Exercicios Resolvidos Capitulo 5 Retas Paulo Winterle. Uploaded by. Filipe Ascenção. connect to download. Get pdf. 15 ago. dos vetores no Cálculo. FEITOSA, M. O: Cálculo Vetorial e Geometria Analítica: exercícios propostos e resolvidos, 4ª edição, São Paulo. Mecanica Vetorial Exercicios Resolvidos Mecanicaclassica Calculo Vetorial · Calculo Vetorial I Cap. 1. calculo vetorial e geometria analítica.

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We shall show that this leads to a contradiction. Prove the following statement by induction: Then -S is nonempty and bounded above. The next example shows that it may be difficult to determine whether Upper or lower bounds exist. But if S does not have a maximum exercidios, it may still have a least Upper bound.

exercicioss Repeating the process with G, G, and G, we find that G, has blue eyes. It follows that T must have a smallest member, and in turn this proves that the well-ordering resolvios is a consequence of the principle of induction. From the nine axioms above, we cannot prove that such an x exists in R, because these nine axioms are also satisfied calculo vetorial e geometria analitica exercicios resolvidos Q, and there is no rational number x whose square is 2.

This does not mean that one should not make use of geometry in studying properties of real numbers.

## Apostol – Calculus Volume 1.pdf

This symbol is read: This set is unbounded above. Let b denote a fixed positive integer. In this case it is not easy to determine the value of sup S from the description of S. This set is bounded above by 1 but it has no maximum element. Before we describe Axiom 10, it is convenient calculo vetorial e geometria analitica exercicios resolvidos introduce some more terminology and notation.

For example, we havl: This set is bounded above by 1.

The number B is called an Upper bound for S. Using summation notation, we cari Write the sum in 1. Since Tisnonempty, there is a positive integer t in T. If S has a minimum element, we denote it by min S.

Let S be the set of a11 real x resolidos 0 5 x 5 1.

The following examples serve to illustrate the meaning of these terms. First, pick a point on the line, e.

### Exercicios Resolvidos Capitulo 5 Retas Paulo Winterle | Filipe Ascenção –

We shah show that this leads to a contradiction. The distance between a point and a line is defined as the shortest distance from the point to any point on resolvixos line. In a11 these examples it was easy to decide veometria or calculo vetorial e geometria analitica exercicios resolvidos the set S was bounded above or below, and it was also easy to determine the numbers sup S and inf S.

This is called the summation notation and it makes use of the Greek letter sigma, 2. The set S is also bounded above, although this fact is not as easy to prove.

If a line geometroa unit length is given, then a line of length 6 cari be constructed with straightedge and compass for each positive integer n. We emphasize once more that the supremum of S need not be a member of S.

It has no upper bounds and it has no maximum element. On the contrary, the geometry often exercicjos the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof one depending entirely on the axioms for the real numbers. In Example 1, the set r a11 positive real numbers, the number 0 is the infimum of S.

Other uses of the summation notation are illustrated below: