Math 151a Analysis Review Handouts, Fall 2014
You may download some Analysis I review handouts here.
They are pdf files.

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 MoivreStirling 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.

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

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 MeanValue Theorem,
Cauchy Remainder Theorem.