Note that a sequence is a type of net, with D being the set of natural numbers directed as usual, namely that n ≺ m means n ≤ m. If α, β ∈ D, then there is γ ∈ D such that α ≺ γ and β ≺ γ.Ī net is a function defined on a directed set. A directed set is a non-empty set D equipped with a partial ordering ≺ satisfying the following three conditions: Note that if we use this definition, then we are not calculating a limit of a sequence, in the usual sense that the students are used to, but rather the limit of a net in the following sense (for the details, see e.g. In that case L is called the length of C. The curve C is said to be rectifiable if the limit L of L n, as n → ∞ and the maximum segment length | P i − 1 P i | → 0, exists. The polygonal line P 0, P 1, P 2, …, P n constructed by joining adjacent pairs of these points with straight lines forms a polygonal approximation to C, having length L n = ∑ i = 1 n | P i − 1 P 1 |. Suppose that we choose points A = P 0, P 1, P 2, …, P n − 1 and P n = B in order along the curve. Let C be a curve in the plane joining A and B. Let A and B be two points in the plane and let | A B | denote the distance between A and B. Adams, 2006 Hass et al., 2017 Stewart, 2015) the concept of curve length is typically defined in the following way.Definition 2.1 The first time students are exposed to arc length calculations of general functions is in introductory calculus courses. For instance, the Greek philosopher Aristotle (384–322 BC) stated the following concerning comparisons of motions along straight lines and along circles: Indeed, over the millennia, many of our greatest thinkers failed to provide satisfying answers to such questions. As teachers, we have to treat these questions seriously, because when pondering over this, the students are placed in very good company. Indeed, it is only natural for them to pose questions such as ‘How can we measure something curved using a straight ruler?’ or ‘What do we really mean when we speak of the length of a curve?’. For many students the transition from understanding straight line measurements to comprehending length measurement of non-linear curves is not so easily accomplished. In middle school, we are presented with the problem of measurement of the circumference a the circle and how to relate this to the length of its diameter. In the early years of schooling we are taught how to measure lengths of straight lines using a ruler and express our findings in appropriate units. Questions such as ‘How tall am I?’ or ‘How long can you jump?’ or ‘How far is it to my friends house?’ arise naturally from them. Even young children are involved in many everyday activities that concern length measurements. One of the first experiences of measurements that we encounter in our lives is that of length.
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