1006 - FUNDAMENTALS OF MATHEMATICS FOR BUSINESS

Subject type: Basic

Coordinator: Núria Masferrer Llabinés

Trimester: First and second quarters

Credits: 8

• Catalan
• Spanish
• English

Check the schedules of the different groups to know the language of teaching classes. Although the material can be in any of the three languages.

The subject “Fundamentals of Mathematics” is conceived as an introductory subject of basic training for the student, as shown by its location in the first year. The course works on the use of mathematical language and the acquisition of working methods that are especially suitable and useful to formalize economic situations.

In particular, the subject develops the fundamental aspects of mathematical calculus in one or more variables (with optimization) and linear algebra that are most used in economics; in this sense, it is therefore an instrumental subject in which mathematical tools are provided that are used, mainly, in economic contexts.

In addition, it should be noted, due to the formative nature of this subject, that logical-deductive reasoning is promoted.

• Master mathematical language as well as algebraic notation and manipulation in the context of univariate calculus.
• Show knowledge of the basic concepts about the real line, real functions, univariate calculus and the properties of basic families of real functions, linear algebra and optimization in various variables.
• Be able to identify and interpret simple mathematical models applied to economics.

In the face-to-face sessions with the whole group, theory sessions will be combined with exercise resolution sessions. The theoretical presentation will include examples that will help the student to solve exercises autonomously.

In the non-contact sessions the students will have to work theoretical-practical knowledge from audiovisual material, documents Online and the material of the face-to-face sessions. The results of this work will be evaluated from questionnaires using the platform moodle or / and with the delivery of projects carried out individually.

FIRST TRIMESTER

0. Preliminaries.

The sets of numbers

Solving equations and inequalities

Solving systems of linear and nonlinear equations

1. Real functions of a real variable.

1.1 Definition, types and properties

Expressions of a function: explicit form and implicit form

Graph of a function

Domain and Path of a function

Operations with functions: sum, product for a scalar, product and quotient

Composition. Properties. Identity function and inverse function

Study of some elementary functions (polynomial, rational, with radical, exponential, logarithmic)

1.2 Limits and continuity

1.2.1 Limits

Definition

Lateral Boundaries

Infinite limits: Vertical asymptotes

Limits to infinity: Horizontal asymptotes

Graphic representation of the boundaries

Calculation of limits. Indeterminacies

1.2.2 Continuity

Definition and equivalent definitions

Types of discontinuity: avoidable, jumping and asymptotic

Continuity Problems

Calculation of the asymptotes of a function: horizontal, vertical and oblique

2. Differential calculus with functions of a variable.

2.1 Derivative of a function at a point: definition

Geometric interpretation of the derivative

Angular Points

Derivative and continuity theorem

Derived function

Function derived from elementary functions (Table of derivatives)

Derivative of operations: sum, product to scale, product, quotient

Derivative of the composition: Rule of the chain

Logarithmic derivation

Successive derivatives

2.2 Applications of the derivative

Calculation of the tangent line at a point

Hospital Rule

Intervals of growth and degrowth of a function

Calculation of extremes (maximums and minimums)

Definition of maximum and minimum

Null derivative theorem

Criteria for determining extremes

Concavity, convexity and inflection points.

The derivative 2ª theorem

Analysis of a function. Complete graphic study.

SECOND TERM

3. Integration.

3.1 Indefinite Integral

Definition. Primitives of a function

Table of immediate integrals

Application of the chain rule in the integration of functions

Properties of the integral

Integration by parts

Integration of rational functions

3.2 Integral defined

Definition. Barrow's rule. Properties

Area calculation

Area between a curve and the axis of bscisses

Area between two or more curves

4. Linear Algebra.

4.1 Matrix

Matrix definition. Order of an array. Square matrices

Transposed from an array. Symmetrical matrices

Operations with matrices

Sum and product for a scalar

Matrix product. Properties

Identity Matrix. Inverse Matrix

4.2 Determinants

Definition. Determinants of order 2 and order 3. Sarrus rule

Complementary deputies and minors

Properties of determinants

Development of determinants applying their properties

Applications of determinants:

Reverse matrix calculation

Solving matrix equations

Range of an array

5. Real functions of two or more variables

5.1 Real functions of two or more real variables

Definition

Graphic representation

Level curves

Mastery of functions of two variables

5.2 Differential calculation of functions of two or more variables

Partial derivatives of a function

Successive partial derivatives. Schwartz's theorem

Compound derivation

5.3 Function ends of two variables

Definition. Highs, lows and saddle points

Determination of extremes. Necessary condition

Singular points

Hessian matrix

Determination of extremes. Sufficient condition

6. Applications of functions to economics

6.1 Optimization with a variable

Highs and lows with applications to the economy

Two variables and an equality constraint.

6.2 Optimization with two variables

Maximum and minimum with applications to the economy

6.3 Optimization with constraints: Linear programming

Concept and formulation

Graphic technique

Matrix formulation

General problem

In general the structure of the week is as follows:

This subject has methodological and digital resources to make possible its continuity in non-contact mode in the case of being necessary for reasons related to the Covid-19. In this way, the achievement of the same knowledge and skills that are specified in this teaching plan will be ensured.

The Tecnocampus will make available to teachers and students the digital tools needed to carry out the course, as well as guides and recommendations that facilitate adaptation to the non-contact mode.

Eliminatory evaluations of the subject will be carried out throughout the two quarters. The final grade will be the weighted arithmetic mean of the grades of the evaluation activities carried out in the first and second quarters. To pass the course, the final grade must be greater than or equal to 5 points out of 10.

The continuous evaluation will take into account the following aspects with the weights indicated (the type of examination will be determined by the indications derived from the PROCICAT plan):

- Three partial exams (P): 60% (if the examinations are carried out in person) or 45% (if the examinations are carried out online).

- Delivery of exercises, evaluation activities and participation (A): 40% (if the examinations are carried out in person) or 55% (if the examinations are carried out online).

Therefore the final note is obtained by applying the formula:

Attendance_note = 0,2 ·P2 + 0,2 ·P3 + 0,2 ·P4 + 0,4 ·A                   or            Note_not_present = 0,15 ·P2 + 0,15 ·P3 + 0,15 ·P4 + 0,55 ·A

On P2 (requires a grade greater than or equal to 4 and eliminates subject) is the mark of the midterm exam that is conducted throughout the first term, and P3 (requires a grade greater than or equal to 4 and removes subject) i P4 (a grade higher than or equal to 4 is required and eliminates subject) are the marks of the partial exams that will be held throughout the second term respectively, and A collects the note of participation of the first and second trimester.

At the end of the exam period of the second term, the student will be able to be examined on the syllabus of the partials that he / she has yet to pass (P2, P3 oh P4). The final grade is calculated with the same formula that is applied in the continuous assessment (a grade greater than or equal to 4 in each is required).

In the recovery period of the second term the student will be able to be examined of the syllabus of the partial ones that remain to him to surpass (a qualification superior or equal to 4 in each is necessary). Students who have not taken the global exams (end of the second term) will not be eligible for the resit exam. The final grade is calculated using the same formula that applies in the continuous assessment.

The note of participation and delivery of exercises (A) it is not recoverable in any case and no grade will be saved from one academic year to another.

Summary of evaluation percentages based on:

A student who did not appear in the first call NO (end of 2nd term) can apply for recovery.

HAEUSSLER, JR., ERNEST, F., RICHARS D. PAUL, RICHARD J. WOOD (2008): Mathematics for administration and economics. Ed Pearson.

GARCÍA, P., NÚÑEZ, J., SEBASTIÁN, A. (2007): Initiation to the university mathematics. Ed. Thomson.

LARSON, HOSTETLER, EDWARDS (2006): Calculus. Eighth edition. Mc Graw-Hill.

LÓPEZ, M. VEGAS, A. (1994): Basic course of mathematics for the economy and the direction of companies. Vol I and II. Ed Pyramid.

STTAN (1998): Mathematics for administration and economics. International Thomson Publishers.

BITTINGER, MARVIN, L. (2002): Calculus for economic-administrative sciences. Seventh education. Ed Pearson.