...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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...The Application of the Profit Function in a Business’s Growth Differential Calculus May 24, 2014 The application of calculus in a business is extremely important since calculus is considered as the study of changes. Its complexity in the study of changes has become one of the humankind’s greatest tool for analyzing changes in the marketplace. The profit function was created with the main purpose for businesses to understand how the changes in revenues and in costs would generate a profit or not profit at all. A profit can be generated when the amount of revenues is higher than the amount of costs. When a business starts, is normal to not gain any profit at all, in fact, most of the time, a business tends to lose money in its first year. However with that information, a business can analyze the money loss of that year and determine any gaps or holes that prevents the maximization of the profit and have a more prosperous result for the following year. For instance, if a company suffer a loss in profit, they can analyze the profit function to determine the main reason of why there was not a positive profit. If they see that the problem of the profit function was a low revenue then they can regulate the sale of products or services and price control, or if the problem lies in the cost function, they can adjust or lower the costs and expenses made by the business. The graph of a profit function can show at what time of the year the company tends to...
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...MATH 1205 | CALCULUS | 3 | A- | ECON 2005 | MICRO PRINCIPLES OF ECONOMICS | 3 | A- | ACIS 1504 | INTRODUCTION TO BUSINESS INFORMATION SYSTEMS | 3 | B+ | ENGE 1114 | EXPLORATION OF ENGINEERING DESIGN | 2 | A | MATH 1206 | CALCULUS | 3 | B+ | PHYS 2305 | FOUNDATIONS OF PHYSICS | 4 | A | AOE 2074 | COMPUTATIONAL METHODS | 3 | B+ | ECON 2006 | MACRO PRINCIPLES OF ECONOMICS | 3 | A- | ECON 3104 | MICRO ECONOMICS THEORY | 3 | B+ | AOE 2104 | INTRODUCTION TO AEROSPACE ENGINEERING | 3 | A- | ESM 2104 | STATICS | 3 | B | MATH 2224 | MULTIVARIABLE CALCULUS | 3 | A- | AOE 3094 | MATERIALS FOR AEROSPACE AND OCEAN ENGINEERING | 3 | B+ | ACIS 2115 | PRINCIPLES OF ACCOUNTING | 3 | A | BIT 2405 | QUANTITATIVE METHODS | 3 | A | AOE 3104 | AIRCRAFT PERFORMANCE | 3 | B+ | ESM 2204 | MECHANICS OF DEFORMABLE BODIES | 3 | B+ | ESM 2304 | DYNAMICS | 3 | A- | ECON 3204 | MACRO ECONOMICS THEORY | 3 | B+ | MGT 3304 | MANAGEMENT THEORY AND LEADERSHIP PRACTICE | 3 | A | AOE 3054 | AEROSPACE EXPERIMENTAL METHODS | 3 | B+ | AOE 3114 | COMPRESSIBLE AERODYNAMICS | 3 | B | AOE 3124 | AEROSPACE STRUCTURES | 3 | A- | AOE 3134 | STABILITY AND CONTROL | 3 | A | MKT 3104 | MARKETING MANAGEMENT | 3 | B+ | FIN 3104 | INTRODUCTION TO FINANCE | 3 | A | AOE 3044 | BOUNDARY LAYER THEORY | 3 | B | AOE 4154 | AEROSPACE ENGINEERING LAB | 1 | A | AOE 4234 | AEROSPACE PROPULSION | 3 | B | AOE 4266 | DESIGN | 3 | A | FIN 3055 | LEGAL ENVIRONMENT OF BUSINESS | 3 | A- | ...
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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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...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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... 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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...As for what I have been informed, Differential Calculus is simply essential in business. It is best recognized by it’s sub-topic, “the obtainment of derivatives”, which basically represents the average change of income of a business. It is something that simply makes the process of calculations on a business much simpler. Knowing more about it makes me think of the thousands of companies that might be using it without knowing its importance more deeply. Differential Calculus’ applications in business are very extensive. It can be so useful that its applications are seem from calculating the cost of public transport to the costs of making a building. The certain amount of things that need to be used for x thing its also well-defined. If it wouldn’t be by this subject there would certainly be an imbalance in most of the calculation methods used in business. As explained before, one of the commonest but most important topics of this subject is the derivatives, which is the topic I chose to explained in detailed and contrast it with the business area as it’s the one I liked the most. The derivatives are a very useful tool when we related to business, since they have the power to make marginal calculations, in other words it finds the rate of change that’s add up when additional total unit, regardless of the monetary amount being considered: cost, income, profit or production. The derivatives are a very useful tool, since by their very nature they allow marginal calculations...
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...organized thought. Apart from my major courses, I have been crazy about attending lectures of Economy Department in my college. The effect of market and economy attracted me a lot. Therefore, I completed macroeconomics, microeconomics and accounting in my sophomore year. In addition, my father is taking managing position in an entrepreneur. Accordingly, I have been interested in business courses and I decide to take financial related courses, especially Actuarial Studies as the direction of my master’s study. I have been interested in numbers since I was a high school student. I felt satisfied even though I had to contribute more than one hour to solve a mathematic problem. I often spent time to think about other methods to solve mathematic problems that my teacher had provided answers. My enthusiasm about mathematics was inspired again when I began my college study. I took some basic mathematics concepts, such as limit, series, calculus and differential coefficient. I also learned some basic theories and the application of related concepts, such as differential coefficient of function of one variable, calculus, partial derivative of function of many variables, differential equation, and Taylor's formula, intermediate value theorem and infinite series which help me to know the nature of function, and the independent vector algebra and space analytic geometry. To be honest, I even made more efforts in the study of mathematics than that in my academic courses. Therefore, I believe that...
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...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 of undetermined coefficients, Solution of simple simultaneous ODE. Module IV: Vector Calculus Scalar and Vector Field, Derivative of a Vector, Gradient, Directional Derivative, Divergence and Curl and their Physical Significance...
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...Math 5616H Midterm 1 with solutions Spring 2013 March 8, 2013 Total 80 points 1. (15 points) Let f (x) and g(x) be real continuous functions on an interval [a, b], such that b b f 2 (x) dx = a a b g 2 (x) dx = 1. Prove that a f (x)g(x) dx ≥ −1, and that a b f (x)g(x) dx = −1 if and only if f ≡ −g on [a, b]. Answer: Since f and g are continuous, so is (f + g)2 , which is therefore integrable. We compute: b b b b b 0≤ a b [f (x)+g(x)]2 dx = a f (x)2 dx+2 a f (x)g(x) dx+ a g(x)2 dx = 1+2 a f (x)g(x) dx+1, so a f (x)g(x) dx ≥ −1. If it is = −1, then the first “≤” must be “=”, so the continuous function [f (x) + g(x)]2 ≡ 0, and f ≡ −g on [a, b]. 2. (25 points) Let α(x) be a strictly increasing function on the interval [0, 1], such that α(0) = 0 and α(1) = 1. Show that the Riemann-Stieltjes integral 1 α(x) dα(x), 0 exists if and only if α is continuous on [0, 1], and evaluate this integral if it is continuous. Answer: Consider any partition P of [0, 1] : P = {0 = x0 , . . . , 1 = xn }. Since α is increasing, Mi := supx∈[xi−1 ,xi ] α(x) = α(xi ) and mi = α(xi−1 ). Then n n U (P, α, α) − L(P, α, α) = i=1 (Mi − mi )∆αi = i=1 (∆αi )2 . Suppose α is continuous; then since [0, 1] is compact, α is uniformly continuous. Thus, for any given ε > 0 there is δ > 0 so that if |x − y| < δ then |α(x) − α(y)| < ε. Hence if P ∗ is a refinement of P which satisfies xi − xi−1 < δ for all i = 1, . . . , n, we have n U (P ∗ ...
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...A Taylor series for the function arctan The integral If we invert y = arctan(x) to obtain x = tan y, then, by differentiating with respect to y, we find dx/dy = sec2 y = 1 + tan2 y = 1 + x2 . Thus we have (ignoring the constant of integration) y = arctan(x) = dx . 1 + x2 (1) If we now differentiate y = arctan(x/a) with respect to x, where a is a constant, we have, by the chain rule, y = 1 a 1 a = 2 . 1 + (x/a)2 x + a2 (2) Thus we obtain the indefinite integral 1 dx = arctan(x/a). x 2 + a2 a (3) The Taylor Series By expanding the integrand in (3) as a geometric series 1/(1 − r) = 1 + r + r 2 + . . ., |r| < 1, and then integrating, we can obtain a series to represent the function arctan(x/a). We use the dummy variable t for the integration on [0, x] and we first write x x arctan(x/a) = a 0 dt 1 = t2 + a 2 a 0 dt 1 + (t/a)2 (4) Substituting the geometric series with r = −(t/a)2 , we find 1 arctan(x/a) = a x ∞ (−t2 /a2 )n dt = 0 n=0 (−1)n x 2n + 1 a n=0 ∞ 2n+1 . (5) The radius of convergence of this series is the same as that of the original geometric series, namely R = 1, or, in terms of x, |x/a| < 1. The series is a convergent alternating series at the right-hand end point x = a; and it can be shown that sum equals the value of arctan(1) = π/4 (as we might hope). Thus we have the nice (but slowly converging) series for π given by π 1 1 1 = 1 − + − + .... (6) 4 3 5 7 By choosing partial sums of (5) we obtain a sequence of Taylor polynomial...
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...Mathematics Syllabus Algebra: Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations. Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots. Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers. Logarithms and their properties. Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients. Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables. Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations. Trigonometry: Trigonometric functions, their periodicity and graphs, addition...
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...ESTABLISHING THE STRESS LOADING THE ELEMENTS [pic] Fig. 1.1 2. MAIN SCREW CALCULUS 2.1. CHOOSING THE MATERIAL It is chosen OL 50 STAS 500/2 [3] PRE-DIMENSIONING CALCULUS The calculus load F= Q·ctgαmin αmin= 30º [pic] Fig. 2.1 F= Q·ctgαmin= 8914·ctg30°= 15439.5 N Calculus of the load Fc, N Fc= β·F= 1.3·15439.5= 20071.3 N β= 1.25 ... 1.3 [3] The thread's inner diameter [pic] [pic] [pic]=100 ... 120 Mpa [3] Choosing the thread It is chosen Tr 20X4 with the dimension in table 24.2 Table 2.1 |Nominal diameter |Pitch |Medium diameter |External diameter |Inner diameter | |d, mm |P, mm |d2=D2,, mm |D4, mm | | | | | | | | | | | | | |D3, mm |D1,mm | |20 |4 |18 |20.5 |15.5 |16 | CHECKING THE SELF-BRAKING CONDITION The thread's declination...
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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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