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GENERAL MATHEMATICS - A.Y. 2012/2013

(code 30062, CLEAM - CLES - CLEF - BIEMF)

DETAILED SYLLABUS

Structures ( 1.1-2, 1.4-5, 1.7, 2.1-3, 4.1, 5.1-4, 5.6, 14.1)

Sets, operations with sets, number sets. The set R of real numbers, its geometric representation.

Operations, order. Intervals. Absolute value, distance, neighbourhoods. The set nR of real vectors

with ncomponents, its geometric representation for 2,3n . Linear operations, linear combinations,

inner product of vectors, order. Norm, distance, neighbourhoods. Interior, exterior, boundary,

isolated, accumulation points. Open, closed, bounded, compact sets. Convex linear combinations,

convex sets.

Functions ( 6.1, 6.3-6, 14.2, 16.1, 16.4-5)

The concept of function. Domain, codomain, range; image, inverse image. Injective, surjective,bijective function. Composite function, inverse function. Operations with functions.

Real functions of one real variable. Natural domain, graph. Examples of elementary functions.

Bounded functions, monotonic functions. Global and local maxima/minima. Convex/concave

functions.

Real functions of nreal variables. Real functions of two real variables: natural domain, graph, level

curves. Bounded functions. Global and local maxima/minima. Convex/concave functions.

Sequences ( 8.1, 8.4-9, 8.11-13)

Sequences of real numbers. Recursively defined sequences. Limits of sequences. Limits from

above, limits from below. Convergent, divergent, irregular sequences. Theorem of uniqueness of

the limit (*). Limits of elementary sequences. Operations with limits, indeterminate forms. Limitsand inequalities. Limits and monotonic sequences. The number e. Calculation of limits.

Comparisons among infinities and among infinitesimals. The symbols ~ and o. Extension to

sequences of real vectors.

Number series ( 10.1-4, 10.6)

The concept of series. The sequence of partial sums. Behaviour of a series; convergent, divergent,

irregular series. Behaviour of the geometric series (*).Necessary condition for convergence. Series

with non-negative terms: regularity theorem, comparison test, asymptotic comparison test,

behaviour of the generalised harmonic series. Series with terms of indefinite sign: simple

convergence and absolute convergence.

Limits of functions and continuity ( 11.1-7, 12.1-5, 16.2, 22.1)

Limits of functions of one real variable (from the left, from the right, bilateral; from above, from

below). Vertical and horizontal asymptotes. Limits of elementary functions. Theorems on limits.

Calculation of limits. Change of variable. Some notable limits. The symbols ~ and o. Continuity for

functions of one real variable. Points of discontinuity. Continuity of elementary functions.

Properties of continuous functions: Weierstrasss theorem, intermediate-value theorem (Darbouxs

theorem), zero-value theorem (Bolzanos theorem).

Limits of functions of n real variables. Continuity for functions of n real variables. Weierstrasss

theorem.

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One-variable differential calculus ( 17.1-12, 19.1-2, 19.4-5, 20.1-3, 21.1, 22.1)

Difference quotient, derivative at a point; geometric meaning. Equation of the tangent line. Left-

hand and right-hand derivative; corners. The derivative function. Relationship between the

existence of the derivative and continuity (*). Differentiability. Relationship between the existence

of the derivative and differentiability. Differential at a point; geometric meaning. Derivatives ofelementary functions. Rules on derivatives. Higher derivatives.

Stationary points. Necessary condition for points of local maximum/minimum: Fermats theorem

(*). Rolles theorem (*). Lagranges mean value theorem (*). Monotonicity test. First sufficient

condition for points of maximum/minimum: local version, global version. Taylors formula and

Maclaurins formula of order n, with Peanos remainder. Convexity test. Second sufficient

condition for points of maximum/minimum: local version, global version. Graph sketching.

Linear algebra (( 3.1-3.6, 13.1-9)

Linear spaces, subspaces. Linear independence, linear dependence. Spanning sets, bases.

Dimension. Matrices. Linear operations with matrices. Row-column product; main properties.

Linear functions (linear functionals, linear operators). Representation theorem (*), representationmatrix, image space, kernel.

Determinant; main properties. Laplaces theorem. Inverse matrix: existence, uniqueness, explicit

writing. Rank: definition as the number of linearly independent rows or columns, its calculation as

the maximum order of non-null minors, Kroneckers algorithm.

Linear systems; matrix writing. Existence of solutions: Rouch-Capellis theorem. Structure of

solutions: homogenous and non-homogenous systems. Determination of solutions: Cramers

theorem (*), Cramers rule, extension to all linear systems.

N-variable differential calculus ( 18.1-3, 19.1, 28.1-4)

Partial derivatives, gradient vector. Differential. Relationship between the existence of partial

derivatives, differentiability and continuity. Stationary points. Unconstrained optimizationproblems. Necessary condition for points of unconstrained local maximum/minimum; Fermats

theorem (*). Constrained optimization problems with one equality constraint. Lagrangean function,

Lagrange multiplier. Necessary condition for points of constrained local maximum/minimum.

Note

A proof is required for all topics marked with (*).

Topics are listed according to the presentation order of the adopted textbook:

Erio Castagnoli, Massimo Marinacci, Elena Vigna, Introduction to Mathematics and Economics,

EGEA draft version (2012).

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Classe 21 N33,N39

Ma 18.00-19.30: 09/10

Ma 16.15-17.45: 25/09 02/10 16/10Ma 16.15-17.45: 20/11 27/11 04/12 11/12

tra Lu 22/10 e Ma 30/10: 2 sessioni da definire

tra Lu 17/12 e Gio 17/01: 2 sessioni da definire

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