Timeline for Why is differential calculus often presented before integral calculus?
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31 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2022 at 4:44 | comment | added | Michael Hardy | $\ldots\qquad y = \text{constant}\times e^{-z^2/2}.$ Finding the area under this curve, and thus finding this constant, are not done by finding the antiderivative of that function. $\qquad4 | |
| Jun 24, 2022 at 4:43 | comment | added | Michael Hardy | Some might be surprised to learn that many integrals can be evaluated without finding antiderivatives. Consider the center of gravity of the northern hemisphere (assuming the earth is a sphere of uniform density). Archimedes showed that this is 5/8 of the way from the north pole to the center of the earth, without finding any antiderivatives. However, the way people think about calculus today is that the most elementary integrals are found by finding antiderivatives. (But in every high-school statistics course one encounters the "bell-shaped curve," whose equation is$\ldots\qquad$ | |
| Jun 22, 2022 at 14:51 | answer | added | Steven Gubkin | timeline score: 3 | |
| Jul 17, 2021 at 11:17 | comment | added | user13234 | I'm kind of shy about this post. Anyway, perhaps there are merits to both approaches. I'm still learning Calculus at the moment. | |
| Jul 14, 2021 at 0:24 | comment | added | Vercassivelaunos | @Adam In my (German) analysis textbook it's actually done: They calculate the integral of the cosine before introducing the FTC, using only trig identities. It's very cumbersome, though, which is probably the point they want to make. | |
| Jul 12, 2021 at 5:20 | vote | accept | CommunityBot | ||
| Jul 9, 2021 at 19:18 | comment | added | Adam | @MikeScott I challenge you to prove the usual set of closed form expressions for antiderivatives without using derivatives. I'm not saying it is impossible. Linear and quadratic functions can be integrated using known results about arithmetic progressions and triangular numbers, respectively. But now try finding $\int \sin(x)dx$ without depending on derivatives at some point. | |
| Jul 9, 2021 at 11:20 | comment | added | Mike Scott | @Adam But you might equally say that anti-integration (which is what differentiation is) requires integrals. And it clearly doesn't. | |
| Jul 8, 2021 at 17:12 | comment | added | Josh Grosso | IIRC Apostol actually does start with integral calculus. | |
| Jul 8, 2021 at 15:01 | comment | added | DKNguyen | @BenVoigt Proofs for sure, but I never ran into the concept of a proof until university, and I only ran into it because I took Math Honours in first year. | |
| Jul 8, 2021 at 15:00 | comment | added | Ben Voigt | @DKNguyen: Proofs (often taught in highschool geometry) require just as much out-of-the-box thinking if not more. | |
| Jul 7, 2021 at 21:36 | answer | added | Andrew Ross | timeline score: 7 | |
| Jul 7, 2021 at 17:42 | comment | added | DKNguyen | @RonJohn That's what I always assumed. Integration requires flexible out of the box thinking quite unlike any math you encounter before university. | |
| Jul 7, 2021 at 16:20 | comment | added | RonJohn | @DKNguyen is that why it's taught before integrals? :D | |
| Jul 7, 2021 at 14:40 | comment | added | RonJohn | I found derivatives a LOT easier than integrals. | |
| Jul 7, 2021 at 14:39 | answer | added | George Menoutis | timeline score: 19 | |
| Jul 7, 2021 at 11:23 | answer | added | guest troll | timeline score: 4 | |
| Jul 7, 2021 at 10:11 | comment | added | tea-and-cake | @MichałMiśkiewicz I recall that Hersh and Davis asserted something like "the integral part of calculus was known to the ancients", in the sense that ancient Greek mathematics used limits of sums to compute areas and volumes, which is pretty close to the way we'd do it using integrals now. Of course the notation and conceptualisation were rather different, but as the results are the same, it seems a not unreasonable claim to make. Of course there was then a millennia-long gap before Newton and Leibniz related the (also ancient) work on derivatives to integrals. | |
| Jul 6, 2021 at 22:48 | answer | added | paul garrett | timeline score: 8 | |
| Jul 6, 2021 at 18:59 | answer | added | alephzero | timeline score: 19 | |
| Jul 6, 2021 at 18:57 | answer | added | DRF | timeline score: 8 | |
| Jul 6, 2021 at 17:09 | comment | added | Michał Miśkiewicz | Can you add some justification for the claim that this is also how calculus was historically developed? As far as I know, integration theory was not developed much before differentiation was introduced, but I certainly don't know all the facts, and I'm curious to know. | |
| Jul 6, 2021 at 16:50 | history | became hot network question | |||
| Jul 6, 2021 at 14:16 | answer | added | Gerald Edgar | timeline score: 19 | |
| Jul 6, 2021 at 14:00 | answer | added | Steven Gubkin | timeline score: 37 | |
| Jul 6, 2021 at 11:43 | comment | added | Adam | b/c Antiderivatives require derivatives. | |
| Jul 6, 2021 at 8:05 | comment | added | user13234 | Also, as mentioned in the math SE post linked above, it seems that many authors prefer to present differential calculus first, whereas a few authors prefer to present integral calculus first. Perhaps there are merits to both approaches. Regardless, maybe this question should be marked closed. | |
| Jul 6, 2021 at 7:54 | comment | added | user13234 | I'm currently skimming an online text Calculus Made Easy: calculusmadeeasy.org. This text seems to start with Leibniz dy/dx notation to begin with rather than the limit and tangent/secant line approach. It presents differential calculus before integral calculus. It seems to tie these concepts together well. Perhaps Leibniz notation and differentials are indeed why. | |
| Jul 6, 2021 at 6:37 | comment | added | J W | See also math.stackexchange.com/questions/245047/… | |
| Jul 6, 2021 at 5:57 | comment | added | Michael Bächtold | All of the first calculus textbooks I know treat differential calculus before integral (L'Hospital, Bernoulli, Euler). With Leibniz notation it also seems clear why: differentials appear in $\int y dx$. | |
| Jul 6, 2021 at 4:28 | history | asked | user13234 | CC BY-SA 4.0 |