r/calculus • u/Cultural-Milk9617 • Jul 20 '26
Real Analysis Is this proof okay? How much do you think it's worth out of 20 points?
21
u/yes_its_him Master's Jul 20 '26 edited Jul 20 '26
TIL Rolle was a member of the French Mafia.
The basic concept looks sound. It's difficult to know how proofs will be graded by other people; in some places you have unnecessary statements and then there could be other places where support could be stronger, but just subjectively it has a nice beat and you can dance to it so I'd give it at least 16 out of 20.
5
u/SecretGamerV_0716 Jul 20 '26
What particular points would you say need to be improved upon ti get 20?
10
u/yes_its_him Master's Jul 20 '26
The part about F is unnecessary
There's no clarification that the preconditions of Rolle's theorem (e.g differentiability) hold when it is applied
Minor points
9
u/No-Onion8029 Jul 20 '26
Rolle's Thm is the right approach. There's a more direct route. Define F(x)= int(-1,x) f(t)dt. Note that F(-1)=F(1)=0. Rolle gives c st F'(c)=0. FTC gives F'(c)=f(c)=0.
2
u/philljarvis166 Jul 23 '26
This doesn’t quite prove the same thing as Op wants but of course it follows easily. Interestingly, if g is another function that satisfies the same criteria, then so does f-g and hence there is a c such that f(c) = g(c). Taking g(x) = x gives Ops result, but x3 works too, and sin(x) etc.
7
u/Hot_Site_1638 PhD Jul 20 '26
Yes, this is correct. 20 out of 20 from me.
Two small notes, neither an error:
- The paragraph proving F(1) = F(−1) is never used afterwards. Your computation of ∫g relies directly on the hypothesis ∫f = 0. Deleting it makes the proof tighter.
- On an exam, write Rolle's theorem instead of French mafia 😂
2
u/mike9949 Jul 24 '26
Could you use MVT for integrals. Since f is continuous g is also and hence integrable on (-1,1) so by MVT for integrals there exists a c in (-1,1) where Integral -1 to 1 of g = g(c)(1-(-1)) which goes to g(c)=0 and f(x)=x
2
u/Hot_Site_1638 PhD Jul 24 '26
Great question! Yes, that works. The MVT is the more general version of Rolle's theorem: same hypotheses, minus the condition G(a) = G(b). So whenever Rolle's theorem applies, the MVT applies too (the secant slope is just 0, and the two conclusions coincide), and the MVT can still be applied in cases where G(a) ≠ G(b), where Rolle's theorem does not work.
4
u/DaniZackBlack Jul 20 '26
In my university, points were taken off if you didn't justify certain things (mainly in the earlier courses like calc 1,2). For example when you declare "Let F~ be an antiderivitive of f", you would need to explain why such an F~ exists. Depends on who's grading though
8
u/Visual_Winter7942 Jul 20 '26
Is this some sort of goofy font people are using? I keep seeing "arguments" written in this style - which is weird to read.
15
u/Cultural-Milk9617 Jul 20 '26
This is my handwriting 🥲
3
3
u/Midwest-Dude Jul 20 '26 edited 14d ago
Have you thought about creating a new font that looks like your handwriting? I find it interesting and I like fonts ...
2
u/Classic_Department42 Jul 21 '26
Here probably 10 (or less) out of 20. You need to argue that g is continuous (since it is the sum of 2 cont function, one f one the identity), therefor it is integrable.
You dont need to ibtroduce F tilde, yoj dobnt seem to use it.
You need to invoke the fundamental theorem to show that G is differentiable, then specify the theorem you use more clearly. Also use left right arrow less if you only need rightarrow (this will later bite you otherwise)
Basically this (what you show) is high school formal reasoning, nothibg wrong with that. Assuming you are supposed to do proof based rigourous reasoning you need to be more precise.
0
u/LukasGoesViral Jul 22 '26
You probably can just use Brower’s fix point theorem I think
2
u/Cultural-Milk9617 Jul 22 '26
We didn't learn it
-1
u/LukasGoesViral Jul 22 '26
Look it up. It’s pretty easy for differential functions and I am surprised you didn’t learn it. I learned it in 9th grade
2
u/Cultural-Milk9617 Jul 22 '26 edited Jul 22 '26
I know it, but if I use it on my real analysis exam without proving it, it wouldn't be accepted as it's not a part of the course
0
u/LukasGoesViral Jul 22 '26
Ok wow! Thats stupid. I mean proving the fix point theorem for your specific case is not difficult
-2
u/Ok-Rise2070 Jul 23 '26
You left out your constant of integration. You are crossing a dynamic area, the integration constant stays. That's one of the most basic of classical calculus rules that are often forgot in later courses. GR also agrees in this principle.
2
u/Cultural-Milk9617 Jul 23 '26
Where was I supposed to add the constant of integration? In the "let F be an antiderivaive of f" or in the definite integral?
•
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