Integers, prime factorization, fractions, decimals, percent, radicals.
Algebraic expressions, polynomial factorization, algebraic equations and inequalities, irrational equations
and inequalities.
Logarithms and their properties, exponential and logarithmic equations and inequalities,
Cartesian Plane, Lines, Circles, and Parabolas

Teaching method: lectures (attendance is strongly recommended).
Learning activities: attendance at lectures and exercises, individual study

Program:
Parte I – Functions in one variable
Definition of a function. Concept of domain of existence of a function and exercises. Operations between
functions. Elementary functions. Inverse function. Euclidian Metric of R.
Limit of a function, Theorem on the uniqueness of the limit, Sign permanence theorem, asymptotes,
calculus of limits, Indeterminate forms
Continuous functions: introduction and first definitions. Theorems on continuous functions. Classification
of discontinuities, Weierstrass Theorem and its conseguences.
Derivative of a function. Tangent Line to the graph of function. Derivatives of elementary functions.
Connection between differentiability and continuity of a function. Derivation rules. De l'Hospital Theorem
and application to computation of limits. Introduction to local/global maxima and minima. Definitions on
maxima and minima points. Critical points and extremum points. The relation between critical points and
extremum points: the Fermat Theorem. The Rolle Theorem, The Lagrange theorem and its application to
study the monotonicity properties of functions. Two sufficient conditions for a critical point to be an
extremum point. Definition of convexity of twice differentiable functions on intervals. Two sufficient
condition for convexity expressed in terms of the first and the second derivative respectively. Study of
functions (polynomial functions, rationals functions, irrationals functions, absolute value functions,
exponential functions, logarithmic functions, piece wise defined functions)
Parte II – Functions in two variables
Topology in R 2 : circular neighborhoods. Domain. Continuity. Level curves. Definition of partial
derivatives. Gradient. Directional derivative. Differentiability. Second order partial derivatives. Hessian
matrix. Sufficient optimality conditions of the second order for critical points.
Introduction to constrained optimization problems (equality constraint): use of the substitution method for
solving constrained optimization problems on a compact region. Method of Lagrange multipliers 

Mathematics for economics, third edition. M.Hoy, J.Livernois, C.McKenna, R.Rees, T.Stengos. The MIT Press.

The exam consists of a written and an oral test.

The exams of Principles of Mathematics will be carried out as follows:
1. A written pre-selection test, lasting 36 minutes, in which you will be asked to perform 12 basic
exercises. You pass the test if you answer at least 6 questions correctly (
correct questions≥6 ). This part gives score in the final written rating.
2. After passing the pre-selection (you will immediately have the score of your test), a written test,
lasting 2 hours, in which you will be asked to perform exercises on the topics of the course. The
written test will take place immediately after the pre-selection test.
3. After passing the written test (the results of which will be published as usual on the elearning
website), an oral test where you will be asked both questions of Theory (definitions, theorems
without demonstrations) and exercises.

The written test is sufficient if the student reaches a total score of 16, which is necessary to be admitted to the oral examination. The oral exam must be held in the same round as the written test. 

The dates of the exams are those specified in the exam site (https://esami.unipi.it/).

In the case you don't pass the oral part (so if you don't pass the exam), then in the new session you have to do the complete exam (preselection+written+oral).