I took Calculus, and I felt I was missing something, but I couldn’t put my finger on it. I much later realized how if you differentiate, and area formula, you get back the original equation. This is the realization that lead Newton to Calculus. Looking back at my Calc books, I don’t think the logic sequence is well explained. There is often a lot of handwaving, and “it’s intuitively obvious”, when it’s not.
Yes, we call this the “fundamental theorem” of calculus, but that’s largely an artifact of how we’ve arranged the logical building blocks in calculus.
If you define a derivative as a limit of a difference quotient, and an integral as a Riemann sum, then the fact that the two are inverses is rather surprising, and requires proof. But if you embrace infinitesimals in the ways that Newton and Leibniz did, then it’s scarcely a “theorem” at all, just a natural property of the objects you’re studying.
All to say: our modern logical arrangement of a subject can sometimes obscure as much as it illuminates!
The Riemann sum, as I recall, is a limit that shows the area is the sum of finer and finer rectangles is equal to the area. My understanding is that Riemann “formally” cleaned up a lot of mathematical loose ends, and tidied up the integral. Archimedes used triangles to find the area of a parabola, which was known in antiquity, and is similar to a Riemann Integral. Prior to Riemann Integral were often called anti-derivatives, because they were reversing the differentiation process.
Does the Riemann Integral show that it is an inverse of differentiation ? Or are there other things with the Riemann integral, that I missed ?
What a masterpiece! I would’ve made the “meh” value the derivative of knowledge, however (essentially equal to the average “learning” or progress, I suppose).
LOVE!
I took Calculus, and I felt I was missing something, but I couldn’t put my finger on it. I much later realized how if you differentiate, and area formula, you get back the original equation. This is the realization that lead Newton to Calculus. Looking back at my Calc books, I don’t think the logic sequence is well explained. There is often a lot of handwaving, and “it’s intuitively obvious”, when it’s not.
Yes, we call this the “fundamental theorem” of calculus, but that’s largely an artifact of how we’ve arranged the logical building blocks in calculus.
If you define a derivative as a limit of a difference quotient, and an integral as a Riemann sum, then the fact that the two are inverses is rather surprising, and requires proof. But if you embrace infinitesimals in the ways that Newton and Leibniz did, then it’s scarcely a “theorem” at all, just a natural property of the objects you’re studying.
All to say: our modern logical arrangement of a subject can sometimes obscure as much as it illuminates!
They definitely obscure, rather than illuminate !!!
The Riemann sum, as I recall, is a limit that shows the area is the sum of finer and finer rectangles is equal to the area. My understanding is that Riemann “formally” cleaned up a lot of mathematical loose ends, and tidied up the integral. Archimedes used triangles to find the area of a parabola, which was known in antiquity, and is similar to a Riemann Integral. Prior to Riemann Integral were often called anti-derivatives, because they were reversing the differentiation process.
Does the Riemann Integral show that it is an inverse of differentiation ? Or are there other things with the Riemann integral, that I missed ?
What a masterpiece! I would’ve made the “meh” value the derivative of knowledge, however (essentially equal to the average “learning” or progress, I suppose).