00679 - Mathematics

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

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

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, 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; 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, present values.

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. Differencial 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.

Chapters: preliminaries, 1, 2, 3, 4, 5, 6, 7.9, 9, 10, 12, 13

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

Chapters: 1, 2,3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.

Lecture notes provided by instructures

Teaching methods

Class lectures

Assessment methods

Final written exam. During the course there are partial written tests. The final test can be replaced by partial tests if the latter are successfully passed. Any midterm test previously accomplished is invalidated by the submission of a final exam. During the exam, textbooks and other teaching materials are not allowed. Laptops, tablets and smartphones must be turned off.

 

Evaluation judgment scale

· <18: nisufficient preparation

· 18-23: sufficient preparation but relating to a limited number of the course contents;

· 24-27: adequate preparation but with some gaps with respect to the course contents;

· 28-30: very in-depth knowledge of all the course contents;

· 30L: excellent knowledge of the course contents.

Teaching tools

Blackboard

Office hours

See the website of Lorenzo Cerboni-Baiardi

See the website of Enrico Smargiassi