BetterExplained: Recent Episodes
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Quick confession? I never fully learned the trig derivatives. Sure, I memorized $\sin' = \cos$ and $\cos' = -\sin $ like everyone else, but the derivative of tangent? Cosecant? Forget it, magic spells. After years of searching, there's a middle ground between tedious derivation and rote memorization. Aha moment: all trig functions change using the […]
If the exponential function $e^x$ is water, the hyperbolic functions ($\cosh$ and $\sinh$) are hydrogen and oxygen. They're the technical, rarely-discussed parts that combine into a famous whole. Admittedly, the hyperbolic functions were tucked into a dark part of my attic. They were defined with strained motivations ("Need yet another way to build a hyperbola?") […]
Like making engineering students squirm? Have them explain convolution and (if you're barbarous) the convolution theorem. They'll mutter something about sliding windows as they try to escape through one. Convolution is usually introduced with its formal definition: Yikes. Let's start without calculus: Convolution is fancy multiplication. Part 1: Hospital Analogy Imagine you manage a hospital […]
You're minding your own business when some punk asks what the integral of $\sin(x)$ means. Your options: Pretend to be asleep (except not in the engineering library again) Canned response: "As with any function, the integral of sine is the area under its curve." Geometric intuition: "The integral of sine is the horizontal distance along […]
The Pythagorean Theorem is often taken as a fact about right triangles. Let's try a broader interpretation: The Pythagorean Theorem explains how 2D area can be combined. Here's what I mean. Suppose we have two lines lying around (the creatively named Line A and Line B). We can spin them to create area: Ok, fun […]