Week #32

I know you're all anxious about how you did on the exam, and I'll do my best to get the exam back to you this week. As always, if you have a personal grading question, you should see me during office hours as soon as you possibly can. The answers are posted in the right sidebar and you should take a look to see how I did the problems (report any errors to me and I'll give you credit). If you're curious to see another college's exam, you might want to see what a freshman calculus exam at MIT looks like, it's given after only six 50 minute lectures (i.e. three weeks of a regular thirteen week academic semester). I have a good deal of faith that many of you could do this exam too, but I'm not sure if you'd find it fair, since we did not do the transcendental functions involving Euler's number (but it was mentioned).

As you know, this week was without homework, mainly because I did not want to burden you with related rate word problems while studying for exam #1. But now that's over, and I want to give you some homework to think over:

• Section 3.8: page 195: problems 5 - 19 odd, 29, and 34.
  
• Section 3.9: page 202: problems 3, 5, 9, 13, and 25.
  

Although I've reviewed the concepts of translating English words into algebraic equation -- especially as it relates to related-rate word problems -- I think many of you will struggle with setting up the proper algebraic equation, however, once you get the proper algebraic equation you're just steps away from answering the question. So please, carefully look at problems 3 and 5 from section 3.9, and then try to imitate what we did in class on problems 9, 13, and 25. Also, although obvious, you should read the textbook, it will help greatly.

CAUTION, PERSONAL ANECDOTE ABOUT THE UPCOMING PRESIDENTIAL ELECTION! READ AT YOUR OWN RISK.

Life is funny, but from an early age I started to realize that our world is in serious trouble. The older I get, the more I see that our ability to reason around simple issues is seriously flawed. We lack (myself included) the framework (small cells that have localized self-interest) required, and have become totalitarian (dualistic about controlling interest outside our frameworks) as a result. Really, our minds are quite parochial in scope, but we've started to think outside our parochial worlds without having a solid foundation to do so.

In America we've been drifting towards mob rule (controlling outside our framework) for some time now, and next week you'll see two massive mobs gathering for an election, and the ugly faces of the unreasoned-mind will push America towards bigger government no matter who wins. Right now, the U.S. Government is even more colossal in size than even Stalin/Mao could have hoped for. It (U.S. Government) controls almost 50% of our collective labor and its people believe that government is the answer to whatever ails them.

What's the point? An interesting study was done in the 1960s by an American sociologist, Daniel Bell, that described a systematic destruction of the single most important connection in any culture (parent child, a parochial framework) has. The Khmer Rouge did the same, but with little regard to 'political correctness', as did most other totalitarian regimes. In America though, we hardly saw it happen.

May You Live in Interesting Times?

Week #23

We've completed the techniques of finding derivatives of a small set of functions and relations. For example, you should be able to easily differentiate many of the functions and relationships that you previously studied, but not all. One big exception is Leonhard Euler's number raised to a variable power, f(x) = e^x, which so happens to have a very strange and rememberable result: f(x) = f'(x).

Next week we'll start up with related rate word problems, and we'll put many of our rules to work. Then it's onto analyzing functions, first using the foundations of pre-calculus, then with a second pass where we'll meticulously refine our understanding of functions using mainly the first and second derivatives.

Oh, almost forgot, as originally advertised you guys will be getting an exam on Friday, October 29, 2004.

Another problem for you guys to ponder, but I don't want this handed in. However, I'm going to post the L^AT_EX file of the solution to see if any of you computer types can figure out how to generate a pdf from it --- send me a pdf of this generated file if you can. I'm just curious. In any case, most professional mathematics is typeset in L^AT_EX and if you plan to study mathematics beyonds the walls of Essex County College, you'll need to rub elbows with the L^AT_EX digerati and I'd strongly suggest learning it. The best part about L^AT_EX is that it's free and is highly portable.

Week #7

Many of you are not bothering to 'vote' in the weekly survey. Please, I'm interested in what you're thinking. As Al Capone once said: "Remember to vote early -- and often." That's supposed to be a joke, but I also want to remind everyone that these are not scientific studies.

We lost last Monday to the legendary Columbus, and I am hopeful that many of you took a closer look at the holiday link in Week #6's post. The story I was told in school was a lot different, but I guess, like most stories told, they're slanted towards benevolence. As my mother once said (I'm sure your's did too): "If you can't say something nice, don't say anything at all."

Here's a short list of quotes to ponder:

"Never attribute to malice that which is adequately explained by stupidity." -- Hanlon

"Misunderstandings and neglect occasion more mischief in the world than even malice and wickedness. At all events, the two latter are of less frequent occurrence." -- Goethe

"You have attributed conditions to villainy that simply result from stupidity" -- Heinlein

Well, I'd also like to believe that the disagreement between Briton and Germany (Newton vs. Leibniz) is an example of national stupidity and not individual malice. Anyway, the madness finally ended, and we're now able to take derivatives of many functions without having to worry about national pride.

Next week we'll move forward, specifically the Chain Rule and Implicit Differentiation. I'd like to think that all this work in finding derivatives will have a payoff, and for me I will use the derivatives as currency to buy information. Like any currency, I need to spend it quickly, for tomorrow it's value may be lost. So yes, we need to find derivatives and then use them quickly, otherwise we may acquire a lot of wealth, but know little of its value.

Week #6

I know some of you are finding the material hard, especially the quizzes. Please continue to work hard, your efforts will pay off. Don't let minor setbacks discourage your desire to learn.

Remember Monday's a holiday!

New bonus questions have been posted, see the sidebar for more information. Number 2 is tough and number 4 is easy (use the definition for absolute value, and divide it into cases), at least I think so. The hint for number 2 is actually given in Example 1 (page 60) of Stewart's textbook. Again, you can work together on these, but you need to write up your own solutions.

At this point, you should be able derive the rules for differentiation that I presented in class. Also, please look at the textbook's proof of the Quotient Rule, and the rules that immediately follow it. The next rule we need to do is the Chain Rule, and for many that will be a toughest one. No proof will be given for the Chain Rule though. We'll also do the derivatives of the trigonometric functions and then finally implicit differentiation. Once the rules are written, we just need to practice them to work out the nuisances that perplex many students when taking derivatives of functions. Rules, however, are a lot easier than using the definition and I think most of you will agree after some practice.

Next week will largely entail working with the rules to find derivatives. If time permits we will use implicit differentiation to solve 'related rate' word problems.

P.S.: The holiday link is from MIT. Many of you may be interested, especially if you like alternative views on commonly held beliefs (this is also a must read if you can trace your roots to España or Taino indian.). However, lest you think the world is divided between evil and good, I leave you with a quote:

"There never was, there never will be, nor is there now, a man who is always blamed, or a man who is always praised."

This translated quote (original text in Pali) is attributed to Siddhartha Gautama in the Dhammapada, but in fact may have been culled from other sources.

Week #5

Quiz number three, as expected, had a mixture of results. Many of you are well prepared for calculus, but others remain unsure of pre-calculus concepts covered on this quiz. Please, if you need help with any pre-calculus topics, I'd really welcome your questions. So, please ask during class or during office hours.

Also, I will try to run weekly anonymous surveys, please consider participating each week.

For those who want extra credit, do problems 1 and 2 on page 125. It's due Monday, October 10 October 04, 2004 at 8:30 a.m.. It's worth 10 points total, and I will just add this to your quiz grades. Even if you're getting perfect scores on all quizzes so far, you should still do this assignment; sort of like an insurance policy. Oh, as usual, you must show all work on this extra credit. I know they're tough problems, but I think many of you can do it. For those that can't, I'll give other extra credits. Everyone, eventually, will find something. Answers to the bonus questions will be posted, in jpeg format, on the right sidebar.

You should have the definition (the one I gave in class) of the derivative down and be able to manipulate your way towards finding the derivatives of any of the following functions: polynomial, rational, root, or combinations and/or compositions of these. Yes, the manipulation may be tough and lengthy. Also, you should have a good sense of what information this derived function (hence, the name derivative) gives about the original function. Two examples of using the derivative were given: instantaneous velocity, and equations of tangent lines at a given point on a function.

Definitions are necessary, but this week we will discuss rules for finding the derivative. All rules given will follow directly from the definition of the derivative.

For those interested in Mathematica, here's the command for taking the derivative of f(x) = 3x² - 2x +7, with respect to x is:

D[3x^2 - 2x + 7, x]

The first part is the function, the second part is the variable that we are differentiating.

I'm not trying to push Mathematica, but the school has a site license for those interested in using it on the school's computers (I believe it's the 3100 labs). Certainly, other software for taking derivatives exists and many calculators are quite capable of taking derivatives. On the TI-89, for example, the command for taking the derivative of f(x) = 3x² - 2x +7, with respect to x is:

d(3x^2 - 2x + 7, x)

I should be getting a copy of MATLAB this week for evaluation, and I will soon post information for those interested. ECC does not have a site license for MATLAB, but for those of you planning on attending NJIT, I'd strongly suggest that you at least familiarize yourself with MATLAB or Mathematica. Both are offered to students at a hefty discount.

On a side note, last week I mentioned Genghis Khan in passing. Certainly, he is a great legend, mostly known for his conquest that extended from the Korean Peninsula all the way to Europe. And yes, like any great leader, he's an interesting character. Why did I mention him? First the arrow (Mongols' weapon of choice), then it's power, used wisely to conquer, in fact he controlled the greatest empire ever. His conquest was fleeting though, all are, but many of the real achievements remain intact. Most know of his ability to subjugate, but it's the softness of his people's hearts that remain his greatest legacy.