User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Exponent Combination Laws
I realized that the proofs of the Laws of Logarithms don't work according to the 2nd proof of Derivative of Natural Logarithm Function. Since I'm the one who added the second proof after the laws of logarithms were put up, I'm trying to add a proof for the laws that aren't circular. While Googling it I found someone who encountered the same problem on a homework assignment, and (lucky for me?) he didn't post his resolution. That is, he found a source for the the definition using $\exp$ as the inverse of the natural log, whereas he wanted to prove it using the definition


 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$

I might wait until next beginning of next year when I do sequences and series and limits at infinity in detail. At the very least, I think I know enough to prove laws of exponents for integer indices, should be a simple proof by induction?, but I don't want to clutter PW with extraneous proofs. Thoughts? My semester just ended so I have some free time until Spring semester. --GFauxPas 12:09, 23 December 2011 (CST)


 * "Extraneous proofs": If it's a result which is interesting enough to document, then it's worth putting up on ProofWiki. One of the first questions you have when putting a site together like this is: "Is this statement too trivial to mention?" If you find yourself asking that question, then you will probably find yourself, sooner or later, making the wrong decision. Thus a vital step in a proof is glossed over, as its proof is "too trivial", and then a mistake creeps in because this point is insuffuciently understood. There's a biblical quote that applies here: something like "[The stone that the builders cast aside has become the keystone]." Hence my (own personal) decision that: there is no mathematical proof which is so simple and trivial it's not worth putting up here. I even believe we can put up a category of "examples" and/or "exercises" (but I haven't bent my brain cell in that direction yet).


 * As for the question about:
 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$
 * Other way about, btw, it should be:
 * $\displaystyle \exp x := \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$
 * (it means: "the $\exp x$ function is defined as being the limit of ..."


 * The aim is to prove that it follows strictly from the definition as given in these pages. Then we can confirm we have no circularity. --prime mover 16:35, 26 December 2011 (CST)
 * It is actually easy to prove that above definition coincides with the unique solution definition (the unique $f$ s.t. $f' = f, f(0)=1$) as, given uniform convergence (which is there on closed intervals), limits and differentiation may be swapped, yielding the desired immediately. I think it will be enough to just show equivalence of the definitions, after which all steps are allowed. Note that proving this equivalence of definitions as I sketched above builds on uniqueness theorems in differential equation theory and elementary calculus, so no dependency on logarithms required. The remaining equivalences I can't pull out of my hat at will.


 * P.m, I agree on your assessment of the exercises and such, compare the three example pages on Definition:Convex Set. Lastly, your quote originates from Psalms 118, verse 22. Also, this is quoted again in the New Testament. Just for completeness' sake. --Lord_Farin 16:57, 26 December 2011 (CST)
 * L_F: Examples: Yes, that page cited is an instance of how examples work - I may spend some time thinking about how to template-ize this. I'm on vacation all this week and my wife's at work, so (if I'm not spending the whole week watching DVDs she doesn't like) I may well think about this. On the other hand I've just got my hands on and, so, a-ha-ha, both are doubtful. --prime mover 17:18, 26 December 2011 (CST)

Problem
Given certain parameters, the problem is to find some rate of change regarding a falling ladder leaning against a wall.

Here is one example:

Suppose a ladder $10$ meters long is leaning against a house. I have slathered grease all over the ground and am pulling the ladder away from the wall in such a way that the top of the ladder is sliding toward the ground. When the bottom of the ladder is $8$ meters away from the wall, at what velocity is the top of the ladder descending?

Solution
The crux of solving the problem is in finding a suitable model.

Define a coordinate system on the $xy$-plane where:


 * The origin is the bottom of the wall of the house.


 * $x$ is the horizontal displacement of the bottom of the ladder


 * $y$ is the vertical displacement of the top of the ladder.


 * $t$ is the amount of time elapsed since I have started my nefarious ladder-pulling scheme.

The assumption is that $x$ and $y$ are implicitly defined differentiable real functions of $t$.

We use the convention that:


 * Horizontal movement to the right is positive


 * Horizontal movement to the left is negative


 * Vertical movement towards the sky is positive


 * Vertical movement towards the ground is negative.

This of course will need a diagram, which I don't feel like making yet.

By this model, the given that the ladder is moving to the right at $8$ meters per second is:


 * $\dfrac {\mathrm dx} {\mathrm dt} = 8$

and the unknown is:


 * $\left.{\dfrac {\mathrm dy} {\mathrm dt}}\right \vert_{x=8} = \ ?$

With the assumption that the wall of the house is perpendicular to the ground, the position of the ladder can be given by the Pythagorean Theorem:


 * $x^2 + y^2 = 10^2$

Differentiating both sides WRT $t$:

To solve for $\left.{y}\right \vert_{x=8}$, we can use the Pythagorean Theorem again: