Download Algèbre et Analyse. Classes Terminales C, D et T by Lebossé C., Hémery C. PDF

By Lebossé C., Hémery C.

Cours conforme aux programmes du eight juin 1966.

Table des matières :

Livre I : Ensembles fondamentaux

Leçon 1 — Ensembles — purposes et fonctions — adjustments ponctuelles
Leçon 2 — Lois de composition — buildings algébriques
Leçon three — L’ensemble N des entiers naturels — examine combinatoire
Leçon four — L’anneau Z des entiers relatifs
    Addition et soustraction
    Multiplication et division
Leçon five — Le corps Q des nombres rationnels
    Opérations sur les rationnels
Leçon 6 — Le corps R des nombres réels
    Le corps R des nombres réels
    Interprétation géométrique des réels
Leçon 7 — Le corps C des nombres complexes
    Interprétation géométrique
    Puissances et racines
Leçon eight — purposes trigonométriques de C — Équations du moment degré dans C — purposes géométriques

Livre II : Arithmétique

Leçon nine — Congruences dans Z — department euclidienne dans N
Leçon 10 — Plus grand commun diviseur — Plus petit commun multiple
Leçon eleven — Nombres premiers — functions aux fractions
Leçon 12 — Numération — Nombres décimaux

Livre III : Étude des fonctions

Leçon thirteen — Fonctions d’une variable réelle — Limites — Formes indéterminées — Fonctions continues
Leçon 14 — Dérivées — Calcul des dérivées — Dérivées successives
Leçon 15 — version des fonctions — Courbes d’équation y = f(x) — Fonctions : y = ax² + bx + c, fonction homographique, y = ax³ + bx² + cx + d; y = ax⁴ + bx² + c
Leçon sixteen — Fonctions : y = (ax² + bx + c)/(a’x + b’), y = (ax² + bx + c)/(a’x² + b’x + c’) — Fonctions irrationnelles — Fonctions diverses
Leçon 17 — Fonctions primitives — Interprétation et functions des primitives
Leçon 18 — Calcul de volumes
Leçon 19 — Suites de nombres réels — Progressions arithmétiques — Progressions géométriques
Leçon 20 — Fonction logarithme népérien — Logarithmes base a — Logarithmes décimaux
Leçon 21 — Fonctions exponentielles — Fonctions e^x et e^(−x) — Fonction a^x — Applications

Livre IV : Applications

Leçon 22 — Équations différentielles — Fonctions vectorielles
Leçon 23 — Calcul numérique — Tables numériques
Leçon 24 — Règle à calcul

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Extra info for Algèbre et Analyse. Classes Terminales C, D et T

Example text

127. 1. Basic tools 25 Proof. If K 0 is strongly pointed, then co (K 0 ∩ S X ) does not contain zero. To see this, suppose we have a sequence x ∈ co (K 0 ∩ S X ) where x → 0 as → ∞. Now x = n n i=1 θ ,i z ,i , where θ ,i ≥ 0, i=1 θ ,i = 1, and z ,i ∈ K 0 ∩ S X for all and i . 3. Now x ≤ 1 for all , so by Alaoglu’s theorem there is a weak* convergent subsequence, which is weakly convergent if X is reflexive. Let x be the weak limit of this subsequence. Then ν, x = lim ν, x (in the subsequence) →∞ n = lim →∞ ≥ lim →∞ θ ,i ν, z θ ,i z ,i i=1 n ,i = 1.

Taking weak limits, ζ , yk / yk → ζ , y ≥ 1, so y ≥ 1/ ζ X > 0. For each k we can choose y k ∈ (x k ) with y k ≤ R. By Alaoglu’s theorem and reflexivity of X, there is a weakly convergent subsequence to which we restrict our attention so that y k y in the subsequence. By convexity of (x k ) for all k, for any 0 ≤ βk ≤ 1 we have y k + βk yk − y k ∈ (x k ). In particular, for a given τ ≥ 0 we can set βk = min (1, τ/ yk ). Then as yk → ∞, for sufficiently large k, yk + τ yk yk − y k ∈ (x k ). Taking weak limits on the left and using hemicontinuity of y +τ y ∈ , we see that (x 0).

This can be solved using methods similar to the simplex method for linear programming. A quick outline of the simplex method Linear programming is the problem of minimizing a linear function subject to linear inequality constraints. This can be put into the standard form: min c T x + d x Ax = b, subject to x ≥ 0. ” We will assume that x ∈ Rn and b ∈ Rm so that A is an m × n matrix (n ≥ m). 127. php 32 Chapter 2. Static Problems simplex tableau, which is a representation of the linear program: b A cT This tableau has a basis B which is a subset of m indexes of the variables x i : B ⊆ {1, 2, .

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