My graph of the day. Inspired by the fact that
\[ \sum_{k=1}^{\infty}{\sin k} \]
is bounded but diverges, I've been exploring other similar sums. Here's the start of
\[ \sum_{k=1}^{\infty}{\tan k} \]
It's not shocking that it looks like \(y = \ln(\|\cos x\| ) \), but I definitely wonder if at some point it blows up down the line!

Today I accidentally learned that limacons are inversions of conic sections!

"I'm not sure if that's a line or if it's two rays", asked a student, validating the instructional design of my introduction to polar coordinates activity.

This makes sense, right? (Trying to better understand arclength in polar.)

Imaginary numbers aren't just mathematical curiosities. They simplify #trigonometry, #calculus, and #geometry, opening up new possibilities in #math and #science. A mathematician explains: https://theconversation.com/taking-a-leap-of-faith-into-imaginary-numbers-opens-new-doors-in-the-real-world-through-complex-analysis-233965

A few weeks back I was complaining about this meme and we had a good discussion about it on here. I ended up making a lesson plan about taxes and my students really liked it so I will share it. (It is for students with a basic understanding of integration. But, you could probably rework it to skip that bit.)

https://drive.google.com/file/d/14xaQ1hMpqy0A68MAbxN49zK4HxK8m2R6/view?usp=sharing

(And if you have feedback let me know!)

There is a point on every #calculus that all other stones may stack upon.

That is #balance.
That is #negotiation.
That is #compromise.

"In #Latin, "calculus" means "small pebble" or "pebble," which is where the mathematical term gets its name from as ancient Romans used pebbles for counting on an #abacus."
#math
#nature
#history

**The Jagged, Monstrous Function That Broke Calculus**

Solomon Adams

_“Weierstrass discovered a function that, according to Ampère’s proof, should have been impossible: It was continuous everywhere yet differentiable nowhere.”_

https://www.quantamagazine.org/the-jagged-monstrous-function-that-broke-calculus-20250123/.

Wrapped up the last day of the semester in #Calculus with a topic that never fails to blow my mind: Fourier Series!

Mechanical Calculator Finds Derivitives - We like mechanical calculators like slide rules, but we have to admit that we had ... - https://hackaday.com/2025/01/06/mechanical-calculator-finds-derivitives/ #derivimeter #derivitive #mischacks #sliderule #calculus

Feels great to do some #calculus after a long #music session. Not sure if it also works the other way round.