MATHEMATICS - 2

This is the first 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.

In "Mathematics I" a particular attention is paid to problems of maximization and minimization of functions of one variable, as preparatory to optimization in multiple variables (discussed in the second part of the course).

At the end of “Mathematics I” students should have acquired the fundamentals of calculus in one variable. Specifically, students should have acquired the following skills.

a) Knowledge and understanding
a.1) Knowledge of basic definitions in calculus in one variable, such as: derivatives, limits, integrals;
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, for functions of one variable: derivatives, limits, integrals (elementary, by parts, by substitution);
b.2) Ability to analyse properties of functions of one variable, such as monotonicity, convexity, behaviour in the long run;
b.3) Ability to compute stationary and inflection points; ability to maximize/minimize a quantity described by a one variable function, particularly when it describes an economic variable;
b.3) Ability to interpret all above properties in examples of economic/managerial vocation.

c) (Lifelong) learning skills
c.1) Improved ability to handle a formal language, to make logic deductions; 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.

Students must be clear of didactic debts (Additional Learning Requirements, ALR).

Topics usually taught at schools of upper secondary education are given as known. In particular: the basics of set theory, real numbers; algebraic operations and properties of real numbers; powers and fractional powers, absolute values; equations and inequalities of first/second order, rational, irrational, exponential and logarithmic type; linear and quadratic functions; exponential and logarithmic functions; elements of analytic geometry: cartesian coordinates, distance of points in the plane, lines, parabolas, ellipses, hyperbolas and their equations and graphs in the plane.

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.

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.

Grading is based on a final written exam, including all topics taught in Mathematics I and Mathematics II, and an optional oral exam.

The written exam consists of 6 problems, 3 of which on the topics of Mathematics I, and 3 on those of Mathematics II, to be solved in 2h30’ overall. Abilities acquired by students are verified by requiring them to solve the problems. Their acquired knowledge is verified by asking them to justify in detail their answers, on the basis of the theoretical results (definitions and theorems).

The exam is closed-notes and closed-book, but students are allowed to use a pocket calculator (scientific calculators computing derivatives and integrals, or plotting graphs, are not allowed).

The oral exam is optional for both the student and the instructor. Students wanting to improve their grade can request to do it (see details in the next section). On the other hand, the teacher can ask students to take a mandatory oral exam, at her sole discretion.

The written exam is also issued in two parts (Part 1, midterm; Part 2, final) corresponding to the halves Mathematics I and Mathematics II, during and after the course time span.

written and oral

More on the Written Exam

The written exam is passed if the student scores at least 8 points in both parts I and II, corresponding to exercises 1-3 a nd exercises 4-6, respectively, and totalling at least 18 points.

The available points (approximately 6 for each exercise) are distributed as follows:

18-20 points for basic questions;
6-8 points for moderately difficult questions;
6-8 points for more complex questions.
If answers are not properly justified, they are worth zero. Therefore, it's important to explain what is being done and why.

Generally, there are 6 points exceeding the 30 corresponding to a full grade. The extra points entitle the student to receive honours. Points are 36 in total, with 18 corresponding to a grade of 18/30 (minimal passing grade) and 30 corresponding to a grade of 30/30.

Samples of Exams with complete solutions and scoring are found in the moodle page of the course.

More on the Oral Exam

Students with 16 or more in the written exam are entitled to an oral exam, if they want to better their grade.

For written exam scores of 18 or higher, the oral exam will focus on theoretical questions.
For written exam scores of 16 or 17, the oral exam will initially consist of 1-2 exercises targeting the weakest areas identified in the written performance. Once reached a passing grade of 18, and if the student chooses to, the oral exam can continue twith theoretical questions.

With the oral exams, the grade can go up/down: usually, the range is -3 to 3 points.
The oral exam is mandatory if the written exam is taken online.

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.