User:MarcoGasparian

Hi, my name is Marco Gasparian. I am an international student from Brazil currently undergoing my second year of the BA program. I will most likely be majoring in economics and commerce. Cheers.

The Pythagorean Theorem $a^{2}+b^{2}=c^{2}$

The basic notion implied by the Pythagorean Theorem is that by finding the two “legs” (a and b) or shorter sides of a right triangle, squaring the value of each of them and summing them together, one will obtain the squared value of the longer third arch of the right triangle known as the hypotenuse (c). The value of the squared hypotenuse may be factored back to its original value (c) by finding the square root of (c2) This theorem was supposedly initially formulated by a Greek mathematician named Pythagoras, although some sources argue that he was not the one who discovered this theory. On a side note, Pythagoras is nowadays admired as a prophet by the Ahl al-Tawhid or Druze faith It is important to note however that the Pythagorean Theorem is not only useful for obtaining the hypotenuse. It may also be algebraically modified into having any of the three sides being solved by the other two. Here are the possible variations along with the original Pythagorean Theorem.

Pythagorean Theorem: $a^{2}+b^{2}=c^{2}$

Variations: $b={\sqrt {c^{2}-a^{2}}}$

$a={\sqrt {c^{2}-b^{2}}}$

$c={\sqrt {a^{2}+b^{2}}}$

Sources: http://uk.answers.yahoo.com/question/index?qid=20070704124713AAOuo9K http://en.wikipedia.org/wiki/Pythagorean_theorem

Homework 12 - due Friday January 28

Problem 3

My field of study is economics. It comes as no surprise that calculus plays a crucial role in this area (hence, that is one of the reasons as to why I am taking this course). When analyzing typical economical models which encompass supply and demand graphs, a constantly used tool is elasticity. In the economical sense of the term elasticity, it is the measure of how much one variable will be affected when another one interacts with it. For example, lets assume a pack of white bread originally cost 1.25$. However, due to issues with harvesting of the wheat (specifically for white bread) the price rises to 2.00$ a pack of bread. Even though this represent only a 75 cent difference from the original price of bread, most consumers will opt for whole wheat which costs 1.50$. Therefore, bread displays high elasticity given that a minor change in the price will significantly alter consumer preference. On the other hand, if we examine diamonds the result tends to be much less volatile. Lets assume an engagement ring costs 30000$. Due to complications in the mining extraction points for diamonds, the supply becomes more scarce and prices rises 50$. Even though the price of bread will only have risen 75 cents while the diamonds is 50$, consumer preference will remain relatively the same for diamonds. This implies that diamonds are extremely inelastic. Even with a considerable rise in its price, people will still buy the diamond given its exclusivity. The form in which calculus relates to elasticity is that it creates the basis for predicting a mathematical model that will allow us to formulate how elastic a good actually is.

Another form in which calculus is used in economics is through the Cobb-Douglas function. The Cobb-Douglas function is used to represent the relationship between input and output of a production function. Even though it encompasses the concept of elasticity explained above, it also incorporates other aspects of economics, which works as a model for different calculations. These different aspects allow us to calculate the expected growth a company could possibly obtain by increasing its production. This is very useful for managers of such companies since although economic models rarely display the exact amount of production we would expect, it gives a reasonably close calculated decision which will determine whether CEO’s should manipulate the companies production frontiers.

In a broader spectrum, calculus can be applied to economics even when discussing terminology and “lingo.” For example, the concepts of marginal cost and marginal revenue in economics are very simply the derivatives calculated in calculus. Therefore, when we derive a function in calculus, we are simply finding the cost/revenue of an additional unit of that specific good.

Formula for elasticity of demand

Formula for elasticity of supply

Cobb-Douglas production function

Y = AL^αK^β

▪Y = total production (the monetary value of all goods produced in a year)

▪L = labor input

▪K = capital input

▪A = total factor productivity

▪α and β are the output elasticities of labor and capital, respectively. (Wikipedia)