Math 412 Advanced Calculus I Review Notes, Spring 2012
You may download Advanced Calculus I review notes (pdf files) below.
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 MoivreStirling 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, 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 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 MeanValue Theorem, Cauchy
Remainder Theorem.