By Hwang A.

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In diesem Lehrbuch finden Sie einen Zugang zur Differenzial- und Integralrechnung, der ausgehend von inhaltlich-anschaulichen Überlegungen die zugehörige Theorie entwickelt. Dabei entsteht die Theorie als Präzisierung und als Überwindung der Grenzen des Anschaulichen. Das Buch richtet sich an Studierende des Lehramts Mathematik für die Sekundarstufe I, die „Elementare research" als „höheren Standpunkt" für die Funktionenlehre benötigen, Studierende für das gymnasiale Lehramt oder in Bachelor-Studiengängen, die einen sinnstiftenden Zugang zur research suchen, und an Mathematiklehrkräfte der Sekundarstufe II, die ihren Analysis-Lehrgang stärker inhaltlich als kalkülorientiert gestalten möchten.

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**Example text**

Velocity is supposed to represent “instantaneous rate of change of position,” but what exactly does this mean? ) The ancient Greek Zeno of Elea discovered the following apparent paradox. ” The stone has a definite location, and is indistinguishable from a stationary stone at the same height. More concretely, imagine an infinitely fast camera that captures literal instants of time. Then there is no way to distinguish a moving object from a stationary object on the basis of a single photograph. But since this argument can be made at every instant of time, there is no difference between moving and standing still!

The subset ∆ is often called the diagonal of X × X. 11 Let X be a set having more than one element. 7, namely by R = (X ×X)\∆ = {(x, y) ∈ X ×X : x = y}. 12 Let X be a set. An equivalence relation on X is a relation, usually denoted ∼, such that • (Reflexivity) For all x ∈ X, x ∼ x. In words, every element is related to itself. • (Symmetry) For all x and y ∈ X, x ∼ y if and only if y ∼ x. Roughly, the relation sees only whether x and y are related or not, and does not otherwise distinguish pairs of elements.

2 says A(ℓ) is true for all ℓ ∈ N, that is, addition is associative. Commutativity is proven by a double application of induction; first show that n + 1 = 1 + n for all n ∈ N (by induction on n), then prove that n + m = m + n for all m, n ∈ N (by induction on m). Associativity is used several times. Consider the statement P (n) : n + 1 = 1 + n. The base case P (1) says the successor of 1 is the successor of 1 (or 1 + 1 = 1 + 1), which is obviously true. Now assume P (k) is true for some k ∈ N; we wish to prove P (k + 1).