Calculus Free Essays

How the calculus was invented? Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Ideas leading up to the notions of function, derivative, and integral were developed throughout the 17th century, but the decisive step was made by Isaac Newton and Gottfried Leibniz. Publication of Newton’s main treatises took many years, whereas Leibniz published first (Nova methodus, 1684) and the whole subject was subsequently marred by a priority dispute between the two inventors of calculus. We will write a custom essay sample on Calculus or any similar topic only for you Order Now Greek mathematicians are credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone’s smooth slope prevented him from accepting the idea. At approximately the same time, Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes of Syracuse developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes’ Quadrature of the Parabola, The Method, Archimedes on Spheres Cylinders. ) It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the time of Newton that these methods were incorporated into a general framework of integral calculus. Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point’s motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983). Before Newton and Leibniz, the word â€Å"calculus† was a general term used to refer to anybody of mathematics, but in the following years, â€Å"calculus† became a popular term for a field of mathematics based upon their insights. The purpose of this section is to examine Newton and Leibniz’s investigations into the developing field of infinitesimal calculus. Specific importance will be put on the justification and descriptive terms which they used in an attempt to understand calculus as they themselves conceived it. By the middle of the seventeenth century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintained Hellenistic mathematics as the starting point for research, Newton, Leibniz and their contemporaries increasingly looked towards the works of more modern thinkers. Europe had become home to a burgeoning mathematical community and with the advent of enhanced institutional and organizational bases a new level of organization and academic integration was being achieved. Importantly, however, the community lacked formalism; instead it consisted of a disordered mass of various methods, techniques, notations, theories, and paradoxes. Newton came to calculus as part of his investigations in physics and geometry. He viewed calculus as the scientific description of the generation of motion and magnitudes. In comparison, Leibniz focused on the tangent problem and came to believe that calculus was a metaphysical explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between the integral and the differential. This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created. Their unique discoveries lay not only in their imagination, but also in their ability to synthesize the insights around them into a universal algorithmic process, thereby forming a new mathematical system. How to cite Calculus, Papers Calculus Free Essays This is a take-home exam. You may consult different sources of information including but not limited to class notes, homework and/or textbook. You may also collaborate with your classmates but you must write your own solutions. We will write a custom essay sample on Calculus or any similar topic only for you Order Now It is fairly obvious when a student is Just copying the work from an external source; if I deem a solution has just been copied I may give you a warning for Academic Dishonesty (ADD for short). Solutions to some of these problems are available elsewhere, if you happen to come cross one of them you should make your best effort to understand it, then write your own using your ideas and understanding of the topics. Question: 2 3 4 5 6 7 8 9 10 Total Points: 25 20 200 Score: Please do not answer the questions in the limited space provided; use scratch paper and attach it to this cover page. Name: Signature: Page 1 of 6 Please go on to the next page†¦ Questions (10 puts) 1. I. Use Roller’s theorem to prove that f x ex. root . 013 xx 2 has at most one real Hint: If has two roots (say a and b) then FAA Feb. O. What does Roller’s theorem say in this situation? (1 5 puts) it. Let f be continuous on a, and differentiable on a, b . Show that there exists c a, b such that the tangent at c, FCC is parallel to the secant through a, FAA and b, Feb. . In other words, show that FAA Equation (1) is known as the Mean Value Theorem formula. Hint: Apply Roller’s theore m on a, to the function G x Feb. FAA Feb. Keep in mind that a, FAA , b and Feb. are constants. 2. True or false. (5 puts) I. Iffy O, f is neither concave up nor concave down around x a. I. It. If is continuous on a, b and c iii. If f is continuous but not necessarily differentiable on O, then the absolute maximum and the absolute minimum off exist. V. If f is differentiable on a, b then it is also continuous on a, b and the absolute maximum and absolute minimum exist. V. If x a corresponds to an inflection point off , then f ii a around x a. A, b is a local maximizes then fix O. O and f ii x changes sign 3. Henry is pulling on a rope that passes through a pulley on a MM t pole and is attached to a wagon. Assume that the rope is attached to a loop on the wagon 2 Ft off the ground. Let x be the distance between the loop and the pole (see figure 1). (10 puts) I. Find a formula for the speed of the wagon in terms of x and the rate at which Henry lulls the rope. We say that x a is a root (or is a zero) off x , if FAA O. We say thatch is a local maximizes if f c is a local maximum. Page 2 of 6 Henry Figure 1: Henry pulling the wagon from problem 3. 10 puts) it. Find the speed of the wagon when it is 12 Ft from the pole, assuming that Henry pulls the rope at a rate of 1. 5 Ft sec. (25 puts) 4. Olav Adagio -a former student of mine- was asked to sketch the graph of a function. Unfortunately Olav often forgets things. Luckily for you, he wrote down some statements. Regarding the function f x , he wrote: * It is only defined on , and it is continuous. * It is strictly positive, except at x 2 and x O where its value is zero. *f 2 2, f 3 1, and f 4 1. 1 when x O. Regarding if x , he wrote: XSL * On the interval (-2, 1) it exists only at those points where g x is well defined. Moreover, it is positive when g is positive; negative when g is negative; and zero when g is zero. * On the interval (1,2) it is identically equal to zero. * On the interval (2,4) it is negative. Lastly, regarding f ii x , he wrote: * On (-2, 1) it exists whenever h x signs on this interval. Is well defined. They also have opposite On (2,4) it changes sign from negative to positive at x 3. Help Olav sketch the graph off . Make sure to clearly identify the local and global extreme as well as the inflection points. 5. A piece of wire 24 CM long is given to you. You can choose to either cut it into two pieces or leave it the way it is. If you decide to cut it, one piece must be bent into the Page 3 of 6 shape of a square, while the remaining one must be bent into the shape of a circle. If you decide not to cut it, you can bend it into either shape. (5 puts) I. Denoting by x the length of the piece of the wire that will be bent into the shape of circle, obtain an expression for the area enclosed by the wire. Make sure that the formula works regardless of whether or not the wire is bent into one or two pieces. (20 puts) it. Find the maximum area that can be enclosed by the wire. Explain how this area can be obtained by specifying the dimensions (ii. , length of sides and/or radius) of the objects to be constructed. The following facts might come in handy: If a square has perimeter. How to cite Calculus, Papers