Mathematics crash course 2014
Instructor:
D.ssa Kinzica Soldano kinzica.soldano@unimi.it
D.ssa Maria Teresa Trentinaglia maria.trentinaglia@unimi.it
Suggested readings:
Knut Sydsaeter, Peter Hammond. Essential mathematics for economic analysis. Pearson, 2008
(3 edition).
Carl Simon, Lawrence Blume. Mathematics for economists. Norton, 1994.
Timetable:
Week 1 (from August 25 to August 29):
Every day from 08.30 to 12.00
Room 20, Via Conservatorio 7
Week 2 (from September 1 to September 5):
Every day from 08.30 to 10.00
Room 20, Via Conservatorio 7
Week 3 (from September 8 to September 12):
Every day from 08.30 to 10.00
On September 8 and 9: room 22, Via Conservatorio 7 (room to be confirmed)
On September 10, 11 and 12: room 20, Via Conservatorio 7
Short course description
The course provides the mathematical tools needed to attend successfully the study of
economic and financial disciplines.
The topics of the course are: sets theory, functions of a real variable, numerical series, limits
and continuity, differential calculus, integral calculus, linear algebra, multivariate functions,
unconstrained optimization.
The exam consists of a written paper, with theoretical questions as well as exercises.
1. Set theory. Introduction to set theory. Cardinality. Numbers. Relations. Necessary and
sufficient conditions: an explanation.
2. Functions of a real variable. Real functions. Odd and even functions. Bounded
functions. Composition of functions. Inverse function. Monotone functions. Maxima
and minima. Convex and concave functions. Polynomial functions. Exponential
functions. Logarithmic functions. Numeric series.
3. Limits and continuity. Definition of limit. Right and left limit. Existence of limits for
monotone functions. Comparison theorem. Operations with limits. Continuous and
discontinuous points. Theorems on continuous functions. Special limits.
4. Differential calculus. Definition of derivative and differential. Left and right
derivative. Geometrical intuition of derivative. Derivative of elementary functions.
1
Continuity and differentiability. Operations with derivatives. Second derivative.
Theorems on differentiable functions. Elasticity of a function.
5. Integral calculus. Concept of indefinite integral. Definite integrals. Fundamental
theorem of integral calculus. Riemann integral. Operations with integrals.
6. Linear Algebra. Vectors and matrices. Linear dependence and independence. Bases.
The vector space R^n. Operations on vectors and matrices. Determinants and rank of
a matrix. Inverse matrix. Linear systems. Theorems on linear systems.
7. Multivariate functions. Continuous functions. Level curves. Partial derivatives. Chain
rule. The gradient vector. Second order derivatives and Hessians. Young’s theorem.
8. Unconstrained optimization. First and second order conditions. Global maxima and
minima. Economic applications.
2