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Chapter 02 differentiation

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Chapter 02 differentiation

  1. 1. Tunis b us i n e ss sc h o o L MATHEMATICS FOR BUSINESS BCOR 120 DIFFERENTIATION
  2. 2. Content <ul><li>Limits of a function </li></ul><ul><li>Continuity of a function </li></ul><ul><li>Derivatives </li></ul><ul><li>Differential </li></ul><ul><li>Local/ Global Optimum </li></ul><ul><li>Convexity and Concavity </li></ul><ul><li>Taylor Polynomials </li></ul>01/02/12
  3. 3. LIMIT AND CONTINUITY <ul><li>1 </li></ul>
  4. 4. LIMIT OF A FUNCTION <ul><li>1.1 </li></ul>
  5. 5. Limits of a Function
  6. 6. <ul><li>The limit L has to be (a finite) number. </li></ul><ul><li>Otherwise we say that the limit does not exist . </li></ul><ul><ul><li>If the limit does not exist: </li></ul></ul><ul><ul><ul><li>f is said to be definitely divergent if L = ± ∞ ; otherwise </li></ul></ul></ul><ul><ul><ul><li>f is said to be indefinitely divergent </li></ul></ul></ul>
  7. 7. Left-side and Right-side limits
  8. 8. <ul><li>We also write </li></ul><ul><ul><li>for the right-side and left-side limits, respectively. </li></ul></ul><ul><li>A relationship between one-sided limits and the limit as introduced in Definition 4.1 is given by the following theorem. </li></ul>
  9. 9. <ul><li>We note that it is not necessary for the existence of a limit of function f as x tends to x 0 that the function value f (x 0 ) at point x 0 be defined. </li></ul>
  10. 10.
  11. 11.
  12. 12. Properties of Limits
  13. 13. Example
  14. 14. Example To avoid indetermination (0/0) we can multiply both terms by
  15. 15. CONTINUITY OF A FUNCTION <ul><li>1.2 </li></ul>
  16. 16. Continuity of a Function
  17. 17. Continuity of a function <ul><li>or, using the notation </li></ul><ul><li>continuity of a function at some point x 0 ∈ D f means that small changes in the independent variable x lead to small changes in the dependent variable y . </li></ul>
  18. 18. Continuous Function BCOR-120 Automn 2011 for all x from the open interval (x 0 −δ, x 0 +δ) the function values f (x) are within the open interval (f (x 0 ) − ε, f (x 0 ) + ε)
  19. 19. Discontinuous Function BCOR-120 Automn 2011 There exist a least an x from the open interval (x 0 −δ, x 0 +δ) For which the function values f (x) is outside the open interval (f (x 0 ) − ε, f (x 0 ) + ε)
  20. 20. When f is discontinuous at x 0 ? <ul><li>The function f is discontinuous at x 0 </li></ul><ul><ul><li>If the one-sided limits of a function f as x tends to x 0 are different; or </li></ul></ul><ul><ul><li>If one or both of the one-sided limits do not exist; or </li></ul></ul><ul><ul><li>if the one-sided limits are identical but the function value f (x 0 ) is not defined; or </li></ul></ul><ul><ul><li>if value f (x 0 ) is defined but not equal to both one-sided limits. </li></ul></ul>BCOR-120 Automn 2011
  21. 21. Types of discontinuities <ul><li>Removable discontinuity: </li></ul><ul><ul><li>f the limit of function f as x tends to x 0 exists but </li></ul></ul><ul><ul><ul><li>(4) the function value f (x 0 ) is different or </li></ul></ul></ul><ul><ul><ul><li>(3) the function f is not defined at point x 0. In this case we also say that function f has a gap at x 0 . </li></ul></ul></ul>BCOR-120 Automn 2011
  22. 22. Example: removable discontinuity BCOR-120 Automn 2011 , since
  23. 23. Example (cont.) BCOR-120 Autumn 2011
  24. 24. <ul><li>Irremovable dicontinuities </li></ul><ul><li>finite jump at x 0 : </li></ul><ul><ul><li>(1) if both one-sided limits of function f as x tends to x 0 exist and they are different. </li></ul></ul><ul><li>Infinite jump at x 0 : </li></ul><ul><ul><li>(2) if one of the one-sided limits as x tends to x 0 exists and from the other side function f tends to (+ or -)infinity </li></ul></ul>BCOR-120 Automn 2011
  25. 25. <ul><li>Pole at point x 0 </li></ul><ul><li>A rational function f = P/Q has a Pole at point x 0 if Q(x 0 )=0 but P(x 0 )≠ 0 </li></ul><ul><ul><li>As a consequence, the function values at x 0 + or to x 0 - tend to either ∞- or +∞. </li></ul></ul><ul><ul><ul><li>The multiplicity of zero x0 of polynomial Q defines the order of the pole: </li></ul></ul></ul><ul><ul><ul><ul><li>In the case of a pole of even order, the sign of the function f does not change at point x 0 ; </li></ul></ul></ul></ul><ul><ul><ul><ul><li>In the case of a pole of odd order, the sign of the function f changes at point x 0 </li></ul></ul></ul></ul>BCOR-120 Automn 2011
  26. 26. <ul><li>Oscillation point at x 0 . </li></ul><ul><ul><li>A function f has an oscillation point at x0 it he function is indefinitely divergent as x tends to x0. </li></ul></ul><ul><ul><ul><li>This means that neither the limit of function f as x tends to x 0 exist not function f tends to ±∞ as x tends to x 0 . </li></ul></ul></ul>BCOR-120 Automn 2011
  27. 27. 01/02/12
  28. 28. One-sided Continuity
  29. 29. Properties of Continuous Functions
  30. 30. Properties of Continuous Functions
  31. 31. Properties of Continuous Functions
  32. 32. Properties of Continuous Functions
  33. 33. Properties of Continuous Functions 01/02/12
  34. 34. Properties of Continuous Functions
  35. 35. DIFFERENCE QUOTIENT AND THE DERIVATIVE <ul><li>2 </li></ul>
  36. 36. Difference Quotients and Derivatives 01/02/12
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  45. 45. 01/02/12 A B A: the set of all points where function f is continuous B: the set of all points where function f is Defferentiable B included A.
  46. 46. DERIVATIVES OF ELEMENTARY FUNCTIONS; DIFFERENTIATION RULES <ul><li>3 </li></ul>
  47. 47. Deriv atives of Elementary Functions- Differentiation Rules
  48. 55. Derivatives of composite and Inverse Functions
  49. 67. DIFFERENTIAL; RATE OF CHANGE AND ELASTICITY <ul><li>4 </li></ul>
  50. 68. Differential, Rate of Change & Elasticity
  51. 70. The differential dy
  52. 76. Example-continued
  53. 83. GRAPHING FUNCTIONS <ul><li>5 </li></ul>
  54. 84. Gra phing Functions 01/02/12
  55. 85. 01/02/12
  56. 86. Monotonicity 01/02/12
  57. 87. 01/02/12
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  59. 89. Extreme Points 01/02/12
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  64. 95. 01/02/12
  65. 106. 01/02/12
  66. 107. Convexity & Concavity 01/02/12
  67. 108. 01/02/12
  68. 109. 01/02/12
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  76. 117. 01/02/12
  77. 118. Limits-revisited 01/02/12
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  90. 131. Graphing Functions 01/02/12
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  99. 140. Taylor Polynomials 01/02/12
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