Closest 3dis to 2dis

[vtk]

I don't remember - how do we prove that cos(x) < sin(x)/x as x approaches 0?

Start with tan(x)>x (I don't remember how best to prove this, but I think you can use the fact that the derivative of tan(x) (which is sec^2(x)) is greater than or equal to 1, to the curve is above a line with derivative equal to 1).

I'm sure there must be a clever proof that cos(x)<sin(x)/x without relying on derivatives.  Unfortunately I often have trouble remembering clever proofs.

Who'd have thought we'd derail a thread so badly over calculus?

[agentsteel53]

I'm sure there must be a clever proof that cos(x)<sin(x)/x without relying on derivatives.  Unfortunately I often have trouble remembering clever proofs.

yes, I feel like using L'Hopital's rule would be circular reasoning.  I don't know the validity here of using Taylor series for sin(x) and cos(x) (as those are generated using derivatives), though doing so does make the proof very easy:

x cos(x) < sin(x)

substitute Taylor series...

x(1-x^2/2!+x^4/4!...) < x-x^3/3!+x^5/5!...

x - x^3/2! + x^5/4! ... < x - x^3/3! + x^5/5! ...

0 < x^3(1/2!-1/3!) - x^5(1/4!-1/5!) + x^7(1/6!-1/7!) ...

as x approaches 0, the most significant term becomes the x^3 term, implying 0 < x^3/3.  since x is positive, this is true.

[agentsteel53]

Start with tan(x)>x (I don't remember how best to prove this, but I think you can use the fact that the derivative of tan(x) (which is sec^2(x)) is greater than or equal to 1, to the curve is above a line with derivative equal to 1).

yep.

tan(x) > x

sin(x)/cos(x) > x

cos(x)/sin(x) < 1/x

cos(x) < sin(x)/x

so if we assume that we know that tan(x) > x and sin(x) < x as x approaches 0 from the positive side, we can use the squeeze theorem to prove that lim(x->0+) sin(x)/x = 1, and probably also lim(x->0+) tan(x)/x = 1 but I will not attempt to prove the second here.

to prove lim(x->0-) sin(x)/x = 1 and lim(x->0-) tan(x)/x = 1, we probably do the exact same proof with certain signs flipped.

[agentsteel53]

Who'd have thought we'd derail a thread so badly over calculus?

I'm surprised I remember most of this.  apart from the infinite series stuff, which I use as back-of-envelope calculations all the time, I had not used most of this since high school and college.

[thenetwork]

What about 3di's that cross or run concurrent, like I-540 in Arkansas which has a 5 Mile duplex with I-40 ?

I think someone mentioned I-87 and I-287 in NY as an example of that.

I can't remember. Are 40 and 540 both signed on the concurrency? Or is it only 40 that's signed?

WB, both are signed. EB, only 40 is signed, except at Exit 7 (540 South)

That sign on the left could qualify under the Worst Of Road Signs thread.  Each set of numbers on the sign shields goes higher than the other.
....and its Clearview (yecch!!!).

[Kacie Jane]

Who'd have thought we'd derail a thread so badly over calculus?

I'm surprised I remember most of this.  apart from the infinite series stuff, which I use as back-of-envelope calculations all the time, I had not used most of this since high school and college.

This thread demonstrated to me why I topped out in college mathematics at right around this level.  Once I started attempting 400-level classes (diff-eqs, etc.), I just totally hit a wall.

[hbelkins]

All of this math is literally giving me a headache.

Yeah, my head hurts. I had all that stuff in high school and promptly forgot it.

[vdeane]

tan(x) > x

Let x = 0.
tan(0) > 0
0 > 0

[Alps]

tan(x) > x

Let x = 0.
tan(0) > 0
0 > 0

No, no, no, lim(tan(x)) as x->0 = 0. I knew a girl named Lim and she was not tan.

[Scott5114]

cos()