Backstory: I had to submit an essay on 'the most significant theory' that I consider...so here's the Theory of the Study of Calculus on Thinking Skills.
Mathematics is a science unlike most, in the sense that determining whether ideas hold or not purely relies on axioms of choice and logical deductions and theorems that follow. The theory of calculus as a relatively modern discipline proves innovative in discovery, rigorous in refinement and applicable in the more tangible sciences, thus making it not just an important theory in Mathematics but also providing key skills to true education.
The conception of both differential and integral calculus have both been very apparent. By using so-called ‘infinitesimal’ quantities, which are too insignificant to make any substantial change to the theory, Newton and Leibniz considered the gradient of the tangent at a point, allowing instantaneous rates of change to be calculated, leading to differential calculus. This is ingenious, as it breaks the conventional norm that gradients need to be between two points, encouraging creativity in thinking, crucial in the study of any discipline. Areas under the curve representing a function f were likewise calculated using infinitesimal rectangles between the curve and the x–axis, and they discovered that rates of change of areas at a point was the value of f at that point, giving rise to integral calculus. This spirit of thinking out of the box and experimenting new approaches to problem-solving is thus the key to extend one’s knowledge of the topic.
Eventually mathematicians questioned the notion of infinitesimals, that so long as it is nonzero, it creates an error between the actual ‘gradient’ at a point and the ‘measured’ gradient. As much as they wanted to be rigorous, they didn’t want to throw the baby out with the bathwater, that is, the key idea of calculus. Thus, the concept of limits was introduced to formalise calculus and allow the ideas to be rigorously defined and used to minimise ambiguity. Diligence and attention to detail is displayed here to maximise clarity in communication, highly vital in a world with increasing emphasis on clarity of communication. Critical thinking is also embodied, where what is taught is not necessarily accepted as blind faith but challenged and made better through a better form.
With a solidified theory of calculus, scientists could then model many processes using differential equations, since most processes involve rates of change (differential calculus) and its reverse (integral calculus). A large portion of Physics involve rates of change from estimations (use of Maclaurin’s series to estimate a quantity with minimal error) to kinematics (velocity as rate of change of displacement and acceleration as rate of change of velocity) and waves (the use of Laplace and Fourier transforms to approximate sinusoidal waveforms) that has allowed them to more efficiently make measurements and calculate predictions. The versatility of calculus has encouraged the act of active learning — application of what is learned at the theoretical framework into sciences and technology that has mankind to do what was previously impossible.
Thus, the theory of calculus not only provides development in the technical sense of clear understanding and consensus in study, but also development in the thinking skills that apply to academia as well as outside of academia. It thus is not only a pursuit for content but a honing of skills that we should all keep developing as we learn continuously.
Mathematics is a science unlike most, in the sense that determining whether ideas hold or not purely relies on axioms of choice and logical deductions and theorems that follow. The theory of calculus as a relatively modern discipline proves innovative in discovery, rigorous in refinement and applicable in the more tangible sciences, thus making it not just an important theory in Mathematics but also providing key skills to true education.
The conception of both differential and integral calculus have both been very apparent. By using so-called ‘infinitesimal’ quantities, which are too insignificant to make any substantial change to the theory, Newton and Leibniz considered the gradient of the tangent at a point, allowing instantaneous rates of change to be calculated, leading to differential calculus. This is ingenious, as it breaks the conventional norm that gradients need to be between two points, encouraging creativity in thinking, crucial in the study of any discipline. Areas under the curve representing a function f were likewise calculated using infinitesimal rectangles between the curve and the x–axis, and they discovered that rates of change of areas at a point was the value of f at that point, giving rise to integral calculus. This spirit of thinking out of the box and experimenting new approaches to problem-solving is thus the key to extend one’s knowledge of the topic.
Eventually mathematicians questioned the notion of infinitesimals, that so long as it is nonzero, it creates an error between the actual ‘gradient’ at a point and the ‘measured’ gradient. As much as they wanted to be rigorous, they didn’t want to throw the baby out with the bathwater, that is, the key idea of calculus. Thus, the concept of limits was introduced to formalise calculus and allow the ideas to be rigorously defined and used to minimise ambiguity. Diligence and attention to detail is displayed here to maximise clarity in communication, highly vital in a world with increasing emphasis on clarity of communication. Critical thinking is also embodied, where what is taught is not necessarily accepted as blind faith but challenged and made better through a better form.
With a solidified theory of calculus, scientists could then model many processes using differential equations, since most processes involve rates of change (differential calculus) and its reverse (integral calculus). A large portion of Physics involve rates of change from estimations (use of Maclaurin’s series to estimate a quantity with minimal error) to kinematics (velocity as rate of change of displacement and acceleration as rate of change of velocity) and waves (the use of Laplace and Fourier transforms to approximate sinusoidal waveforms) that has allowed them to more efficiently make measurements and calculate predictions. The versatility of calculus has encouraged the act of active learning — application of what is learned at the theoretical framework into sciences and technology that has mankind to do what was previously impossible.
Thus, the theory of calculus not only provides development in the technical sense of clear understanding and consensus in study, but also development in the thinking skills that apply to academia as well as outside of academia. It thus is not only a pursuit for content but a honing of skills that we should all keep developing as we learn continuously.
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