37292 - 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 (30 hours) 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.

PLEASE NOTE: It is of FUNDAMENTAL IMPORTANCE that all students have a perfect knowledge of the topics covered in the introductory mathematics course as:

a) they will not be re-explained in the general mathematics course;

b) without an excellent command of these topics, the possibility of understanding the contents of General Mathematics is nil.

Course content - Calculus and Linear Algebra (90 hours)

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 on Difference Equations and Eigenvalues and Eigenvectors will be provided by the Professor.

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: 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 atteinable in case of a perfect exam is 32.

The test assessment grid will be as follows:

· <18 insufficient

· 18-23 sufficient

· 24-27 average/good

· 28-30 very good

· 30 cum laude excellent/outstanding

If your total is <=30 score, your score corresponds to your mark. If your score is >30, then you get 30 cum laude.

The exam of the first (summer) session can be taken in 3 steps: a first midterm exam (after 1/3 of the course, during the mid-term session of January/February) with a duration 1 hour, a second partial exam (after 2/3 of the course, during the session of April) with a duration of 1 hour on the second part the course, and a third midterm exam of duration 1 hour on the third part of the course during the first call of session of June/July. In occasion of the third partial exam, students who have not taken the partials can only take the total exam (duration 3 hours).

NOTE: ALL CLABE STUDENTS ARE ALLOWED TO TAKE MIDTERM EXAMS AND NOT ONLY FIRST YEAR STUDENTS.

During the exam, students are not allowed to use calculators. Textbooks and other teaching materials are not allowed.

NOTE: THOSE WHO COMMIT FRAUD DURING THE EXAM WILL NOT BE ABLE TO TAKE THE EXAM FOR A FULL A.A. UNTIL THE CORRESPONDING CALL OF THE FOLLOWING YEAR AND THE TEST AT THAT POINT WILL BECOME ORAL AND WILL INCLUDE THE SOLUTION OF EXERCISES, THE ENUNCIATION AND DEMONSTRATION OF ALL THEOREMS RELATING TO THE TOPICS SHOWN IN THE COURSE'S PROGRAM AND CONTAINED IN THE BOOKS.

Grade rejection

The only grades that can be rejected without any communication from the student are those of the first and second mid-term exams.

Teaching tools

Slides

Blackboard

Office hours

See the website of Gian Luca Tassinari