Math 151a Analysis Review Handouts, Fall 2014

You may download some Analysis I review handouts here. They are pdf files.

1. Real Numbers
   • Real Number System: Introduction, Fields, Ordered Sets, Ordered Fields, Real Numbers, Extended Real Numbers.
   • Sequences of Real Numbers: Sequences and Subsequences, Convergence and Divergence, Monotonic Sequences, Limits and e, Wallis Product Formula, De Moivre-Stirling Formula, Limit Superior and Limit Inferior, Cauchy Criterion, Contracting Sequences.
   • Sums of Real Numbers: Infinite Series, Geometric Series, Series with Nonnegative Terms, Series and e, Series with Nonincreasing Positive Terms, Alternating Series, Absolute Convergence, Root and Ratio Tests, Dirichlet Test.
   • Sets of Real Numbers: Closure, Closed, and Dense, Completeness, Connectedness, Sequential Compactness.

2. Functions and Regularity
   • Functions, Continuity, and Limits: Functions, Continuity, Extreme-Value Theorem, Intermediate-Value Theorem, Limits of a Function, Monotonic Functions.
   • Differentiability and Derivatives: Differentiability, Derivatives, Differentiation, Local Extrema and Critical Points, Intermediate-Value and Sign Dichotomy Theorems, Concave and Convex Functions.
   • Mean-Value Theorems and Their Applications: Lagrange Mean-Value Theorem, Lipschitz Bounds, Monotonicity, Convexity, Error of the Tangent Line Approximation, Convergence of the Newton Method, Error of the Taylor Polynomial Approximation, Cauchy Mean-Value Theorem, l'Hospital Rule.
   • Cauchy and Uniform Continutity: Cauchy Continuity, Uniform Continuity, Sequence Characterization of Uniform Continuity, Bounded Domains and Uniform Continutity, Continuous Extensions.

3. Riemann Integrals and Integrability
   • Riemann Integrals: Partitions and Darboux Sums, Refinements, Comparisons, Definition of the Riemann Integral, Convergence of Riemann and Darboux Sums, Darboux Partitions Lemma.
   • Riemann Integrable Functions: Integrability of Monotonic Functions, Integrability of Continuous Functions, Linearity and Order for Riemann Integrals, Nonlinearity, Restrictions and Interval Additivity, Extensions and Piecewise Integrability, Lebesgue Theorem, Power Rule.
   • Relating Integration with Differentiation: First Fundamental Theorem of Calculus, Second Fundamental Theorem of Calculus, Integration by Parts, Substitution, Integral Mean-Value Theorem, Cauchy Remainder Theorem.