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Words 1124

Pages 5

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, product and composition. To differentiate, apply the differentiation rule corresponding to the last construction.

5. Rules for Constructions

SUM: LINEARITY: PRODUCT: RECIPROCAL: QUOTIENT: CHAIN (for compositions): D( f + g) = D f + Dg D(a f + bg) = aD f + bDg D( f · g) = D f · g + f · Dg D(1/ f ) = −D f / f 2 g · D f − f · Dg g2 D( f ◦ g) = (D f ) ◦ g · Dg dy dy du = dx du dx D( f /g) = 1…...

...SOME DIFFERENTIAL CALCULUS YOU MAY NEED 1. THE FIRST DERIVATIVE 1.1 For a power function y = f(x) = Axm , dy/dx = Amxm-‐1 This holds for ALL values of m. If m= 0, i.e.-‐ y = Ax0 (which is equal to A) dy/dx = A0x0-‐1 = 0. So the derivative of a fixed (constant) quantity is zero Examples: a. For a linear function of a single variable: y = f(x) = ax dy/dx = a Thus for a cost function C(q) = 4q, the marginal cost dC/dq = 4 b. For an affine (linear with intercept) function: y = f(x) = c + mx dy/dx = m (note that the c vanishes) For a cost function with a fixed cost, C(q) = 180 + 3q, dC/dq = 3 c. We can also get a function of mixed higher order polynomials y = f(x) = a + bx + cx2+ dx3 dy/dx = b + 2cx + 3dx2 For a cost function C(q) = 150 + 34q3 + 20q2, ......

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...this may not seem to be that hard of a problem but for me these types of problem are difficult. I wanted to pick a problem that pertained to my major, which is Mechanical Engineering but that did not work out to well for me on the last project because I ended up treating the problem like an engineering problem and not a calculus problem, so with that being said that’s one reason that I picked this problem. The other reason that I choose this problem is that it seemed interesting to me and it is a story problem. For me personally I struggle with story problems and have difficulty comprehending what the problem is asking me to find. So let me tell what the problem is and then I will explain how I think it is related to the curriculum and a real world phenomenon: A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m3/min, how fast is the water level rising when the water is 30 cm deep? This problem is related to the curriculum because it is about linear approximation and differentials (Stewart). We covered this section is 3.9 in our text book and I thought that this was a changeling section. Now for a real life situation well we can use the math and the problem solving skills that are involved in this problem and apply them to many different situations that involve a rate of change for filling a......

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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......

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...Section 1.2 (Page 87) (Calculus Book): 14, 23, 26, 29, 30, 31, and 32 14.��������→�� ���� +���� −����+�� ���� −����+�� ���� + ���� − ���� + �� = ������ �� ��→�� �� − ���� + ���� − �� − ���� + �� ���� − ���� + ������ − ���� − ���� + �� = ������ �� ��→�� �� �� − �� + �� �� − �� − �� �� − �� ���� �� − �� + ���� �� − �� − �� �� − �� = ������ ��→�� ���� + �� − �� �� − �� ���� �� − �� + ���� �� − �� − �� �� − �� = ������ ��→�� �� − �� �� − �� �� − �� ���� + ���� − �� = ������ ��→�� �� − �� �� − �� ���� + ���� − �� − �� = ������ �� ��→�� �� + ���� − �� − �� = ������ ��→�� �� �� + �� − �� �� + �� �� �� + �� − �� �� + �� �� + �� �� − �� �� + �� �� − �� �� + �� �� + �� �� = = �� + �� �� + �� �� ��+�� ���� −���� = ������ ��→�� = ������ ��→�� 23 ������ ��→�� = ������ ��+�� ��→�� ��+�� ��−�� ⟹ ������ ��→�� �� �� �� = = = ������������������ ∴ ���������� ����������′ �� ���������� �� − �� �� − �� �� ��−�� ���� −����−�� 26 ������ ��→�� = ������ ��→�� ��−�� ���� −����+����−�� = ������ ��−�� ��→�� �� ��−�� +�� ��−�� Page | 1 = ������ ��→�� �� − �� �� − �� −�� �� = = = �� − �� �� + �� �� − �� �� + �� �� × �� �� ∴ ���������� �������� ������ ����������. �� − �� ��−�� �� �� = ������������������; 29������ ��→�� ��−�� ��−�� = ������ ��→�� = ������ ��→�� ��−�� ��+�� ��−�� = ������ ��→�� �� + �� = �� + �� = �� + �� = �� ��−�� 30������ ��→�� ��− �� = ������ �� ��......

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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......

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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......

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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......

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...LRBI Checklist Differential Reinforcement Differential Reinforcement ifferential reinforcement is the reinforcement of one form of behavior and not another, or the reinforcement of a response under one condition but not another. Differential reinforcement uses positive reinforcement to differentiate or separate appropriate student behavior from inappropriate behavior by increasing one while decreasing the other. Definition D Things to Do Select behaviors to be decreased or increased. Select alternative, incompa tible or communicative behavio , rs to be taught. Determine time interval. Set criterion. Ignore inappropriate behav iors. Monitor the student’s performance. Level 1: Positive Interaction Procedures 1 LRBI Checklist Differential Reinforcement There are six strategies that comprise or make up differential reinforcement: Select behaviors to be decreased or increased. Select behaviors to be decreased or increased. These behaviors should be objectively defined and must be observable and measurable. For DRO, DRI, DRA, and DRC, there will be a specific behavior to decrease, such as talk-outs, hitting, not following directions, out of seat, disrupting other students, or tantrums. For DRH, there will be a specific behavior to increase, such as contributing in class, positive social interactions, getting to class on time, or task completion. D R H RL D problem behaviors Level 1: Positive Interaction Procedures ...

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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......

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...Product differentiation For Larson to separate themselves from the household battery names such as Energizer and Duracell they must focus on marketing and pricing strategies, which should include investment into differentiating their product from those competitors to create a competitive advantage. According to (n.d.) “Product differentiation can be achieved in many ways. It may be as simple as packaging the goods in a creative way, or as elaborate as incorporating new functional features.” (Product differentiation, para. 2). Since there is stiff competition in the battery industry Larson finds it difficult to differentiate its product. If Larson chooses to differentiate its product they will have more pricing power and could even have the possibility of creating somewhat of a monopoly. There are several ways in which Larson can seek to differentiate its products, some include: Improving the product itself by improving reliability, quality, performance, and styling. In some cases differentiation does not involve making changes to the product at all, but instead creating a new advertising campaign, sales promotions, or even making changes to the lackluster packaging can help differentiate their product. Additionally, differentiation can be created by educating customer about the difference between products through product branding and advertising. After operating in Germany for over 15 years Larson has built its brand using unattractive packaging and humorous commercials......

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...for my masters here at NMSU. In order to meet the prerequisites for the MBA program, I needed to take calculus and I’m glad I did because I wouldn’t have otherwise. Everything I learned in calculus challenged me to think critically and in another sense that I hadn’t known before. Being able to apply calculus to optimize a business’s performance, whether it be through current performance or future expectations is huge for any business. (Saiz, 2001) Since I can remember I’ve always felt like I am supposed to help my dad’s business despite it not really being my passion. I have found it to be difficult to work in that kind of environment. My dad’s business has done really well but it’s not something I love doing and honestly I don’t want anything given to me because of my dad’s successes, I want to do it on my own. The knowledge I have learned through college and the amount I have grown to be independent has made me realize that I can do anything. I have since decided to start my own in business in something I am extremely passionate about. Using calculus to determine rates to charge potential customer for optimal profits can really help my business get started. I think many people that start businesses jump into it without having structure and often times fail. I have found many tools and resources such as calculus to be very beneficial to the success of a business. Calculus will help my business predict maximum sales for a given service and help determine minimum costs......

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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......

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...Student Solutions Manual for SINGLE VARIABLE CALCULUS rS al e SEVENTH EDITION DANIEL ANDERSON University of Iowa Fo JEFFERY A. COLE Anoka-Ramsey Community College N ot DANIEL DRUCKER Wayne State University Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher except as may be permitted by the license terms below. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, ISBN-13: 978-0-8400-4949-0 ISBN-10: 0-8400-4949-8 Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd. e © 2012 Brooks/Cole, Cengage......

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...This page intentionally left blank A Course in Financial Calculus A Course in Financial Calculus Alison Etheridge University of Oxford CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521890779 © Cambridge University Press 2002 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 eBook (EBL) ISBN-13 978-0-511-33725-3 ISBN-10 0-511-33725-6 eBook (EBL) ISBN-13 ISBN-10 paperback 978-0-521-89077-9 paperback 0-521-89077-2 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface 1 Single period models Summary 1.1 Some deﬁnitions from ﬁnance 1.2 Pricing a forward 1.3 The one-step binary model 1.4 A ternary model 1.5 A characterisation of no arbitrage 1.6 The risk-neutral probability measure Exercises Binomial trees and discrete parameter......

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...Kellie Aosley8 Recent Hedical school &a&ate "CALCULUS FOR THE UTTERLY CONFUSED has proven to be a wonderful review enabling me t o move forward in application of calculus and advanced topics in mathematics. I found it easy t o use and great as a reference for those darker aspects of calculus. I' Aaron Ladeville, Ekyiheeriky Student 'I1am so thankful for CALCULUS FOR THE UTTERLY CONFUSED! I started out Clueless but ended with an All' Erika Dickstein8 0usihess school Student "As a non-traditional student one thing I have learned is the Especially in importance of material supplementary t o texts. calculus it helps to have a second source, especially one as lucid and fun t o read as CALCULUS FOR THE UTTERtY CONFUSED. Anyone, whether you are a math weenie or not, will get something out of this book. With this book, your chances of survival in the calculus jungle are greatly increased.'I Brad &3~ker, Physics Student Other books i the Utterly Conhrsed Series include: n Financial Planning for the Utterly Confrcsed, Fifth Edition Job Hunting for the Utterly Confrcred Physics for the Utterly Confrred CALCULUS FOR THE UTTERLY CONFUSED Robert M. Oman Daniel M. Oman McGraw-Hill New York San Francisco Washington, D.C. Auckland Bogoth Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Library of Congress Cataloging-in-Publication Data Oman, Robert M. Calculus for the utterly confused / Robert M.......

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