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 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.
-
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.
-
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.