User:YihongChen
Hi, I'm Yihong. I'm a first year student in the faculty of arts planning to switch into commerce.
Pythagorean Theorem
The Pythagorean theorem is a very useful theorem named after Pythagoras. Although it is named after him, it was also known by Indian, Greek, Chinese and Babylonian mathematicians before he was born. The theorem states that the square of the hypotenuse in a right angle triangle is equal to the sum of the squares of the other two sides. The theorem is typically represented as a^2+b^2=c^2, where c is the length of hypotenuse and a and b are the other two sides. Knowing this, you can find out the length of any side of a right angle triangle if you know the length of the other two sides. This theorem is used in everyday applications. An example is: a window cleaner needs to find out how long a ladder to get to clean windows that are 10 feet above the ground. He has to place the ladder 10 feet away from the building in order to avoid flowers. 10^2+10^2=200, which is the square of the length of the ladder needed. The square root of 200 is aabout 14.142, so the window cleaner needs a ladder that is 15 feet. The relevance and usefulness of the Pythagorean theorem in many applications has been recognized in Greece, Japan, San Marino, Sierra Leone, and Surinam with postage stamps depicting Pythagoras and the Pythagorean theorem as well as a Ugandan coin released in 2000 in the shape of a right triangle and an image of Pythagoras and the Pythagorean theorem. This theorem has proven to be and will continue to be very practical and useful in many applications.
--YihongChen 02:29, 20 September 2010 (UTC)
Calculus provides the means by which economists solve problems. One important topic in economics is elasticity. Elasticity measures the responsiveness of a function to changes in parameters and is the ratio of the percent change in one variable to the percent change in another variable. We can use basic calculus to calculate elasticity. When we are given a formula such as Z = f(X), elasticity = (percentage change in Z) / (percentage change in Y). (percentage change in Z) / (percentage change in Y) = (dZ / dY)*(Y/Z) where dZ/dY is the partial derivative of Z with respect to Y. Thus, elasticity of Z with respect to Y = (dZ / dY)*(Y/Z)
Calculus is also often used to examine functional relationships in economics, such as the relationship between the dependent variable income and independent variables, such as education and experience. If average income increases as amount of education and work experience increase, then a positive relationship exists between the variables. Derivatives in calculus are identical to the economic concepts of marginalism, which examines the change in an outcome that results from a single-unit rise in another variable, such as the average change in income relative to a single year’s rise in education and/or experience. Marginal changes relate to an important principle in economics: that people tend to think at the margin. According to Harvard economist Greg Mankiw, economists use the term “marginal changes” to describe small, incremental changes, such as incremental changes in work hours or factory output. If benefits (B) and costs (C) depend on the level of an activity (x), then the derivative of B with respect to x represents marginal benefit and the derivative of C with respect to x represents marginal cost. If revenue depends on the quantity of a good sold, then the derivative of revenue with respect to quantity represents marginal revenue. If production depends on labor input, then the derivative of production with respect to labor input represents the marginal product of labor. If resource cost depends on labor input, then the derivative of resource cost with respect to labor input represents the marginal resource cost of labor.
Calculus can help business managers maximize their profits and measure the rate of increase in profit that results from each increase in production. The firm increases its profits as long as marginal revenue exceeds marginal cost. For a given product, the slope is defined as the rate of change in the Y variable (total cost of a product for example) for a given change in the X variable (Q, or units of the good). Taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good. We can not only evaluate costs at a particular level, but also see how our marginal costs are changing as we increase or decrease our level of production. The change in marginal cost or change in slope can be calculated by taking the second derivative.
Finally, calculus provides a means for determining the amount of interest paid over the life of a loan. It is helpful for the interest rates for a number of things, including mortgages, loans and advances.
