MATHEMATICS - 2

This is the second of the two parts of a compulsory first year course in Mathematics. It aims at providing all fundamental tools in Calculus and Linear Algebra used in economic and managerial disciplines, along with examples and models of economic vocation.

A particular attention is paid to problems of maximization and minimization of functions of several variables and to simple but relevant application in Economics.

At the end of “Mathematics II” students should have acquired the fundamentals of calculus in multiple (mainly, two) variables. Specifically, students should have acquired the following skills.

a) Knowledge and understanding
a.1) Knowledge of basic definitions in calculus in multiple variables and linear algebra;
a.2) Interpretation of the above definitions in terms of geometric properties, supported by a span of crucial examples.

b) Ability to apply knowledge and understanding
b.1) Ability to compute/plot, for functions of multiple variables: domains, level curves, partial derivatives, stationary points, limits (along restrictions), linear approximations, tangent planes;
b.2) Ability to maximize/minimize a two-variable function, using: concavity/convexity properties; restriction on a compact subdomain (Weierstrass Theorem); Lagrange’s method;
b.3) Ability to apply the above tools to examples of economic/managerial vocation.

c) (Lifelong) learning skills
c.1) Improved ability to handle formal language, make logical deductions; and enhance rigorous rational thinking;
c.2) Improved ability to translate a problem into formal terms, solve it, and interpret the solution in terms of the original problem.

The contents and comprehension of "Mathematics I" and all of its prerequisites is given for granted.

In MODULE I:

a) Derivative of a function
a.1) Definition and geometric interpretation. Rules of differentiation.
a.2) Linear approximation.
a.3) Increasing/decreasing functions.
a.4) Rates of change and applications to economic examples.
a.5) Derivative of the inverse

b) Limits
b.1) Definition. Operations with limits. Indefinite forms.
b.2) L'Hopital’s rule.
b.3) Notable limits. Comparison of infinities of different strengths.
b.4) Change of variables in limits
b.5) One sided limits.

c) Continuous functions
c.1) Definition and examples
c.2) Necessary and sufficient conditions of continuity via left/right limits. Application to piecewise defined functions.
c.3) Intermediate value Theorem and applications.
c.4) Left/Right derivative. Sufficient conditions of differentiability via left and right limits.
c.5) Continuity versus differentiability.

d) Optimization
d.1) Definiton of maximum and minimum point. Stationary points.
d.2) First and second order conditions of optimality.
d.3) Weierstrass Theorem. Optimization on a compact interval.
d.4) Economic examples

e) Concavity/Convexity
e.1) Convex sets.
e.2) Epigraph, hypograph. Definition of convex and concave functions.
e.3) Necessary and sufficient conditions of convexity
e.4) Inflection points. Convexity and second derivatives.
e.5) Examples of notable concave functions in Economics.

f) Integration
f.1) Rules of integration, antiderivatives.
f.2) The Riemann integral, definite integrals. Fundamental theorem of integral calculus. Integral functions.
f.3) Economic examples
f.4) Improper integrals.

In MODULE II:

a) Functions of many variables
a.1) Subsets of R^n: interior/boundary points, open/closed sets, bounded sets, compact sets.
a.2) Natural domains and their representation in the plane. Graphs.
a.3) Level curves and their representation in the plane.

b) Partial derivatives
b.1) First-order partial derivatives
b.2) Second-order partial derivatives, Hessian matrix.
b.3) Linear approximation; tangent plane.

c) Continuity and Differentiability in R^n
c.1) Definition of a continuous function.
c.2) Continuity vs differentiability. Functions of class C^1. Examples and counterexamples.
c.3) Stationary points.

d) Unconstrained optimization in R^2
d.1) Definition of maxima/minima, local and global, in R^2.
d.2) First-order conditions of optimality.
d.3) Concave/convex functions; second-order conditions of optimality.
d.4) Inflection points.

e) Implicit functions
e.1) The chain rule
e.2) The implicit function theorem and applications

f) Constrained optimization in R^2
f.1) Weierstrass theorem. Application to examples.
f.2) Lagrange multipliers method for a function of 2 variables, subject to 1 equality constraint.
f.3) Economic applications: maximization of production with budget constraints; minimization of expenditure with production constraints.

g) Linear Algebra
g.1) Vectors and linear independence.
g.2) Matrices and operations on matrices.
g.3) Determinants. Expansion by cofactors.
g.4) Invertible matrices. Inverse of a matrix.

h) Linear Systems
h.1) Gaussian elimination. Rank of a matrix.
h.2) Rouché-Capelli Theorem.
h.3) Solution of a linear system by Gaussian elimination.
h.4) Application to economic/managerial problems.

K. Sydsaeter, P. Hammond and A. Strom, Essential Mathematics for Economic Analysis, Pearson.

In addition, lecture slides, homework and solved exams are made available on the webpage of the course, the university e-learning platform moodle.unive.it.

The final assessment for the course is integrated with that of the Mathematics I module, according to the procedures described below.

The examination consists of a written test and an oral examination. The written test specifically assesses the ability to apply the mathematical tools covered in the course to the solution of exercises and problems; the oral examination assesses the theoretical understanding of the topics, the ability to discuss the written test, and the correct use of mathematical language.

The written test includes all the topics covered in Mathematics I and Mathematics II. It consists of 6 problems, 3 relating to the topics of Mathematics I and 3 relating to the topics of Mathematics II, to be solved in a total time of 2 hours and 30 minutes. The skills acquired by students are assessed through the solution of the proposed problems; theoretical knowledge is assessed by requiring students to justify their answers in detail on the basis of the relevant theoretical results, such as definitions and theorems.

During the written test, the use of electronic devices of any kind, including calculators, is not allowed.

The oral examination takes place in the days immediately following the written test. In order to be admitted to the oral examination, students must obtain at least 16 points in the written test. The oral examination begins with a discussion of the written test and may then extend, if necessary, to the other topics covered in the course.

written and oral

The instructor is responsible for ensuring the authenticity and originality of all examinations and coursework. In cases of suspected academic misconduct, an additional on-site assessment may be required during the exams, which may differ from the standard format.

The score for the written examination ranges from 0 to 30 points and represents the grade out of thirty. Additional, more complex questions, for a maximum total of 6 points, may be added to allow for the possible award of honours.

The ordinary 30 points are distributed as follows:

* 22-24 points for basic questions;
* 6-8 points for questions of moderate difficulty;
* 6 points for more complex questions.

Answers that are not adequately justified will receive no credit. It is therefore important to explain clearly what is being done and why.

In order to be admitted to the oral examination, students must obtain at least 8 points in the first part, corresponding to Mathematics I (exercises 1-3), and 8 points in the second part, corresponding to Mathematics II (exercises 4-6).

At the instructor’s discretion and if conditions allow, the written examination may be replaced by two partial tests, in addition to the official exam sessions, to be taken before the first official exam session. The first partial test takes place immediately after the end of the Mathematics I module, and the second at the end of the Mathematics II module. The two tests follow the same assessment criteria as the final written examination. The partial tests have the dual purpose of encouraging regular study during the course and allowing students to divide their study workload, while capitalising in advance on the results achieved.

The oral examination mainly has the function of confirming the grade obtained in the written examination and, on average, may change that grade by an interval ranging from -3 to +3 points. This indication should be understood as an approximate description of what may happen, not as a formal rule for assigning the final grade.
In particular, should serious discrepancies emerge between the assessment of the written examination and that of the oral examination, the assessment of the oral examination will prevail, without any point constraints.

The topics of the course are conveyed using face-to-face lectures, practice sessions, and weekly office hours after class.
In particular, during the course time, office hours are held in public. Students may come and ask questions or simply sit and listen to other students’ questions and to the instructor’s answers. A further discussion is also possible by appointment.

The topics discussed in class are supported by materials made available for download on the webpage of the course https://moodle.unive.it/course/view.php?id=4882 , including:
a) the complete set of slides/lecture notes;
b) weekly sets of homework;
c) a list of previous exams, all completely solved
d) all relevant information about the course, and real-time updates.

Students should register in the related course on the university e-learning platform moodle.unive.it.

Accessibility, Disability, and Inclusion
Accommodation and support services for students with disabilities and students with specific learning impairments

Ca' Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding support services and accommodation available to students with disabilities. This includes students with mobility, visual, hearing, and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). If you have a disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.