In t r a t ionbyp a r t s................................ 9 7 8 Factorization o± polynomials, Integration by partial ±ractions 100 8 .1 L on gd i s ,r t 1 0 0 8.2 ²actorization over C ............................... 1 0 2 8.3 ²actorization over R 1 0 3 8 .4 Th ep a r t lf r a c t iond e compo s i t ion . ...................... 1 0 5 8 .5 In t r a t iono fr a t a lfun c t s......................... 1 0 6 9 Inverse Functions, log, exp, arcsin, ... 110 9 .1 Inv e r s c t s ................................. 1 1 0 9.2 The natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9 .3 Th eexpon t c t 1 1 4 9 .4 a r c s ,a r c co s r c t ,e ta l............................ 1 1 9 9 .5 Au s e fu lsub s t i t .............................. 1 2 0 9 .6 App end ix : L ’Hop i t a l sRu l e........................... 1 2 1 10 Taylor’s theorem, Polynomial approximations 124 1 0 .1T a y lo rpo lyn om l 1 2 4 10.2 Approximation to order n 1 2 7 1 0 .3T a y r sR ema ind e o rmu la. .......................... 1 3 0 10.4 The irrationality of e 1 3 5 2

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11 Uniform convergence, Taylor series, Complex series 136 11.1 Infnite series oF Functions, convergence . . . . . . . . . . . . . . . . . . . . . 136 1 1 .2T a y lo rs e r i e s....................................
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## This note was uploaded on 02/22/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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calculus_notes - Notes on Calculus by Dinakar Ramakrishnan...

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