Short Syllabus:

1. Sets and Numbers: sets and subsets, numbers, functions

2. Sequences, Series and Limits: definition of a sequence, limit of a sequence, properties of sequences, series

3. Function theory: Continuity of functions, applications, intermediate value theorem

4. Differentiability, rules of differentiation, mean value theorem, local maxima and minima, concavity and convexity, function analysis, optimization.

5. Taylor series formula and approximation.

6. Linear Algebra: systems of linear equations, matrix theory, determinants and inverse matrix, Cramer’s rule

7. Integration: indefinite and definite integral, properties of integration, techniques of integration, improper integral.

If some additional time is left

8. Multivariate calculus: continuity and differentiation of functions, local maxima and minima.

Detailed Syllabus:

Logic and Sets

Propositions, theorems and proofs. Negation of a proposition, equivalent propositions. Logical implication and fundamental property of the implication. Sum and product of propositions.
Sets, inclusion relation, set rapresentation with Eulero-Venn diagrams, quantifiers and propositions using quantifiers. Intersection, union, difference. Complement operation and cartesian product. Properties of the set operations. Distributive properties and De Morgan rules.

Numeric Sets

Natural numbers, sum and product of natural numbers. Neutral element and inverses. Relative numbers and rational numbers. Existence of incommensurables with proof that the diagonal of the unit square is not rational. Irrational and real numbers. Consistency and density of the real numbers. Intervals. Bounded sets, upper bounds and lower bounds. Least upper bound, greatest lower bound, maximum and minimum. Accumulation points and isolated points. Internal and external points. Interior, exterior and boundary of a subset of real numbers. Open and closed sets.

Functions and Properties

Function definition and function representation. Domain, codomain and range. Image and counterimage. Graph of a function, piecewise-defined functions. Symmetry and even and odd functions. Increasing and decreasing functions. Elementary functions: affine functions, absolute value, power functions with integer and rational exponent, power functions with irrational exponent, exponential and logarithmic functions, trigonometric functions. Determining the domain of a function. Function transformations: shifting, stretching, reflecting, sum, product and composition. Inverse functions. One-to-one and invertible functions. Determining the graph and equation of the inverse.

Limits and Continuity

Definition of limit and examples. Left and right limits, infinite limits and limits at infinity. Precise definition of limit. Properties of limits. Indeterminate forms and computing limits. Limit theorems: comparison, squeeze (with proof) and sign persistence theorems (with proof). Uniqueness of limit (with proof). Order of infinity, substitution principle. Continuity, continuity from the left and from the right. Continuity of basic functions, continuity of composition (with proof). Intermediate value and root existence theorems (with applications). Continuity of the inverse function (with counterexample).


The tangent problem, definition of derivative. Left and right derivatives, higher order derivatives. Necessary condition of differentiability (with proof). Nondifferentiable functions. Differentiation formulas (with proofs). Global and local maximum and minimum. Extreme value theorem. Fermat’s Theorem (with proof). Finding global extreme points. Rolle’s Theorem. Lagrange Theorem (with proof). Differentiability of the inverse function. Differentiability and monotonicity (with proofs). Differentiability and local extremes. Concavity and inflection points (with proofs). Necessary and sufficient conditions for the existence of inflection points (with proofs). Curve sketching. Higher order sufficient conditions for the existence of local extremes and inflection points. De l’Hospital Theorem. Linear approximation and differential. Taylor approximation.


Definite integration. Riemann Sums and definition of the definite integral. Integrable and non integrable functions. Properties of the definite integral. Indefinite integration and antiderivatives. The fundamental theorem of calculus (with proof). Integration formulas. Integration by part and substitution.

Introduction to Linear Algebra

Matrices, scalars, submatrices and vectors. Matrix and vector operations. Linear combinations, systems of generators. Matrix-vector product and scalar product. Properties of the matrix-vector product and scalar product.

Systems of linear equations, consistency and uniqueness of solutions. Elementary row operations and equivalency theorem. Row-Echelon form and Reduced Row-Echelon Form. Gauss theorem and Gauss-Jordan elimination. Rank and nullity of a matrix. Uniqueness and existence of solutions of a linear system. Cramer’s Theorem.

Span of a set of vectors, generators and generating sets. Linear dependence and independence. Minimality of a system of generators. Linear independence and rank. Basis.


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