CM111A–Calculus I
        Compact Lecture Notes
                 ACCCoolen
                 Department of Mathematics, King’s College London
                 Version of Sept 2011
                                                                                                                    2
              1 Introduction                                                                                        5
                 1.1   Abit of history ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    5
                       1.1.1   Birth of modern science and of calculus
                               Stage I, 1500–1630: from speculation to science ... . . . . . . . . . . . . .        5
                       1.1.2   Birth of modern science and of calculus
                               Stage II, 1630–1680: science is written in the language of mathematics! .            8
                       1.1.3   Birth of modern science and of calculus
                               Stage III, around 1680: how to speak the language of mathematics! . . .              9
                 1.2   Style of the course    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  12
                 1.3   Revision of some elementary mathematics          . . . . . . . . . . . . . . . . . . . . .  13
                       1.3.1   Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     13
                       1.3.2   Powers of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .      15
                       1.3.3   Solving quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . .       16
                       1.3.4   Functions, inverse functions, and graphs       . . . . . . . . . . . . . . . . . .  16
                       1.3.5   Exponential function, logarithm, laws for logarithms         . . . . . . . . . . .  18
                       1.3.6   Trigonometric functions      . . . . . . . . . . . . . . . . . . . . . . . . . . .  20
              2 Proof by induction                                                                                22
              3 Complex numbers                                                                                   25
                 3.1   Introduction and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      25
                 3.2   Elementary properties of complex numbers . . . . . . . . . . . . . . . . . . . . .          26
                 3.3   Absolute value and division      . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  27
                 3.4   The complex plane (Argand diagram) . . . . . . . . . . . . . . . . . . . . . . . .          28
                       3.4.1   Complex numbers as points in a plane . . . . . . . . . . . . . . . . . . .          28
                       3.4.2   Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     29
                       3.4.3   The exponential form of numbers on the unit circle . . . . . . . . . . . .          31
                 3.5   Complex numbers in exponential notation . . . . . . . . . . . . . . . . . . . . .           33
                       3.5.1   Definition and general properties       . . . . . . . . . . . . . . . . . . . . . .  33
                       3.5.2   Multiplication and division in exponential notation . . . . . . . . . . . .         34
                       3.5.3   The argument of a complex number . . . . . . . . . . . . . . . . . . . . .          35
                 3.6   De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       37
                       3.6.1   Statement and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       37
                       3.6.2   Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    38
                 3.7   Complex equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       39
                                                                                                                      3
              4 Trigonometric and hyperbolic functions                                                              41
                  4.1  Definitions of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . .        41
                       4.1.1    Definition of sine and cosine . . . . . . . . . . . . . . . . . . . . . . . . .      41
                       4.1.2    Elementary values      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  44
                       4.1.3    Related functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     44
                       4.1.4    Inverse trigonometric functions      . . . . . . . . . . . . . . . . . . . . . . .  45
                  4.2  Elementary properties of trigonometric functions . . . . . . . . . . . . . . . . . .         48
                       4.2.1    Symmetry properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       48
                       4.2.2    Addition formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     50
                       4.2.3    Applications of addition formulae . . . . . . . . . . . . . . . . . . . . . .       51
                       4.2.4    The tan(θ=2) formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . .       53
                  4.3  Definitions of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . .       54
                       4.3.1    Definition of hyperbolic sine and hyperbolic cosine . . . . . . . . . . . . .        54
                       4.3.2    General properties and special values . . . . . . . . . . . . . . . . . . . .       55
                       4.3.3    Connection with trigonometric functions . . . . . . . . . . . . . . . . . .         57
                       4.3.4    Applications of connection with trigonometric functions . . . . . . . . . .         57
                       4.3.5    Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . .      58
              5 Functions, limits and differentiation                                                                62
                  5.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     62
                       5.1.1    Rate of change, tangent of a curve . . . . . . . . . . . . . . . . . . . . . .      62
                       5.1.2    Finding tangents and velocities – why we need limits . . . . . . . . . . .          63
                  5.2  The limit     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  66
                       5.2.1    Left and right limits    . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  66
                       5.2.2    Asymptotics - limits involving infinity . . . . . . . . . . . . . . . . . . . .      67
                       5.2.3    When left/right limits exists and are identical . . . . . . . . . . . . . . .       68
                       5.2.4    Rules for limits of composite expressions . . . . . . . . . . . . . . . . . .       69
                       5.2.5    Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      70
                  5.3  Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     72
                       5.3.1    Derivatives of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .    72
                       5.3.2    Rules for derivatives of composite expressions       . . . . . . . . . . . . . . .  73
                       5.3.3    Derivatives of implicit functions . . . . . . . . . . . . . . . . . . . . . . .     76
                       5.3.4    Applications of derivative: sketching graphs       . . . . . . . . . . . . . . . .  79
              6 Integration                                                                                         80
                  6.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     80
                       6.1.1    Area under a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      80
                       6.1.2    Examples of integrals calculated via staircases . . . . . . . . . . . . . . .       83
                       6.1.3    Fundamental theorems of calculus: integration vs differentiation . . . . .           88
                                                                                                                4
                      6.1.4   Indefinite and definite integrals, and other conventions       . . . . . . . . . . 90
                 6.2  Techniques of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  91
                      6.2.1   List of elementary integrals and general methods for reduction . . . . . .       92
                      6.2.2   Examples: integration by substitution . . . . . . . . . . . . . . . . . . . .    94
                      6.2.3   Examples: integration by parts . . . . . . . . . . . . . . . . . . . . . . .     96
                      6.2.4   Further tricks: recursion formulae . . . . . . . . . . . . . . . . . . . . . .   98
                      6.2.5   Further tricks: differentiation with respect to a parameter       . . . . . . . . 100
                      6.2.6   Further tricks: partial fractions . . . . . . . . . . . . . . . . . . . . . . . 102
                 6.3  Some simple applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
                      6.3.1   Calculation of surface areas    . . . . . . . . . . . . . . . . . . . . . . . . . 106
                      6.3.2   Calculation of volumes of revolution . . . . . . . . . . . . . . . . . . . . . 107
                      6.3.3   Calculation of the length of curves     . . . . . . . . . . . . . . . . . . . . . 109
             7 Taylor’s theorem and series                                                                   112
                 7.1  Introduction to series and questions of convergence      . . . . . . . . . . . . . . . . 112
                      7.1.1   Series – notation and elementary properties      . . . . . . . . . . . . . . . . 112
                      7.1.2   Series – convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . 113
                      7.1.3   Power series – notation and elementary properties . . . . . . . . . . . . . 114
                 7.2  Taylor’s theorem     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
                      7.2.1   Expression for the coefficients of power series . . . . . . . . . . . . . . . . 117
                      7.2.2   Taylor series around x = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 119
                      7.2.3   Taylor series around x = a . . . . . . . . . . . . . . . . . . . . . . . . . . 120
                 7.3  Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
                      7.3.1   Series expansions for standard functions     . . . . . . . . . . . . . . . . . . 121
                      7.3.2   Indirect methods for finding Taylor series . . . . . . . . . . . . . . . . . . 122
                 7.4  L’Hopital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
             8 Exercises                                                                                     125