[RE] Choices

What we make of our time alive is ultimately decided based off of what choices you decided to choose over another. Every single moment your are deciding to do something over another which is shaping the events that happen to you throughout your lifetime. This statement completely amazes me.  -Adam Redfield, quoted from "mbahkung"

In mbahkung, Redfield speaks about the recent death of his grandfather. At first thought, I figured the post to be about life of his grandfather sprinkled with personal stories about their times together. Interestingly enough as I read, I came upon this quote about choices. While the quote is intended to spark conversation about choices and destiny - how decisions we make in the present make our lives in the future - this section brought up thoughts of parallel universes.

The concept of parallel universes is complex and imprecise. With such complexity, there are several different perspectives (or interpretations) of parallel universes. Some of these models are more stories told in science fiction, rather than physics or cosmology. Overall, parallel universes are more theories and speculation - seeing that we cannot prove its existence (or lack thereof). So how can a definition sum up an idea with infinite possibilities? Hypothetically, all of time, space, energy, and matter are compiled by all the parallel universes lined up together.

So depending on whom you're talking to, parallel universes may be called multiverses, alternate universes, or metaverses. What separates these terms? Really, (as far as I understand) it's interpretation.

Alternate universes (which are most common in sci-fi applications) take a person's timeline and change one aspect. It could be anything from a different job, family, hometown, or whatever that changes. Then all the events that surround this one aspect are created around this new change. In this case, the alternate universe is created around an arbitrary point in a person's timeline. Each alternate universe then operates independently from each other "universe."

On the other hand, a multiverse takes a position in a person's current timeline where there is a choice. Depending on which direction (choice) the person chooses, a separate multiverse is branched apart from the other options. Each multiverse, then, operates in unison with the other multiverses in a person's timeline with continuous branches of universes each time a decision has to be made. This can be called a many-worlds perspective because, while each coexists simultaneously, the person can only interact with multiverse (otherwise a collapse or paradox would occur).

Relating back to what Adam said, deciding which choice a person chooses over another shapes his (or her) multiverse. However, I wonder what would happen if he chose the other path. How would it be cool to step into a different multiverse and see how one event could alter a personal timeline...? That would be the cosmic perspective.

[FR] Indeterminate Forms (cont.) and l'Hospital's Rule

So a while back, I covered a lesson on Indeterminate Forms. We discussed two common indeterminate quotients (0/0 and ∞/∞) and their troubles when determining limits. Is there a solution...? Yes. The solution is called l'Hospital's Rule.

Find the limit of [x - 49] / [(√x) - 7] as x ---> 49, where √ represents the square root function.


Normally, by the Direct Substitution Property: [49 - 49] / [√(49) - 7] = 0 / [7 - 7] = 0/0

Notice the Direct Substitution Property fails. It gives no means to solve this limit in this form. Algebraic manipulation (such as multiplying by the a / a form of the denominator's conjugate) or attempting to rewrite the equation might work in some cases. However this is a situation where such methods also do not work. Not to mention, they are time consuming and prone to arithmetic errors.

However, l'Hospital's rule can be used to solve this limit.

L'Hospital's Rule:
Suppose the functions f and g are differentiable and g '(x) ≠ 0 on an open interval I that contains a (except possibly at a). Supose that f(x) = 0, x ---> a and g(x) = 0, x ---> a or that f(x) = ± ∞, x ---> a and g(x) = ± ∞, x ---> a (i.e. we have an indeterminate form of type 0 / 0 or ∞ / ∞). Then f(x) / g(x) = f '(x) / g '(x), x ---> a.

This rule states that the quotient of a limit that results in an indeterminate form can be solved by taking the derivatives of the numerator and the denominator and evaluating the limit at x.

Caution: The limit as x ---> a of f(x) / g(x) must be an indeterminate form before applying l'Hospital's Rule.
Caution: L'Hospital's Rule takes f ' and g ' independently. You need not to use the Quotient Rule when formulating the limit.
Caution: The indeterminate form must be a quotient of 0 / 0 or ∞ / ∞

If we apply l'Hospital's Rule on the example:

f (x) = x - 49 ---> f '(x) = 1 - 0 = 1
g(x) = (√x) - 7 = [x ^ (1/2)] - 7 ---> g '(x) = (1/2) x ^ (-1/2) + 0 = 1/(2√x)

Thus: the limit of [x - 49] / [(√x) - 7] as x ---> 49 becomes the limit of 1/(1/2√x) as x ---> 49

Algebraic maniuplation gives us: 2√x as x ---> 49
                                                2√49 = 2 ∙ 7 = 14

How does this work? Well...needless to say, there is a long extensive proof that shows how this property is true. Typing it out would be an extremely cumbersome process, so I leave it up to you if you want to look more into l'Hospital's Rule.

One cool property of l'Hospital's rule is that so long as an indeterminate quotient results as the limit, the process can be repeated multiple times.

[CE] The Genomic Revolution

TED Talks: Richard Resnick - Welcome to the genomic revolution (Video)




The world of science is changing fast. Fields like genetics and computer engineering change so quickly that textbooks are updated and rewritten regularly. What saddens me so much is that high school science classes (namely biology, chemistry, and physics) need to meet up with certain standards and content needs to be standardized. These kinds of classes haven't changed in centuries. It's not to say that the classes can change, but all the fundamentals have been tested (and retested) on numerous accounts. Laws and theorems can't change (otherwise, they would be laws or theorems anymore). And there are only so many ways to motivated, prove, and explain concepts. Many times, new editions of textbooks are created solely to rewrite examples that are unclear or change numbers around in exercises. I could guess that if science teachers could explore the ever so fast moving world of modern science, classes could be a lot more interesting to a whole bunch more people.

This is exactly what TED talks do. I have the ability to learn something different and new that I haven't the opportunity in class.

In this talk, RESNICK informs us about a new approach to analyzing and decoding the human genome. The fundamental instructions that code each of us humans are all tucked away in the combination of four proteins sequenced in a seemingly endless chain - a chain so long that it would take 5 boxes of paper to type out all the genetic code in a single human.

The process of decoding the genome was expensive and time consuming. The completion of the Human Genome Project spanned over thirteen years and billions of US Dollars. While genetic testing for medicinal and forensic use is limited to specific genomes, the HGP wanted to collect data from all 22,000 - 25,000 genes and map out their individual distinctions.

It seems now, however, that this entire process has changed.... What will scientists come up with next?

[RE] Wasting Food

Here I've got a quote from a post on the Blog of Vivian Nguyen. She brings up the topic of wasting food.

[W]asting food [is] pretty much disrespectful. It [is] rude to not eat or finish your food. The food on your plate costs money and you are fortunate to have food to eat. […] I also get ticked off when people just toss their food away like it's worthless. It might not seem like a big deal, but it bothers me. In my opinion, throwing away precious food is like throwing away money in the trash.

While I can say that wasting food is never a good thing to do, I have conflicted views regarding the act as "disrespectful". I figure each person should be able to gauge how much s/he is able to eat. Measure out your ingredients accordingly; feel free to deviate from a recipe and cut it in half. I think that's half the fun - experimenting with food, cooking, and science.

If you're cooking for a family dinner and cannot gauge the appropriate amount of food to prepare, cook the food and execute proper portion control. Once you present the food at the dinner table, only take what you know you can eat. You always have the opportunity to get some more; don't be inclined to collect all your food on the first pass. This way, no food is trashed simply because someone decided to pick up a fillet of fish s/he couldn't finish and any food that is disposed isn't done so in worthlessness.

Many leftover food items can be refrigerated to be used in another dish, or reheated again for a separate meal.

What if this thought is one of the contributing factors of obesity in America? Because everyone is told to eat everything on his plate or she should not waste the precious food because it is seen as throwing away money...

The topics may not be completely unrelated - but I'm not a nutritional anthropologist, so it's not my place to speak of the matter.

So what attracted me to respond to this post?
It actually was a quote to which Vivian was responding that caught my eye:

We had to eat every single grain of rice as a symbol of thanks to the farmers and the workers that had to plant, collect, and produce the grain.

It's a fair point. Eating all the rice is a sign of gratitude for the farmers that toil daily to provide the grains for your meal, and I respect that. However, different cultures finds that having empty bowls with all the rice eaten (or whatever meal was prepared) is a sign of disrespect to the host/hostess. It is a sign that s/he did not make enough food to serve everyone. Conversely, eating all the food can be seen as respectful in that the meal was delicious and everyone enjoyed it.

So.... in the end, cultural mixes and personal perspectives can blur the matter of wasting food. Changing the methodology in which someone prepares or serves food may help limit the amount of food trashed.

[FR] The Chain Rule

What is the Chain Rule? Without any knowledge of differential calculus, it will be impossible to explain. However, for those who know derivatives and rules of differentiation, we can start this lecture.

One set of the basic differentition rules is that of sine and cosine.

d/dx [sin(x)] = cos(x) and d/dx [cos(x)] = -sin(x)

But what happens when the argument x has a coefficient? Or what if the argument is a sum or difference?

The first instinct would be to simply leave the argument in place and take the derivative:

d/dx [sin(3x)] = cos(3x)

But this is incorrect.

Remember how derivatives could be described as a rate of change? Consider this example:
*****
You are given a metal bar of consistant density. You are asked to determine the rate at which the bar expands or contracts due to heat. The metal bar is being cooled at a rate of 5 °C per minute and that the bar expands 2 centimeter per °C.

Using simple dimensional analysis, you can conclude that the bar is expanding at -10 cm/°C. (Negative because the bar is being cooled). So, how did you do it? Well you multiplied the two rates! And that is the foundation of the chain rule.
*****
Now (hopefully) I've motivated you calculus students to understand that given two rates of change for a single function, the total rate of change can be found by simply multiplying the two rates. Let's refer back to the example:

We know that the rate at which sin(x) changes d/dx [sin(x)] = cos(x). And we know the rate at which 3x changes d/dx 3x = 3.

Keeping the argument consistant, d/dx [sin(3x)] = cos(3x) ∙ 3. The derivative (rate of change) of the entire function is the product of its two individual rates of change.

The actual proof of the Chain Rule is long and complex. But the actual theorem says:
**********
If g is differentiable at x and f is differentiable at g(x), then the composite function F = f ○ g defined by F(x) = f(g(x)) is differentiable at x and F ' is given by the product

F '(x) = f '(g(x)) ∙ g '(x)

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

 dy/dx = dy/du ∙ du/dx

**********

One more example will be done with an argument of sums.

Find d/dx [cos(13x + x^2)], where x^2 represents the square of x.

Using the Chain Rule definition:
f (x) = cos(x)          and          g(x) = 13x + x^2
f ' (x) = -sin(x)        and         g '(x) = 13 + 2x

F '(x) = -sin(13x + x^2) ∙ (13 + 2x)

3-Space Grapher

(a)

(b)

(c)

These are all graphical representations of the function f(x, y) = (xy)/(x^2 + y^2).

The (a) and (b) are produced by Wolframalpha and (c) is generated by Microsoft Mathematics.

Fig. (a) is called a contour map or level surface. It takes a 3-space surface and compresses it into a 2-space diagram. Areas of darker red/brown correspond to the negative z values of the function. A 3-space model on a contour map is the only stable way to represent a function of three variables w = F(x, y, z)

Fig. (b) and (c) are 3-space representations of the functions. If you had a piece of modeling clay and were told to mold it into the function f, you would generate this shape.