A tour of single variable calculus

1. Single Variable Calculus – A Tour Dr. Abdulla Al-Othman

2. Part1: Differentiation Motivation, Key Ideas, Theory Calculus 1 2

3. Gradients Also called slopes Calculus 1 3

4. A Line has a Single Slope Calculus 1 4 y ∆y ∆x x

5. A Curve has Many Slopes Calculus 1 5 y Many Slopes x

6. Example: Slopes of y=x2 Calculus 1 6 y c b The slope increases as we move in the direction of the arrows a x

7. Measuring the Many Slopes of a Curve: Calculus 1 7 Gottfried Leibniz 1646-1717 Isaac Newton 1642-1726

8. Calculus 1 8 y a, f(a) a x Objective: To find the Slope of the Curve at Point (a , f(a))

9. Be Observant Calculus 1 9 جلال‌الدین محمد رومی “If thou wilt be observant and vigilant, thou wilt see at every moment“

10. The Key Observation of Newton and Leibniz Calculus 1 10 Slope =l1 y Slope =l2 We make h smaller, and the limit of the slopes as it approaches 0, if it exists, will be the slope of the tangent line Slope=l3 Slope =ln a, f(a) a+h3 a+hn a x a+h1 a+h2

11. Calculus 1 11 The key idea translated into the language of mathematics

12. Differentiability and Continuity How they’re related Calculus 1 12

13. Differentiable => Continuous Calculus 1 13 “In Mathematics as in Design, Perfection is almost always Simplicity.” Abdulla Al-Othman

14. Proof: Calculus 1 14

15. Continuous Does Not ===> Differentiable Calculus 1 15

16. A Continuous non Differentiable Function Calculus 1 16 y x

17. Some Useful Rules No new ideas involved Calculus 1 17

18. The List: Calculus 1 18

19. Properties of Differentiable Functions cont… Calculus 1 19

20. The Proofs Calculus 1 20 “What is now proven was once only imagined.” William Blake

21. The Sum Rule Calculus 1 21

22. Extracting the Constant Rule Calculus 1 22

23. The Product Rule Calculus 1 23

24. The Quotient Rule Calculus 1 24

25. Cont…. Calculus 1 25

26. The Chain Rule Calculus 1 26

27. The Inverse Function Rule Calculus 1 27

28. Differentiation of Polynomials Calculus1 Calculus 1 28

29. محمد بن موسى الخوارزمي Calculus 1 29

30. Calculus 1 30 The Algorithm

31. Examples: Calculus 1 31

32. Calculus 1 32 Example1: Calculating the Derivative of y=c

33. Calculus 1 33 Example2: Calculating the Derivative of y=x

34. Calculus 1 34 Example3: Calculating the Derivative of y=x2

35. Calculus 1 35 Example4: Calculating the Derivative of y=x3

36. Tabulating our results, do you see a pattern? Calculus 1 36

37. Key Result1 (Our first building block) Plug and Play Calculus 1 37

38. Polynomial Differentiation: Plug and Play Calculus 1 38

39. Trigonometric Functions Review Calculus 1 39

40. Basics Calculus 1 40 y 1 θ x

41. Sine Unit CIRCLE SINe 41 Y 1 1 θ θ X -1 1 X 0 π/2 π 2π 3π/2 -1 -1 Calculus 1

42. Cosine Unit CIRCLE: Cosine: 42 Y 1 2π π/2 3π/2 π θ θ X -1 1 X 0 -1 Calculus 1

43. Tan=Sine/Cos ine Calculus 1 43 π/2 π -π -π/2 0 θ

44. Sin -1 sINe : Sine-1 Calculus 1 44 The Sine function is 1:1 over this range so it admits an inverse 1 π/2 θ π/2 0 -π/2 -1 1 -π/2 -1

45. Cos -1 cos: cos-1: 45 The Sine function is 1:1 over this range so it admits an inverse π 1 π/2 θ π/2 0 -1 1 π -1 Calculus 1

46. Tan -1 tan : Tan-1 46 The tan function is 1:1 over this range so it admits an inverse θ π/2 π/2 θ -π/2 0 -π/2 Calculus 1

47. Wikepedia Animation Calculus 1 47

48. Important Formulae Calculus 1 48

49. The Proofs Calculus 1 49 "Everything should be made as simple as possible, but not simpler." Einstein

50. Rule1 : Calculus 1 50 A1 1 1 A2 β α

51. Cont… Calculus 1 51 B3 1 1 B2 B1 β α

52. Cont.. Calculus 1 52

53. Rule2: Calculus 1 53

54. Rule3 : Calculus 1 54

55. Rule 4 : Calculus 1 55

56. Differentiation of Trigonometric Functions Calculus 1 56

57. The Formulae Calculus 1 57

58. Two Important Limits: Calculus 1 58

59. d/dx (sin(x)) Calculus 1 59

60. d/dx (cos(x)) Calculus 1 60

61. d/dx tan(x)) Calculus 1 61

62. d/dx (sec(x)) Calculus 1 62

63. Derivatives of Exponentials and Logarithms Calculus2 Calculus 2 63

64. Derivatives of Exponentials: Calculus 2 64

65. A Special Limit Calculus 2 65

66. Geometric Proof Calculus 2 66 1 0

67. Proofs : Calculus 2 67

68. Derivatives of Logarithmic Functions: Calculus 2 68

69. Proofs: Calculus 2 69

70. The Mean Value Theorem Taylor Series Expansions Calculus 1 70

71. Local Max and Min Calculus 1 71

72. Local Max and Min Calculus 1 72 Local Max Y Local Min X b a Stationary Points

73. Theorem 1 Calculus 1 73

74. Proof: Calculus 1 74

75. Stationary Points Calculus 1 75 Y Local Max Local Min b a X Point of Inflexion

76. Roll’s Theorem (Prelude to the Mean Value Theorem) Calculus 1 76

77. Idea Calculus 1 77 Case 2: Max (or Min) occurs at an interior point ξ Case 1: Max and Min occur at end points, so function is constant. f(x) f(x) a b x a b x

78. Theorem not true if f(a) ≠ f(b) Calculus 1 78 f(x) f(x) a b x a b x

79. Proof: Calculus 1 79

80. The Mean Value Theorem Calculus 1 80

81. The Picture: Calculus 1 81 b, f(b) a, f(a) ξ a b

82. Proof: Calculus 1 82

83. Taylor’s Theorem Calculus 1 83

84. The Idea: Calculus 1 84

85. The Idea cont: Calculus 1 85

86. The Proof (optional): Calculus 1 86

87. The Proof (Optional): Calculus 1 87

88. The Proof (optional): Calculus 1 88

89. The Proof (Optional) cont..: Calculus 1 89

90. Applications: exp(x) Calculus 1 90

91. Applications: Calculus 1 91

92. Applications: sin(x), cos(x) Calculus 1 92

93. Applications: sin(x), cos(x) cont Calculus 1 93

94. Applications: sin(x), cos(x) cont Calculus 1 94

95. Applications: Calculus 1 95

96. Applications: Cont Calculus 1 96

97. The Binomial Theorem: Calculus 1 97

98. Applications: Proving the Binomial The0rem Calculus 1 98

99. Applications: Binomial Cont.. Calculus 1 99

100. Application: L’Hopitals Rule Calculus 1 100

101. Application: L’Hopitals Rule Calculus 1 101

102. Exercises: Calculus 1 102

103. Introduction to the Theory of Integration Calculus 1 Dr. Abdulla Al-Othman

104. Part 2: Integration Motivation, Key Ideas, Theory Calculus 1 104

105. The Ideas of Riemann Bernhard Riemann, 1826-66 Calculus 1 105

106. Objective : To Find Area S Under a Curve f Calculus 1 106

107. The Key Idea Calculus 1 107

108. The Key Idea Translated into the Language of Mathematics Calculus 1 108

109. The Fundamental Theorems of Calculus Calculus 1 109

110. The First Fundamental Theorem of Calculus (FFT) Calculus 1 110

111. The First Fundamental Theorem of Calculus cont (FFT) Calculus 1 111

112. The First Fundamental Theorem of Calculus (FFC) Proof (Optional) Calculus 1 112

113. The Second Fundamental Theorem of Calculus Calculus 1 113

114. The Second Fundamental Theorem of Calculus (SFC) Proof (Optional) Calculus 1 114

115. Examples Calculus 1 115

116. Example 1: Calculus 1 116

117. Example 2: Calculus 1 117

118. Example 3: Calculus 1 118

119. Properties of the Integral Calculus 1 119

120. Calculus 1 120

121. The Anti Derivative Or Primitive Calculus 1 121

122. Definition: Anti Derivative Calculus 2 122

123. Techniques for Finding the Anti Derivative Method of Substitution Method of By Parts Method of Partial Fractions Calculus 2 123

124. 1) Technique of Substitution Calculus 2 124

125. Proof Calculus 2 125

126. The Cook Book Approach Substitution (Plug and Play) Calculus 2 126

127. The Cook Book Approach ( Reverse Substitution) Calculus 2 127

128. Exercises: Calculus 2 128

129. Partial Solutions to Selected Examples: Calculus 2 129

130. Partial Solutions to Selected Class Examples cont..: Calculus 2 130

131. Partial Solutions to Selected Class Exercises cont: Calculus 2 131

132. Exercises: Trigonometric Integrals Calculus 2 132

133. Examples cont… Calculus 2 133

134. Solutions to Selected Exercises: Calculus 2 134

135. Cont… Calculus 2 135

136. Cont… Calculus 2 136

137. Proofs… Calculus 2 137

138. Trigonometric Substitution Motivating Example Calculus 2 138 y x t

139. Example Cont… Calculus 2 139

140. Example Cont… Calculus 2 140 a x u

141. Further Examples: Integrals Using Trig Substitution and Trig Identities Calculus 2 141

142. Further Examples cont… Calculus 2 142

143. Cont… 143 y u 2 Calculus 2

144. 2)The Technique of “By Parts”: Calculus 2 144

145. Proof: Calculus 2 145

146. Cont… Calculus 2 146

147. Calculus 2 147 The Cook Book Approach (Plug and Play)

148. Exercises: Calculus 2 148

149. Exercises Cont: Solutions Calculus 2 149

150. Cont.. Calculus 2 150

151. Cont… Calculus 2 151

152. 3) Partial Fractions Calculus 2 152

153. 3) The Technique of Partial Fractions..cont Calculus 2 153

154. 3) The Technique of Partial Fractions..cont Calculus 2 154

155. 3) The Technique of Partial Fractions..cont Calculus 2 155

156. Exercises: Calculus 2 156

157. Differential Equations An Introduction Calculus 1 157

158. Definition A Differential Equation is an equation which contains at least one derivative of an unknown function. Calculus 1 158

159. Examples Calculus 1 159

160. Solution of Differential Equations A solution to a differentiable equation is a relation between the variables involved which is: Free From Derivatives Is Consistent with the Differential Equation Calculus 1 160

162. 5 is called a Partial Differential Equation

165. Equation 3 is of degree 3

167. General Solutions Calculus 1 165

168. General Solution Cont… Calculus 1 166

169. Boundary Conditions Calculus 1 167

170. Techniques for Solving Order One Ordinary Differential Equations Separating Variables Integrating Factor Changing Variables Calculus 1 168

171. Separating Variables Technique Calculus 1 169

172. Example Calculus 1 170

173. Integrating Factor Technique (Linear Equations of Order 1) Calculus 1 171

174. Example1 Calculus 1 172

175. Example2 Calculus 1 173

176. Example3 Calculus 1 174

177. Technique of Changing the Variable Calculus 1 175 Occasionally an unpleasant looking Differential Equation can be converted into something more manageable by making a change of variable. Unfortunately, it is seldom easy to think of an appropriate change

178. Example1: Calculus 1 176

179. Example2: Calculus 1 177

180. Exercises: Calculus 1 178

181. Exercises cont… Calculus 1 179