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Program

Updated A.Y. 2022-2023

Natural numbers, Integers, rational numbers and the properies of operations. Decimal and fractional representations of rational numbers. The solution of the equation x^2=2 and the Real numbers. Decimal representation of real numbers. 

 Axioms of real numbers: Axioma of operations, Axioms of ordering and Axiom of completeness. The meaning of the exaiom of completeness (examples of separating points). The set of rational numbers does not satisfy the axiom of completeness (example). Set theory: quantifiers For all, Exists, Exists only one, Not Exist, Subsets. Union of sets, Intersection of sets, Subtraction of sets, the complementary of a set. Intervals as subsets of real numbers: open, closed, not open nor closed, bounded, unbounded. 

 Implications: if-then, if and only if. Structure of a Theorem (Hypotheses-Thesis). Distance in R and the absolute value. Functions: general definition and examples. Real function of a real variable, domain of a function, range of a function, graph of a function, plot of a function.  

 Computation of the domain and the range of a function, even/odd functions, increasing functions. 

Increasing/decreasing functions, Injective functions. 

 Surjective functions, bijective functions, invertible functions, Computation of the inverse of a function. Linear functions. 

Quadratic functions, absolute value function, power functions with natural and real exponent, exponential functions, logarithmic functions. Composite functions, plots of a few composite functions (absolute value). 

 Injective, surjective functions, composite functions

Sequences: definition, limit of a sequence, convergent sequence, divergent sequence, verification of a limit with the definition. 

Sequences that do not admit a limit, definition of a subsequence, Theorem on convergence of sequences and subsequences, Theorem: Uniqueness of the limit, Absolute value theorem, Operations with limits. 

Operations with limits, Practical rules for limits of sequences, Undetermined forms, Notable limits: the exponential sequence, Notable limits: the hierarchy of infinity.

Notable limits of type n sin(1/n) (Statement and Proof). Monotonic sequences: definition. 

Theorem on convergence of monotonic sequences (statement and intuition). The Euler sequence. Convergence of the Euler sequence (statement and intuitive proof). Definition of rela numbers as limits of sequences. Geometric Sum (definition and proof of the sum). Geometric Series (Definition and proof of convergence).  

 Limits of functions: mathematical definition of Finite limit at a point. Left and right limits. Piecewise define functions. Theorem of existence of the limit via left and right limits (statement). Mathematical definition of Finite limit at infinity, definition of hoirizontal asymptote. Mathematical definition of Infinite limit at a point, definition of vertical asymptote

 Mathematical definition of infinite limit at infinity (4 cases). Computations of limits of undetermined forms via mathematical operations. Notable limits. 

Notable limits: Examples. Asymptotes: Exercises. Continuity of a function at a point (Definition), Contnuity of a function in a set (Definition), discontinuity points (classification).

 Maximum and minimum of a function, Weierstrass theorem Intermediate zero theorem. Examples and exercises. 

Derivatives: intuition, mathematical and geometric meaning, definition. Derivatives of elementiary functions (proof with the definition), Theorem: Derivatives of composite functions (only statement) and of the inverse function. Derivatives of arcsin, arccos, arctan.  

Non-differentiability points: definitions. Continuity and differentiability: Differentiable implies continuous (statement and proof), Continuous does not imply differentiable (examples). First order Taylor appoximation and the error (absolute error and error in percentage). Increasing and decreasing functions. Necessary and sufficient condition for monotonicity of differentiable functions (statement). Stationary points (definition). Fermat Theorem or 1st order necessary condition for local maxima and minima (statement).  

Local maxima and minima of non-differentiable functions: the example of the absolute value. Concavity and convexity (general definition). Necessary and sufficient condition for cancavity/convexity of differentiable functions (statement and geometric intuition). Definition of higher order derivatives. Necessary and sufficient condition for cancavity/convexity of twice differentiable functions (statement). Definition of Inflection points. 2nd order sufficient condition for local maxima and minima (statement). De l'Hopital rule (two theorems, statements). 

Taylor expansions and approximatation of purely irrational number via Taylor expansions.

Objectives

Knowledge and understanding of the basic principles that are useful for deep comprehension of the main concepts of mathematical analysis, starting from the foundations of topology. Applying knowledge and understanding of mathematical analysis to economic modelling with a rigorous approach.

Teaching method

Lectures (ITA: Lezioni frontali) and tutorials (ITA: esercitazioni)

Books

The Notes of the course. 

Further readings

Carl P. Simon and Lawrence Blume 

Mathematics for Economists  – 2010