I went into the thing expecting an exclusive focus on agriculture, but permaculture's scope is more expansive than that. In a way, it's like systems theory for beginners with an emphasis on ecology. Taken broadly, it might be a lifestyle blueprint, a movement that began as the brainchild of a couple of Australian horticulturists/engineers/environmentalists (David Holmgren and Bill Mollison) who advocated dropping off the grid, moving out into the country, and living in self-sufficient homesteads. Taken more narrowly, permaculture is the methodology devised for designing and arranging the components of this homestead -- or farmland, garden, residence, etc., or any devices they might employ. (My project involved a chicken tractor and movable chicken enclosures specifically designed for the elevated beds at this place's biointensive garden. If chickens relentlessly eat greens and leave dung everywhere, why not put them to work in the fallow beds, managing the weed population and fertilizing the soil?)

Most of the books about permaculture tools and philosophy enjoying high circulation today were penned by Holmgren, including the one used in this course:

*Permaculture: Principles and Pathways Beyond Sustainable Design*. I can't say it's a bad read -- it's informative, well-argued, and very thorough -- but it often gets on my nerves for the same reason cited by the Amazon reviewer who gave it its only one-star rating:

*Yes, Mr. Holmgren, you can be a male Western scientific materialist and still want to create a sustainable environment and society for your children.*Unless you're already fond of literature with a pompous New Age reactionist tone, you're gonna get rankled. And as you'd presume from a book whose back cover blurb reads "do mainstream concepts of sustainability dodge the critical issue of global energy peak?" there's a

*lot*of fire and brimstone.

But it's to be expected: composing such a comprehensive and persuasive guide to a set of intellectual principles and methods requires that one not only be an adherent, but a hardliner. When Holmgren writes about the necessity of readjusting the global lifestyle with concern for sustainability and ecological neutrality, he doesn't play tee-ball. Do you drive a car on a daily basis? Pat yourself on the back, jackass -- you're part of the biggest problem facing humanity. Do you eat red meat? You stupid, selfish pig. Are you the type of person who throws out plastic cups without a second thought?

**Fuck. You.**

Well, Holmgren doesn't actually

*castigate*his audience like that -- but it may certainly seem so if you're the kind of person who's always driven a car, eaten beef, and thrown plastic cups in the trash, and doesn't see any urgent need to change his life in order to live up to a set of standard imposed by some pedantic Australian hippie he's never met.

I mean, yeah -- trust me, it irritates me as much as you. It hits that same grating, holier-than-thou pitch as the people who whine at me about my smoking. If you're a smoker, you know

*exactly*what kind of person I'm talking about: that preachy fuck with the nasal voice squawking "oh smoking is so

*stupid,*why would you ever start to begin with? and it's such an

*expensive*habit!" like you're not aware of how much a pack of cigarettes costs and how much more easily you're running out of breath than before. But what the hell business is it of theirs? Why should you have to

*explain*yourself to them? And when you try to explain to them that you're aware of the risks but

*enjoy*it too much to quit, they either look at you're scum or otherwise start

*pitying*you: "oh it's really so

*sad*to to see someone so young and bright and nice do something so

*awful*to themselves for no good reason." But you

*have*reasons! You have lots of reasons! Is there one single thing that smoking doesn't enhance? When hasn't it made your day to life easier and more delicious? How would you ever get through your day without its help as a stress reliever? What's the

*point*of coffee and beer without cigarettes? If you've got the same predilections as me, you wonder how the hell you'd get

*any*work done without cigarettes -- writing while smoking causes the most brilliant and fitting words to jump out of the pen and onto the page without you even

*asking*them to. Think of how much your working habits would suffer if you quit, and how long the disruption would last! And when you finally admit that yeah, you're planning on quitting eventually, the implacable fucker still isn't shutting his noise hole. "Why not

*now?*Quit while you're

*young!*Smoking

*ages*you, you know that?" Blah blah blah bitch bitch blah. Like they're on some kind of mission. Like his own happiness

*depends*on his successfully persuading you to give up something you enjoy.

But yeah, these assholes are absolutely right. Smoking is a toxic habit.

Those environmentalist douches who announce "I DON'T OWN A CAR I RIDE A BIKE" at any conversational mention of automobile ownership and take every chance to ruin your mood with all their gloomy doomy peak oil talk? They've got a point, too.

Impertinent fuckers. All of them. Doesn't make them any less correct, though.

Need we belabor this analogy with more words about how smoking cigarettes as harmful a habit to maintain as the proliferation of the affluent, oil-dependent "western" lifestyle" is dangerous to the long-term tenability of global civilization? Or could you have just inferred that's where we were headed?

Before I go looking for sources to cite, would it be pressing your patience to mention how the smoker's tobacco habit is really only deadly to himself, and then point you toward some statistics about the average United States citizen's carbon footprint, the correlation between global (over)population and oil consumption, the concurrent rises of India and China's GDPs, consumer cultures, and emissions rates, and the effects of atypical regional temperatures on agricultural output?

Alright. Sorry, sorry...I'll stop. Just sayin', though.

Of course -- as someone who still drives his car two miles to the convenience store and still smokes, who am I to prescribe your business?

For the record, though, I've had three cigarettes in the last fourteen days. Three months ago that number would have probably been somewhere between 100 and 140.

Shit. I'm still doing it, aren't I?

On a totally unrelated note, my on-and-off battle with calculus seems as much as a losing fight as ever. If anyone can give me step-by-step instructions for solving the following problem, you will have my profound gratitude. I'll even mail you a drawing of a chicken, if you'd like.

I don't know enough about permaculture to say anything intelligent, so I'll skip to the calculus! Sorry for the plain-ASCII notation.

ReplyDeleteI assume 'a' is some known value, and the difficulty is just with differentiating f(x)? If so, the key is the chain rule.

Define a temporary variable u = 3-x.

You can write your function in terms of u, f(u) = 2 * u^(-1/2).

Now you have f in terms of u, but you want df/dx. Multiply and divide by du (which doesn't change anything, since du/du = 1) and then rearrange: df/dx = df/dx * du/du = df/du * du/dx.

Getting df/du is easy using the power rule: 2 * (-1/2) * u^(-3/2) = -u^(-3/2).

du/dx is also easy: -1

So, df/dx = (-u^(-3/2) * (-1) = u^(-3/2).

Now, just substitute the definition of u back in to get the answer in terms of x: df/dx = (3-x)^(-3/2).

If you don't have math software, you can check yourself using Wolfram Alpha: http://www.wolframalpha.com/input/?i=differentiate+2%2Fsqrt%283-x%29 . In fact, if you click on the 'show steps' button to the right of the derivative, you'll see essentially the same thing as what I've written.

I've never used u-substitution on a derivative problem before, but that's definitely a way to do it!

ReplyDeleteI'd simplify the problem with exponents. Turn it into:

2*(3-x)^(-1/2)

The 2 is a constant that isn't affected by taking the derivative, so just focus on the (3-x) part. Use the CHAIN RULE to take the derivative. d/dx[2*(3-x)^(-1/2)] = 2*(-1/2)*(3-x)^(-3/2)*(-1) =

(3-x)^(-3/2)

Did you catch it? We took the derivative of (3-x)^(-1/2) just like we were taking the derivative of x (multiply it by the value of the exponent, subtract 1 to actual exponent), except right at the end we multiplied it by the derivative of the INSIDE of (3-x), that is to say, -1.

You see the same thing if you take the derivative of (5-x^2)^4:

4 * (5-x^2)^3 * (-2x)

Remember, the derivative of a constant is just zero. The problems can be as convoluted as you want! 17(x^3+3x^2+2x+5)^(10000) becomes:

17*10000*(x^3+3x^2+2x+5)^(9999)*(3x^2+6x+2)

That's it really -- remove the confusing fractional element by turning it into an exponent (in situations where you can't just turn it into an exponent, that's where you'll have to understand the quotient rule), then use the chain rule to find the derivative. Unless you're confused about the f'(a) part in which case... hell if I know.

Well, I was talking about the chain rule too, not u-substitution. My line, df/dx = df/du * du/dx,

ReplyDeleteisthe chain rule expressed in Leibniz notation. I think that notation makes it clearerwhythe mechanics of the chain rule work the way they do.But u-substitution and the chain rule are very similar processes. You can even think of u-substitution as the chain rule in reverse.

Yeah...here's the thing. The textbook hasn't actually gotten into *any* of that yet. The only derivative formula it's given me so far is this one:

ReplyDeletehttp://upload.wikimedia.org/wikipedia/en/math/4/2/c/42cf4f4861ae1266b13104c4115e7b5d.png

I don't feel like I can move on in the text until I can solve this sort of problem with that formula, and I just. can't. do it.

Oops. Wikipedia image transparencies. Trying again:

ReplyDeletehttp://i.imgur.com/aQzzu.png>

Is it *possible* to solve that problem with only this formula? I'm driving myself crazy.

Gah. Is your textbook available online, by any chance? I'd like to see what they expect you to know at this point. You can get the derivative that way ... kind of.

ReplyDeleteSubstitute your f(a) into the definition of the derivative and you get the expression 2/h*[(3-a-h)^(-1/2) - (3-a)^(-1/2)].

Factor out (3-a)^(-1/2) to get (skipping some algebra ...) 2/h*(3-a)^(-1/2)*[(1-h/(3-a))^(-1/2) - 1].

Now, for small x, (1-x)^(-1/2) can be expanded in an infinite series as 1 + x/2 + (3/8)*x^2 + ... .

Keep just the first two terms and substitute them into our expression (where x = h/(3-a)): 2/h*(3-a)^(-1/2)*[1 + (1/2)*h/(3-a) - 1] .

In the expression in square brackets the +1 and -1 terms cancel. In the remaining term the 2/h and the h/2 cancel, and you're left with (3-a)^(-3/2). (Tadaa!)

It's way too late so I might be missing something obvious, but I think that's how you'd have to approach it. The thing is, that series expansion comes from a Taylor series -- which is based on derivatives! Which is why I'm curious to see the textbook; there might be another approach to infinite series that I'm forgetting.

But there's no reason to find complicated derivatives with this approach, unless you like pain. If you've got the idea that you're making ever-closer approximations of the slope of the curve (as h -> 0), then just move on.

Jeez, this nonsense again? Asking you to find complex derivatives with the limit approach is ridiculous. I agree with Matt, it's better to move on to the next chapter and learn how you're 'actually' going to be taking derivatives for the rest of your calculus career. I commend you for your thoroughness though.

ReplyDeleteGood call. Flipped 15 pages ahead to "hey, here's a list of shortcuts that will make everything fifty times easier!" The problem crumbled in 90 seconds and I nearly threw the book out the window.

ReplyDeleteThanks for your help, gentlemen! Who wants drawings of chickens?

Onward and upwards!

ReplyDeleteI cede to John my share of any chickens.