Calculus 1

1. Functions (a verbal rule, a table, a graph, algebraic representation, a machine, an object )
2. IVPs, Rate Equations and Mathematical Models (S-I-R, Newton's Law of Cooling)
3. Euler's Method
4. Successive Approximations (Limits)
5. Derivative (definition of, using difference quotients, differentiability, local linearity)
6. Rules of Differentiation (chain rule, product rule, quotient rule, elementary functions)
7. Applications of Derivatives (Newton's Method, L'Hopital's Rule, curve sketching)
8. Optimization (critical points, inflexion points, extremum points)
9. Partial Derivatives (Functions of Two Variables)
10. Taylor Polynomials (Microscope Approximation)

Calculus 2

1. Integration (techniques: Integration By Parts, Integration By Substitution, elementary anti-derivatives, partial fractions, tables)
2. Taylor Approximation (general formula)
3. Fundamental Theorem of Calculus: Part1, Part2, Part 3
4. Accumulation functions (graphing the antiderivatives)
5. Riemann Sum: Def of the integral, evaluation.
6. Series: Taylor, Power, infinite series.  Convergence tests: absolute ratio, integral, root, last term (zero divergence), alternating series test
7. Numerical Approximation of integrals: Trapezoid rule, Simpson's rule.
8. improper integral: 1. infinity as bounds, 2. undefined somewhere in the boundary.
9. Fourier series.
10. Applications of integration: length of curve, average value of integral, area between curves

Multivariable Calculus

1. Multiple Integration evaluation changing order setting up as a multiple integral
2. Coordinate systems spherical cylindrical polar
3. Vector fields and Gradient fields
4.  Curl, DIV, and Gradient 
5. R^n (n-dimenional space), equations of planes and lines in multidimensional space 
6. Fundamental theorem of Line Integrals 
7. Line Integrals 
8. Level Curves, surfaces and tangent planes 
9. Jacobian (Taylor's Approximation of multivariable function)
10. Multivariable constrained optimization, Lagrange multipliers, at max/min grad f =0 test to see if min, max or saddle point
11. Greens theorem

Linear Systems

1. Fundamental Theorem of Linear Algebra; back of the Linear Book)
2. Inverse Matrix; when it exists (determinant is no zero), how to find it (Gauss- Jordan elimination), uses, properties of inverses
3. Singular System of Equations; when determinant equals zero (either infinite solutions or no solutions)
4. Determinants; computation, Cramer’s Rule
5. Eigenvalues/Eigenvectors; how to find them, uses
6. Vector Spaces
7. Basis, Dimension
8. Linear Independence
9. Matrices and Vectors; multiplication, addition, transpose, dot product, orthogonality.
10. Gaussian Elimination; solving for x when Ax = b.

Discrete Math

1. Propositions and Connectives --Truth Tables, Logical Equivalences, Tautologies and Contradictions, Quantifiers--Universal, Existential
2. Sets and Connectives--Power Sets, Cartesian Products, Union, Intersection, Symmetric Difference, Countability, Cardinality, De Morgan’s Laws for propositions and sets
3. Methods of Proof: Contrapositive, Contradiction, Induction, Direct
4. Integers--Divisibility, Division Algorithm, Euclidean Algorithm, gcd, lcm, Modular Arithmetic and Equivalence
5. Relations--Transitive, Symmetric, Reflexive, Antisymmetric, Closure, Matrix Representations, Equivalence Relations--Classes/Partitions
6. Functions--Domain, One-to-one, Onto, Bijections, Compositions, Inverses, Fibonacci Sequence
7. Counting Techniques--Combinations, Permutations, Product and Sum Rules, Tuples, Pigeonhole Principle, Principle of Inclusion and Exclusion, Counting onto functions
8. Binomial Theorem, Pascal’s Triangle and relations to combinations.
9. Partial Orders--Hasse Diagrams, Max, Min, glb, lub, Lattices
10. Graphs--Undirected and Directed Graphs--Circuits, Paths, Connectivity, Adjacency Matrix, Incidence Matrix, Multigraphs, Graph Isomorphism, Euler Circuits and Paths.
11. Review Theorems and their proofs.