# PRELIMINARY COURSE ON MATHEMATICS FOR ECONOMIC APPLICATIONS

# PRELIMINARY COURSE ON MATHEMATICS FOR ECONOMIC APPLICATIONS

Numbers, absolute and relative. Elementary arithmetical operations (addition, subtraction, multiplication, division) and their properties. Prime factor decomposition. Fractionary calculus.

Powers with integer exponent and their properties. Infinite decimal representations, periodic (with reduction to fractionary form) and nonperiodic. Irrational numbers most commonly used.

Powers with fractionary exponent. Calculus of radicals.

Equations and inequalities of first degree in a single variable. Special algebraic products.

Euclid's postulate for geometry. Parallel straight lines with a common transverse line; corresponding, opposed, alternate internal (external) angles. Thales' theor.

Disjoint, adjacent, overlapping, coincident segments ed angles. Sharp, obtuse, square, flat angles. Sum and difference of angles. Complementary, supplementary, esplementary angles.

Triangles and their properties. Congruence and similarity. Bisectrices, medians, heights. Equilateral and isosceles triangles. Pytaghoras and Euclid theor.

Angles with vertex on a circle and on its center and their relation.

Polygones. Parallelograms and their properties.

Area of triangles, parallelograms, trapezoids, regular polygones, disks.

Volumes of cubes, parallelepipedes, prisms and pyramides with basis of known area, balls.

Cartesian coordinate systems on the line and on the plane. Absolute and relative measure of oriented segments. Coordinate change by origin displacement.

Analytic representation of elementary geometric transformations (symmetries with respect to the axes, to the origin, to the bisectrix of quadrants I and III; clockwise and counterclockwise rotations by 90°).

Cartesian equation of straight lines in the plane: general and explicit form. Angular coefficient.

Mutual position of two straight lines: parallelism (including the special case of coincidence), incidence (including the special case of orthogonality).

Linear systems of two equations in two unknowns and their geometric interpretation.

The standard parabola through the origin with vertical axis (i.e., the graph of the square function). Parabolas obtained by the standard one through vertical expansion/contraction, or vertical symmetry. Parabolas with vertical axis in general form (defined by an arbitrary trinomial of second degree in a single variable), and formulas for the coordinates of their vertex.

Second degree equations in one variable, and points of a parabola on the horizontal axis.

Circles, their cartesian equation given center and radius and determination of center and radius from the cartesian equation.

Mutual position of straight lines and parabolas with vertical axis, and of straight lines and circles. Equation of tangent lines.

Complete solution of second degree inequalities in a one variable.

Goniometric circle, measure of angles in radiants, and definition of the elementary trigonometric functions sine, cosine, tan. Values achieved at particular angles.

Fundamental goniometric identities, associated angles. Addition and subtraction, duplication and bisection, prostapheresis, Waring formulas.

Definition and application of the inverse trigonometric functions arcsin, arccos, arctan. Direct evaluation of the inverse trigonometric functions in particular cases.

Simple trigonometric inequalities.

The set /R/ of the real numbers as a complete ordered field.

Topology of /R/. Neighbourhoods. Internal, external, boundary, isolated, accumulation points of subsets of /R/. Open and closes sets. Bolzano-Weierstrass theor.

Limits and operations on them. The number e. The extended real line.

Continuity. Types of discontinuity. Weierstrass and Darboux theor.

Computation of derivatives. Rolle, Lagrange theor.

Integral calculus (substitution, parts, decomposition, fundamental theor.)

I will try to allow the students some fine tuning of their standard

mathematical background, in order to enhance in them a suitable and

convenient beginning with their ordinary coursework, particularly (but

non only) in relation to the course: mathematics for economic

applications.

1. Review of elementary propositional calculus, logic of predicates, sets and relations.

2. Review of differential and integral calculus for real functions of a single real variable.

3. Review of analytic geometry in three dimensions.

4. Review of partial differentiation and implicit function theorems.

Your own good old calculus textbook

T.Apostol, Calculus, John Wiley, New York 1967, vol. I, chap. 12 and 13.

S.Afriat, Logic of choice and economic theory, Clarendon Press, Oxford 1987, app. VI.1, pp. 487-497

Notes (on topic 1) by the teacher.

Lectures in class. Daily homework, to be handed in, corrected, discussed,

and graded.

Daily homework grades. Final paper

Office hours in the first semester 2014-15 - from October 8th to December 19th, 2014 (included):

Monday 4 p.m. - 6 p.m. s. Niccolò room 209

Friday 8 a.m. - 10 a.m. s.Francesco room 11