At Urban Academy, our commitment to academic excellence extends to challenging our students with practical applications of mathematical concepts. This year, students undertook the dual challenge of studying Calculus 12 alongside Pre-Calculus 12, delving deep into functions, differentiation, integration, and real-world applications.
Designing Thrilling Roller Coasters
As students developed a repertoire of polynomial, rational, and transcendental functions in Pre-Calculus, they also learned to differentiate and integrate functions in Calculus 12. Differentiation involves tracking the steepness or slope of different functions; integration involves determining the area under the curve of a function.
Armed with their understanding of slope and area under curves, they embarked on a creative endeavour: designing a roller coaster function that met safety standards while delivering an exhilarating ride. The challenge was to create a continuous (connected at all points) and differentiable function, ensuring smooth transitions and thrilling drops without exceeding steepness limits.
Over the course of a month, students perfected their designs, employing calculations to ensure continuity and differentiability at every connection point. The result? Roller coasters that were as challenging to create as they were exciting to ride. A few designs are shown below; needless to say, these were more challenging to create than meets the eye!
Calculating Volume with Root Vegetables
The students also applied their understanding of volume of revolution using definite integration to calculate the volume of a root vegetable of choice. They imported a photo of their vegetable into Desmos and created a piecewise function to trace their vegetable using regression. Then, they used volumes of revolution (shell method) to find the volume of their vegetable. They were able to verify their measured and calculated results in class using water displacement.
Fostering Mathematical Competencies
Both abstract thinking and making connections to the real world are important mathematics competencies that are developed through the British Columbia Kindergarten through Grade 12 curriculum. Through their culminating work, our Calculus students have shown that they have achieved both and are thoroughly prepared for any relevant post-secondary studies or personal explorations of mathematics in the future.
