Course abstract

• Polynomials: Interpolation, Taylor's formula, Polynomials with integer values, Polynomials in several variables, Counting monomials.
• Counting Principles: Basic methods, the Pigeonhole principle, the Binomial theorem, Permutations, Graphs, Recurrence relations, Bijective proofs.
• Functions: Continuous functions, the Intermediate value property and its applications, Fixed points, Linear transformations of the plane, the Derivative and dilations, Higher order derivatives and the binomial theorem, Polynomial approximation to functions.
• Matrices: Matrices and transformations, Multiplication vs composition, Determinants as dilation factors, Polynomials applied to matrices, Matrices in probability theory. Matrices in Polynomial interpolation, the Vandermonde determinant.
• Conservation laws: Invariants of transformations, Discrete transformations, and applications, Transformations in Euclidean Geometry - circle inversions.
• Elementary number theory: Modular arithmetic, Divisibility, Prime numbers.
• Exploratory project suggestions.