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Calculus

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A tour of single variable calculus

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Calculus

  1. 1. Single Variable Calculus – A Tour<br />Dr. Abdulla Al-Othman<br />
  2. 2. Part1: Differentiation<br />Motivation, Key Ideas, Theory<br />Calculus 1<br />2<br />
  3. 3. Gradients <br />Also called slopes<br />Calculus 1<br />3<br />
  4. 4. A Line has a Single Slope<br />Calculus 1<br />4<br />y<br />∆y<br />∆x<br />x<br />
  5. 5. A Curve has Many Slopes<br />Calculus 1<br />5<br />y<br />Many Slopes<br />x<br />
  6. 6. Example: Slopes of y=x2 <br />Calculus 1<br />6<br />y<br />c<br />b<br />The slope increases as we move in the direction of the arrows<br />a<br />x<br />
  7. 7. Measuring the Many Slopes of a Curve:<br />Calculus 1<br />7<br />Gottfried Leibniz 1646-1717<br />Isaac Newton 1642-1726<br />
  8. 8. Calculus 1<br />8<br />y<br />a, f(a)<br />a<br />x<br />Objective: To find the Slope of the Curve at Point (a , f(a))<br />
  9. 9. Be Observant<br />Calculus 1<br />9<br />جلال‌الدین محمد رومی<br />“If thou wilt be observant and vigilant, thou wilt see at every moment“ <br />
  10. 10. The Key Observation of Newton and Leibniz <br />Calculus 1<br />10<br />Slope =l1<br />y<br />Slope =l2<br />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<br />Slope=l3<br />Slope =ln<br />a, f(a)<br />a+h3<br />a+hn<br />a<br />x<br />a+h1<br />a+h2<br />
  11. 11. Calculus 1<br />11<br />The key idea translated into the language of mathematics<br />
  12. 12. Differentiability and Continuity<br />How they’re related<br />Calculus 1<br />12<br />
  13. 13. Differentiable => Continuous<br />Calculus 1<br />13<br />“In Mathematics as in Design, Perfection is almost always Simplicity.” Abdulla Al-Othman<br />
  14. 14. Proof:<br />Calculus 1<br />14<br />
  15. 15. Continuous Does Not ===> Differentiable<br />Calculus 1<br />15<br />
  16. 16. A Continuous non Differentiable Function<br />Calculus 1<br />16<br />y<br />x<br />
  17. 17. Some Useful Rules <br />No new ideas involved<br />Calculus 1<br />17<br />
  18. 18. The List:<br />Calculus 1<br />18<br />
  19. 19. Properties of Differentiable Functions cont…<br />Calculus 1<br />19<br />
  20. 20. The Proofs<br />Calculus 1<br />20<br />“What is now proven was once only imagined.” <br />William Blake<br />
  21. 21. The Sum Rule <br />Calculus 1<br />21<br />
  22. 22. Extracting the Constant Rule<br />Calculus 1<br />22<br />
  23. 23. The Product Rule <br />Calculus 1<br />23<br />
  24. 24. The Quotient Rule <br />Calculus 1<br />24<br />
  25. 25. Cont….<br />Calculus 1<br />25<br />
  26. 26. The Chain Rule <br />Calculus 1<br />26<br />
  27. 27. The Inverse Function Rule<br />Calculus 1<br />27<br />
  28. 28. Differentiation of Polynomials<br />Calculus1<br />Calculus 1<br />28<br />
  29. 29. محمد بن موسى الخوارزمي <br />Calculus 1<br />29<br />
  30. 30. Calculus 1<br />30<br />The Algorithm <br />
  31. 31. Examples:<br />Calculus 1<br />31<br />
  32. 32. Calculus 1<br />32<br />Example1: Calculating the Derivative of y=c<br />
  33. 33. Calculus 1<br />33<br />Example2: Calculating the Derivative of y=x<br />
  34. 34. Calculus 1<br />34<br />Example3: Calculating the Derivative of y=x2<br />
  35. 35. Calculus 1<br />35<br />Example4: Calculating the Derivative of y=x3<br />
  36. 36. Tabulating our results, do you see a pattern?<br />Calculus 1<br />36<br />
  37. 37. Key Result1 (Our first building block) <br />Plug and Play<br />Calculus 1<br />37<br />
  38. 38. Polynomial Differentiation:<br />Plug and Play<br />Calculus 1<br />38<br />
  39. 39. Trigonometric Functions <br />Review<br />Calculus 1<br />39<br />
  40. 40. Basics<br />Calculus 1<br />40<br />y<br />1<br />θ<br />x<br />
  41. 41. Sine <br />Unit CIRCLE<br />SINe<br />41<br />Y<br />1<br />1<br />θ<br />θ<br />X<br />-1<br />1<br />X<br />0<br />π/2<br />π<br />2π<br />3π/2<br />-1<br />-1<br />Calculus 1<br />
  42. 42. Cosine<br />Unit CIRCLE:<br />Cosine: <br />42<br />Y<br />1<br />2π<br />π/2<br />3π/2<br />π<br />θ<br />θ<br />X<br />-1<br />1<br />X<br />0<br />-1<br />Calculus 1<br />
  43. 43. Tan=Sine/Cos ine<br />Calculus 1<br />43<br />π/2<br />π<br />-π<br />-π/2<br />0<br />θ<br />
  44. 44. Sin -1<br />sINe : <br />Sine-1<br />Calculus 1<br />44<br />The Sine function is 1:1 over this range so it admits an inverse<br />1<br />π/2<br />θ<br />π/2<br />0<br />-π/2<br />-1<br />1<br />-π/2<br />-1<br />
  45. 45. Cos -1<br />cos: <br />cos-1:<br />45<br />The Sine function is 1:1 over this range so it admits an inverse<br />π<br />1<br />π/2<br />θ<br />π/2<br />0<br />-1<br />1<br />π<br />-1<br />Calculus 1<br />
  46. 46. Tan -1<br />tan : <br />Tan-1<br />46<br />The tan function is 1:1 over this range so it admits an inverse<br />θ<br />π/2<br />π/2<br />θ<br />-π/2<br />0<br />-π/2<br />Calculus 1<br />
  47. 47. Wikepedia Animation<br />Calculus 1<br />47<br />
  48. 48. Important Formulae<br />Calculus 1<br />48<br />
  49. 49. The Proofs<br />Calculus 1<br />49<br />"Everything should be made as simple as possible, but not simpler." Einstein<br />
  50. 50. Rule1 :<br />Calculus 1<br />50<br />A1<br />1<br />1<br />A2<br />β<br />α<br />
  51. 51. Cont… <br />Calculus 1<br />51<br />B3<br />1<br />1<br />B2<br />B1<br />β<br />α<br />
  52. 52. Cont.. <br />Calculus 1<br />52<br />
  53. 53. Rule2: <br />Calculus 1<br />53<br />
  54. 54. Rule3 :<br />Calculus 1<br />54<br />
  55. 55. Rule 4 :<br />Calculus 1<br />55<br />
  56. 56. Differentiation of Trigonometric Functions<br />Calculus 1<br />56<br />
  57. 57. The Formulae<br />Calculus 1<br />57<br />
  58. 58. Two Important Limits: <br />Calculus 1<br />58<br />
  59. 59. d/dx (sin(x))<br />Calculus 1<br />59<br />
  60. 60. d/dx (cos(x))<br />Calculus 1<br />60<br />
  61. 61. d/dx tan(x))<br />Calculus 1<br />61<br />
  62. 62. d/dx (sec(x))<br />Calculus 1<br />62<br />
  63. 63. Derivatives of Exponentials and Logarithms<br />Calculus2 <br />Calculus 2<br />63<br />
  64. 64. Derivatives of Exponentials:<br />Calculus 2<br />64<br />
  65. 65. A Special Limit<br />Calculus 2<br />65<br />
  66. 66. Geometric Proof<br />Calculus 2<br />66<br />1<br />0<br />
  67. 67. Proofs :<br />Calculus 2<br />67<br />
  68. 68. Derivatives of Logarithmic Functions:<br />Calculus 2<br />68<br />
  69. 69. Proofs:<br />Calculus 2<br />69<br />
  70. 70. The Mean Value Theorem<br />Taylor Series Expansions<br />Calculus 1<br />70<br />
  71. 71. Local Max and Min<br />Calculus 1<br />71<br />
  72. 72. Local Max and Min<br />Calculus 1<br />72<br />Local Max<br />Y<br />Local Min<br />X<br />b<br />a<br />Stationary Points<br />
  73. 73. Theorem 1<br />Calculus 1<br />73<br />
  74. 74. Proof:<br />Calculus 1<br />74<br />
  75. 75. Stationary Points <br />Calculus 1<br />75<br />Y<br />Local Max<br />Local Min<br />b<br />a<br />X<br />Point of Inflexion<br />
  76. 76. Roll’s Theorem (Prelude to the Mean Value Theorem)<br />Calculus 1<br />76<br />
  77. 77. Idea<br />Calculus 1<br />77<br />Case 2: Max (or Min) occurs at an interior point ξ<br />Case 1: Max and Min occur at end points, so function is constant.<br />f(x)<br />f(x)<br />a<br />b<br />x<br />a<br />b<br />x<br />
  78. 78. Theorem not true if f(a) ≠ f(b)<br />Calculus 1<br />78<br />f(x)<br />f(x)<br />a<br />b<br />x<br />a<br />b<br />x<br />
  79. 79. Proof:<br />Calculus 1<br />79<br />
  80. 80. The Mean Value Theorem<br />Calculus 1<br />80<br />
  81. 81. The Picture:<br />Calculus 1<br />81<br />b, f(b)<br />a, f(a)<br />ξ<br />a<br />b<br />
  82. 82. Proof:<br />Calculus 1<br />82<br />
  83. 83. Taylor’s Theorem<br />Calculus 1<br />83<br />
  84. 84. The Idea:<br />Calculus 1<br />84<br />
  85. 85. The Idea cont:<br />Calculus 1<br />85<br />
  86. 86. The Proof (optional):<br />Calculus 1<br />86<br />
  87. 87. The Proof (Optional):<br />Calculus 1<br />87<br />
  88. 88. The Proof (optional):<br />Calculus 1<br />88<br />
  89. 89. The Proof (Optional) cont..:<br />Calculus 1<br />89<br />
  90. 90. Applications: exp(x)<br />Calculus 1<br />90<br />
  91. 91. Applications: <br />Calculus 1<br />91<br />
  92. 92. Applications: sin(x), cos(x) <br />Calculus 1<br />92<br />
  93. 93. Applications: sin(x), cos(x) cont<br />Calculus 1<br />93<br />
  94. 94. Applications: sin(x), cos(x) cont<br />Calculus 1<br />94<br />
  95. 95. Applications: <br />Calculus 1<br />95<br />
  96. 96. Applications: Cont <br />Calculus 1<br />96<br />
  97. 97. The Binomial Theorem:<br />Calculus 1<br />97<br />
  98. 98. Applications: Proving the Binomial The0rem<br />Calculus 1<br />98<br />
  99. 99. Applications: Binomial Cont..<br />Calculus 1<br />99<br />
  100. 100. Application: L’Hopitals Rule<br />Calculus 1<br />100<br />
  101. 101. Application: L’Hopitals Rule<br />Calculus 1<br />101<br />
  102. 102. Exercises:<br />Calculus 1<br />102<br />
  103. 103. Introduction to the Theory of Integration <br />Calculus 1 <br />Dr. Abdulla Al-Othman<br />
  104. 104. Part 2: Integration<br />Motivation, Key Ideas, Theory<br />Calculus 1<br />104<br />
  105. 105. The Ideas of Riemann <br />Bernhard Riemann, 1826-66<br />Calculus 1<br />105<br />
  106. 106. Objective : To Find Area S Under a Curve f<br />Calculus 1<br />106<br />
  107. 107. The Key Idea <br />Calculus 1<br />107<br />
  108. 108. The Key Idea Translated into the Language of Mathematics<br />Calculus 1<br />108<br />
  109. 109. The Fundamental Theorems of Calculus<br />Calculus 1<br />109<br />
  110. 110. The First Fundamental Theorem of Calculus (FFT) <br />Calculus 1<br />110<br />
  111. 111. The First Fundamental Theorem of Calculus cont (FFT) <br />Calculus 1<br />111<br />
  112. 112. The First Fundamental Theorem of Calculus (FFC) Proof (Optional)<br />Calculus 1<br />112<br />
  113. 113. The Second Fundamental Theorem of Calculus<br />Calculus 1<br />113<br />
  114. 114. The Second Fundamental Theorem of Calculus (SFC) Proof (Optional)<br />Calculus 1<br />114<br />
  115. 115. Examples<br />Calculus 1<br />115<br />
  116. 116. Example 1:<br />Calculus 1<br />116<br />
  117. 117. Example 2:<br />Calculus 1<br />117<br />
  118. 118. Example 3:<br />Calculus 1<br />118<br />
  119. 119. Properties of the Integral<br />Calculus 1<br />119<br />
  120. 120. Calculus 1<br />120<br />
  121. 121. The Anti Derivative<br />Or Primitive<br />Calculus 1<br />121<br />
  122. 122. Definition: Anti Derivative<br />Calculus 2<br />122<br />
  123. 123. Techniques for Finding the Anti Derivative<br />Method of Substitution<br />Method of By Parts<br />Method of Partial Fractions<br />Calculus 2<br />123<br />
  124. 124. 1) Technique of Substitution<br />Calculus 2<br />124<br />
  125. 125. Proof<br />Calculus 2<br />125<br />
  126. 126. The Cook Book Approach Substitution (Plug and Play)<br />Calculus 2<br />126<br />
  127. 127. The Cook Book Approach ( Reverse Substitution)<br />Calculus 2<br />127<br />
  128. 128. Exercises:<br />Calculus 2<br />128<br />
  129. 129. Partial Solutions to Selected Examples:<br />Calculus 2<br />129<br />
  130. 130. Partial Solutions to Selected Class Examples cont..:<br />Calculus 2<br />130<br />
  131. 131. Partial Solutions to Selected Class Exercises cont:<br />Calculus 2<br />131<br />
  132. 132. Exercises: Trigonometric Integrals<br />Calculus 2<br />132<br />
  133. 133. Examples cont…<br />Calculus 2<br />133<br />
  134. 134. Solutions to Selected Exercises:<br />Calculus 2<br />134<br />
  135. 135. Cont…<br />Calculus 2<br />135<br />
  136. 136. Cont…<br />Calculus 2<br />136<br />
  137. 137. Proofs…<br />Calculus 2<br />137<br />
  138. 138. Trigonometric Substitution Motivating Example<br />Calculus 2<br />138<br />y<br />x<br />t<br />
  139. 139. Example Cont…<br />Calculus 2<br />139<br />
  140. 140. Example Cont…<br />Calculus 2<br />140<br />a<br />x<br />u<br />
  141. 141. Further Examples: Integrals Using Trig Substitution and Trig Identities<br />Calculus 2<br />141<br />
  142. 142. Further Examples cont…<br />Calculus 2<br />142<br />
  143. 143. Cont…<br />143<br />y<br />u<br />2<br />Calculus 2<br />
  144. 144. 2)The Technique of “By Parts”:<br />Calculus 2<br />144<br />
  145. 145. Proof:<br />Calculus 2<br />145<br />
  146. 146. Cont…<br />Calculus 2<br />146<br />
  147. 147. Calculus 2<br />147<br />The Cook Book Approach (Plug and Play)<br />
  148. 148. Exercises:<br />Calculus 2<br />148<br />
  149. 149. Exercises Cont: Solutions<br />Calculus 2<br />149<br />
  150. 150. Cont..<br />Calculus 2<br />150<br />
  151. 151. Cont…<br />Calculus 2<br />151<br />
  152. 152. 3) Partial Fractions<br />Calculus 2<br />152<br />
  153. 153. 3) The Technique of Partial Fractions..cont<br />Calculus 2<br />153<br />
  154. 154. 3) The Technique of Partial Fractions..cont<br />Calculus 2<br />154<br />
  155. 155. 3) The Technique of Partial Fractions..cont<br />Calculus 2<br />155<br />
  156. 156. Exercises:<br />Calculus 2<br />156<br />
  157. 157. Differential Equations<br />An Introduction<br />Calculus 1<br />157<br />
  158. 158. Definition<br />A Differential Equation is an equation which contains at least one derivative of an unknown function.<br />Calculus 1<br />158<br />
  159. 159. Examples<br />Calculus 1<br />159<br />
  160. 160. Solution of Differential Equations<br />A solution to a differentiable equation is a relation between the variables involved which is:<br />Free From Derivatives<br />Is Consistent with the Differential Equation<br />Calculus 1<br />160<br />
  161. 161. Comments:<br />Calculus 1<br />161<br /><ul><li>Y=sin x is a solution to 1
  162. 162. 5 is called a Partial Differential Equation
  163. 163. All the others are called Ordinary Differential Equations</li></li></ul><li>Orders and Degrees<br />Calculus 1<br />162<br /><ul><li>The Order of a Differential Equation is the order of the derivative of the highest power.
  164. 164. The Degree of a Differential Equation is the algebraic degree with which the derivative of the highest order appears in the equation</li></li></ul><li>Examples:<br />Calculus 1<br />163<br /><ul><li>Equations 1 and 6 are first order all others are second order
  165. 165. Equation 3 is of degree 3
  166. 166. All others are of degree1</li></li></ul><li>In this Course :<br />Calculus 1<br />164<br />Calculus 2<br />
  167. 167. General Solutions<br />Calculus 1<br />165<br />
  168. 168. General Solution Cont…<br />Calculus 1<br />166<br />
  169. 169. Boundary Conditions<br />Calculus 1<br />167<br />
  170. 170. Techniques for Solving Order One Ordinary Differential Equations<br />Separating Variables<br />Integrating Factor<br />Changing Variables<br />Calculus 1<br />168<br />
  171. 171. Separating Variables Technique<br />Calculus 1<br />169<br />
  172. 172. Example<br />Calculus 1<br />170<br />
  173. 173. Integrating Factor Technique (Linear Equations of Order 1)<br />Calculus 1<br />171<br />
  174. 174. Example1<br />Calculus 1<br />172<br />
  175. 175. Example2<br />Calculus 1<br />173<br />
  176. 176. Example3<br />Calculus 1<br />174<br />
  177. 177. Technique of Changing the Variable<br />Calculus 1<br />175<br />Occasionally an unpleasant looking Differential Equation can be converted into something more manageable by making a change of variable. <br />Unfortunately, it is seldom easy to think of an appropriate change<br />
  178. 178. Example1:<br />Calculus 1<br />176<br />
  179. 179. Example2:<br />Calculus 1<br />177<br />
  180. 180. Exercises:<br />Calculus 1<br />178<br />
  181. 181. Exercises cont…<br />Calculus 1<br />179<br />

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