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File: Calculus Pdf 168706 | Calculus
economics 101a section notes gsi david albouy notes on calculus and optimization 1 basic calculus 1 1 denition of a derivative let f x be some function of x then ...

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            Lecture Notes in Calculus
                RazKupferman
              Institute of Mathematics
              TheHebrewUniversity
                July 10, 2013
       2
                 Contents
                 1   Real numbers                                                                      1
                     1.1   Axiomsoffield . . . . . . . . . . . . . . . . . . . . . . . . . . .          1
                     1.2   Axiomsoforder(astaught in 2009) . . . . . . . . . . . . . . . .             7
                     1.3   Axiomsoforder(astaught in 2010, 2011) . . . . . . . . . . . . .            10
                     1.4   Absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . .      13
                     1.5   Special sets of numbers . . . . . . . . . . . . . . . . . . . . . . .      16
                     1.6   TheArchimedeanproperty . . . . . . . . . . . . . . . . . . . . .           19
                     1.7   Axiomofcompleteness . . . . . . . . . . . . . . . . . . . . . . .          23
                     1.8   Rational powers . . . . . . . . . . . . . . . . . . . . . . . . . . .      37
                     1.9   Real-valued powers . . . . . . . . . . . . . . . . . . . . . . . . .       42
                     1.10 Addendum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         42
                 2   Functions                                                                        43
                     2.1   Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .     43
                     2.2   Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     49
                     2.3   Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     50
                     2.4   Limits and order . . . . . . . . . . . . . . . . . . . . . . . . . . .     63
                     2.5   Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     67
                     2.6   Theoremsabout continuous functions . . . . . . . . . . . . . . .           70
                     2.7   Infinite limits and limits at infinity . . . . . . . . . . . . . . . . .     79
                     2.8   Inverse functions    . . . . . . . . . . . . . . . . . . . . . . . . . .   81
                     2.9   Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . .       85
                 ii                                 CONTENTS
                3   Derivatives                                                                   91
                    3.1   Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   91
                    3.2   Rules of differentiation . . . . . . . . . . . . . . . . . . . . . . .   98
                    3.3   Another look at derivatives . . . . . . . . . . . . . . . . . . . . . 102
                    3.4   Thederivative and extrema . . . . . . . . . . . . . . . . . . . . . 105
                    3.5   Derivatives of inverse functions . . . . . . . . . . . . . . . . . . . 118
                    3.6   Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
                    3.7   Taylor’s theorem    . . . . . . . . . . . . . . . . . . . . . . . . . . 123
                4   Integration theory                                                           133
                    4.1   Definition of the integral   . . . . . . . . . . . . . . . . . . . . . . 133
                    4.2   Integration theorems    . . . . . . . . . . . . . . . . . . . . . . . . 143
                    4.3   Thefundamental theorem of calculus . . . . . . . . . . . . . . . . 153
                    4.4   Riemannsums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
                    4.5   Thetrigonometric functions      . . . . . . . . . . . . . . . . . . . . 158
                    4.6   Thelogarithm and the exponential . . . . . . . . . . . . . . . . . 163
                    4.7   Integration methods . . . . . . . . . . . . . . . . . . . . . . . . . 167
                5   Sequences                                                                    171
                    5.1   Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
                    5.2   Limits of sequences . . . . . . . . . . . . . . . . . . . . . . . . . 172
                    5.3   Infinite series  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
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