37292 - Mathematics

Learning outcomes

This course aims to introduce students to differential calculus: limits, derivatives, extremes of a function and convexity, the qualitative study of a function, integral calculus: definitions and properties, integration techniques and differential equations and elements of linear algebra: linear systems, linear applications and matrices.

Course contents

A preliminary tutorial (10 hours) covers a number of introductory topics, including elementary set theory, sets of real numbers, polynomials, linear and quadratic equations and inequalities, systems of inequalities, absolute value and rational inequalities, Cartesian coordinate system, basic analytic geometry, basic concepts and definitions about functions, elementary functions (power, exponential and logarithmic), exponential and logarithmic equations and inequalities.

Course content:

Linear Algebra (10 hours)

Linear algebra: matrices and their properties, binary operations (algebraic addition and multiplication) and unary operations on matrices (transposed matrix, rank, determinants and their properties, inverse matrix). Resolution of linear systems, analysis of the existence of solutions of a system of linear equations. Gauss elimination method and Cramer’s Rule.

Differential and Integral Calculus of Real Functions (40 hours)

One-variable functions: basic definitions, graphs and elementary functions (linear, quadratic, polynomial, rational, irrational, power, exponential, logarithmic, absolute value). Odd and even functions. Composite functions. Inverse functions.

Limits and continuity.

Differentiation of one-variable functions: tangents and derivatives, rules of differentiation, chain rule, higher-order derivatives.

Derivatives in use: implicit differentiation and economic examples, differentiation of the inverse function. Linear and quadratic approximations, Taylor's formula, continuity and differentiability, intermediate-value theorem, De L’Hôpital’s Rule.

Single-variable optimization: local and global extrema, stationary points and first-order condition, simple tests for extreme points, extreme points for concave and convex functions, second-order derivative and convexity, inflection points, study of the graph of a function, asymptotes.

Sequences and series; arithmetic and geometric series.

Integration: the Riemann integral and its geometrical interpretation; primitives and indefinite integrals, Fundamental Theorem of Integral Calculus. Rules and methods of integration: immediate integrals, integration by parts, integration by substitution.

Integration applied to probabilistic distributions: Normal Distribution and Cumulative Distribution Function.

Readings/Bibliography

K. SYDSÆTER, P. HAMMOND, A. STRØM, A. CARVAJAL. Essential Mathematics for Economic Analysis, 5th Edition. Pearson, 2016.

Teaching methods

Class lectures. During the class lectures (as well as in the additional exercise classes) each topic will be illustrated by examples and worked-out exercises.

Assessment methods

Written exam (in presence 3 h): students have to solve different exercises on the course topics. To each exercise a given maximum number of point is associated, and to get it the student has to solve correctly the exercise and all the steps must be justified. The theoretical maximum number of points in case of a perfect exam is 32. The test assessment grid will be as follows:

Mark range:

• 18-19: knowledge of a very limited number of topics covered in the course and analytical skills that emerge only with the help of the teacher, expressed in an overall correct language;
• 20-24: knowledge of a limited number of topics covered in the course and ability to autonomous analysis only on purely executive matters, expression in correct language;
• 25-29: good knowledge of a large number of topics covered in the course, ability to make independent choices of critical analysis, mastery of specific terminology;
• 30-30 cum laude: excellent knowledge of the topics covered in the course, ability to make autonomous choices of critical analysis and connection, full mastery of specific terminology and ability to argue and self-reflection.

The exam of the first session consists of 2 steps: a first midterm exam,during the mid-term session of November, which lasts 2 hours, a second partial exam with duration of 2 hours. During the other sessions, the full test lasts 3 hours.

NOTE: ONLY FIRST YEAR STUDENTS ARE ALLOWED TO TAKE MIDTERM EXAMS.

Teaching tools

Slides

Online presentations

Software Geogebra

Blackboard

Office hours

See the website of Donatella Giuliani