In our AP Calculus class, we have learned about derivatives and their applications. During this time, we specifically learned how to find the minimum and maximum of functions, and to apply them to real-life problems in optimization and minimization. These insights are used in everyday life to optimize the use of materials, profits, and measurements, as well as, to minimize materials and dimensions.
Using our previous knowledge in mathematics, we were able to solve analytical exercises for the use of derivatives in volume optimization in specific geometric figures. By completing the procedures, calculations, and conclusions, we realized that there was a much easier way to fully understand the problems, and to understand the relationship that each measure has with the other. Taking this into account, we created three simulations in GeoGebra, which can be seen in the video shared with the community as evidence of our work.
In this process, we went from the written problem to a 3D simulation with sliders that allow us to see how the dimensions of the geometric figure change and how it is rotated through an axis to form the total figure (In exercise 31.b). This process can also be called a solid revolution. In our assigned exercises we had to optimize the volume of a prism based on some initial measurements, which will be seen in the first simulation, with the process of assembling the box, and in the other exercise where we optimized the volume of a cylinder. The latter was created by folding a rectangular piece of material and the volume of another cylinder that used the same rectangle of material, but with measures of height and radius, which will be evidenced in the explanatory video. In the last exercise, measurements of the same sheet of paper had to be optimized, and in the first exercise, the measurements had to be optimized to maximize the volume within the limitations of the given material. This was, overall, a really interesting activity in which we interpreted mathematical problems more tangibly to understand the relationship between the real world and mathematics. We feel proud of the evidence we provided and the applications of real-life problems in our AP Calculus class!
Jose Luis Zamora High School Mathematics Teacher
Valeria Calderón and Juan Diego Acuña Twelfth Grade AP Calculus Students
Résumé :
En el desarrollo del tema de razones relacionadas realizado en la clase de AP Cálculo, se trabajaron algunos ejercicios con enfoque hacia la optimización de material de construcción, capacidad (volumen) y estructura externa de algunos recipientes utilizados en la vida cotidiana. El propósito inicial era realizar una explicación matemática de la solución óptima de cada situación problemática. La siguiente muestra constituye en un trabajo de calidad dado que los estudiantes utilizaron como complemento a su trabajo, simulaciones del proceso de la solución en la plataforma GeoGebra. En esta, se evidencia el proceso de construcción de los modelos 3D de diferentes figuras presentes en las situaciones planteadas y se evidencia la relación visual con el tema. Esta simulación representa cómo cambian el área de los polígonos, volumen y otras medidas al alterar cualquier variable, haciendo visualmente fácil de entender y clara la forma de identificar el punto de optimización de cada una de las situaciones. ¡Nos sentimos orgullosos de la evidencia que presentamos y las aplicaciones de los problemas de la vida real en nuestra clase de Cálculo AP!
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