75660 - CALCULUS AND LINEAR ALGEBRA

Conoscenze e abilità da conseguire

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.

Contenuti

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, power, exponential and logarithmic, trigonometric); a preliminary discussion on 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; a more rigorous perspective on limits and continuity, 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.

Integration: the Riemann integral and its geometrical interpretation as area; primitives and indefinite integrals, fundamental theorems of integral calculus; rules and methods of integration: integration by parts, integration by substitution, improper integrals.

Integration in economics: continuous compounding and discounting, present values; 'stock' and 'flow' variables; probability in economics and finance; a glimpse at differential equations: separable and linear differential equations.

Infinite sequences and series; convergence criteria, geometric series; a glimpse at difference equations.

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.

Testi/Bibliografia

Essential Reference

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).

Further readings

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

Metodi didattici

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.

Modalità di verifica e valutazione dell'apprendimento

Written exam.

The duration of the written exam is one hour and thirty minutes. The exam of the first (summer) session can be taken in two steps: a midterm exam (after 1/3 of the course, during the mid-term session of January/February) and a final exam (also called second midterm or second partial exam). The mid-term exam is weighted 40% in the final grade, the second partial exam is weighted 60%. During the exam, students are allowed to use a pocket scientific calculator. Textbooks and other teaching materials are not allowed. Laptops, tablets and smartphones must be turned off.

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

Students can reject the grade obtained at the exam once. To this end, he/she must email a request to the instructor within the date set for registration. The instructor will confirm reception of the request within the same date.

Rejection is intended with respect to the whole exam (whose grade is a weighted average of the grades obtained in the midterm exam and the final exam). If the grade is rejected, the student must retake the full exam (consisting of both parts). The only grade that can be rejected without any communication from the student is the one of the first mid-term: in this case the student can either take the second mid-term or sit the full exam (thus automatically losing the grade obtained in the first mid-term).

Students sitting the first mid-term can take the second mid-term only on the first examination date set for the full exam (right at the end of the integrated course) or on the following call. A student can sit the second mid-term only once; if he/she fails or rejects the grade obtained, he/she will have to resit the full exam and will lose the grade obtained in the first mid-term.

Strumenti a supporto della didattica

Blackboard, slides.

Orario di ricevimento

Consulta il sito web di Roberto Dieci