June 15, 2019
June 15, 2019
October 19, 2019
Visual and Intuitive Explanations to Chain, Product and Quotient Rules
Today’s students are exposed to information presented in visual, intuitive and concise ways. They expect explanations for why a subject is important and relevant, as well as for its potential use. This is most pertinent in math courses that are usually taught with little or no connection to other disciplines. In order to adapt to students’ new learning preferences, efforts must be made to further modify teaching methods.
This paper focuses on introducing three concepts in calculus, namely Chain Rule, Product Rule and Quotient Rule by linking them to daily experiences using relevant and analogy-based examples that can be used prior to delving into purely mathematical explanations. The examples are meant to help in understanding the material, and therefore we use discrete values that can help in developing good intuition for the different rules.
The paper details many examples. Among them: (a) Chain Rule: --Inflating a balloon: Change in the volume of a constantly inflated (or deflated) balloon depends on the change in its radius which changes as a function of time.
(b) Product Rule --Number of working hours in a company: A manufacturing company has X factories, Y people in each, and each one works Z number of hours per year. The change in the total yearly working hours is due to change in number of factories, change in number of people in each factory, and the change in number of working hours per employee.
(c) Quotient Rule: --Taxis and passengers in NY: A metropolitan area published a report about the number of taxis and passengers during the 2015, 2016 and 2017 years, proudly stating that both the number of taxis (“g”) and the number of passengers (“f”) grew consistently. The city also published the average number of passengers per taxi, showing a decline in the ratio. Each function is a function of time. To show the change in the number of passenger per taxi over time, the change over time of the ratio of f/g must be calculated. In this example we show a specific numerical example that shows how to calculate the change in f/g, followed by taking the case to the limit and derive the actual formula for Quotient Rule.
The material in this paper is “work in progress.” In the past, when using the above examples (and many others), students have demonstrated better, clearer understanding of difficult concepts. Even though this was not an official assessment, based on similar experience that was gained and assessed by the author multiple times in other engineering related subjects (Control Systems, Digital Signal Processing, Computer Algorithms, and Physics), it is believed that the approach has a great potential.
Raviv, D. (2019, June), Visual and Intuitive Explanations to Chain, Product, and Quotient Rules Paper presented at 2019 ASEE Annual Conference & Exposition , Tampa, Florida. 10.18260/1-2--33540
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