...Calculus is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot. Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations...
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...Calculus From Wikipedia, the free encyclopedia This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus [show]Integral calculus [show]Vector calculus [show]Multivariable calculus Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits,functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches,differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science,economics, and engineering and can solve many problems for which algebra alone is insufficient. Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus...
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...Project Gutenberg EBook of Calculus Made Easy, by Silvanus Thompson This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Calculus Made Easy Being a very-simplest introduction to those beautiful methods which are generally called by the terrifying names of the Differentia Author: Silvanus Thompson Release Date: October 9, 2012 [EBook #33283] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK CALCULUS MADE EASY *** Produced by Andrew D. Hwang, Brenda Lewis and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) transcriber’s note Minor presentational changes, and minor typographical and numerical corrections, have been made without comment. All A textual changes are detailed in the L TEX source file. This PDF file is optimized for screen viewing, but may easily be A recompiled for printing. Please see the preamble of the L TEX source file for instructions. CALCULUS MADE EASY MACMILLAN AND CO., Limited LONDON : BOMBAY : CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK : BOSTON : CHICAGO DALLAS : SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO CALCULUS MADE EASY: BEING A VERY-SIMPLEST...
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...Many gifted students such as myself accredit Gottfried Leibniz to be the precursor of their impending demise. Mr. Leibniz is the curator of Calculus, the idol of integrals, the devil of derivatives. Calculus is the study of change, and since it’s inception in the 17th century, it has changed the world. I also believe it to be the keystone to changing our future. Studies and general common sense show that our world is quickly deteriorating, and although judgements vary, it is no secret we will soon be evicted from Earth. Our future relies in physics, as it is our only foundation for understanding the world outside our world, and Calculus is our foundation for understanding our sole gateway. Physics would be just a game if it weren’t for Calculus, and we need the higher level of physics to comprehend what is outside our atmosphere and galaxy. Once the day approaches where humanity’s existence is futile and we are being shipped off to our new home in some foreign galaxy, Calculus will be our intellectual voucher to save humanity and all of it’s progression since our conception. Yes, I like all, have suffered through its limits and fundamental theorems, but I, unlike all, see the value in the deed. People love to hate it, but what I’ve learned while racing at the highest level, is that you need to embrace the struggle, and use that struggle to achieve something greater than yourself. I’ve lost entire days of my life studying for seemingly pointless tests, struggling to grasp optimization...
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...the prologue and chapter one of “The Calculus Diaries”, my perspective on calculus and its concepts have changed. “The Calculus Diaries” describes the history of calculus, such as who discovered it, when it was discovered, and how it can be used in everyday life. It starts out by describing the story of Archimedes, who invented devices to help fend off the Roman Empire from invading Syracuse. He was considered to be the first person to describe calculus concepts. The author describes that the two main concepts that make up calculus are the derivative and the integral. The author also describes his personal conflicts with calculus in the past and what it took for him to overcome his hatred for calculus and math in general. The...
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...AP Calculus was the most difficult of all the classes. This class was a struggle to get through, but I was able to pull through. When most people first think of the term mathematics or the word “Calculus,” they don’t get too excited. Most people tend to say “I hate math!” or the big one, “When are we ever going to use it in our lives.” Calculus meant one needed to be prepared to keep up. There was no time to lose in this class. Struggling in class definitely made it harder to learn. I knew right then I had to do something in order to and get help so as not to be left behind. My analysis as to why it was such a difficult subject would be the fact that I was not intellectually prepared to go into such convoluted math problems, though it was...
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...Derivative=limf(x+change in x) –f(x) R=x*p P=R-C Change in x Limits: Point Slope form: y-y,=m(x-x,) Hole (removable discontinuity) Jump: Limit does not exist Vertical Asymptote: Limit does not exist Walking on graph at x=#, what is the y-value? Find the equation of a tangent line on f(x)=1/x at (1,1) Ex1: lim x^2+4x+3 = (-1)^2 +4(-1) +3 =0 point: (1,1) f(x)=x^-1 m=-1 = -1 = -1 = m x-1 x+1 -1+1 0 m=f’(x) f’(x)=-1x^-2 (1)^2 1 ------------------------------------------------- y-1=-1(x-1) y-1=-x+1 = y=-x+2= EQUATION Product Rule: f’(x)=u’v+v’u Quotient Rule: f(x)=h(x)g’(x)-g(x)h’(x)=lowd’high-highd’low [h(x)]^2 bottom^2 Chain Rule: derivative of the outside(leave inside alone)*derivative of the inside Implicit Differentiation: (1) take derivative of each term normally, if term has y on it, we will multiply it by y’ Critical Points: (1) Find f’(x); (2) Set f’(x)=0, solve it; (3) Plot points on # line; (4) Test points around the points in step 3, by plugging them into derivative. If positive: up If negative: down; (5) Write our answer as an interval Max and Mins (relative extrema): (1) Do all the up and down stuff from 3.1; (2) If you went up then down you have a max; if you went down then up you have a min; (3) Label the points (x,y) for max and mins to get the y, go back to f(x) Ex2: f(x) =1/4x^4-2x^2 a) Find the open intervals on which the function is increasing or...
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...There are many reasons why I want to take AB Calculus next year. The first reason why I would like to take AB Calculus is that this course will challenge me and help me excel in college when I take other calculus classes. In college, I will be pursuing a major/degree in business, and AB Calculus will lay a solid math foundation for me to excel in college. I will prepare over summer break, with a tutor if necessary, to get caught up with the Honors Pre-Calculus class and make sure that I am ready for AB Calculus when school starts. I missed getting into AP Calculus by one point and I do not want to be in a slow-paced math class my senior year because of this one point. I do not believe that one point should dictate what class I should be placed...
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...Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) f (x) d For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), d dx , d f (x), f (x), f (1) (x). The dx “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy whereas f (x) is the value of it at x. If y = f (x), then Dx y, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f , “∆ f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy Historical note: Newton used y, while Leibniz used dx . About a century later Lagrange introduced y and ˙ Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. 4. Operating Principle Many functions are formed by successively combining simple functions, using constructions such as sum...
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...Applications of Derivatives So we know how to find and solve for derivatives. But what is the significance? Derivatives can allow us to do many things, including: * Find rate of change (slope) of a curve * Determine the motion of a particle * Find the maximum and minimum values of the function in an interval * Analyze graphs and functions * Sketch curves by hand * Optimization (solving applied minimum and maximum problems But before we get into these topics, let’s go over some basic definitions and theorems we will use. definitions extrema | max / min values in a function | local max / min | peaks and valleys within a curve | absolute max / min | the greatest and lowest value a function obtains, either on its entire domain or closed interval | critical point | an x value for which f’(x) | theorems extreme value theorem | if the curve is continuous on a closed interval, then a max/min exist | theorem 3.2 | extrema can only occur at critical points | mean value theorem | assume f(x) is continuous on [a,b] and differentiable on (a,b). Then there exists at least one value c in (a,b) such that f’(c) = fb-f(a)b-a | * finding the rate of change of a curve solving derivatives allow us to find the slope at a particular point on a curve. let’s start with a basic problem. find the slope of the curve y=4x2 at x = 3. at the right, we can see that the slope of the curve at x = 2 is 16. * determining the motion of a particle ...
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...Seminar-1 Article Analysis: Why do we study Calculus? Name: Salman 1. * This article is very interesting, I came to know about a lot of great contributors to our life like Newton, Pluto, Aristotle, and Leibnitz and their famous work .In a summary of this article would say that I came to know how different field of science and economy benefit from the calculus. Economics, physics, Astronomy and General Science all these field of study have huge impact of Calculus; they need help of Calculus in one way or the other way. * Primary Topics: Kepler’s laws: 1. The orbits of the planets are ellipses, with the sun at one focus point 2. The velocity of a planet varies in such a way that the area covered out by the line between planet and sun is increasing at a constant rate 3. The square of the orbital period of a planet is proportional to the cube of the planet's average distance from the sun. (Reference: Article: Why do we study Calculus?) * Numbers are uncountable and we can measure the change of them with respect to time 2. I found these topics covered from the material of the first seminar, they were the applications of the material topics. * Change in one variable in respect to the other variable * Rates of change * Limits * Graphs and distance of one point from the another point I found that the knowledge of these points was applied for the calculations of the things discussed in this article like velocity of an object, planets...
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...CALCULUS I (MATH 156) SUMMER 2013 — FINAL EXAM JULY 26, 2013 Name: This exam consists of 8 questions and 1 bonus question. Show all your work. No work, no credit. Good Luck! Question Points Out of 1 18 2 20 3 18 4 15 5 8 6 8 7 14 8 7 9(Bonus) 7 Total 115 18 points 1. For the function f (x) = x3 − 6x2 + 9x − 3 (a) find f (x). (b) determine all the critical points of f. (c) find the intervals where f is increasing and where it is decreasing. (d) classify each critical point as relative maximum or minimum. (e) Find f (x). (f) Find the intervals where the graph of f is concave up and concave down. (g) Determine the inflection points. Page 2 20 points 2. Evaluate the following limits: (a) lim x2 − 4x + 4 x→2 x3 + 5x2 − 14x (b) lim x2 x→0 cos 8x − 1 (c) lim x − 8x2 x→∞ 12x2 + 5x (d) lim e3x − 1 x→0 ex − x (e) lim x2 e−x x→∞ Page 3 18 points 3. Find the following indefinite integrals: (a) 3 cos 5x − √ + 6e3x dx x (b) √ 4x dx x2 + 1 (c) x2 + √ x x−5 dx Page 4 15 points 4. Evaluate the following definite integrals: 1 (a) 0 x4 + 3x3 + 1dx e2 (b) 1 (ln x)2 dx x π 4 (c) 0 (1 + etan x ) sec2 xdx Page 5 8 points 5. Sketch and find the area of the region that lies under y = ex and above the x axis over the interval 0 ≤ x ≤ 7. 8 points 6. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand...
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...The Application of Matrix and Calculus in Business Submitted To: Ms. Farzana Lalarukh Associate Professor Department of finance University of Dhaka No NAME ID 01 Md. Shezanur Rahman 16-011 02 Morjina Begum 16-061 03 Ishrat Amin 16-119 04 Md. Liakot Akbar 16-121 05 Farah Tasneem 16-163 Ms. Farzana Lalarukh Associate Professor Department of Finance University of Dhaka Subject: Submission of Term paper on “The Application of Matrix and Calculus in Business” We are pleased to submit this term paper on “The Application of Matrix and Calculus in Business” as a preliminary requirement of fulfillment of this course of our BBA program. Throughout the study, we have tried with the best of our capacity to conciliate as much information as possible. We would like to thank you for assigning us such a responsibility and helping us on different aspects of the report. We hope you will appreciate the sincere effort. Yours sincerely Farah Tasneem (On behalf of the group) ID: 16-163 Section: A It is a great honor for us to submit this...
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...MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Find the x coordinates of all relative extreme points of[pic]. |A)[pic] |B) [pic] |C) [pic] |D) [pic] |E) [pic] | [pic] First find the derivative of the function[pic], f ’(x): |[pic] |= |[pic] |apply power rule of differentiation | | |= |[pic] |simplify | | |= |[pic] |finish simplifying by first factoring | | | | |out GCF | | |= |[pic] |next factor the trinomial factor, | | | | |leaving the final simplified form of | | | | |the derivative | Set[pic]and solve for x to find critical point(s): When the derivative is set to zero, [pic]; thus, this implies each factor could be equal to zero...
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...At the beginning of my senior year I had set in mind what classes I would be taking this year, this including the two I was most looking forward to AP Calculus and Art. To my surprise when I received my schedule was that it was not at all what I imagined considering and most important to me was that I did not get Art as I requested. When I met up with Ms. Cruz I was disappointed to find out that there is no space in the World Art 1 classes and she set me up with the second best, service learning for Ms. Feury, the art teacher, for her World Art 2 class. As a shocking a delightful surprise Ms.Feury hesitantly but certainly placed me as a student in her World Art 2 class without me having any previous experience with it. Aside from Art I had...
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