We’re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature’s way of saying “This has lots of energy. We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process. Just a few words on “rigor”.ĭid you know we don’t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because “Sure, it works in practice. I can feel the math pedants firing up their keyboards. That’s just not happening with your velocity equation. That ring/circle thing we made? You could build it out of several pipe cleaners, separate them, and straighten them into a crude triangle to see if the math really works. I prefer starting with physical, visual examples because it’s how our minds work. That’s great, but it can be hard to relate: honestly, how often do you know the equation for velocity for an object? Less than once a week, if that. Many calculus examples are based on physics. And sometimes the little things are easier to work with. This is a recurring theme in calculus: Big things are made from little things. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings. This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Yowza! The combined area of the rings = the area of the triangle = area of circle! Calculus lets us start with $\text (r) (2 \pi r) = \pi r^2$, which is the formula for area! But most of us learn these formulas independently. Don’t these formulas seem related in some way? It all fits together.Ĭalculus is similarly enlightening. You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). You understand why drugs lead to resistant germs (survival of the fittest). My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. If you want more Calculus topics covered, let me know which ones.I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education.Ĭalculus relates topics in an elegant, brain-bending manner. The Method of Undetermined Coefficients.Exact Equations and Integrating Factors.Homogeneous Differential Equations ( Homogeneous Functions).First Order Linear Differential Equations.In our world things change, and describing how they change often ends up as a Differential Equation: an equation with a function and one or more of its derivatives:
#Calculus for dummies series#
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