... Calculus From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). | It has been suggested that Infinitesimal calculus be merged into this article or section. (Discuss) Proposed since May 2011. | Topics in Calculus | Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus | Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | 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 modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the...
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...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 considered to have been founded 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, lambda calculus, and...
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...Academic Year 21250 Stevens Creek Blvd. Cupertino, CA 95014 408-864-5678 www.deanza.edu 2015 - 2016 Please visit the Counseling Center to apply for degrees and for academic planning assistance. A.A.T./A.S.T. Transfer Degree Requirements 1. Completion of all major requirements. Each major course must be completed with a minimum “C” grade. Major courses can also be used to satisfy GE requirements (except for Liberal Arts degrees). 2. Certified completion of either the California State University (CSU) General Education Breadth pattern (CSU GE) or the Intersegmental General Education Transfer Curriculum (IGETC for CSU). 3. Completion of a minimum of 90 CSU-transferrable quarter units (De Anza courses numbered 1-99) with a minimum 2.0 GPA (“C” average). 4. Completion of all De Anza courses combined with courses transferred from other academic institutions with a minimum 2.0 degree applicable GPA (“C” average). Note: A minimum of 18 quarter units must be earned at De Anza College. Major courses for certificates and degrees must be completed with a letter grade unless a particular course is only offered on a pass/no-pass basis. Associate in Science in Business Administration for Transfer A.S.-T. Degree The Business major consists of courses appropriate for an Associate in Science in Business Administration for Transfer degree, which provides a foundational understanding of the discipline, a breadth of coursework in the discipline, and preparation...
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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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...Curriculum & Scheme of Examination APPLIED MATHEMATICS - I Course Code: BTC 101 Credit Units: 04 Course Objective: The knowledge of Mathematics is necessary for a better understanding of almost all the Engineering and Science subjects. Here our intention is to make the students acquainted with the concept of basic topics from Mathematics, which they need to pursue their Engineering degree in different disciplines. Course Contents: Module I: Differential Calculus Successive differentiation, Leibnitz’s theorem (without proof), Mean value theorem, Taylor’s theorem (proof), Remainder terms, Asymptote & Curvature, Partial derivatives, Chain rule, Differentiation of Implicit functions, Exact differentials, Tangents and Normals, Maxima, Approximations, Differentiation under integral sign, Jacobians and transformations of coordinates. Module II: Integral Calculus Fundamental theorems, Reduction formulae, Properties of definite integrals, Applications to length, area, volume, surface of revolution, improper integrals, Multiple Integrals-Double integrals, Applications to areas, volumes. Module III: Ordinary Differential Equations Formation of ODEs, Definition of order, degree & solutions, ODE of first order : Method of separation of variables, homogeneous and non homogeneous equations, Exactness & integrating factors, Linear equations & Bernoulli equations, General linear ODE of nth order, Solution of homogeneous equations, Operator method, Method...
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...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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...MONEERAH INTEGRATED SCHOOL Merila, Ubaldo Laya, Iligan City S.Y. 2013- 2014 A Requirement in Mathematics IV: Calculus Controversy: Leibniz vs. Newton by Noronsalih Ali, Jra Submitted to Ms. Moneerah A. Bint- Usman Dedication I would like to dedicate this research to my adviser, teacher Monie. And to all of the people who inspired me, especially my parents for their support and to God for giving me enough knowledge to make this study successful. Acknowledgement Abstract This research explores more about the history of the two Mathematicians and how did they invent calculus with the same idea. This is a study about a controversy in Mathematics where Sir Isaac and Gottfried von Leibniz were involved. It tackles about who was the real father of calculus and who gets the credit of inventing it. Inside this paper, the researcher will also discuss a brief summary about Calculus, and short biography of the Mathematicians that were involved in this matter. Many people debates about this matter and we will also tackle some of it in this study. Introduction: So who really invented calculus first? Was it Sir Isaac Newton or Gottfried von Leibniz? Well let's do some investigation. There is no doubt about it that Newton and Leibniz made great mathematical breakthroughs but even before they began studying Calculus there were other people such as Archimedes and Euclid who discovered the infinite and infinitesimal. Much of Newton and Leibniz's work...
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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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...I've been asked this same question many times before and it has always been the hardest for me to answer. My enthusiasm for the sciences and mathematics didn't emerge in one crystallizing moment or epiphany, but rather evolved over time. Curiosity has always been a defining quality of mine, probably ever since I was a young boy watching "The Magic School Bus" and reading "ZooBooks". A general yearning for discovering new things, regardless of topics or delivery via books, television, or teacher, was the first step to spark my interest in science. Later as a middle school student, I became much more aware that I enjoyed and excelled in science and math more than any other subjects. Entering into my middle school's math and science accelerated program allowed me to further develop my interest in science as I took Honors Biology and spent my summer between 8th and 9th grade volunteering for Cornell’s Cooperative Extension conducting water sampling for effluent matter at local bays. It was also at this point that I started to give thought to what careers I may want to pursue, specifically in a scientific field. In my past three years of high school, I've taken great initiative to enrich my scientific experience and identify which fields directly interest me. I became a member and now president of my school's selective science research program, attended lectures at Stony Brook University and started ready Scientific News. Reading about Physics made me inquisitive about the unknown...
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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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...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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...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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...University of Connecticut Department of Mathematics Math 1131 Sample Exam 1 Spring 2014 Name: This sample exam is just a guide to prepare for the actual exam. Questions on the actual exam may or may not be of the same type, nature, or even points. Don’t prepare only by taking this sample exam. You also need to review your class notes, homework and quizzes on WebAssign, quizzes in discussion section, and worksheets. The exam will cover up through section 3.2 (product and quotient rule). Read This First! • Please read each question carefully. Other than the question of true/false items, show all work clearly in the space provided. In order to receive full credit on a problem, solution methods must be complete, logical and understandable. • Answers must be clearly labeled in the spaces provided after each question. Please cross out or fully erase any work that you do not want graded. The point value of each question is indicated after its statement. No books or other references are permitted. • Give any numerical answers in exact form, not as approximations. For example, one-third 1 is 3 , not .33 or .33333. And one-half of π is 1 π, not 1.57 or 1.57079. 2 • Turn smart phones, cell phones, and other electronic devices off (not just in sleep mode) and store them away. • Calculators are allowed but you must show all your work in order to receive credit on the problem. • If you finish early then you can hand in your exam early. Grading - For Administrative Use Only Question:...
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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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...Computer Assignment Use Wolfram Alpha or any other technology to answer the questions below. Copy all relevant answers into this Word document. Save the Word document and send it to me via email attachment. Do NOT forget to type your name into the document, and include in your responses the commands you used to get the answers. 1. Consider the function f(x)=(e^2x-1)/x Find the limit of f(x) as x approaches zero. 2. Define the function Find the derivative of that function. Find f’(0.67) (the first derivative at 0.67). What does that mean for the function f at the point? Find f’’(0.67) (the second derivative at 0.67). What does it mean for the function f at that point? Find all points where the derivative is zero. A) B) C) D) 3. Define the function Find the derivative of the function and use Wolfram Alpha to confirm your answer. Find all points where the derivative is zero and classify them as local extrema, if possible Determine if f is increasing (going up) or decreasing (going down) between the points found in (b) A) B) Local extreme’s are listed C) Increasing 4. Find the following integrals: a) b) 5. Find the area between the graph of f(x) = (x2 – 4) (x2 - 1) and the x axis. Note that one simple definite integral won’t do it, you will need to carefully determine where the function is positive and negative and integrate accordingly, perhaps using multiple steps. 6. Solve the system...
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