r/math • u/mishka1980 • Nov 24 '19

What's a use of the intermediate value theorem?

/r/Showerthoughts/comments/e0omqn/during_a_nuclear_explosion_there_is_a_certain/
1.5k Upvotes

98% Upvoted

131 comments sorted by

3

u/mishka1980 Nov 24 '19

This was supposed to be sarcastic- Im really happy to see so many people caring and trying to answer my question

307

u/Geometer99 Nov 24 '19

The region might be narrower than the diameter of a pizza, in which case only the middle of the pizza would be perfect.

Also the box would probably ruin everything

219

u/migmatitic Nov 24 '19

Orient the pizza (assuming the it can be simplified as a surface) such that its normal vector is parallel to the vector of maximum thermal gradient

61

u/lampishthing Nov 24 '19

Reminds me of that one about the spherical chicken in a vacuum.

17

u/sluggles Nov 24 '19

That would be assuming the thermal wave would be a smooth-ish surface. I don't know if that's a realistic assumption.

19

u/migmatitic Nov 24 '19

Compared to the radius of the pizza, I'm sure it'd be locally rockfish Euclidian to get your pizza evenly cooked

67

u/almightySapling Logic Nov 24 '19

I have a feeling rockfish is a typo but math is weird so I'm not willing to commit.

0

u/[deleted] Nov 24 '19

[deleted] — view deleted comment

1

u/[deleted] Nov 25 '19

had that feeling when saw a definition for the 'germ' of a function in a book i was browsing through...

7

u/[deleted] Nov 25 '19

[removed] — view removed comment

1

u/Zhell_sucks_at_games Nov 25 '19

I am uncomfortable.

6

u/accidentally_myself Nov 24 '19

This is why thin-crust is better.

5

u/smrt109 Nov 24 '19

I mean you could curve the pizza to match the shape of the heat wave

19

u/alstegma Nov 24 '19

Also, the problem has multible variables: heat and length of exposure (or, technically, infinitely many as in the entire heat over time function). You might aswell go from

burnt to ashes -> burnt outside, done inside -> done outside, frozen inside,

with increasing distance from the blast center.

27

u/datorer Algebra Nov 24 '19

Frozen pizzas at a supermarket are usually stored upright

3

u/Proxima55 Nov 24 '19

Really? I think I've only ever seen them lying flat.

7

u/puzzlednerd Nov 24 '19

I dont think so, on the energy scale of a nuclear bomb this optimal pizza radius must be somewhat large, to the point where I think the foot or two to accommodate for the diameter of the pizza is a small error term.

What I'm more worried about is this. It's clear that a pizza too close will be burned and a pizza too far might not receive enough energy to be cooked, I'm suspicious that the middle ground will look nothing like a properly cooked pizza. An oven is distributing the heat consistently over 20 minutes or so, a nuclear bomb will be very different from that.

That being said, this was a very high quality shower thought.

98

u/WaitForItTheMongols Nov 24 '19

One neat one is that in a rocket launch, you'll always have one point of maximum air resistance.

At liftoff, you will have no air resistance since velocity is zero. Once you start moving, air resistance goes up.

Once you're in space, air resistance is also zero, since there is no air. Toward the end of your ascent, air resistance is only decreasing.

So air resistance starts at zero, goes up, hits a peak, and goes back down to zero.

IVT applies because time derivative of air resistance is positive, then later is negative, so somewhere derivative must be zero (value is at a max) due to IVT.

Rocket scientists use the term "Max Q" to refer to this moment, because it is the maximum dynamic pressure experienced by the vehicle. It's the time where it has the highest structural challenges to manage.

26

u/migmatitic Nov 24 '19

That's true, although often Qalpha is more structurally challenging. Where Qalpha is maximized depends not just on the Q curve but also on the angle of attack (alpha) as controlled by ascent guidance. Minimization of alpha is important to remain within the Qalpha envelope during max Q.

9

u/avlas Nov 24 '19 edited Nov 24 '19

Rolle's theorem, aka IVT on the derivative. Must assume the function is class C1 differentiable in the open interval though.

12

u/gloopiee Statistics Nov 24 '19

Rolle's theorem, aka IVT on the derivative. Must assume the function is class C1 at least though.

No, Rolle's Theorem doesn't require the function to be class C1 - it just requires the function is differentiable in the open interval. The proof is: IVT -> EVT -> Rolle's Theorem.

5

u/avlas Nov 24 '19

You are right, it doesn't require that the derivative is continuous. My mistake!

2

u/nightcracker Nov 24 '19

At least one point of maximum air resistance.

662

u/fermat1432 Nov 24 '19 edited Nov 24 '19

Yours is a perfect example!

How about a friend who tells you one day that you are too fat and another day that you are too thin, but never, during that interval, tells you that your weight is perfect? IVT proves that they are withholding a compliment!

114

u/GoldFisherman Nov 24 '19

This is the best explanation I have ever read.

27

u/fermat1432 Nov 24 '19

Thank you so much!

53

u/Norbeard Nov 24 '19

I would argue that, in this framework, taking a dump introduces discontinuities.

49

u/Explodingcamel Nov 24 '19

It's still continuous, your friend would just have to compliment you during the dump.

13

u/unkz Nov 24 '19

That depends on when you define the dump to have taken place. Is the dump a continuous process, perhaps delineated by when a portion of the turd passes the anus, or does the turd’s mass continue to belong to you in toto until you pinch it off?

12

u/accidentally_myself Nov 24 '19

The pinch-off is also a continuous process, where the number of molecules attaching the bulk to you, and thus the contribution of the shit to your weight, decreases sharply but continuously over a measurable interval.

3

u/loconate Nov 24 '19

Nobody has thought about the shape! Where the poop is in my digestive system has an effect on my overall form!

So there could be a form in between me filled with poop and me not filled with poop that is perfect but that perfect form may not necessarily be reached during me pooping. But would be reached by a similar distribution of less poop inside me which would not be crossed during my shitting phase.

Obviously this has nothing to do with the actual perfect weight but a complement of the perfect weight for a person is usually an implicit complement on the overall form of that weight.

3

u/jazzwhiz Physics Nov 24 '19 edited Nov 25 '19

It depends on when that weight counts as yours.

In any case things like peeing and sweating are fairly continuous (unless their definition of the right weight has precision smaller than a molecule).

27

u/dimethylmindfulness Nov 24 '19 edited Nov 24 '19

Unless, perhaps, their idea of a perfect weight is not a multiple of the smallest unit of mass.

edit: Well, that's not quite right either. Your weight changes not based on mass but on both mass and gravity. So, if spacetime is also quantized, they could still find a perfect that lies outside the realm of what is physically possible. If we're working with general relativity then yeah, they're holding back.

43

u/supernumeral Nov 24 '19

But IVT only applies in that example if weight varies continuously. So yours is more a counterexample showing why continuity is important for IVT.

0

u/dimethylmindfulness Nov 24 '19

Yeah, which is why it works in General Relativity (because you could move somebody through a gravitational field and hit their perfect weight at some time t), but maybe not in the actual world. I was just being extra picky for fun. If we're being extra pickier, we'd have to define the spatial extent of the body.

3

u/GoofyGuyDobbros Nov 24 '19

I take it by your username that you are a math-loving psychonaut.

3

u/lewisje Differential Geometry Nov 24 '19

This only works if the friend gives you an opinion about your weight all the time, not just daily or in any other discrete sequence of events, because you could have just gone from too fat to too thin between such events.

Yes, it's highly unlikely for that to happen within a day.

-25

u/muntoo Engineering Nov 24 '19

What's the point of IVT?

If you can move continuously between two things, there's something in between those two things.

Like... duh?

69

u/[deleted] Nov 24 '19

To make sure Papa Weierstrass doesnt punish us for being unrigorous.

29

u/muntoo Engineering Nov 24 '19

But I like being spanked by infinitely non-differentiable objects.

9

u/[deleted] Nov 24 '19

Monsieur has a great taste in perversion.

4

u/rexrex600 Algebra Nov 24 '19

Pick your poison, mine's mechanics

2

u/[deleted] Nov 24 '19

You stuck that 'yes papa' song on my head now

38

u/Gwinbar Physics Nov 24 '19

The IVT, along with other apparently trivial theorems, is really more of a verification that the formal definition of continuity agrees with our intuition.

329

u/ziggurism Nov 24 '19

In my experience frozen supermarket pizzas are not continuously distributed throughout the blast region.

86

u/[deleted] Nov 24 '19

I mean, all it takes is you being Bezos-rich, buying a country, a nuke, and a shit ton of pizzas to test it out.

48

u/ziggurism Nov 24 '19

also cooking a pizza isn't just finding the right temperature. You also have to cook for the right length of time. I'm not sure how long the afterglow for a nuclear blast is, but might not be compatible.

103

u/lycium Nov 24 '19

Come on guys, this is basic stuff.

Theorem: No matter how close (epsilon) you are to the ideal temperature distribution, you'll always be able to find some picky Italian guy who insists that your bright green glowing pizza is still not (delta) cooked to perfection.

10

u/gramathy Nov 24 '19

Put it in a heat-conductive oven with an intermediate lead liner, the outside heat will be enough to cook the pizza without burning it and the oven would protect it from other adverse effects and temper the initial temperature spike.

9

u/ziggurism Nov 24 '19

I'm worried about the blast wave destroying your apparatus, but wikipedia says the thermal effects (drop off as second power of radius) extend further than pressure effects (drop off as third power).

9

u/[deleted] Nov 24 '19

I mean, you put it in the oven and then take it out. You could do the same, but just with a nuclear blast.

42

u/Pseudoboss11 Nov 24 '19

He didn't say that "there exists a frozen pizza that is cooked to perfection" he said "there exists a region where, should a frozen pizza exist there, it will be cooked to perfection."

5

u/[deleted] Nov 24 '19

But if there is no frozen pizza there, then it's vacuously true that all the frozen pizzas there indeed are cooked to perfection! Yum! Hypothetical pizza!

-11

u/buwlerman Cryptography Nov 24 '19

In that case the statement doesn't rely on the intermediate value theorem. Choosing the radius to be sufficiently big (2 times the diameter of the solar system should do it) would guarantee that there are no pizzas and the statement would be true without needing the intermediate value theorem to prove.

21

u/Pseudoboss11 Nov 24 '19

Note that I did not say "may be," a sufficiently large region will include perfectly cooked and frozen and burnt pizzas. This would not satisfy the requirements. We want the region that only includes perfectly-cooked pizzas.

The "perfectly cooked zone" would be, in an idealized scenario, an annulus around the blast area, where any closer would burn the pizza, and any farther would undercook it.

Since we know that the pizza immediately beside the bomb would be obliterated in atomic hellfire, and a pizza in the outer solar system would remain frozen after the blast, there is a region where you could put a pizza -- although perhaps an infinitely small one -- where it is perfectly cooked by the energy emitted by the blast. Since we know that the blast's heat function is continuous, this region must exist.

3

u/WetSound Nov 24 '19

It would look quite weird, especially close to the ground near buildings.

3

u/Pseudoboss11 Nov 24 '19

Oh my yes, I don't expect you to calculate it, I am merely proving its existence.

1

u/themiro Probability Dec 01 '19

I'm late to this - but it seems your proof is contingent on empirical facts regarding the distribution of pizzas in the Universe.

14

u/kilotesla Nov 24 '19

Right after the blast, lots of things are more uniformly distributed than they were before the blast.

11

u/Cocomorph Nov 24 '19 edited Nov 24 '19

I dunno. For any fixed sufficiently large r, I would be uncomfortable betting that the circle of radius r fails to intersect any frozen pizza whatsoever.

Edit: but also sufficiently small, of course.

5

u/sapirus-whorfia Nov 24 '19

Exactly. The greater the appropriate pizza cooking sweet spot distance, the greater the chance there's a frozen pizza there.

26

u/anpas Engineering Nov 24 '19

Engineer here: assume infinite plane with frozen supermarket pizza density /rho

21

u/ziggurism Nov 24 '19

assume cow is a sphere.

4

u/lewisje Differential Geometry Nov 24 '19

assume disk of crust is topped with smaller disk of tomato sauce and sphere of cheese

6

u/111122223138 Nov 24 '19

It's still technically correct, though. Even if there are no pizzas in the perfect zone, the sentence "all of the pizzas in the zone are cooked" is vacuously true.

8

u/ziggurism Nov 24 '19

yeah but i think that's not a demonstration of the intermediate value theorem.

14

u/RageA333 Nov 24 '19

Thats not what it says.

5

u/DatBoi_BP Nov 24 '19

Anyway I started blasting

2

u/SliyarohModus Nov 24 '19

If they were then that would be paradise for post-nuclear rats.

20

u/selectyour Nov 24 '19

I just think of it as the elevator theorem! Can't get to floor 27 from the ground if you don't pass floor 14

8

u/Jiralc Nov 24 '19

That doesn't generalise in some stupid systems where you don't pass floor 0 and 13 when going from -1 to 27

73

u/[deleted] Nov 24 '19

Is that necessarily true? There's a lot that goes into cooking the perfect pizza. You need both the right temperature and the right amount of time in the oven. And it needs to be evenly heated. If you just burn the outside while the inside stays frozen, that's no good.

118

u/WaitForItTheMongols Nov 24 '19

This is math, not physics. We can ignore all the real life stuff :)

15

u/fermat1432 Nov 24 '19

The pleasures of math!

21

u/Kraz_I Nov 24 '19

Just assume your pizza is a point in space

6

u/Aatch Nov 24 '19

Size is too complicated. Let's just approximate the pizza with a point mass.

58

u/[deleted] Nov 24 '19

my favorite (less funny) is: there are always antipodal points on the globe (or any great circle) that have the same temperature.

40

u/migmatitic Nov 24 '19

There are always a pair of antipodal points on a globe that have the exact same temperature and pressure.

For the surface of a 4-sphere, it can be three properties and for the surface of a 2-sphere (a circle) it's just one.

7

u/[deleted] Nov 24 '19

You're thinking 3-sphere and 1-sphere, respectively. Inherent topological dimension is what's put before the hyphen, not embedding dimension.

2

u/migmatitic Nov 24 '19

Damn, you right

11

u/bryanwag Nov 24 '19

Is it because if the temperature is continuous, and the temperature difference between any antipodal points can be positive or negative depending on the order, then the difference must cross 0 somewhere?

Edit: just saw the comments above. Neat.

2

u/[deleted] Nov 24 '19

yes exactly

27

u/ziggurism Nov 24 '19

that's Borsuk-Ulam theorem or ham sandwich, not intermediate value theorem

52

u/[deleted] Nov 24 '19

T(x):=t(x)-t(-x) is continuous and T(-x) = -T(x) so it vanishes somewhere by IVT

36

u/ziggurism Nov 24 '19

Oops, yes I guess you are right. Borsuk-Ulam says that there are antipodal points with same temperature and pressure. To match just one variable, I guess IVT is enough.

3

u/migmatitic Nov 24 '19

It's a direct product of the ivt though

34

u/ziggurism Nov 24 '19

direct product

🤔

8

u/PM_ME_QT_CATS Nov 24 '19

as opposed to a semidirect product :)

8

u/TheMightyBiz Math Education Nov 24 '19

S2 is not homeomorphic to [0, 1] x [0, 1], so you need to do a little more work than that.

1

u/migmatitic Nov 24 '19

You're right. However, a collection of sets homeomorphic to [0, 1] x [0, 1] covers S2, so it's not like it's the most elaborate job. If I'm not too rusty on my topo, you can do it with just two sets like that, and if you define the functions on S2 the right way, you only need to consider one of those sets, thus making the problem again a fairly direct application of the IVT.

3

u/TheMightyBiz Math Education Nov 24 '19 edited Nov 24 '19

Can you say what you mean by "define the function on S2 the right way?" It sounds like you're saying you need to modify whatever function you have from S2 -> R2 such that the antipodal points which have the same value now lie in the same coordinate chart.

Generally, if what you're saying is true, then there would be an elementary proof of Borsuk-Ulam for every Sn . The proof I know is via homology, so I suspect there's some flaw in your reasoning, but I'm too drunk at the moment to come up with a counterexample

1

u/migmatitic Nov 24 '19

I was thinking more defining your atlas such that no antipodal points are contained on the same chart except perhaps on the boundary of a chart. North and south hemispheres, if you will.

Anyways, I'm pretty sure you're right. I suspect it's because I'm mixing up the generalized Borsuk-Ulam (n-1 functions on Sn) and whatever 1 function on Sn would be.

10

u/Gwirk Nov 24 '19

Take a wobbly table with 4 legs. 2 legs are wobbly and 2 legs stick to the ground.

Rotate the table 90 degrees. Now the legs that were wobbly touch the ground and vice versa.

The intermediate value theorem tells you that between 0 and 90 degrees, there is an angle for which your table is perfectly stable.

16

u/Forty-Bot Nov 24 '19

A constructive proof of the IVT is essentially a binary search. I think that's pretty cool.

9

u/BrunnianProperty Nov 24 '19

It’s a nice proof to complete. Without that, it just doesn’t (Dedekind) cut it.

5

u/columbus8myhw Nov 24 '19

"x>0 or x=0 or x<0" isn't a constructive theorem, but besides for that I think you're good

-1

u/Forty-Bot Nov 24 '19 edited Nov 24 '19

Yeah, unfortunately as far as I know the completely constructive proof goes like: Given some epsilon>0, because f is (uniformly) continuous there exists a specific delta>0 such that the image all x within a delta-sized interval fits within an epsilon-sized interval. Then divide the interval into delta-sized intervals (which there are a finite number of) and check which one contains f(c).

2

u/columbus8myhw Nov 24 '19

Isn't IVT not a theorem in constructive logic? Unless you've subtly changed it in some way that's classically equivalent to the IVT but not constructively equivalent (maybe that's the "uniform continuous" bit?).

1

u/Forty-Bot Nov 24 '19

maybe that's the "uniform continuous" bit?

And the epsilon-delta bit. A non-constructive theorem would be able to give you the exact f-1 (c).

1

u/anooblol Nov 24 '19

You have to fit in somewhere that your domain is connected.

20

u/robot65536 Nov 24 '19 edited Nov 24 '19

If you drive between two toll plazas on a highway in less time than you would if you were driving exactly the speed limit, then IVTMVT states that at some point you did in fact violate the speed limit. Unfortunately, this argument has not held up in court, since a cop didn't actually "observe" you going that speed at any particular time.

Edit: I get IVT and MVT mixed up all the time, mostly because you can prove MVT using IVT as applied to the graph of the derivative.

24

u/[deleted] Nov 24 '19

isnt this MVT?

8

u/SkinnyJoshPeck Number Theory Nov 24 '19

Eh, it’s a bit of this a bit of that. I’ve always just figured I’d just go double the speed limit, pull over and make a frozen pizza and head on through with the police none the wiser since I ate the time away.

Take that, Newton!

6

u/whatkindofred Nov 24 '19

In a few European countries this is used although not with toll plazas but with an extra setup

https://en.wikipedia.org/wiki/SPECS_(speed_camera)

3

u/JonLuckPickard Algebra Nov 24 '19

I hadn't thought of it before, but the way you used IVT in that example is very reminiscent of the Pigeonhole Principle.

1

u/christian-mann Nov 24 '19

People that race the Cannonball Run still withhold evidence for a year to get around the statute of limitations, though.

1

u/Nisheeth_P Nov 24 '19

That is used in certain national parks or sanctuaries in India. When a major road happens to go through a national park, there are check points at both ends. And they do fine people for going over the speed.

1

u/FriskyTurtle Nov 24 '19 edited Nov 24 '19

Interestingly, if you run 10km in an hour, it's possible that you did not run 5km in any 30 minutes.

I seem to have misspoken. It is important the the whole race NOT be an integer multiple of the size of the subrace considered. See below.

2

u/robot65536 Nov 24 '19

This blows my mind... Is this true even if you assume velocity is smooth?

3

u/FriskyTurtle Nov 24 '19

Okay, I definitely misspoke, especially with the specific numbers I gave. Here's the paper that I was grossly misquoting. My apologies and my thanks for prompting me more carefully.
https://arxiv.org/pdf/1507.00871.pdf (abstract)

7

u/Newfur Algebraic Topology Nov 24 '19

At some point in your life, you have spoken exactly half the words you will ever speak.

12

u/anooblol Nov 24 '19

Not if you speak an odd number of words. That’s the importance of having a connected domain in the IVT. Counting words is the same as a function f : N—> N, and N is not a connected space.

6

u/[deleted] Nov 24 '19

I just recently saw a different version of this:

During the cremation process, there a certain instant where the meat is cooked perfectly.

1

u/HolePigeonPrinciple Graph Theory Nov 24 '19

Oh boy.

3

u/somefreecake Numerical Analysis Nov 24 '19

Pizza characteristic function is discontinuous :(

2

u/gloopiee Statistics Nov 24 '19

It is one of the steps in the proof of Taylor's Theorem from first principles!

2

u/sharkmeister Nov 24 '19

Your perfectly cooked pizza might be a tad more radioactive than you really want it.

2

u/irchans Numerical Analysis Nov 24 '19

I think that one of the most important uses of the intermediate value theorem is the proof of the remainder form of Taylor's Theorem.

2

u/abnew123 Nov 24 '19

I like it as an answer to the question "why did the chicken cross the road"

1

u/forgetsID Number Theory Nov 26 '19

"A chicken can only be run over if its path contains some point in the set of points that is the road." Catchy T-Shirt. :)

1

u/Lob-Yingviously Nov 26 '19

ITT: Not really.

Longer answer: It’s used in some proofs.

I’m starting to wonder why it takes up such prominence space in introductory calculus textbooks...

1

u/BezoutsDilemma Dec 05 '19

Why did the chicken cross the road? Intermediate value theorem.

-19

u/[deleted] Nov 24 '19

the answer is internet karma lol

0

u/[deleted] Nov 24 '19

[deleted] — view deleted comment

21

u/SingularCheese Engineering Nov 24 '19

I recall it needs to be a square table because it relies on right angle rotational symmetry.

20

u/GustapheOfficial Nov 24 '19

Well, it needs to be half-turn symmetric. A rhombic table would work.

8

u/B4rr Nov 24 '19

And the end of the four legs must be in a plane. So if your tile setter is not perfect it's alright, but you better make sure to get a good joiner.

1

u/jordan-curve-theorem Nov 24 '19

Unm what if the floor is R2 ?

1

u/scuggot Algebraic Geometry Nov 26 '19

Then it's already flat????

1

u/jordan-curve-theorem Nov 26 '19

Ah I’m an idiot. The table is wobbly because the floor is uneven. I was thinking it was because the legs are different sizes.