Facoltà di Economia

Aggiornato A.A. 2021-2022

Detailed Program

You can find below a detailed program which will be update on a daily basis during the teaching period.

Sep 15, 2021 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. Examples and Esercises.

Sep 16, 2021 (Practical) Reminder on equations and inequalities (polynomials, rational exponential logarithmic and trigonometric)

Sep 16, 2021 Axioms of real numbesr: 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. Examples and Esercises.

Sep 17, 2021 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. Examples and Esercises.  

Sep 22, 2021 Computation of the domain and the range of a function, even/odd functions, increasing functions. Examples and Esercises.

Sep 23, 2021 (Practical) Computations of the domain, even/odd functions, increasing functions. Examples and Esercises.

Sep 23, 2021 Increasing/decreasing functions, Injective functions. Examples and Esercises. 

Sep 24, 2021 Surjective functions, bijective functions, invertible functions, Computation of the inverse of a function. Linear functions. Examples and Esercises.

Sep 29, 2021 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). Examples and Esercises.

Sep 30, 2021 (Pratical) Injective, surjective functions, composite functions, plots of composite functions (|g(x)|, -g(x), g(-x), g(x+a), g(x)+a)

Sep 30, 2021 Sequences: definition, limit of a sequence, convergent sequence, divergent sequence, verification of a limit with the definition. Examples and Esercises.

Oct 01, 2021 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. Examples and Esercises. 

Oct 01, 2021 (Practical) Trigonometric functions

Oct 06, 2021 Operations with limits, Practical rules for limits of sequences, Undetermined forms, Notable limits: the exponential sequence, Notable limits: the hierarchy of infinity. Examples and Exrcises. 

Oct 07, 2021 (Practical) Limits of sequences: verication with the definition, Applications of Theorem of Subsequences, Comparison Theorem and Absolute Value Theorem. Computation of limits.  

Oct 07, 2021 Notable limits of type n sin(1/n) (Statement and Proof). Examples and Exercises. Monotonic sequences: definition. 

Oct 08, 2021 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).  Examples and Exercises

Oct 12, 2021 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. Examples.  

Oct 13, 2021 Mathematical definition of infinite limit at infinity (4 cases). Computations of limits of undetermined forms via mathematical operations. Notable limits. Examples and exercises.  

Oct 14, 2021 (Practice) Notable limits of sequences, Series, Limits of functions.

Oct 14, 2021 Notable limits: Examples. Asymptotes: Exercises. Continuity of a function at a point (Definition), Contnuity of a function in a set (Definition), discontinuity points (classification). Examples and exercises. 

Oct 15, 2021 Maximum and minimum of a function, Weierstrass theorem Intermediate zero theorem. Examples and exercises. 

Oct 20, 2021 Derivatives: intuition, mathematical and geometric meaning, definition. Derivatives of elementiary functions (proof with the definition), Theorem: Derivatives of composite functions (only statement). Examples and exercises. 

Oct 21, 2021 (Practice) Continuity of functions, Asymptotes, Computation of derivatives, Equation of the tangent line. 

Oct 21, 2021 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).  

Oct 22, 2021 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). Examples and Exercises. 

Nov 4, 2021 (Practice) Applications of derivatives: Increasing/decreasing functions, Concave/convex functions, Maxima minima and inflection points, full study of a function. 

Syllabus in short

• Real Numbers
• Real Functions
• Limits of Sequences
• Numerical Series
• Limits of Functions and Continuous Functions
• Derivatives
• Applications of Derivatives to the study of Functions
• Taylor/McLaurin polynomials

Teaching Material

It is strongly recommended to take notes during lecttures.

Slides of Lectures and Exercises will be provided by the Professor. Slides do not substitute books; they only provide a rough indication of the material covered in classes.

Suggested books:

Knut Sydsaeter, Peter Hammond and Arne Strom, Essential Mathematics for Economic Analysis

Carl P. Simon, Lawrence Blume, Mathematics for Economists

Available on the webpage of the course and on Teams:

Slides of the course

Additional Exercises

Available on Moodle:

Self-check tests are uploaded on a weekly basis

Aggiornato A.A. 2019-2020

Lecture by lecture Program

17/10/2019

Sequences: monotonic sequences. Existence of the limit for monotonic sequences. 

Euler sequence: definition, properties (without proofs), Limit of the Euler sequence. Exercises. 

Subsequences. Necessary and sufficent condition for the existence of the limit of a sequence. 

Recursive sequences: Existence of the limit.

22/10/2019

Again on recursive sequences: Existence of the limit and its computation. Use of the Induction Principle. 

Series. Geometric, Harmonic and Euler series: conditions for convergence

24/10/2019

Idea of limits. Four case studies: Finite limit at a point, infinite limit at a point, Finite limit at infinity, Infinite limit at infinity. Discussion of the definition on function plots. 

Notable Limits and Undetermined forms. 

Exercises on computations of limits for undetermined forms 0/0.

5/11/2019

Finite limit at a point: Definition and verification of a limit using the definition.

Infinite limit at a point: Definition and verification of a limit using the definition. Vertical asymptotes. 

Finite limit at infnity: Definition and verification of a limit using the definition. Horizontal asymptotes.

Exercises on computations of limits. 

7/11/2019

Infinite limits at infinity: Definition and verification of a limit using the definition.

Oblique asymptotes.

Notable limits and Undetermined forms: exercises.

Continuity.

Classification of discontinuities.   

12/11/2019

Theorems on continuous functions (sum, product ratio and composition of continuous functions is continuous).

Weierstrass theorem: statement and meaning.

Intermediate value theorem: statement and meaning.

Theorem, on existence of zeros: statement and meaning. Examples and exercises.

Derivatives: Definition as limit of the incremental ratio.

Interpretation as the slope of the tangent line to the graph of a function at a given point.

Discontinuity of derivatives: angle point, cusp point, inflection point with vertical tangent. 

14/11/2019

Differentiability and continuity (A differentiable function is also continuous).

Example of a continuous function that is not differentiable.

Derivatives of elementary functions.

Operations with derivatives.

Theorems on differentiable functions: Rolle, Lagrange (statement and interpretation).

Differentiability and monotonicity.

Local extrema: local minima, local maxima, inflection points with horizontal tangent.   

19/11/2019

De L'Hopital rule

Convex and concave functions

Second-order conditions for maxima and minima

Derivative of the inverse of a function (exponential and trigonometric functions)

26/11/2019

Taylor formula: derivation and applications. 

Program (The final program will only be available at the end of the course)

Monotone sequences. Conditions for convergence of monotonic sequences. The Euler sequence. Subsequences. Recursive sequences: Existence of the limit.

Series: formal definition. Necessary condition for convergence. Convergence of armonic series. Convergence of geometric series. 

Limits of real functions of one real variable: Definition and uniquness of the limit.  Continuity: definition and fundamental theorems. Formal proof of the continuity of the square root and of the sin functions. Characterization of the discontinuity points of a function: jumps and removable (or type-I) and Infinite (or type-II) discontinuities.

Derivative of a real functions of one real variable. Necessary condition for derivability (continuity of the function).
Counter-examples of continuous functions that are not derivable. Continuity Theorem of a derivative of a function (the derivative of a function cannot have jump discontinuities). Example of a function with discontinuous first derivative with type-II discontinuity. Maxima and minima of a function: necessary and sufficient conditions.

Fundamental theorems on derivatives: Rolle, Lagrange. Convex and concave functions. Second-order conditions for maxima and minima. De L'Hopital rule. 

Taylor formula: derivation and applications to the approximation of irrational numbers with rational numbers.

Complete study of a function: sketch graph.

Objectives

Knowing and understanding the basic principles of mathematical analysis, starting from the foundations of topology. Applying those principles to econocmic modeling with a rigorous approach.

Teaching method

Lectures and tutorials