Academic Year 2019/2020

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Business and Economics (cod. 8965)

Learning outcomes

At the end of the course the student will be capable of using the techniques of Linear Algebra; furthermore he will have acquired a working knowledge of First Year Calculus, together with the related applications in Finance and Economics.

Course contents

Course contents

A preliminary tutorial covers a number of introductory topics (so-called precalculus), including elementary set theory, sets of real numbers, complex 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, trigonometric functions.

Course content - Calculus and Linear Algebra

Introduction to the course and crash review of preliminary mathematical notions

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, elasticities; continuity and differentiability, intermediate-value theorem, 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; convergence criteria; geometric series; Taylor's series. Sequences and series in financial mathematics.

Difference equations. Linear, first order, autonomous difference equations. Steady state and convergence analysis. Linear, first order, non autonomous, difference equations. Difference equations in financial mathematics.

Integration: the Riemann integral and its geometrical interpretation; primitives and indefinite integrals, fundamental theorems of integral calculus. Rules and methods of integration: immediate integrals, integration of rational functions, integration by parts, integration by substitution. Improper integrals.

Integration in economics: continuous compounding and discounting.

Differential equations. First order differential equations. Linear, first order, autonomous differential equations. Steady state and convergence analysis. Linear, first order, non-autonomous differential equations. Differential equations with separable variables. Differential equations in financial mathematics.

Linear algebra: vector spaces, bases and dimension; matrices and their properties, matrix operations, rank and determinant; linear maps and associated matrices, systems of equations, existence of solutions, cases of one solution and infinitely many solutions, Gaussian elimination, inverse of a matrix and Cramer's rule; eigenvalues and eigenvectors.

Multi-variable calculus: partial derivatives with two variables, geometric interpretation; partial elasticities; chain rules, implicit differentiation along a level curve; functions of more variables, gradient, differentials and linear approximations; economic applications.

Multi-variable optimization; maxima, minima and saddle points; tests based on second derivatives; constrained optimization and Lagrange multipliers.

Readings/Bibliography

R.A. ADAMS, C. ESSEX. Calculus, a complete course, 9th Edition, Pearson, 2018.

K. SYDSÆTER, P. HAMMOND (with A. STRØM). Essential Mathematics for Economic Analysis, 4th Edition. Pearson, 2012 (a student solutions manual is freely available from the publisher's website).

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

Compulsory written exam and additional optional oral exam both on-line.

The exam of the first (summer) session can be taken in two steps: a first midterm exam (after 1/3 of the course, during the mid-term session of January/February) with a duration 1 hour and 20 minutes, and a second partial exam (on the second and the part of the course, during the session of June/July) with a duration of 2 hours. In this case, the first mid-term exam is weighted 40% in the final grade, the second partial exam is weighted 60%. In occasion of the second partial exam, students who have not taken the first mid-term exam can only take the total exam (duration 3 hours). The student who took the first partial can take the second partial either on the June call or on the July call (but only one time).

During the exam, students are allowed to use a pocket scientific calculator. Textbooks and other teaching materials are not allowed.The student must have a web-cam to allow the identification of the students and to perform the regular surveillance.

The written exam aims at testing the student's ability to correctly and effectively apply the basic and advanced techniques learned in the course to specific problems in calculus and linear algebra. The written exam consists of a number of short exercises (routine exercises about basic concepts and calculations) and one or two more challenging 'review problems' (for instance, the complete study of a one-variable function, a problem of constrained optimization in two variables, the general and particular solutions of a differential equation, problems on matrices and linear systems depending on a parameter, etc.). Such review problems generally include questions of different levels of difficulty as well as connections to the several economic applications illustrated in the textbook.

Grade rejection

The only grades that can be rejected without any communication from the student is the one of the first mid-term exam.

Teaching tools

Blackboard

Office hours

See the website of Gian Luca Tassinari

SDGs

Quality education Decent work and economic growth

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.