GENERAL MATHEMATICS

Single discipline educational activity

Course Sheet

Academic Year of enrolment:

2019/2020

Type of Course:

Basic

Optional/Mandatory:

Compulsory

Degree:

Bachelor's Degree Programme in TOURISM ECONOMICS AND TOURISM MANAGEMENT

Disciplinary Sector:

MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES

Credits:

Programme Year:

Professor and Collaborators:

RUSSU Paolo

Hours of classroom activity:

Prerequisites:

Literal equations, use of brackets. Polynomials. The difference of squares. Expressions with fractions. The solution of identity and equations. First and second-degree algebraic equations. First and second-degree algebraic inequalities. Inequalities with fractions. Radicals, inequalities with radicals.

AF code:

40000525

Assessment Methods

Verification of learning:

The exam consists of a written exam. Two alternative modalities are available:
1. three partial written exams.
2. whole written exam.
The students have the opportunity to take three partial exams:
a) the first partial written exam (40 minutes) is scheduled during the course and concerns the drawing of function graphs and simple transformations of elementary functions that deals with the topics described in modules 1 and 2 (see the teacher's website on uniolbia.it). The partial exam is considered passed if at least 2 exercises will be performed correctly;
b) the second partial written exam(60 minutes) is scheduled during the course and concerns the resolution of exercises related to modules 3,4,5, (see the teacher's website on uniolbia.it). The partial exam is considered passed if at least 4 exercises will be performed correctly;
c) the third partial exam(40 minutes) is scheduled at the end of the course and concerns of exercises related to module 6 (see the teacher's website on uniolbia.it). The test is considered passed if at least 2 exercises will be performed correctly.
The final mark is the weighted average of the achieved grades, rounded up, of the three partial marks, according to the formula vf = 0.3 * v1 + 0.3 * v2 + 0.4 * v3, where vi are the marks given to the three partial written exams. All students who take both partial exams and get a final mark of at least 18 points have passed the exam.
Students who do not pass one of the three partial exams, will have to take the whole written exam that is based (see example teacher site on uniolbia.it):
a) 8 exercises (3 points each) of which 3 concerning modules 1 and 2; 4 exerciss concerning modules 3,4 and 5; 1 exercise concerning the module 6 with reference to the domain of the functions of several variables or partial derivatives or optimization unconstraint.
b) 1 exercise (6 points) referred to the constrained maximization, with the Lagrange multipliers method.
All the exercises both in the partial exames and in the whole exam will be exclusively mathematical without reference to particular application problems (even if these will be illustrated during the course).

Teaching period

Second trimester

Objectives

The course provides the basic notions of mathematical analysis useful for the analysis and understanding, through mathematical models, of economic phenomena.
T.U. C1-C2. KNOWLEDGE AND UNDERSTADING: Graph of elementary functions and their transformations in the Cartesian plane. LEARNING SKILL: knowing how to evaluate simple situations and translate them into a linear and non-linear mathematical model; be able to process information and use calculation methods; knowing how to construct simple problem solving procedures with linear and non-linear relationships. LINK / APPLYING KNOWLEDGE AND UNDERSTADING: linear and non-linear models used in Economics, in the Business and in Statistics: functions of cost, revenue, profit, demand functions, supply and price of equilibrium, linear regression, simple interest rate, compound interest rate.

T.U. C3.KNOWLEDGE AND UNDERSTADING: average variation rate from a table, from a formula, incremental ratios, instantaneous variation rate, slope of the tangent equaled to the rate of variation, calculation of the derivative by algebraic way, analyze the continuity / discontinuity of a function both algebraically and graphically, asymptotes. LEARNING SKILL: knowing how to recognize algebraically and graphically when the continuity / discontinuity domain of a function (even at times). knowing recognize from the graph of a function if the function is derivable; knowing how to calculate simple limits of linear and non-linear functions (finite and infinite); knowing how to enunciate a theorem and recognize the necessary / sufficient conditions expressed by a theorem. LINK / APPLYING KNOWLEDGE AND UNDERSTADING: non-linear models used in Economics: cost, revenue and marginal profit.
T.U.C4.KNOWLEDGE AND UNDERSTADING: Acquire the calculation techniques related to the derivation for the product and the ratio of functions, chain rule; right tangent of an explicit and implicit function. LEARNING SKILL: knowing how to draw graphs of a function from that of its derivative and vice versa; knowing how to use the concept of derivative in the sphere of geometry and economics; know how to model simple phenomena with increasing or decreasing trend
over time. LINK/ APPLYING KNOWLEDGE AND UNDERSTADING: Cobb-Douglas production function
T.U. C5. KNOWLEDGE AND UNDERSTADING: maximum and minimum of a function; unlimited domain; singular point; concavity and convexity of a function; Fermat's theorem and theorems on second order n / s conditions. LEARNING SKILL: being able to calculate the extreme points of a function; given the function graph to recognize the extremes (local and global) points, the concavity and convexity domain; knowing how to apply the theorems of the first and second order conditions on the excellent; to qualitatively succeed in drawing the graph of a function using the acquired tools; knowing how to draw the graph of a function given the properties of it (max / min, concavity / convexity, domain and asymptotes). LINK / APPLYING KNOWLEDGE AND UNDERSTADING: minimize the average cost and consumption of raw materials, maximize revenue and profit; acceleration of sales.
T.U. C6. KNOWLEDGE AND UNDERSTADING: domain of multi-variable functions, surface analysis, constrained domains, Lagrange multiplier method; level curves. LEARNING SKILL: being able to calculate with the Lagrange multiplier method the optimum of a 2-variables function with an equality constraint; extrema problem of 2-variables function as function of one variable. LINK / APPLYING KNOWLEDGE AND UNDERSTADING: profit maximization with a Cobb-Douglas function (consumer utility).

Contents

T.U. C1) Linear functions and models
Functions numerical and algebraic viewpoints
Functions graphical viewpoints
Linear functions
T.U. C2) Nonlinear functions and models
Quadratic functions and model
Exponential functions and models
Logarithmic functions and models
Functions type y=(a+bx)/(c+dx)
T.U. C3) The derivative
Average Rate of Change
Derivatives: geometric approach
Derivatives: numerical and graphical viewpoints
A first application: marginal analysis
Limits and continuity: numerical and graphical
viewpoints
Limits and continuity: an algebraic approach

T.U. C4) Techniques of differentiation
Derivatives of Powers, Sums, and Constant
Multiples
The product and quotient rules
The chain rule
Derivatives of logarithmic and exponential
functions
Implicit differentiation
T.U. C5) Applications of the derivative
Maxima and Minima
Higher order derivatives and analyzing
graphical
Derivatives of composite functions
T.U. C6) Functions of several variables
Functions of several variables from numerical
and algebraic viewpoints
The domain of functions of two variables
Partial derivatives
Maxima e Minima
Maxima e Minima with constraints

Recommended Bibliography

Angelo Guerraggio Matematica per le Scienze Perason.
Other reference text: Stefan Waner, Steven R. Costenoble Strumenti quantitativi per la gestione aziendale Apogeo.
Teacher notes on the site in www.uniolbia.it.

Teaching Methods

Face to face lessons are supported by teacher's notes.
Teacher's Lecture Notes that can be downloaded from the University web site www.uniolbia.it
Use of computer software (open source) in the classroom, in particular with reference to the drawing of the graphs and to the problems of optimizing the
functions with several variables (Wolfram Alpha).
Weekly classes taken by the tutor will be devoted to the solution of exercises.
Traditional lectures together with practice exercises that are highly recomended.

Contacts/More Information

Office hours: during the semester of the course,
the teacher will meet students to help them to solve problems and to give them all the information they need after lessons concerns the solution of exercises; in the semester where no lessons will be held, the office hours will be announced.
Availability to provide a tutoring service also in English for Erasmus students or in mobility and availability of teaching materials and bibliographic references.
Further information about office hours, please consult
www.edisea.uniss.it and/or www.http://polo.uniolbia.it/
Contact MAIL russu@uniss.it Tel. 079213016/0789642184

GENERAL MATHEMATICS

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GENERAL MATHEMATICS