Need Calculus Help

[~Jimbo_69~]

Hello. I am working on a problem, and cannot seem to figure it out. In advance, any help is greatly appriciated:

Which is bigger? (I am only allowed to solve algebraically)

[wildrice]

What do mean by solving this algebraically?
Like no calculator to solve this?

[wildrice]

Is this is even calculus?

All I can think of is that number to the power of a smaller number. (i.e 3^5) is smaller than vice versa (5^3)
3^5 = 243
5^3 = 125

Therefore.
e = 2.718
pie = 3.14
so e^pie = 23.14
pie^e = 22.4

[Audio_Rookie]

I didnt do this in calculus. I did this in linear algebra though.....

[~Jimbo_69~]

Ya...This needs to be algebraic, as in no numbers... Kinda setting it up as y(1)=y(2)...and using derivatives from there...Can it be done algebraically?

[Horatio]

what's the exact question? The way it is here, any derivative you take is 0 because all you have are contants

I have an idea but I can't attach the image; shoot me an e-mail at [email protected] and I'll send it to you

Greg

[Audio_Rookie]

where does it say he has to take the derivative?

[~Jimbo_69~]

You don't have to take the derivative...It was just an idea that I thought may work...

[lint]

I think the question is worded poorly. y is used as the variable in both equations, but they are not meant to be equal.

Tell me if this is what the question is asking:

y = e^pi

x = pi^e

Which is larger? y or x?

[wildrice]

OO..I think i got it. One sec.

[wildrice]

This was my attempt for now. I didn't quite know where to go from there.

[no_joke]

I gave it a go... This is a tough question. This is a calc question but I tried to approach it "logically," so there's probably a more precise way of doing it. I wanna try to figure it out though so what chapter/unit is this question from?

[Seanith]

Originally posted by wildrice
This was my attempt for now. I didn't quite know where to go from there.

Thats the way I would have went about it.. take ln of both sides.

[turbotrip]

Originally posted by wildrice
This was my attempt for now. I didn't quite know where to go from there.

you get e ln PI = PI next, but i doubt this is the right way cuz y1 cant equal y2

[V6-BoI]

Hmm... I don't think you can solve this problem using algebra, or derivatives. I mean, nothing is changing, e, and PI are both constants. If you try to take the derivative of each y, you will get 0. Are you sure there isn't an x value in the equation somewhere? And what calculus is this for anyways?

[gram]

e to the pie=23.14069263277


pie to the e=22.4591577183610



:thumbsup:

[wildrice]

Originally posted by gram
e to the pie=23.14069263277


pie to the e=22.4591577183610



:thumbsup:

:banghead: :banghead: :banghead: :banghead:

How come all of US didn't think of that!:rolleyes:

[Phy]

Using wildrice's attempt

pi^e = e^pi

ln(pi^e)=ln(e^pi)

ln(pi^e)= pi

y = pi^e

ln(y) = pi

y = e^pi

ln(e^pi) = pi

pi = pi

pi^e = pi^e

y = y

That doesn't change the fact that it's completely broken at the start anyway. pi^e != e^pi, therefore y != y.

Edit: Probably shouldn't use this... proving a statement x=y by using the statement x=y in your proof is bogus.

[gram]

Here is the full equation and how it makes sense:



e = 2.7182818284590452353602874713527
pi = 3.1415926535897932384626433832795


in the 2 equations substitute the variables


You can get the value of e by doing the following in a scientific calc:

punch in "1" then hit the "inv" button and then hit "ln" (natural logarithm)


So easy




It's so easy:thumbsup: :thumbsup: :thumbsup:

[AsianCaucasian]

Originally posted by gram
Here is the full equation and how it makes sense:



e = 2.7182818284590452353602874713527
pi = 3.1415926535897932384626433832795


in the 2 equations substitute the variables


You can get the value of e by doing the following in a scientific calc:

punch in "1" then hit the "inv" button and then hit "ln" (natural logarithm)


So easy




It's so easy:thumbsup: :thumbsup: :thumbsup:

... he's asking for an algebraic method ie. no values.

[gram]

you get e ln PI = PI next, but i doubt this is the right way cuz y1 cant equal y2



IS NOT CORRECT!


ln e^PI=PI



IS CORRECT:thumbsup:

[wildrice]

But that still doesn't show which is bigger!

[gram]

Give me like 10 minutes...I'll have an answer for you that makes sense....this is a pain in the ass to answer the way you want it.


Gram

[gram]

you need to show that the derivative of

x/lnx is greater than 0 for x>e

then you can rearrange the problem as follows:

e^pi > pi^e

take the ln of both sides

ln(e^pi) > ln(pi^e)

pi ln(e) > e ln(pi)

divide both sides by ln(pi) and ln(e)

pi / ln(pi) > e / ln(e)

so if you look at the equation y = x/ln(x), then if the derivative of that is positive for x>e, and, given that pi>e, then you know the slope of the equation must be positive past x>e, meaning that y(pi) must be greater than y(e).

therefore

if we have the original equation

y(x) = x/ln(x)
y(pi) must be bigger than y(e) if the slope of y(x) is positive for x>e

i'm not upto deriving x/ln(x) for you. haha

[gram]

well, here's the proof that the derivative of x/ln(x) is >0 when x>e

i'm using an apostrophe to denote derivative

general equation, (x/z)' = (x'z - xz')/z^2

so (x/ln(x))' = ( x'ln(x) - x(ln(x))' ) / (lnx)^2

solving for derivatives

(lnx - x(1/x))/(lnx)^2

simplifies to

(lnx - 1)/(lnx)^2

lnx is always positive, so squaring it, it will still be positive.

lnx > 1 for x>e

so the top half will always be positive when x>e, so the entire derivative will always be positive when x>e




There....damn this question::thumbsup:

[turbotrip]

Originally posted by gram
you get e ln PI = PI next, but i doubt this is the right way cuz y1 cant equal y2



IS NOT CORRECT!


ln e^PI=PI



IS CORRECT:thumbsup:

why isnt that right? we were upto ln PI ^e = PI and isnt it a log property that u can move the exponent out? i cant really remember it now but i think that was one, so then u move the E to the front

[lint]

Originally posted by turbotrip


why isnt that right? we were upto ln PI ^e = PI and isnt it a log property that u can move the exponent out? i cant really remember it now but i think that was one, so then u move the E to the front

It's not right because the two equations aren't meant to be equal to each other.

the question is not:
y = e^pi
y = pi^e

Which is bigger. That doesn't even make sense.

It should be:
y1 = e^pi
y2 = pi^e

Which is bigger, y1 or y2?

Your "solution" was acheived by making y1 = y2, which is incorrect because you should be trying to show that either y1 > y2 or y1 < y2 is true.

[turbotrip]

^oh haha no i didnt start the equation i just took wildrice's equation another step further

[no_joke]

Originally posted by gram
well, here's the proof that the derivative of x/ln(x) is

so the top half will always be positive when x&gt;e, so the entire derivative will always be positive when x&gt;e
....

There....damn this question::thumbsup:

Nice work, buddy. Just curious, are you in the Math program in university or college? Or am I just easily dazzled by intro calculus math problems?

Anyways, I'm still kinda confused. So, if:
let f(x) = x / ln(x), f'(x) = ( ln(x) - 1 ) / (ln(x))^2;
then f'(x) > 0 for x > e

and if r > s, where {r,s >= e}
then f(r) > f(s) and f'(r) > f'(s)

Is the justification for f(Pi) > f(e) because f'(Pi) > f'(e)? And this happens to apply for this particular function?

To everyone else: Yes, I like math a bit more than the average person. And I'm not in Math myself.

[gram]

Nice work, buddy. Just curious, are you in the Math program in university or college? Or am I just easily dazzled by intro calculus math problems?

Anyways, I'm still kinda confused. So, if:
let f(x) = x / ln(x), f'(x) = ( ln(x) - 1 ) / (ln(x))^2;
then f'(x) > 0 for x > e

and if r > s, where {r,s >= e}
then f(r) > f(s) and f'(r) > f'(s)

Is the justification for f(Pi) > f(e) because f'(Pi) > f'(e)? And this happens to apply for this particular function?

To everyone else: Yes, I like math a bit more than the average person. And I'm not in Math myself.

sorry, no math program.... just UBC physics undergrad program ;)

off the top of my head i couldn't tell you what the f' function looks like, but so long as we can show that it is positive for x>e, then any values after x=e must get bigger (since the slope is ALWAYS positive). since we already have the axiom pi>e, we can be guaranteed that f(pi)>f(e)

it's not so much a matter of f'(pi)>f'(e) as much as it is the fact that f'(x) > 0 for x>e

in general though, you can't really say f(pi)>f(e) because f'(pi)>f'(e), because the function COULD be wildly different between e<x<pi, and the instantaneous slope might still be bigger at pi than at e, but the absolute value may not be.

i hope that helps hahaha

[no_joke]

Okay, I get it... I was mistaken in my previous post.