Course Overview
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Course Synopsis
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This course focuses on The Real Number System Sequences and Series Limits Continuity and Differentiability Riemann Integration.
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Course Learning Outcomes
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Upon successful completion of this course you should be able to understand
- Develop a solid understanding of fundamental mathematical structures including set theoretic statements real and complex number systems and ordered sets. They will be able to apply these concepts to analyze and manipulate mathematical expressions.
- Acquire a rigorous understanding of limits continuity and differentiability of realvalued functions. They will be able to prove various theorems related to limits sequences and functions. Emphasis will be placed on developing logical and rigorous proofs to support mathematical claims.
- Analyze and evaluate the convergence or divergence of sequences and series. They will demonstrate proficiency in identifying special sequences understanding the concept of subsequences and applying various tests to determine the convergence or divergence of sequences and series.
- Learn and apply advanced calculus concepts including the derivative of a function the BolzanoWeierstrass theorem the Mean Value Theorem Riemann sums and Riemann integrals. They will demonstrate the ability to apply these concepts to solve problems and they will be able to provide proofs for theorems related to differentiability and integrability.
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Course Calendar
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Week 01
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1
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Informal Introduction to the course
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4
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Some Properties Of Positive Integers
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6
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Finite and Infinite Sets: initial segments & Cardinality
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7
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Finite and Infinite Sets: Dedekind Infinite Set
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9
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The Set Of Rational Numbers
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10
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The Gaps In the Rational Numbers: p^2 =2 solution
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11
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Two Sets Of Rational With No largest And Smallest Numbers
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Week 02
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13
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Lower and Upper Bounds
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14
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Supremum And Infimum Examples
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15
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The Least Upper Bound Property
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16
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The Completeness Axiom
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17
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The Achimedean Property
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18
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The Set Of Rationals Is Dense In R
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19
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The Set Of Rationals Is Not Complete
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20
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The Set Of Irrationals Is Dense In R
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21
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Field Properties Some Conclusions
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22
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The Triangular Inequality
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24
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The Extended Real Number System
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25
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The Principle Of Mathematical Induction
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26
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The Principle Of Mathematical Induction: Examples
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Week 03
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27
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The Principle Of Mathematical Induction: Examples Continue
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28
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The Principle of Mathematical Induction II
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29
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The Principle of Mathematical Induction II: Examples
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30
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Some Concepts From Set Theory
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32
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Open & Closed Sets (Continue)
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33
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Limit Points, Boundary Points, Closure (Definitions)
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34
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Open Coverings With Examples
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36
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Closed Sets & Limit Points
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37
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Bolzano-WeierstrassTheorem
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40
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Sequences Divergent To Infinity
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Week 04
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42
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Convergent Sequence Is Bounded
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43
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Convergent Sequences (Continue)
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Assignment
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45
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Monotonic Sequences (Continue)
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46
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Some Special Sequences
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47
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Some Special Sequences (Continue)
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48
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Theorem On Algebra Of Limits
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51
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Convergence Of Subsequences
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52
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Sequences & Subsequences (Continue)
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53
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Equivalent Definition Of Limit Point Of Set
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Week 05
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54
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Theorem: Sequences & Subsequences
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55
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Limit Superior & Inferior
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56
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Limit Superior & Inferior (Continue)
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57
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Cauchy Sequence Of Real Numbers
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58
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Cauchy's Convergence Criterion
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60
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Infinite Series: Examples
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61
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Infinite Series (Continue)
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62
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Cauchy's Convergence Criterion For Series
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64
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Necessary Condition For Convergence of Infinite Series
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65
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Results About Convergent Infinite Series
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66
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Series Of Non negative Terms
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Quiz1
Week 06
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67
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The Comparison Test: Theorem
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68
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The Comparison Test Examples
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69
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Infinite Series: Theorem
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70
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Infinite Series : Theorem Applications On Examples
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71
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Infinite Series: Logarithmic Series
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72
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Infinite Series: Examples (Continue)
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73
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Infinite Series: Irrational Number 'e'
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74
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Infinite Series: Number 'e' (Continue)
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75
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Infinite Series: Number 'e' Is Irrational (Proof)
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76
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Infinite Series: Theorems
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77
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Infinite Series: Theorem Application
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78
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Infinite Series: Corollary & Example
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79
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The Ratio Test Theorem
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80
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The Ratio Test (Continue)
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Week 07
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81
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The Ratio Test : Examples
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82
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Infinite Series: Raabe's Test
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83
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Cauchy's Root Test Theorem
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84
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Absolute Convergence Of Series
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85
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Absolute Convergence Of Series (Continue)
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86
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Dirichlet's Test For Series
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87
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Dirichlet's Test For Series: Examples
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90
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Alternating Series & Alternating Series Test
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91
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Grouping Terms In A Series
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92
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Grouping Terms In A Series (Continue)
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93
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Rearrangements of series
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94
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Addition And Multiplication Of Series
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95
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Convergence Of Product Of Two Series
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Quiz2
Week 08
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97
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Convergence Of Power Series
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98
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Radius & Interval Of Convergence Of Power Series
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99
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Power Series: Radius Of Convergence (Continue)
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100
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Power Series: Examples
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Mid Term
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101
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Introduction To Functions
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102
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Arithmetic Operations On Functions
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104
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Limits: A Formal Definition
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106
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Limits: Uniqueness Of Limit
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107
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Theorems About Limits
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108
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Limits: One Sided Limit
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Week 09
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109
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Limits: One Sided Limit (Continue)
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110
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Limits: Example And Relation Between One Sided Limits
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117
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Piecewise Continuity Jump discontinuity
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118
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Examples: Sum Difference Continuity
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119
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Removable Discontinuity
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Quiz3
Week 10
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121
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Composite Function: Theorems
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122
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Continuity Of Composite Functions: Example
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124
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Bounded Functions: Theorems
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125
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Bounded Functions: Theorems (Continue)
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126
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Bounded Functions: Examples
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127
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Intermediate Value Theorem
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129
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Uniform Continuity Examples
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130
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Uniform Continuity: Theorems
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132
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Theorem about monotonic functions
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133
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Limits Inferior and Superior
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134
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Monotonic Functions Theorems
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Week 11
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135
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Monotonic Functions Theorems (Continue)
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136
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Monotonic Functions Theorem Examples
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137
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Derivative Of A Function
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138
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Derivative Of A Function: Examples
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139
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Geometrical Interpretation Of The Derivative
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140
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Differentiability: Lemma
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141
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Differentiability implies continuity
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142
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Differentiability: Examples
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Week 12
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143
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Algebra of Differentiable Functions
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145
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Derivatives: Examples
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146
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One Sided Derivatives
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147
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One sided derivatives Example
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148
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Extreme Values Of A Function
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149
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Extreme Values Of A Function (Continue)
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152
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Generalized Mean Value Theorem
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153
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Applications Of Mean Value Theorem
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Week 13
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156
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The Indeterminate Forms
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157
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Indeterminate Forms: Examples
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158
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The indeterminate Forms Examples
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159
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The Indeterminate Form for Infinity into Infinity
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160
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Other Indeterminate Forms
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161
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Other Indeterminate Forms: Examples
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163
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Taylor Polynomials: Theorems
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164
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Taylor Polynomials: Examples
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165
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Application To Finding Local Extremum
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166
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Local Extremum Examples
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Quiz4
Week 14
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170
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Riemann Integral: Examples
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171
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Riemann Integral: Theorem
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172
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Riemann Integral: Upper And Lower Sums
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173
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Riemann Integral: Upper And Lower Sums Theorem
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174
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Riemann Integral: Upper And Lower Sums Examples
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175
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Riemann Integral: Lemma
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176
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Riemann Integral:Theorems (Continue)
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177
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Existence Of The Riemann Integral Theorem
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178
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Riemann Integral: Lemmas
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179
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Riemann Integral: Lemmas & Theorems
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Week 15
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180
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Riemann Integral: Lemmas & Theorems (Continue)
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181
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Riemann Integral: Integrability of Bounded Functions
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182
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Riemann Integral: Integrability of Bounded Functions (Continue)
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183
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Riemann Integral: Theorem About Monotonicity Of Function
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184
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Properties Of The Integrals
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185
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Properties Of The Integrals: Theorems
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186
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Properties Of The Integrals: Theorems (Continue)
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187
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Properties Of The Integrals: Theorem 3.3.3
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188
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Properties Of The Integrals: Theorem 3.3.4
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189
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Properties Of The Integrals: Theorem 3.3.5
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190
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Properties Of The Integrals: Theorem 3.3.6
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191
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First Mean value Theorem of Integrals
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192
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Properties Of The Integrals: Theorem 3.3.8
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193
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Properties Of The Integrals: Theorem 3.3.9
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194
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Properties Of The Integrals: Theorem 3.3.10
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195
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Properties Of The Integrals: Theorem 3.3.11
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196
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Properties Of The Integrals: Examples
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Week 16
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197
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Fundamental Theorem Of Calculus And Related Theorems
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198
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Fundamental Theorem Of Calculus
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199
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Integration By Parts Theorem
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200
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Second Mean Value Theorem
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201
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Change Of Variables Theorem
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202
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Change Of Variables: Examples
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Final Term
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