A person who posts an article like this is telling me they have neither written nor read their article. Or their own code base, for that matter. So this person had problems in a code base he didn't write, didn't read either, it seems, and now wants all of us to read an article about this zombie code base, except he also didn't write or read over that article before posting it.

To the author: Your article can be rewritten in a few sentences like this: "I vibe coded a code base and asked Claude to add tons of unit tests. I didn't read the code or the tests; just wanted it to work. But I either didn't know to ask to wire the tests into my CI/CD pipeline, didn't know what that was, or did know but was too lazy to check that it happened. Anyway, I sure learned my lesson: What happens in localhost:3000 stays in localhost:3000. Gotta wire up your .github/workflows/cicd.yml file to pull in your tests. Otherwise it really doesn't do much for you. Lesson learned!" And you know what? As banal as that would have been, I would have enjoyed that article so much more. Just take 5 minutes man. Five freaking minutes.

Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.

I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.

Let M_n := Mary's age now = 24, A_n := Ann's age now, and let M_p, A_p be the ages of Mary and Ann at a certain point in the past. We have that M_n = 24 = 2(A_p) from "Mary is 24 years old. She is twice as old as Ann was", immediately implying that A_p = 12. At the point in time p when M_p, A_p were the ages of Mary and Ann, respectively, we have that M_p = A_n from "when Mary was as old as Ann is now". Also, let x := | M_n - M_p | = | A_n - A_p | be the amount of time that has passed between the point p in time and now. Since M_n > M_p and A_n > A_p by construction, we can drop absolute values, yielding M_n = M_p + x, A_n = A_p + x. Substituting, we have M_p = 24 - x, and A_n = 12 + x, yielding x = 24 - M_p = A_n - 12, yielding A_n = 36 - M_p. But, since M_p = A_n, we can write 2(A_n) = 36 yielding A_n = 18.

This problem, however, completely ignores the fact that Ann boarded an interstellar spaceship 6 years ago at a reasonable fraction of the speed of light c. To account for this missing detail, we need to use Lorentz factors. Since Ann was the one traveling in space, we have to adjust her current age by calculating A_n = A_p + x * sqrt(1 - (v/c)^2)), where v is Ann's velocity Given that the problem is missing the critical detail of how fast Ann is hurtling towards the outer boundary of the universe, we really can't calculate their age at all. Nonetheless, given the completely reasonable and plausible assumption that Ann has been traveling at 95% of the speed of light because spaceships totally can do that, Ann is obviously about 13.87 years old. So 18 years old is clearly the wrong answer.

Note that the fact that Ann is not aging as much due to traveling at an enormous velocity does NOT change Mary's age. Mary remains exactly twice as old as Ann was at the given point in the past, so she's 24 and 6 years have passed from her perspective. Only Ann's age changes. Obviously.

I suggest re-engaging with your friends and family. It really will help you figure out what you want out of life and of this ongoing venture. Doing that will help you realize what really matters to you. Perhaps you'll find yourself reinvigorated and will dive back into the work even harder than before. Maybe you'll just have a clearer picture of what your exit strategies might be. Either way, you'll be glad you reconnected with the people who care about you. It's more important to do that than any startup ever could be.

Anyway, wishing you best of luck.

1. Current month's usage * applicable rates; + 2. Estimated future usage for the month * applicable rates.

And Item 1 obviously requires proper data persistence, but Item 2 is just a projection. If they don't have Item 1 correct, AWS's whole system is in question, but I don't think that's the issue. I'm going to guess now -- looking forward to reading the root cause analysis -- that the problem is that someone decided to get too fancy with the estimates, and built a latent requirement that all prior estimates for the month must be available to compute the current estimate. Without estimates working, no estimates are available, and some denominator in an averaging or smoothing or normalizing function goes to 0; then everyone's estimated bill explodes without bound (subject to floating-point arithmetic) resulting in trillion-dollar estimates.