# Math 412 Advanced Calculus I Review Notes, Spring 2012

They cover much, but not all of the material from Math 410.

• Reals 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.
• Series of Real Numbers: Infinite Series, Geometric Series, Series with Nonnegative Terms, A Series for 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 Continuity: Cauchy Continuity, Uniform Continuity, Sequence Characterization of Uniform Continuity, Sequential Compactness and Uniform Continuity, Continuous Extensions, Characterization of Cauchy Continuity.

• Riemann Integrals and Integrability
• Riemann Integrals: Partitons 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.