Calculus: The Raw FormulaBy John Kuczmarski |
|||||||||||||||||||||
| Calculus, according to Webster's dictionary is defined as: "a method of computation or calculation in a special notation (as of logic or symbolic logic)"(1).This, unfortunetly provides one with very little insight on the study of calculus. Calculus, basically, is the branch of mathematics that goes over properties of different functionalthat deal with the concepts of limits, derivatives, and integrals (2). Its sole purpose, in the most rudimentary form, is to describe the precise way in which changes in one variable relate to changes in another. For example, car acceleration responds to the way the flow of gasoline is controlled. Similarly, the level of a drug in a person's blood stream corresponds to the controlled variable of the dosage and timing of its prescription. By using calculus, one can use the understandings of certain controlled variables to better understand and predict the outcome of uncontrolled variables. In the sciences, if one is working with variables that are linked by way of chance or randomness, statistics is usually used to understand the variables better. However, in situations where "where a deterministic model is at least a good approximation, calculus is a powerful tool to study the ways in which the variables interact" ( 3). | |||||||||||||||||||||
| The history and roots of calculus can be traced back as far as Archimedes. Some of the main precursory philosophers and mathematicians before the explosion of calculus in the 17th century movement of the Scientific Revolution were Kepler, who approximated volumes of solids of revolution with the sum of numerous thin layers; Galileo, who recognized that the area under the time/velocity curve represented the distance that something traveled;and Fermat, who had methods on finding maximum's and minimum's of polynomial functions and how they relate to the modern method of finding a derivative of a function and setting it equal to zero. Fermat was also known as the creator of differential calculus.(4) Then in the 17th Century by means of a coalescence of concepts and understandings of mathematics, Newton and Leibniz created the calculus that we know today (5). | |||||||||||||||||||||
| Calculus will help me see the world in a totally new perspective. It will teach me the methods and the mathematically language that I must know "to understand how the wind blows, how the waters flow, how the sun shines, how music reaches your ear, how the planets cycle through the heavens, and much more."(6)The calculus that we know and use today can be applied in a myriad of different ways in the real world. One area of calculus involves the concept of wavelets. Wavelets make it possible to mathematically capture and store data of images and signals at much less data than previously required. "Wavelets allow unprecedented image-compression ratios at landmark speeds" (8). Because of this, areas such as astronomy, acoustics, nuclear engineering, image processing, speech synthesization, and earthquake prediction are being constantly drowned in knew revelations and advancements (7). Calculus is also used to reconstruct synthesized sound into computer data. This computer data takes up much less space, and thus can be stored and transferred more easily. The FBI requires a huge amount of storage for all of its fingerprint data, which contains "34 million cards-a figure comparable to 18 stacks as tall as the Empire State Building in New York City"(9). This storage is accomplished by means of calculus compression algorithms. Calculus is used to study how air, blood, and water flows around vehicles and boats, hoping to increase speed and/or lower resistance. Moreover, calculus can be integrated into Bjerke's principle, stating that any weather prediction can be concluded by examining and measuring certain variables in an instant of time, to more efficiently predict hurricanes, tornados, and weather in general. This is so beneficial because the amount of money required to prepare for a hurricane along a 300 mile coastline is roughly 50 million dollars. Better predictions will result in less false preparations, and, consequentially, enormous economics savings. Combined with physics, calculus is used in weather prediction as well. Calculus, combined with the chaos theory of finding regularities into things that appear irregular and chaotic, can be used to predict and analyze the chaotic pulsation's of human heart tissue. This application of calculus could be used to prevent heart attacks and understand heart cancer in greater depth. Finally, calculus and advanced mathematical analysis can be used in deep space exploration and to discover new planetary systems. Calculus is needed to expand and discover new frontiers in many fields of everyday life. If one is interested pursuing a profession in the field of business, accounting, astronomy, acoustics, nuclear engineering, image processing, speech synthesization, meteorology, and pretty much any other profession, an understanding of calculus is necessary. The applications of calculus are endless. | |||||||||||||||||||||
| One of the primary concepts of calculus is the limit. The limit is analogous to taking the square root of a number: "limit is to functions as square root is to numbers" (10). So just as one can take the square root of a number, one can take the limit of a function. Limits always involve a point of interest on the function. In studying limits, one hopes to better understand the behavior of the function for all points around that point of interest.
|
|||||||||||||||||||||
|
Another key aspect to understanding calculus the derivative. After one has learned to take a limit, taking a derivative can be approached. A point on a graph where there is a limit are called "points of differentiability"(11). If the point x on function f is differentiable , then it is said that the derivative of the function is at x The derivative is the primary tool that is used to calculate rates of change and slopes of tangent lines in graphs. The derivative can be defined f' of function f can be defined as a function whose slope at x is the slope of the tangent line at the point of the graph of f(x), or a derivative can be interpreted as the instantaneous rate of change of the function f at point x. The key concept behind a derivative is to take the difference of a point in time x and after a short while, h, where one will be, f(x+h): (f(x + h) - f(x))/h. If you take the limit of the function as h approaches zero, then you have derived the function at the value x, and the new function is the derivative.
The above illustration of a graph may help you better discern what a derivative is. The variable x is any number along the X axis. The variable h is a distance (the distance from x). It follows that the distance from x to (x + h) is h. That is called the run of the triangle. You get the rise by figuring out the function at x, then evaluating it again, this time at (x + h), and subtracting one from the other. So, the distance between f(x) and f(x + h) is the rise of our triangle. The slope of the segment that we used to form our triangle is the rise divided by the run. The picture shows us that one is finding the length of a segment ( the green brace). It is easy to see that if we made smaller triangles, we would get a more precise measurement of the slope. This is why we make h--->0. One of the most basic examples of a derivative being used in everyday life involves the equation that models the change of a population as being proportional to the number of individuals in the population. In symbols, if P(t) represents the number of individuals in a population at time t, then the so-called exponential growth model is: dP/dt = k P. Another example of how a derivative can be used in the real world is to figure out, say, when the part of an airplane is under the most stress. A function that shows the stresses on a particular airplane part with respect to time in flight is: f(t) = -t2 + 2 t + 3
The derivative is a slope in its most simple terms. If you have a function f(x), then its derivative (f'(x)) would be the slope of the line that is tangent to some curve of the graph of f(x). If the graph is a straight line, or linear function, then the derivative would simply be the slope of the graph, which leads to the realization that "derivative of a straight line is everywhere equal to its slope"(12). A derivative is also a rate. It is simply a rate that can change constantly with time or with some other variable. Because the world is full of variables that have constantly changing rates, understanding derivatives will allow you to understand the world more clearly. The derivative allows one to find the slope of a tangent line of the graph, but what about finding the area of a graph? |
|||||||||||||||||||||
| One of the major problems of calculus is trying to find the area of a continuous function of the interval from point a to point b. By the seventeenth century mathematicians could calculate areas of functions, parabolas, and spirals using a method called "method of exhaustion"(14), created by Archimedes, which was the earliest form of integration. The problem with the method of exhaustion was that it had different procedures and calculation for every different problem - it wasn't universal. Newton and Leibniz created the antiderivative, or the indefinite integral, method for finding areas of polygons. These mathematicians realized that they could find the derivative of the function A(x). An indefinite integral can be defined as follow: "If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x) = F(x) + C,for all x in I where C is a constant" (15). This is a general list of integral rules (the "§" sign stands for the integral sign):
An integral can be used in many ways. One of the ways that an integral can be applied is to find the volume of a solid object that is revolving around a line. The solid is generated by the region. The axis of revolution is the line about which the revolution takes place. The problem of solving for a sold revolving around a line is best illustrated by the following above image. There are many ways to solve for this volume, but one of the most common is the disc method."To find the volume of a solid of revolution with the disc method, us the formula : Volume = V = pi § [R(x)]2 dx" (16), and examine the data below. |
|||||||||||||||||||||
| So calculus sounds pretty cool. Really cool, actually - compressing data, predicting weather, saving people from hurricanes, finding volumes of crazy shapes, exploring the depths of space - the list goes on and on. So what am doing learning pre-calculus, when I could be learning this awesome calculus stuff? Well, precalculus provides one with all of the necessary steps to understand the intriguing world of calculus. Pre-calculus gives one insight on the basics of functions - linear, exponential, polynomial, logarithmic, radical, trigonometric, greatest integer, and c step functions. Without a basic understanding of how to add, subtract, multiply and divide these functions, calculus would be impossible because calculus expounds on these basic concepts of pre-calculus. Pre-calculus is the like the arithmetics of functions instead of numbers. A question one should ask themselves is "Could algebra be possible without an understanding of arithmetic - how to add, subtract, multiply, and divide numbers?". If precalculus is arithmetic, then calculus would be algebra There are also some parallels between calculus and its predecessor, pre-calculus. Asymptotes, a topic in precalculus, are strikingly similar to limits in calculus.
All of the precalculus functions can be graphed with a graphing calculator. If one knows the formulas and fully understands the concepts of using calculus - limits, derivatives ,and integrals for starters, then they can compute it on the calculator. Use the link below to solve any function!
|
|||||||||||||||||||||
| Thank you for attempting to expand your conscious mind by exploring the world of calculus, feel free to use the bibliography below as a list links to further your journey into the unknown.....
|
|||||||||||||||||||||
| This page was created by John Kuczmarski of Lincoln Park High School. To e-mail the webmaster click the link below
|
|||||||||||||||||||||
|
|||||||||||||||||||||
is the problem for finding the limit of the function f(x)= x2 - x + 1 as x approaches 2 (the point of interest. To solve this limit one would simple plug in 2 for x and find that the limit of the function as x approaches 2, is 3.
