First fundamental theorem of calculus: How do engineering students interpret and apply it? / Primer teorema fundamental del cálculo: ¿cómo lo interpretan y aplican estudiantes de ingeniería?

Abstract

The first fundamental theorem of calculus relates differential and integral calculus, one of its important aspects according to Bressoud (2011) is that it shows the existence of two ways of calculating an integral: with the limit of a Riemann sum and by an antiderivative. Larsen, Marrongelle, Bressoud and Graham (2017) indicate that calculus is a barrier to the academic progress of many students and that there is a need for research that seeks to develop proposals for instruction to improve the understanding of its concepts. Therefore, with the idea of carrying out this type of research in the future, the present study seeks to identify the common interpretation of the first theorem of calculus and whether it is useful in solving contextual problems. Answering these questions will provide some elements to develop a proposal for instruction. This study involved 18 students between the ages of 18 and 21 from engineering careers at a university located in Mexico City, who had completed a calculus course. The instrument was a set of three problems, in two, we propose contextual situations that can be solved by applying the first fundamental theorem of the calculus or by performing integration and derivation operations (one situation is about the ratio of change of the volume of water contained in a tank, with respect to time, where water falls to a variable ratio; the other is about the ratio of change of the volume of water contained in a cylindrical tank with respect to the height of water). In the last problem, the same type of situation is posed in abstract form: If F(x) = f t dt ! ! , obtain F′(x), justify your answer.