Abstract
Three perspectives on the concept of derivatives at school are commonly applied: the geometric, the algebraic–analytical and the application-oriented. Often the geometric perspective is used for introducing the concepts of calculus on given curves as graphs of a function. However, students’ experience with tangents as global support tangent lines, e.g., tangents to circles, differs from the new interpretation of a tangent as a straight line that locally has the same slope at the point of osculation as the function at the same point. These different perspectives on tangents are a well-described problem for students starting on differential calculus. The algebraic–analytical perspective seems to be more complicated than the geometric perspective, as it deals with the concept of limit, which is not mandatory in all curricula in its rigorous form, at least in German classrooms today.
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Notes
- 1.
Catenary: Which curve describes a chain freely suspended at its ends? While Galileo still suspected that it was a parabola, Bernoulli showed that the curve can be represented by the hyperbolic cosine. Brachistochrone: In a vertical plane, two points, \(\mathrm{A}\) and \(\mathrm{B}\), are given. Which curve describes the path of a ball that rolls from \(\mathrm{A}\) to \(\mathrm{B}\) in the shortest time possible?
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Stoffels, G., Witzke, I., Holten, K. (2022). Comparison: Differential Calculus Through Applications. In: Dilling, F., Kraus, S.F. (eds) Comparison of Mathematics and Physics Education II. MINTUS – Beiträge zur mathematisch-naturwissenschaftlichen Bildung. Springer Spektrum, Wiesbaden. https://doi.org/10.1007/978-3-658-36415-1_17
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