[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.

[FR] Indeterminate Forms

Indeterminate forms are algebraic expressions obtained when taking a limit. This most often occurs when the limit of dependent variable x approaches either 0 or ∞. However, that is not to say that algebraic expressions in which the dependent variable approaches 0 or ∞ are indeterminate forms.

-----
For example: the limit as x approaches ∞ of 1/x is 0.

You might be thinking, "How does dividing 1 by ∞ make 0?"

Consider the function 1/x and make x increase to ∞.

x = 1                 1/x = 1/1 = 1
x = 10               1/x = 1/10 = 0.1
x = 100             1/x = 1/100 = 0.01
x = 1000           1/x = 1/1000 = 0.0001
x = 10000         1/x = 1/10000 = 0.00001
...

Notice as we take x to be sufficiently large, the value of 1/x decreases to 0.
-----

Problems arise when limits of fractions form a type 0/0 or ∞/∞. While the unknowing mathematician would say a/a is 1, he forgets the facts that you cannot divide by 0 and ∞ is more a direction rather than a value or constant, which is why these are limits.

So why does 0/0 or ∞/∞ give us a problem?
Consider 0/0:
If the numerator approaches 0 faster (that is, at a greater rate) than the denominator, then the limit becomes 0 itself.

However, if the denominator approaches 0 faster than the numerator, then the limit becomes ∞.

Two opposite sides of the spectrum.

The concept is the same for ∞ in that if the numerator approaches faster, then the limit is ∞, and if the denominator approaches faster, than the limit is 0.

[RE] PG&E - One year later

On 2010 September 09 at about 18:00, one of the Pacific Gas & Electric (PG & E) 30 inch diameter gas pipelines exploded into flames. The resulting fire killed eight people and injured dozens more. While I am uncertain of exact numbers, I am sure that a few hundred people had their homes destroyed and required relocation. Even after all the main gas sources were switched off, the fire continued to burn for several hours into the next morning.

After that, PG & E took time to reevaluate the pipeline system and probably increased inspection and laid down firmer enforcement on regulations. Many people were displaced because of the exploded pipeline. And as a consequence, PG & E is now facing many of these people in court.

Damien doesn't think suing PG & E is the best approach:

I see no point in both parties wasting the time, effort, money, and any other resources on going to trial with this, unless the only way these people are able to get compensation is by suing.

Damien suggests that instead of making this a political thing and going to court to sue PG & E, why does PE & E not simply compensate victims of the explosion? Maybe "compensation" cannot be as simple. What exactly is the value of a life? What must PG & E compensate to those families of the eight people who perished in the fire?

Is there a numeric or monetary value of a human life? Some value to give to those families to compensate them?

Not only does PG & E have to worry about these cases in court, but I'm sure that government agencies are investigating (or have investigated), too. I wouldn't be surprised if PG & E received fines or penalties for failing to meet certain standards or pass proper inspections. Hopefully this is an example that will continue to remind us that disasters like this can still happen...

[CE] Bad Science?

Ben Goldacre: Battling Bad Science (Video)



Epidemiologist Ben GOLDACRE uses this TED talk to explore the faults of solely relying on popular magazines, newspapers, television commercials, etc. when looking at medications.

A newspaper or health magazine may say, "according to a new study...," but how reliable is that "scientific study?" How can we be sure that all the data presented in this study is unbiased and accurate? I'm guessing that the short answer is: We can't.

Pharmaceutical companies and governments spend billions of dollars researching and developing new medications, vaccines, and health supplements. Then, the manufacturers need to spend a few years to do controlled studies before these medicines can go on market. After so much money is invested in these products, they don't want to present the general public with unconvincing results. While not all companies use biased scientific test practices, it is difficult to prove otherwise.

GOLDACRE mentions the placebo effect and sugar pills. Control groups are used as a benchmark to see what improvement (if any) the medication does. The most common way to control a study is to administer the individuals of part of the study a placebo. In the case of pills, capsules, and tablets, the placebo is a sugar pill - which should have no effect on an individual (unless s/he is diabetic, I guess). For the control, theoretically nothing should happen at all. However, if the placebo responds positively and in the same manner as the medicated subjects (the placebo effect), then the administration of the medication is not any more advantageous than that of a simple sugar pill, in which case the medication has relatively no beneficial properties.

[BC] Common Application Draft Essay 01

Because I am in the process of applying to several colleges that use the Common Application, I decided to draft the essay required by Common Application for all the colleges which accept it. Of the six essay topics, I have chosen:

A range of academic interests, personal perspectives, and life experiences adds much to the educational mix. Given your personal background, describe an experience that illustrates what you would bring to the diversity in a college community, or an encounter that demonstrated the importance of diversity to you.

During the summer of 2011, I enrolled into Berkeley City College for an eight week chemistry class. I took two community college classes during prior summers. However this was my first comprehensive university-level chemistry course, an intense class that many college students would not endeavor during the summer semester. This class condensed sixteen weeks of laboratory experiments, reports, lectures, notes, exams, and quizzes into eight short weeks. While I walked out of the class with an “A,” the laboratory experiments and formal college setting are invaluable experiences that I will bring to the diversity in a college community.

The AP Chemistry class teaches the first two semesters of introductory university chemistry. After taking AP Chemistry, I gathered this eight week chemistry course would be easy since it covers only the first semester. A week later, I noticed the expansive lack of laboratory work incorporated in AP Chemistry. Moreover, the professional laboratory reports were unlike any assignment completed in high school science classes. Simple worksheets and packets evolved into complex ten-page reports with extensive details and calculations. College-level chemistry experiments require the utmost precision and numerous trials to qualify as a complete experiment. All these aspects compacted in two- or three-day experiments with up to three experiments in one week. The amount of laboratory work surpasses any amount of writing I have completed for a class.

The class structure in college classes varies greatly from the high school analogs. The entire lecture section of the class consisted of only lecture; there were no in-class assignments or group discussions. The time in lecture was spent solely on the professor lecturing new material or reviewing homework and exam questions. I experienced a different atmosphere in this college class. The lecture room is different

Student-teacher interactions

What do you think?

I was skimming the Internet and I came upon this :

No one chooses his or her country of birth, so it's foolish to be proud of it.

It's one of those questionaires that ask you to rate the statement on a scale 1 to 5, wherein 1 is "I disagree" and 5 is "I totally agree" with the statement.


This isn't intended to be one of my required Blog posts, but for those of you who are following me - or just happen to stumble upon this post - feel free to comment your thoughts or ideas about this. I'd be interested to see what you have to say.


If you feel particularly lazy and just want to post your rating 1 through 5, feel free to do so. I don't mind either way.

[FR] Calculus III: Functions of multivariables and vectors

So this year I am taking four courses in high school and one course at the community college. This special part-time enrollment in a college class is a unique experience that I am happy to take with me to university next year.  I have taken several college courses before, but this is the first time I am taking a course during the academic school year.

The class is Calculus III (third semester of calculus, if that wasn't obvious) and it's similar to Calculus I (single variable calculus) with conceptual topics except instead of having a single variable, there are multiple variables.

What does this exactly mean?
The question may be difficult to answer depending on the amount of mathematics experience you have. For those with an Algebra II background (maybe even Algebra I), the function f with respect to x is represented f(x).

This means, for every one input x, there is a respective output, usually y. The dependent variable is responsive to only one independent variable. A simple x, y chart can numerically represent this concept and a graph can visually represent it.

For a function of two variables, a single output (now called z) responds to two independent variables x and y. Instead of f(x), the two variable analog represents a function f with respect to x and y as f(x, y).

Note that the letter f only represents the name of the function. The function can be named any letter (or symbol for that matter), but by convention generic functions are usually named f, g, and h.
For word problems, it may be logical to say the function A represents Area with height h and width w. So f(x, y) becomes A(h, w). This clarifies both clarifies the dependent and independent variables.

So if Calc I finds tangents, derivatives, and integrals in two dimensions, my class Calc III finds tangents, derivatives, and integrals in three dimensions. Instead of finding the area under a 2-space graph, we get to find the area under a 3-space curve. It is quite an interesting class.

[RE] Google Reader Search: Chemistry

This week, instead of searching through hundreds of posts my friends publish on their respective blogs, I decided to use the search feature on Google Reader to find a post which (might) interest me. Being that I am declaring a chemical engineering major, I decided to enter chemistry into the search bar.

Funny enough, I found two of the first five search results were posts that I had published. To my dismay, the majority of "chemistry" posts did not quite talk about the chemical science. A few posts talked about chemistry as in interaction between people or mutual attraction toward people. Others talked about their chemistry classes - like how they failed a test or quiz, or switching around classes to enroll in HP. Not exactly the type of posts I am neither interested in reading nor responding.

Needless to say, I did find a post to respond. Titled "My dream job," Justin Guan shares his dream of becoming a marine biologist. He seems highly educated about the processes of studying to be a marine biologist.

To be a marine biologist, [sic] I have to take basic biology, zoology, chemistry, physics, biometrics, mathematics, and statistics [...]

 He lists a dozen different programs, colleges, and universities to enroll which would provide him with the education he needs to study marine biology. Naturally, all of these institutions are located either near the coast or have installed some form of artificial fishery to study marine life. They have locations all over the US, from California to New York.

For a degree, I would need a bachelor for a starting degree, and then I would later on need a PhD.

I'm glad to see that Justin has already planned to pursue a PhD - a difficult endeavor, for sure. 


While Justin doesn't mention any specific interest in which branch of marine biology he likes best (be it plants, mammals, fish, plankton, reptiles, fungi, or whatnot), it is obvious that he has high hopes to study in the oceans.

[CE] Nuclear Radiation in Fukushima

Japan's Fukushima 'worst in history' (Video)



Two weeks ago on 2011 September 11 was not only the 10-year anniversary of the September 11 Terrorist Attack but the 6-month anniversary of the 2011 Tōhoku Earthquake and the following tsunami. My current event post of last week focused on this tsunami - one of the worst events Mother Nature has poured upon us.

Unfortunately, the 15,000 casualties caused by the powerful wall of crashing water isn't the end. Nuclear plants are emitting dangerous radiation into the atmosphere. So what happened exactly? The short answer: the tsunami damaged diesel fuel generators that help cool the nuclear reactors that operate the cooling system. Heat builds up in the cylinder which eventually pressurizes due to the expanding, boiling water. This video can hopefully better explain the basics of this dangerous situation.



As this second video says, one solution is to release the steam through controlled releases; however, background radiation is also released in this process. This small radiation release is a small sacrifice for the amount of radioactive material that would evolve from a complete nuclear meltdown.

Over the past six months, the "controlled" radiation has spread throughout Japan. Nuclear analysts calculate the concentration of radiation that has dispersed to the Japanese capital of Tokyo nears those recorded during the Chernobyl disaster. Scientists disagree whether the levels of radiation are safe enough for people to remain in the city. With the capital city housing more than 13 million citizens and millions more in the surrounding areas, it is impractical to evacuate them all.

One thing on which scientists can agree is that high levels of radiation are particularly more harmful for the elderly, children, and pregnant women. Of course, it takes no scientist to understand that concentrated radiation contact for pregnant women or breastfeeding mothers leads to birth defects, genetic mutation, and unhealthy development.

Millions of citizens surrounding the nuclear plants are asked to evacuate or relocate. Sadly, many do not have the financial means to do so. The best they can do is to avoid going outdoors and avoid eating local, radiation infested foods.

[FR] Welcome School Year of 2012

[This post probably should have been the second post on this Blog…. However, the timing didn’t work out so well, so now this is Post № 07. Had this been done as scheduled, I’m sure it would have answered some of the questions I’ve received.]

Two years ago I was introduced to the world of Blogging. Had I not been in Mr. Sutherland's English 2(P) course, I would most likely be void of the Blogging world today. Now, after a one year hiatus from Alameda High, Sutherland has returned and I am lucky enough to be enrolled in his English 4(P) course.

For a new  year, I have decided to clear the then-Rare Finds blog to make room for this upcoming year's assignments/posts. I think I appropriately matched the blog title "Prof. Christopher's Lecture Notes" with the URL. You may find former posts from Rare Finds at Uncle's Rare Finds <http://uncleclam.blogspot.com> (Jackie Chan Adventures reference) This blog (I assume) will remain untouched. It is meant to be like an old archive – a preservation of the historical posts of 2009 and 2010. I want to start this year with a clean slate, literally, by starting with a clean blog.

I hope that my writing has improved since my first post in 2009 September and will continue to develop as this year progresses. As I’ve written many times before: writing is a form of communicating, and communication is invaluable; without communication, sharing ideas and thoughts would be virtually impossible.

I guess some of the stuff about the clean slate isn’t quite as true since six posts have already been published. However, I hope the analogy still has the same effect.