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Functions and their curves
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
An asymptote to a curve is defined as a straight line to which the curve approaches as the distance from the origin increases. Alternatively, an asymptote can be considered as a tangent to the curve at infinity.
Reasoning about geometric limits
Published in International Journal of Mathematical Education in Science and Technology, 2021
Andrijana Burazin, Ann Kajander, Miroslav Lovric
Working with limits often involves geometry; for instance, students are asked to reason from graphs of functions (constructed by hand or by using software), and ‘read’ or determine the value of the limit of a function. This approach comes with its own set of problems. Kolar and Cadez (2012) provide evidence that both students’ and teachers’ intuitions, for instance about a geometric object ‘approaching’ another geometric object (such as the graph of a function ‘approaching’ its asymptote), can interfere with their understandings of the limit process. In part, this is due to the fact that the phrases that often appear in narratives about limits (‘to approach’, ‘to come closer and closer to’, etc.) have somewhat ambiguous meanings in everyday life (Kolar & Cadez, 2012; Monaghan, 1991; Oehrtman, 2009).
University students turning computer programming into an instrument for ‘authentic’ mathematical work
Published in International Journal of Mathematical Education in Science and Technology, 2020
Chantal Buteau, Ghislaine Gueudet, Eric Muller, Joyce Mgombelo, Ana Isabel Sacristán
In this paper we investigate programming technology use in the context of pure or applied mathematical investigation, referring to the theoretical frame of the instrumental approach (Rabardel, 2002). In brief, the instrumental approach in the context of mathematics education provides a lens to describe how a student learns mathematics in a technology-rich environment. In an activity with a mathematical goal (e.g. determine the infinite limit of a function), the student learns to use an artefact (e.g. with a calculator plot the graph of a function, adjust the window for large values of x; etc.), and learns mathematics at the same time (e.g. ‘a graph with a horizontal asymptote indicates that the function has an infinite limit’). The instrumental approach proposes a model of these intertwined processes – we elaborate more in Section 3.
Synthesis and characterization of (1 − x)(La0.6Ca0.4MnO3)/x(Sb2O3) ceramic composites
Published in Phase Transitions, 2019
N. Amri, M. Nasri, M. Triki, E. Dhahri
The magnetization curves show a typical transition for all materials, at the TC Curie temperature, from a low temperature FM state to a high temperature PM state. TC value is derived from the dM/dT curves and it remains unchanged around 264 K for all the samples which is compatible with results in [15,16]. It could be found also using the tangent method (intersection of the tangent at the point of inflection of the curve with its horizontal asymptote). The inflexion point of the temperature derivative of magnetization was used to determine TC values which are found to be 264 K for x(LCMO)1 − x/(Sb2O3)x (x = 0.00, 0.07 and 0.12) samples. The observed constant of TC is related to the unchanged exchange interaction between Mn3+-Mn4+. The lack of impact on TC by Sb2O3 doping level refers to its intrinsic property that depends on the ferromagnetism inside the grain [17]. The observed constancy of TC also indicates that stoichiometry of LCMO phase within the grains remains essentially stable as Sb2O3 is not accommodated within the perovskite structure and it occupies only the boundaries of the LCMO grains which matches with the results obtained from XRD and SEM micrographs.