Math question

[sivullinen]

Hey flist,

how would you solve 2we^(w^2)-2e^w=0?

(And just to make sure, it is the f'(w) of a function f(w)=e^(w^2)-2e^w, right?)

I hope there's at least one math geek among you :S

ps. I usually love this kind of math problems, but for some reason now I can't do it :(

Comments

[ladyphoenixia]

What you do is you set a new variable, say, y=e^w

Then you just have to solve y^2-2y=0
So y = 2 or 0
e^w = 2 or 0
w = ln2 or ln0 (but ln0 doesn't exist, o'course)

...it's been a while. Stupid maths. Evil subject.

[mara_202]

But if you take y^2 = (e^w)^2, don't you get e^(2w) instead of e^(w^2)?

[sivullinen]

So if y=e^w, is e^(w^2) y^w? I'M SORRY I'M HORRIBLE WITH EXPONENTS. :P

[mara_202]

Um... *long silence* I'm not really all that good with exponents either. ^^;; I think that's true, but I'm not 100% sure...

[mara_202]

...maths geek here.

I have no idea how to solve this, though. I suppose you're not allowed to solve it graphically? (The derivative is correct, by the way.) The only solution I see is the obvious one, w = 1 (2*1*e^1-2*e^1=0), but I can't get there by calculating it.

Sorry I can't be of more help. :S

Still love your icon. :P

[sivullinen]

The only solution I see is the obvious one, w = 1

Yeah... that's what I get too, when I count the zero point with my calculator. And because this is only part of a problem (the real problem is finding the max/min for a f(x,y,w) function), if I can't figure it out I'll just do the rest of the problem with no explanation how I got w=1, but... I love algebra, so I would like to know how to it :D Plus, a perfectionist when it comes to math

I have tried like three different ways, but always get stuck. *sigh*

Thanks for help, anyway! Maybe our teacher made a mistake giving us that kind of function, if no one can figure it out! :D

[aiwritingfic]

I've forgotten most of my math, but. Can you solve wy^(w-1) = 1? If so, here is the work:

2we(w^2) = 2e^w
we^(w^2) = e^w
Let e^w = y
we^(ww) = wy^w = y
wy^(w-1) = 1

ETA: Will direct some math people your way. Also, try math_help?

ETA2: Image here:

[sivullinen]

Can you solve wy^(w-1) = 1?

Uh, I don't think so? I have tried solving w(e^(w^2-w))=1, which is like the same thing but with e^w instead of y, right? *sigh*

I'm starting to think, since no one hasn't figured it out, that our teacher made some kind of a mistake giving us that function :P (Then again, it HAS an answer, w=1, so we should be able to solve it, right?)

Thanks for help (and for directing others here)! :D I'll be sure to post the solution for you, on Wednesday at the latest when I have that class ^^

[aiwritingfic]

*laughing* I was going to tell you that w=1, actually, but I couldn't tell you how I got it except that it was the trivial answer. ^_^;;;;;;;;; (Because 1 makes a LOT of things easy to answer.)

ETA: I have tried solving w(e^(w^2-w))=1, which is like the same thing but with e^w instead of y, right?
Looks like a yes to me, which just looks like it's going around in circles. ^_^;

ETA2: Perhaps the teacher doesn't want you to calculate the answer, but wants you to do it using graphs? Since you need the max/min anyway.

[sivullinen]

No, we never would use graphs, so that can't be right answer... *scratched head*

If this collective mind can't figure out the answer, I think I'll soon give up trying to do it by myself :D

[macey_muse]

Solved! (here from Ai, btw ^.^)

Right, so, 0 = 2w*e^(w^2) - 2e^w = 2w(e^w)^w - 2e^w

0/2 = e^w*(e^(w-1) - 1) (factorizing the e^w, and the 2's pointless with a 0 result)

Which proves e^(w-1) = 1

w-1 = ln|1| = 0

w = 1

Which, using my graphical calculator, is the only solution for w as a member of the real numbers.

[macey_muse]

Image, because writing with ^'s confuses me:

[macey_muse]

...waitasec, mace, that's wrong. I scribbled out a 'w' I shouldn't've. If I hadn't, I would have ended up with w + ln|w| = 1 which isn't helpful. Grr.

It continues down to w = e^(1-w) which is -simpler-, but if I don't change now I'll be late for work, so I can't chase further. That'll teach me to leap before looking! M'sorry ~ *blushing*

[sivullinen]

Oh, I already got excited, but yeah, then I noticed the same mistake you did :D (Don't worry - I've tried like three different and always did some stupid mistake with this >__<)

Thanks for help! :) I'll just use w=1 as a fact, probably, and leave it for the teacher to explain why it is so, or something.

(ICON LOVE)

[thephoenixboy]

Directed here by aiwritingfic.

Looking through the other comments, my method was much the same as shown on her picture, with a similar result. The only solution I could find algebraic was w=1 - which could very well be the only solution. I then stuck into into my maths package on my laptop. It's possible that I coded it wrong, but either way it didn't come out with anything more helpful so we could well be right.

If you're using that eqn to find max/min, you're not expecting a whole string of answers for an eqn involving e.

Sorry I couldn't be of more help

[sivullinen]

The only solution I could find algebraic was w=1 - which could very well be the only solution.

Yup, it is. So, did you have some actual way for getting it? Or was it just a conclusion reached by looking at it, or something?

Thank you, in any case! ^^

[thephoenixboy]

Ok

Let y=e^w cos it makes it easier to look at

2wy^w-2y=0
Divide by 2
w(y^w)-y=0
Take out a factor of y
y(wy^(w-1)-1)=0
So either y=0 or w(y^(w-1))=1

Replace y by e^w
e^w=0 never happens, so no solutions from this factor

Rearranging the other equation
y^(w-1)=(1/w)
By recognition, if you take w=1, y^(w-1)=1 because anything to the power 0 is equal to 1. Also 1/1=1, so w=1 is a solution.

I've played around with the second factor some more and can tell that there aren't any other integer solutions but off the top of my head I can't prove that there aren't any non-integer solutions.

[macey_muse]

Argh yes, of course ~ I got to w = e^(1-w) (slightly different route, and destination) which of course, w has to equal 1. Anything bigger and RHS gets fractional, smaller and we get weird roots of e and strangeness. (Although, lack of -proof- makes me twitch)