Mechanical Calculator Finds Derivatives

January 6, 2025 by Al Williams 12 Comments

We like mechanical calculators like slide rules, but we have to admit that we had not heard of the Ott Derivimeter that [Chris Staecker] shows us in a recent video. As the name implies, the derivimeter finds the derivative of a function. To do that, you have to plot the function on a piece of paper that the meter can measure.

If you forgot calculus or skipped it altogether, the derivative is the rate of change. If you plot, say, your car’s speed vs time, the parts where you accelerate or decelerate will have a larger derivative (either positive or negative, in the decelerate case). If you hold a steady speed, the derivative will be zero.

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Human Civilization And The Black Plastic Kitchen Utensils Panic

December 19, 2024 by Maya Posch 98 Comments

Recently there was a bit of a panic in the media regarding a very common item in kitchens all around the world: black plastic utensils used for flipping, scooping and otherwise handling our food while preparing culinary delights. The claim was that the recycled plastic which is used for many of these utensils leak a bad kind of flame-retardant chemical, decabromodiphenyl ether, or BDE-209, at a rate that would bring it dangerously close to the maximum allowed intake limit for humans. Only this claim was incorrect because the researchers who did the original study got their calculation of the intake limit wrong by a factor of ten.

This recent example is emblematic of how simple mistakes can combine with a reluctance to validate conclusions can lead successive consumers down a game of telephone where the original text may already have been wrong, where each node does not validate the provided text, and suddenly everyone knows that using certain kitchen utensils, microwaving dishes or adding that one thing to your food is pretty much guaranteed to kill you.

How does one go about defending oneself from becoming an unwitting factor in creating and propagating misinformation?

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Making Floating Point Calculations Less Cursed When Accuracy Matters

March 14, 2024 by Maya Posch 27 Comments

Inverting the earlier exponentiation to reduce floating point arithmetic error. (Credit: exozy)

An unfortunate reality of trying to represent continuous real numbers in a fixed space (e.g. with a limited number of bits) is that this comes with an inevitable loss of both precision and accuracy. Although floating point arithmetic standards – like the commonly used IEEE 754 – seek to minimize this error, it’s inevitable that across the range of a floating point variable loss of precision occurs. This is what [exozy] demonstrates, by showing just how big the error can get when performing a simple division of the exponential of an input value by the original value. This results in an amazing error of over 10%, which leads to the question of how to best fix this.

Obviously, if you have the option, you can simply increase the precision of the floating point variable, from 32-bit to 64- or even 256-bit, but this only gets you so far. The solution which [exozy] shows here involves using redundant computation by inverting the result of e^x. In a demonstration using Python code (which uses IEEE 754 double precision internally), this almost eradicates the error. Other than proving that floating point arithmetic is cursed, this also raises the question of why this works.

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Integration Taught Correctly

February 26, 2024 by Al Williams 31 Comments

[Math the World] claims that your calculus teacher taught you integration wrong. That’s assuming, of course, you learned integration at all, and if you haven’t forgotten it. The premise is that most people think of performing an integral as finding the area under a curve or as the “antiderivative.” However, fewer people think of integration as adding up many small parts. The video asserts that studies show that students who don’t understand the third definition have difficulty applying integration to real-world problems.

We aren’t sure that’s true. People who write software have probably looked at numerical integration like Simpson’s rule or the midpoint rule. That makes it pretty obvious that integration is summing up small bits of something. However, you usually learn that very early, so you’re forgiven if you didn’t get the significance of it at the time.

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Calculus Made Easy In The Car

July 29, 2022 by Al Williams 24 Comments

If you had the traditional engineering education, you’ve made your peace with calculus. If you haven’t, you may have learned it on your own, but for many people, calculus has a reputation for being super difficult. While some of the details can be very tricky, the core concepts are actually simple and [Mathologer] has a very simple explanation along with some good graphics that can help you get started on calculus mastery if you’ve been putting it off. Using a car on the highway as the prototypical example, he covers quite a bit of ground in the 30 minute video that you can see below.

Of course, this isn’t a unique idea that calculus is actually simple. The video credits the great book “Calculus Made Easy” that we’ve talked about before. That 100-year-old (and then some) book has a similar approach to the topic.

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Hackaday Links: July 10, 2022

July 10, 2022 by Dan Maloney 9 Comments

We always like to call out a commercial success stemming from projects that got their start on Hackaday.io, and so we’re proud to announce the release of MAKE: Calculus by Joan Horvath and Rich Cameron, a book that takes a decidedly different approach to teaching calculus than traditional courses. Geared to makers and hackers, who generally tend to have a visual style of learning, the book makes heavy use of 3D-printed models to illustrate the relationships between functions. The project started five years ago as a 2017 Hackaday Prize entry, and resulted in a talk at the 2019 Supercon. Their book is now available for preorder, and might be a great way to reacquaint themselves with calc, or perhaps even to learn it for the first time. Continue reading “Hackaday Links: July 10, 2022” →

Talking Head Teaches Laplace Transform

August 9, 2020 by Al Williams 25 Comments

Most people who deal with electronics have heard of the Fourier transform. That mathematical process makes it possible for computers to analyze sound, video, and it also offers critical math insights for tasks ranging from pattern matching to frequency synthesis. The Laplace transform is less familiar, even though it is a generalization of the Fourier transform. [Steve Bruntun] has a good explanation of the math behind the Laplace transform in a recent video that you can see below.

There are many applications for the Laplace transform, including transforming types of differential equations. This comes up often in electronics where you have time-varying components like inductors and capacitors. Instead of having to solve a differential equation, you can perform a Laplace, solve using common algebra, and then do a reverse transform to get the right answer. This is similar to how logarithms can take a harder problem — multiplication — and change it into a simpler addition problem, but on a much larger scale.

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