Separable differential equation

2.1.1.23] Solve




Solution:














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[Reference: 5]

Real life {*_*}

It is only two weeks into the term that, in a calculus class, a student raises his hand and asks: "Will we ever need this stuff in real life?"

The professor gently smiles at him and says: "Of course not - if your real life will consist of flipping hamburgers at MacDonald's!"

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Source:

[2]

A drink too many

A cone-shaped paper drinking cup is to be made to hold 27 cu cm of water. Find the height and radius of the cup that will use the smallest amount of water. [4]

This is one of the problems which, by taking the usual road, can make the solution more complicated than the problem.

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Source:

[4]
Problem 36, page 338.

Horseplay {*_*}

A mathematician and a stock broker go to the races to bet on horses. The broker suggests a bet of $10,000. That's too much for the mathematician's taste: First, he wants to understand the rules, have a look at the horses, etc.


"Don't worry", the broker says. "I know an empirical algorithm that allows me to find the number of the winning horse with absolute certainty."


This does not convince the mathematician.


"You are too theoretical!" the broker exclaims and puts his $10,000 on a horse.
The horse comes in first - making the broker even richer than he already is. The mathematician is baffled.


"What is your algorithm?" he wants to know.


"It's rather easy. I have two children, three and five years old. I add up their ages and bet on that number."


"But three plus five is eight - and that horse had number nine!"


"I told you that you're too theoretical! Didn't I just experimentally prove that my calculation is correct?!"


Source:


[2]

Losing weight

Most stocks in the US and in the Philippines are taking a beating recently. The biggest drops (so far) were early this March. The past few days have mellowed but still there are some lingering erratic movements.

To wise investors, there's always a good opportunity even when the stock drops. For one, the stocks become cheap during this slide and so buying is better.

While at this, I thought of posting this problem 4.65 from page 305 of Introduction to the Practice of Statistics, 5^th edition, by David S. Moore and George P. McCabe. This is a simple problem on stocks using basic statistical probability. We have our final exam these days so this was also part of my review for Statistics.

4.65/305). You buy a hot stock for $1,000. The stock either gains 30% or loses 25% each day, each with probability 0.5. Its returns on consecutive days are independent of each other. You plan to sell the stock after two days.

a) What are the possible values of the stock after two days, and what is the probability for each value? What is the probability that the stock is worth more after two days than the $1,000 you paid for it?

b) What is the mean value of the stock after two days?

The problem can be solved starting with a tabular presentation below:

For a), the possible outcomes are:

There are 4 possible outcomes and so, each outcome has a probability of 1 of 4 or 25%. After day 2, note that there are two $975 coming from outcome 2 and 4. Hence, the probability of getting $975 after day 2 is the sum of the probability of outcome 2 and 3. This is 0.25 + 0.25 = 0.5 or 50%.

There is only one outcome of getting more than $1000 after two days. This is 1690 in outcome 1 which has probability 25%.

The mean value of the stock, say µX, is the summation of the all individual outcome with its specific probability. This is the same as the weighted value with probability serving as the weights.

There is a small gain on the average which is possible if one has several investments of this kind. However, if there is only one account, the law of large number is not at work. Given only one stock with an equal chance of losing 25% or gaining 30%, the chance of losing is actually greater. As shown in the table, there are three possible losing outcomes (out of four) that would shrink the stock below $1000.

In other words, this stock has a losing chance of 75%. Not a good one. In the stock market, there are several companies with better lose-gain attribute.

Historical daily performance can be checked from the company profile. This simple statistical analysis is only one of the tools that can be used as guide in investing. In particular, this gives an insight on what may happen should one redeem or sell the stock.

Statistics is a good exercise to keep our stocks and other investments in good shape.

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[written 03.14.2007]

Anti-terrorism {*_*}

In a speech to a gathering of mathematics professors from throughout the United States, George W. Bush warned the academics not to misuse their position to force their often extremist political views on young Americans. "It is my understanding", the president said, "that you are frequently teaching algebra classes in which your students learn how to solve equations with the help of radicals. I can't say that I approve of that..."

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Source:

[3]

Root of the problem: Negative infinity

I came across this interesting problem which is quite simple in appearance yet, at the same time, it possesses some qualities of a quiz bee material.

The solution starts with removing the radical by multiplying the expression with its conjugate. From here, there are algebraic routes to the same destination of simplifying the expression which has become a fraction. The denominator now contains the other half of the conjugate.

Here, the gerater trick lies on the concept of absolute value.

The x-squared inside the radical sign can be extracted as x. If x goes to negative infinity, x must be negative. Therefore the root of this term must be negative.

Here is another interesting part. If the true value of x is negative becasue x approaches negative infinity, it is very tempting to over-emphasize the absolute value concept by changing the sign of all x including all the other terms with no root extraction required. Again, one may argue, if x goes to negative infinity as indicated in the limit, then each x in the whole expression is supposed to be covered by that condition. It means x is less than zero or negative. Hence, it is so tempting to rewrite the expression as:

Note that, when simplifying towards the end, x is factored out from the denominator and it cancels with the x in the numerator. As this happens, the rewriting of x outside the radical sign to reflect its true sign becomes somewhat moot.

On the other hand, all x outside the radical sign before extraction, in this case, the terms -2x in the numerator and the first term x in the denominator, are not normally rewritten in texts that I've seen at least, to reflect the true location of x in the number line. There must be some good reason. Before going there, consider:

In this second case, when one changes the sign of all x terms by invoking the blanket effect of absolute value as x goes to negative infinity, the limit is positive 3. On the contrary, when absolute value is invoked only to the roots of extracted term, the limit is negative 3.

It is wise therefore to confine the absolute value effect on root extraction. The other terms outside the radical sign may appear the way they do because they may have acquired their sign from previous mathematical operations. This shouldn't discount that they are essentially negative from the start when the condition says the limit goes to negative infinity. Likewise, when the expression is yet to be simplified, changing the sign of all the other unextracted x can lead the expression to wayward paths during succeeding mathematical operations.

The limit going to positive infinity makes similar problems very easy. On the other hand, when it goes to negative infinity, the concept of absolute value plays a very important role. If not taken into consideration very well, the answer usually becomes either zero or infinity which is wrong.

In the first case, the answer is negative 1.

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