Topic 1: Functions and sets
- Sets.
- The concept of a generic element of a set.
- Mathematical notation.
- Numeric sets.
- Real intervals.
- Implications.
- Quantifiers.
- Proof by contradiction.
- Real functions of a real variable:
- General concepts.
- Graph,
- Properties.
- Algebraic operations.
- Composition, inversion.
Elementary functions and their graphs.
Operations on graphs:
- Translation.
- Dilation.
- Absolute value symmetries.
- Calculation of function domains (reviewing algebraic and transcendental equations and inequalities).
Topic 2: Limits
- Functional limits.
- Left-hand and right-hand limits.
- Continuity.
- Infinities and infinitesimals of higher and lower order.
- Indeterminate limits, asymptotic expressions.
- Notable limits.
- Calculation of limits of polynomials.
Topic 3: Derivatives
- Derivation.
- Differentiation.
- Equivalence between these two concepts.
- Differential.
- The concept of local linearization.
- Tangent line to the graph of a function at a point.
- Derivatives of elementary functions.
- Lagrange's theorem and L'Hôpital's rule.
- Orders of infinities and infinitesimals.
- Higher-order derivatives.
- Concavity and convexity.
- Taylor expansion of a function.
- Application to the calculation of indeterminate limits.
- Graphical analysis of functions.
- The concept of optimization and concrete examples.
Topic 4: Integrals
- Introduction to integrals,
- Riemann construction,
- Antiderivatives,
- Fundamental theorem of calculus and
- Mean value theorem for integrals.
- Calculation of integrals
- Immediate.
- Semi-immediate.
- By substitution.
- By partial fraction decomposition.
Numerical approximation:
- The Cavalieri-Simpson formula.
- Applications.
Topic 5: Sequences and series
- Mathematical induction.
- Sequences.
- Numerical series.
- Series of non-negative terms.
- Convergence tests.
- Sequences and series of functions.
- Power series.
- Fourier series.
Personal notes: