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### Chapter 02 differentiation

1. Tunis b us i n e ss sc h o o L MATHEMATICS FOR BUSINESS BCOR 120 DIFFERENTIATION
2. Limits of a Function
3. Left-side and Right-side limits
4. Properties of Limits
5. Example
6. Example To avoid indetermination (0/0) we can multiply both terms by
7. Continuity of a Function
8. 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)
9. 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)
10. Example: removable discontinuity BCOR-120 Automn 2011 , since
11. Example (cont.) BCOR-120 Autumn 2011
12. 01/02/12
13. One-sided Continuity
14. Properties of Continuous Functions
15. Properties of Continuous Functions
16. Properties of Continuous Functions
17. Properties of Continuous Functions
18. Properties of Continuous Functions 01/02/12
19. Properties of Continuous Functions
20. Difference Quotients and Derivatives 01/02/12
21. 01/02/12
22. 01/02/12
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29. 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.
30. Deriv atives of Elementary Functions- Differentiation Rules
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38. Derivatives of composite and Inverse Functions
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50. Differential, Rate of Change & Elasticity
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52. The differential dy
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58. Example-continued
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65. Gra phing Functions 01/02/12
66. 01/02/12
67. Monotonicity 01/02/12
68. 01/02/12
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70. Extreme Points 01/02/12
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87. 01/02/12
88. Convexity & Concavity 01/02/12
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98. 01/02/12
99. Limits-revisited 01/02/12
100. 01/02/12
101. 01/02/12
102. 01/02/12
103. 01/02/12
104. 01/02/12
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106. 01/02/12
107. 01/02/12
108. 01/02/12
109. 01/02/12
110. 01/02/12
111. 01/02/12
112. Graphing Functions 01/02/12
113. 01/02/12
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115. 01/02/12
116. 01/02/12
117. 01/02/12
118. 01/02/12
119. 01/02/12
120. 01/02/12
121. Taylor Polynomials 01/02/12
122. 01/02/12
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### Editor's Notes

1. This notion deals with the question of which value does the dependent variable y of a function f with y = f (x) approach as the independent variable x approaches some specific value x0?
2. In the above definition, the elements of sequence { x n } can be both greater and smaller than x 0 . In certain situations, only the limit from one side has to be considered. In the following, we consider such one-sided approaches, where the terms of the sequences are either all greater or all smaller than x 0 .
3. If we apply Theorem 4.2, part (4), and separately determine the limit of the numerator and the denominator, we find that both terms tend to zero, and we cannot find the limit in this way. Therefore, we rationalize the numerator by multiplying the numerator and the denominator by √ x + 2 and obtain: