Calculus Part 2

In STEM++, we are trying to make learning fundamental concepts more fun. One of our recent efforts involved how to convey the power of Calculus in everyday applications. Here are a few examples of teaching practical calculus.

Application: Ball drop

The first is a ball drop experiment. Imagine a ball is dropped from the top of a building that is 1000 ft high. How long will the ball take to hit the ground? I’ve realized that even elementary school kids can solve this given the right background. Depending on the grade level, here are the concepts I had to go through:

High school: concept of speed / distance / acceleration, relationship between area and integration

Middle school: plotting (cartesian only), simple equations, area (+ above high school concepts)

Elementary school: basics of angles, fractions, variables (+ above middle and high school concepts)

For the fastest kid in elementary school, the whole exercise took about 4 hours. The approach – what I call “top-down” – started with a problem statement, defined the need for certain concepts, and then went about learning those concepts. In fact, this process worked so well that I’m now extending this to most of the material we will put out in Thorro Learning.

Application: Draining Water Tank

Imagine a tank filled with water, with a hole at the bottom [1]. We need to find the rate of change of volume V. Toricelli’s law states that the rate of change in volume is proportional to the square root of the water depth d. We can start with a linear model and improve assumptions as we get more observations. A simple way to demonstrate the experiment is to take an empty bottle that is large enough (>1 liter) and drill a hole on its side, fill with water, and check the height every second as the water drains.

[1] Farmer and Gass, Physical Demonstrations in the Calculus Classroom, The College Math Journal (1992)

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I’ve always been a fan of calculus. It is impressive how this field of math has been used in everything from electromagnetism to economics. So my natural instinct to calculus has always been that it i