chasing the limit 📊

now for some background, my math teachers in high school never let me double up and take ap math classes like i did with history or science, so i didn't learn anything about derivatives or integrals then.

i thought the subject was fun though. i always saw my math classes as a kind of game, and i'd watch math content online to see if i could learn more cool tricks for tests or mathlete competitions.

i remember watching this extra credits history of math vid on calculus with riemann sums used to approximate the area under a curve.

i misunderstood video and thought that the "infinitely many rectangles" was some type of theoretical concept that we approached till we got a good enough result like some kind of brute force calculation compared to the deserved pretension of euclid's infinite primes proof i would learn in transitions later on.

so when our first unit in calc ii covered newton's method, an algorithm to find zeroes by brute force, my disappointments were being met.

then we moved on to riemann sums with little rectangles used to calculate the area under a curve. we played around with leftbound, rightbound, and then trapezoidal slices. all finite, all leaving behind little shavings of area that were uncounted.

this is the part where i thought we'd learn some algorithm equation or even pull out numpy or matlab to get the right amount of deltas needed to meet an arbitrary percentage amount.

but no! way cooler than that, our professor made us write out a generic riemann sum equation for n many slices. then we just did what we'd do in calc i with derivatives: apply a limit of n approaching infinity.

if we actually ran the limit, you'd get a deadly undefined quantities that made the whole thing meaningless, but we were very strategic with our algebra, slowly as simplifying our expression, massaging out these knots of boundlessness into convergent quantities. we even used the same equations i had proved by mathematical induction just a trimester before…

and we got a result! the original equation was \(x^2\) from \([0,1]\) equaling a rational, clean \(\frac{1}{3}\). woahhhh.

then the real mic drop: our professor washing off the sigma summation and delta x at the top of the whiteboard to the integral and diff symbols instead¹.

\[ \lim_{n\to\infty} \sum_{i=1}^n f(c_i) \Delta x \Rightarrow \int_a^b f(x) d x \]

\[ c_i = a+i \Delta x \]

\[ \Delta x = \frac{b-a}{n} \]

we had left the ancient greeks, both in terms of letters and ideas. and my mind was blown.

"after the math comes the aftermath"

– cormac mccarthy, the passenger

this was the most important lecture for me and just about every other engineer in that room. calculus is the cornerstone of all money-making stem fields. i use this theorem each time i'm calculating volumes of 3d shapes² with vectors or divining percent chance with probability density functions or earning cash money with continuous-compounding annuity equations.

but more than that was the inspirational effect it had.

what we actually proved that day wasn't some analytical treatise on the fundamental theorem of calculus; we didn't prove the connection between riemann sums to the antiderivative or really talk about all the caveats needed to make the definition hold.

what we did do though, with the sticks and stones they give professors teaching engineering students just enough calc to not collapse a bridge, was to use the same knowledge we had before but compose it in such a way to solve a novel problem: the essence of mathematics.

and this was the real cornerstone of education i learned that day. that all these rules and limits around us are liquid. that they can be shaped arbitrarily or cut down altogether if you just have the wishful thinking³ to escape into something new.

i've got a ton of respect for the professors i had in my math department, but this one i had for calc ii was so fun. she's a real showman, a magician with a phd. she was also one of the russian émigrés that taught me in my time at my uni, so she had an internationalist sense of humor…

a particular lecture i remember was when we were deep in the class. we were doing some nonsense applied practice problem of an integral being used to calculate the amount of work done for a physics problem.

i remember we got out our textbooks and read aloud the problem: a soviet weight lifter lifts 300 kg up 0.33 m for 60 s while an american one lifts 1000 lb up 3 in for 5 s. these numbers are all just from my own memories, but this was the setup more or less (and based on actual history.)

she picked on me for never taking physics ("i took physics since i was a little girl in russia!"), so she had me guess for the class who would win just based on vibes.

we landed on the moon, so naturally i guessed the american.

and we do the calculation of one on the whiteboard then do the other ourselves in our notes… and it's not even close. the soviet won by over an order of magnitude, the kind of difference you'd see in five-year-plan quotas.

and she says to me in front of my cohort, "in my class ангел, the russian always wins."

FIN.