| CHAPTER 1 Real Numbers And Monotonic Sequences |
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Sequences, weakly increasing/decreasing, strictly increasing/decreasing, monotonic, upper/lower bound, bounded above/below, bounded, Completeness Property
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| CHAPTER 2 Estimations and Approximations |
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Using inequality laws, absolute vales, and triangle inequalities to approximate values; (Revised) Completeness Property
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| CHAPTER 3 The Limit of a Sequence |
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A formal definition for the limit of a sequence, convergent/divergent, uniqueness of limits, infinite limits, the limit of an.
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| CHAPTER 4 Error Term Analysis |
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Error terms, analyzing how quickly a sequence converges, Applications to the geometric sequence.
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| CHAPTER 5 Limit Theorem for Sequences |
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Limits of sums, products and quotients. Algebraic operations for infinite limits. Squeeze theorem for finite and infinite limits. Location theorems. Using subsequences to show that a limit does not exist.
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| CHAPTER 6 The Completeness Property |
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Nested Intervals Theorem, cluster points, Bolzano-Weierstrass Theorem, Cauchy sequences, Cauchy convergence, Completeness properties for sets, infimum, supremum.
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| CHAPTER 7 Infinite Series |
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Sequences and Series. Linearity of convergent series. Test for Divergence. Comparison and Limit Comparison tests. Absolute convergence and the Ratio and Root tests. Integral test. Alternating Series test.
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| CHAPTER 8 Power Series |
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Computing radius and interval of convergence. Adding and multiplying power series.
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| CHAPTER 9 Functions of one Variable |
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Operations on functions (addition, subtraction, multiplication, division, composition). Increasing/decreasing/monotonic functions. Even/odd/periodic/inverse functions.
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| CHAPTER 10 Local and Global Behavior |
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Local properties (bounded, positive, increasing, etc.). Completeness property for functions on an interval. Supremum and infimum.
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| CHAPTER 11 Continuity and Limits of Functions |
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The formal definition of the limit of a function and continuity. Limits of sums, products and quotients. Squeeze theoerms and location theorems. Using sequences to show that a function is not continuous.
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| CHAPTER 12 The intermediate value theorem |
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Bolzano's Theorem and applications (finding zeros of f(x)), Intermediate Value Theorem, Inverse Function Theorem.
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| CHAPTER 13 Continuous Functions on Compact Intevals |
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Sequentially compact, continuous functions on a compact interval are bounded, Extreme Value theorem, continuous functions map compact intervals to compact intervals.
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| CHAPTER 14 Differentiation: Local properties |
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Limit definition of derivative, differentiation rules, local extreme values, Fermat's theorem.
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| CHAPTER 15 Differentiation: Global properties |
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Rolle's Theorem, Mean Value Theorem and applications, Cauchy's Mean Value Theorem, L'Hospitals' Rule.
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| CHAPTER 16 Linearization and Convexity |
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| CHAPTER 17 Taylor Approximations |
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| CHAPTER 22 Sequences and Series of Functions |
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