Continuity, Differentiability, and Integrability
The derivative of a function is the rate of change of that function. It shows how fast or how slow the function is changing. This can be useful in determining things such as instantaneous rates of change, velocity, acceleration and maximum profits. A good way to explain the concept of a derivative is to do it graphically. To illustrate, think of a drag car race. The track is only ¼ of a mile long, or 1320 feet. The dragster crosses the finish line in six seconds. How fast was the dragster going when it crossed the finish line? The dragster traveled 1320 feet in 6 seconds, so the average speed of the dragster is 1320 divided by 6 which equals 220 feet per second, or 150 miles per hour. The following graph represents the dragster’s position function as the red curve. The position function for the dragster is 36 2/3 x^2. The green line is the secant line connecting the dragster’s starting point and end point. The slope of this secant line is the average speed of the dragster, 220 feet per second, or 150 miles per hour.
This only tells us the average rate of speed. To find the exact speed the car was going when it crossed the finished line, approximate it by finding the slope of the line between the point we want, in this case, (6, 1320), and a point closer to it along the curve. As we move the secant line closer and closer, the slope changes and gets closer and closer to the instantaneous speed. By doing this, mathematicians discovered that the slope of the line tangent to the point is the instantaneous rate of change. So to find the speed of the dragster as it crossed the finished line, we can use the slope of the line tangent to the point of the finish line. Th...
... middle of paper ...
...al., 2008, p. 372).” The proof of this theorem is beyond the scope of this course and will not be included here. This does not mean that a discontinuous function is not integrable. For instance, take the piecewise function f(x)= {█(0 for x≤0@1for x >0)┤. The area under the curve for any value less than 0 is simply 0, where the area under the curve for any x greater than 0 is 1 times that value of x. This function, while discontinuous, is still integrable. The graph of this function is shown here.
In conclusion, a differentiability implies continuity, but continuity does not guarantee differentiability. Also, continuity implies integrability, but integrability does not guarantee continuity.
Works Cited
Weir, M. D., Hass, J., & Giordano, F. R. (2008). Thomas’ calculus early transcendentals media upgrade (11th ed.). New York, NY: Pearson Education, Inc..
Then, he characterizes this rule as something that always and necessarily follows. Also, this rule must make the
Kant, Immanuel, and Friedrich Max (Indologe) Müller. "Doctrine 1/The Element of Transcendentalism." Critique of Pure Reason: In Commemoration of the Centenary of Its First Publication. London: Macmillan, 1881. 37-59. Print
One of the most popular arguments for the existence of God belongs to St. Thomas Aquinas. In his “Five Ways” writing, Aquinas discusses five arguments for the existence of God. Thomas Aquinas uses what is known as a posteriori argument, which uses information from experience to form the argument. It is important to note that Aquinas was a strong believer in God to begin with, as he was Catholic. He is recognized as a Saint by the Catholic Church. His argument for the existence of God was controversial at the time of its writing, but is now highly regarded as the definite philosophical explanation of the Catholic faith. Aquinas felt that it was only necessary to have five arguments for the existence of God in order to prove his existence. The first way discusses the Argument from Motion. The second talks about the Argument from Efficient Cause. The third regards the Argument to Necessary Being. The fourth explains the Argument from Gradation, and the fifth is about the Argument from Design. For this paper, I am going to analyze Aquinas’ second way, the Argument from the Efficient Cause.
The gradient of the graph tells us whether the different rate curves have the same relation, meaning if they have a similar rate of reaction. Reactions can take place in a variety of customs; they can bee steep or steady. The steeper the slope, the faster the reaction takes place. The steadier the slope, the slower the reaction takes place. Aim:
Directional derivative: Directional derivative represents the instantaneous rate of modification of the function. It generalizes the view of a partial derivative.
Classification of Derivatives: Derivatives are classified in terms of their payoffs and as exchange traded and over the counters.
Wippel, John. The Metaphyiscal Thought of Thomas Aquinas. (Washington, District of Columbia: The Catholic University of America Press, 2000). Print.
Bardi, Jason Socrates. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time. New York: Thunder's Mouth Press, 2006.
Descartes, R. and Sutcliffe, F. 1968. Discourse on method and the meditations. Harmondsworth, England: Penguin Books.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26. Retreived from: http://math.coe.uga.edu/olive/EMAT3500f08/instrumental-relational.pdf
Mr. Keating, the new English teacher at Welton Academy, is the epitome of transcendentalist values. He devoutly embraces the idea of nonconformity, a key aspect of transcendentalism. At the start of the first English class, he instructs his students to tear out the introductory portion of their textbook because he disagrees with the content. This captures the attention of his student immediately differentiating him from the other professors at Welton and their orthodox teaching styles. Emerson, a famous poet who led the Transcendentalist movement of the mid-19th century, said, “Whoso would be a man, must be a nonconformist.... Nothing is at last sacred but the integrity of your own mind.” Further emphasizing the importance of individuality, Mr. Keating takes his student out to the courtyard and asks them to walk in their own exclusive speed and style, independent of how everyone else is walking. When inquired by Mr. Nolan, the Headmaster at Welton, as to what exactly he was doing, Mr. Keating replies that he w...
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
... resultant speed and, by the definition of the tangent, to determine the angle of which the object is launched into the air.
The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world. A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced. Mathematicians are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest richness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof.