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[Powder Miner]

Well, I'm more than a bit confused about something for my calculus midterm tomorrow -- something I thought I had learned is being contradicted by a midterm preparation question and answer set the teacher made - but it looks like the answer he have may be flawed.
Does the sandwich/squeeze theorem ensure that for the following piecewise equation:

Code: [Select]

f(x)={xsin(1/x) x≠0
     {0         x=0f'(0)=0?

He said no -- based on the fact that, using the definition lim(h->0)(sin(1/h)) has no value, but he used sin(1/h) in that statement, NOT hsin(1/h) and I think that might well be a critical difference. I'm not sure though, because this is hard as all hell to input into something like wolfram alpha and the things available on the internet are absurdly contradictory -- people on sites like stack exchange saying that no, it doesn't exist, but a document from a different teacher AT MY VERY UNIVERSITY for THIS VERY COURSE talking about the sandwich theorem potentially indicating it is 0

[MagmaMcFry]

The derivative of a function f at 0 is lim_h->0(f(0+h)/h). Inserting the definition of this particular f gives lim_h->0(h*sin(1/h)/h) which simplifies to lim_h->0(sin(1/h)). This is where the sin(1/h) comes from.

[Powder Miner]

Oh, that makes sense.
Does this also apply to (x^2)sin(1/x)?
I guess it would.

[MagmaMcFry]

It applies similarly, but x²sin(1/x) does in fact have a well-defined derivative at 0.

[Powder Miner]

What is it that makes the sandwich theory apply to x²sin(1/x) but not xsin(1/x)?

[Powder Miner]

The odyssey of preparing for my midterm tomorrow continues, and now I desperately need to shore up my primary weak point: related rates.

This is an excerpt from the practice problems (not the actual test problems), and I need help with number 5.
The rate I have is dy/dt and the one I want is dθ/dt, so I took θ=sin^-1(y/h) and went to dθ/dt=1/(sqrt(1-(y(dy/dt)/h)^2)*((dy/dt)/h)) thanks to the chain rule and h being a constant 10
Then dθ/dt=1/(sqrt(1-(sqrt(75)*1/10)^2)*((1)/10))=1/(sqrt(1-75/100)*1/10=1/sqrt(1/4)*1/10=2*1/10=1/5
The answer is 1/20. What is my folly?

[Reelya]

Maybe you're meant to use tan^-1 here not sin^-1

The value of h=10 is not a constant, it changes as y moves. The parallel distance to the board of 5 is the constant.

[Powder Miner]

oh, my god
that did it
i blew so much effort into that and it was just that one base thing wrong -- that made it so SIMPLE

A. thank you
B. this is why i hate related rates

[hops]

So I have a weird question:

Riddle: Assume decimal representation, say that a digit sum is the sum of all the digits in a number. What is the smallest natural number n whose digit sum and the digit sum of its next number n+1 are both divisible by 10?

I want to prove that n must have at least one 9 in its digits. I don't know how to explain why, but I have a hunch that's true.

I also want to prove that n exists.

I want to prove that n doesn't exist.

[TheBiggerFish]

So you can't prove that n doesn't exist.  Because there is an n that satisfies.

[MagmaMcFry]

Smallest positive example: 18999999999.

[hops]

How do you work that out? I was bruteforcing it for a computer science assignment but it's taking a billion years to find an answer which was why I got curious.

[MagmaMcFry]

When you increment a number, its digit sum changes by +1, or by -8, -17, -26, etc. The "smallest" change divisible by 10 is a change by -80. So either the digit sum goes from 80 to 0 (which isn't possible), or from 90 to 10, and the smallest number with digit sum 90 that satisfies this property is 18999999999.

[hops]

This is a stupid question but how do I prove that a sequence is bounded? I always just find the limit before, but for some sequences that doesn't really work. I have absolutely no idea how to do this, because it's not like I can use standard induction because the assumption is useless?

[Dozebôm Lolumzalìs]

Umm... take the absolute value of the sequence, find some M such that a_n < M for all n?

Also why would induction be useless?