Integral Calculus
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Notes on Calculus II Integral Calculus Miguel A. Lerma
Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus. -
Extended Instruction in Business Courses to Enhance Student Achievement in Math Lessie Mcnabb Houseworth Walden University
Walden University ScholarWorks Walden Dissertations and Doctoral Studies Walden Dissertations and Doctoral Studies Collection 2015 Extended Instruction in Business Courses to Enhance Student Achievement in Math Lessie McNabb Houseworth Walden University Follow this and additional works at: https://scholarworks.waldenu.edu/dissertations Part of the Other Education Commons, and the Science and Mathematics Education Commons This Dissertation is brought to you for free and open access by the Walden Dissertations and Doctoral Studies Collection at ScholarWorks. It has been accepted for inclusion in Walden Dissertations and Doctoral Studies by an authorized administrator of ScholarWorks. For more information, please contact [email protected]. Walden University COLLEGE OF EDUCATION This is to certify that the doctoral study by Lessie Houseworth has been found to be complete and satisfactory in all respects, and that any and all revisions required by the review committee have been made. Review Committee Dr. Calvin Lathan, Committee Chairperson, Education Faculty Dr. Jennifer Brown, Committee Member, Education Faculty Dr. Amy Hanson, University Reviewer, Education Faculty Chief Academic Officer Eric Riedel, Ph.D. Walden University 2015 Abstract Extended Instruction in Business Courses to Enhance Student Achievement in Math by Lessie M. Houseworth EdS, Walden University, 2010 MS, Indiana University, 1989 BS, Indiana University, 1976 Doctoral Study Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Education Walden University March 2015 Abstract Poor achievement on standardized math tests negatively impacts high school graduation rates. The purpose of this quantitative study was to investigate if math instruction in business classes could improve student achievement in math. As supported by constructivist theory, the students in this study were encouraged to use prior knowledge and experiences to make new connections between math concepts and business applications. -
Notes Chapter 4(Integration) Definition of an Antiderivative
1 Notes Chapter 4(Integration) Definition of an Antiderivative: A function F is an antiderivative of f on an interval I if for all x in I. Representation of Antiderivatives: If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x) = F(x) + C, for all x in I where C is a constant. Sigma Notation: The sum of n terms a1,a2,a3,…,an is written as where I is the index of summation, ai is the ith term of the sum, and the upper and lower bounds of summation are n and 1. Summation Formulas: 1. 2. 3. 4. Limits of the Lower and Upper Sums: Let f be continuous and nonnegative on the interval [a,b]. The limits as n of both the lower and upper sums exist and are equal to each other. That is, where are the minimum and maximum values of f on the subinterval. Definition of the Area of a Region in the Plane: Let f be continuous and nonnegative on the interval [a,b]. The area if a region bounded by the graph of f, the x-axis and the vertical lines x=a and x=b is Area = where . Definition of a Riemann Sum: Let f be defined on the closed interval [a,b], and let be a partition of [a,b] given by a =x0<x1<x2<…<xn-1<xn=b where xi is the width of the ith subinterval. -
Calculus Formulas and Theorems
Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. -
Business Mathematics and Personal Finance 236.2021
Business Mathematics and Personal Finance EXAM INFORMATION DESCRIPTION Exam Number This assessment is designed to represent the standards of 236 learning that are essential and necessary for all students. The Items implementation of the ideas, concepts, knowledge, and skills will create the ability to solve mathematical problems, analyze 53 and interpret data, and apply sound decision-making skills. This Points will enable students to implement the decision-making skills they must apply and use these skills in a hands-on manner to 60 become wise and knowledgeable consumers, savers, investors, Prerequisites users of credit, money managers, citizens, employees, employers, inventors, entrepreneurs, and members of a global NONE workforce and society. Recommended Course Length EXAM BLUEPRINT ONE SEMESTER National Career Cluster STANDARD PERCENTAGE OF EXAM BUSINESS MANAGEMENT & 1- Decision Making and Goals 3% ADMINISTRATION 2- Problem Solving 19% FINANCE 3- Income and Career Preparation 17% 4- Money Management 20% Performance Standards 5- Saving, Investing, and Retirement 17% INCLUDED (OPTIONAL) 6- Sales and Marketing Discounts 12% 7- Home/Business Ownership 5% Certificate Available 8- Macro and Microeconomics 7% YES 9- Depreciation and Book Value (Optional) 10- Inventory (Optional) www.precisionexams.com Business Mathematics and Personal Finance 236.2021 STANDARD 1 Students will use a rational decision-making process to set and implement financial goals. Objective 1 Explain how goals, decision-making, and planning affect personal financial choices and behaviors. 1. Discuss personal values that affect financial choices (e.g., home ownership, work ethic, charity, civic virtue). 2. Explain the components of a financial plan (e.g., goals, net worth statement, budget, income and expense record, an insurance plan, a saving and investing plan). -
13.1 Antiderivatives and Indefinite Integrals
ANTIDERIVATIVES Definition: reverse operation of finding a derivative Notice that F is called AN antiderivative and not THE antiderivative. This is easily understood by looking at the example above. Some antiderivatives of 푓 푥 = 4푥3 are 퐹 푥 = 푥4, 퐹 푥 = 푥4 + 2, 퐹 푥 = 푥4 − 52 Because in each case 푑 퐹(푥) = 4푥3 푑푥 Theorem 1: If a function has more than one antiderivative, then the antiderivatives differ by a constant. • The graphs of antiderivatives are vertical translations of each other. • For example: 푓(푥) = 2푥 Find several functions that are the antiderivatives for 푓(푥) Answer: 푥2, 푥2 + 1, 푥2 + 3, 푥2 − 2, 푥2 + 푐 (푐 푖푠 푎푛푦 푟푒푎푙 푛푢푚푏푒푟) INDEFINITE INTEGRALS Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol f (x) dx The symbol is called an integral sign, The function 푓 (푥) is called the integrand. The symbol 푑푥 indicates that anti-differentiation is performed with respect to the variable 푥. By the previous theorem, if 퐹(푥) is any antiderivative of 푓, then f (x) dx F(x) C The arbitrary constant C is called the constant of integration. Indefinite Integral Formulas and Properties Vocabulary: The indefinite integral of a function 푓(푥) is the family of all functions that are antiderivatives of 푓 (푥). It is a function 퐹(푥) whose derivative is 푓(푥). The definite integral of 푓(푥) between two limits 푎 and 푏 is the area under the curve from 푥 = 푎 to 푥 = 푏. -
Antiderivatives Math 121 Calculus II D Joyce, Spring 2013
Antiderivatives Math 121 Calculus II D Joyce, Spring 2013 Antiderivatives and the constant of integration. We'll start out this semester talking about antiderivatives. If the derivative of a function F isf, that is, F 0 = f, then we say F is an antiderivative of f. Of course, antiderivatives are important in solving problems when you know a derivative but not the function, but we'll soon see that lots of questions involving areas, volumes, and other things also come down to finding antiderivatives. That connection we'll see soon when we study the Fundamental Theorem of Calculus (FTC). For now, we'll just stick to the basic concept of antiderivatives. We can find antiderivatives of polynomials pretty easily. Suppose that we want to find an antiderivative F (x) of the polynomial f(x) = 4x3 + 5x2 − 3x + 8: We can find one by finding an antiderivative of each term and adding the results together. Since the derivative of x4 is 4x3, therefore an antiderivative of 4x3 is x4. It's not much harder to find an antiderivative of 5x2. Since the derivative of x3 is 3x2, an antiderivative of 5x2 is 5 3 3 x . Continue on and soon you see that an antiderivative of f(x) is 4 5 3 3 2 F (x) = x + 3 x − 2 x + 8x: There are, however, other antiderivatives of f(x). Since the derivative of a constant is 0, 4 5 3 3 2 we can add any constant to F (x) to find another antiderivative. Thus, x + 3 x − 2 x +8x+7 4 5 3 3 2 is another antiderivative of f(x). -
Department of Economics 1
Department of Economics 1 Both economics degrees require that students complete 120 credits Department of towards the degree. All Economics majors are subject to the university and college's residency requirements. Additionally, students in the Economics Economics majors (B.A. or B.S.) must earn at least 1/2 of their required economics (EC) credits while enrolled in the curriculum, and students must complete at least one-half of the required economics credit hours The Department of Economics provides a broad-based education with (EC courses) at NC State University. a specialization in economic theory and applications. Students can select the Bachelor of Arts in Economics degree, with a focus on liberal Economics Department arts courses, or the Bachelor of Science in Economics, with a focus on Poole College of Management courses in business, mathematics, statistics, and science. The economics 4102 Nelson Hall programs are flexible, and, with thoughtful advance planning, students Raleigh, NC 27695 can easily pursue an economics degree along with a minor, or even a Phone: (919) 515-5565 second major, in another academic area. https://poole.ncsu.edu/economics/ Economics students can develop their understanding of economic Lee Craig issues in a variety of areas including: econometrics, game theory, Department Head health economics, industrial organization, international economics, labor Alumni Distinguished Professor economics, money and financial institutions, public finance, resource and environmental economics. Faculty A degree in economics provides rigorous analytical training with a Department Head broad understanding of the workings of the global economic system. Its flexibility allows students to tailor their education to specific interests Lee A. -
Appendix: TABLE of INTEGRALS
-~.I-- Appendix: TABLE OF INTEGRALS 148. In the following table, the constant of integration, C, is omitted but should be added to the result of every integration. The letter x represents any variable; u represents any function of x; the remaining letters represent arbitrary constants, unless otherwise indicated; all angles are in radians. Unless otherwise mentioned, loge u = log u. Expressions Containing ax + b 1. I(ax+ b)ndx= 1 (ax+ b)n+l, n=l=-1. a(n + 1) dx 1 2. = -loge<ax + b). Iax+ b a 3 I dx =_ 1 • (ax+b)2 a(ax+b)· 4 I dx =_ 1 • (ax + b)3 2a(ax + b)2· 5. Ix(ax + b)n dx = 1 (ax + b)n+2 a2 (n + 2) b ---,,---(ax + b)n+l n =1= -1, -2. a2 (n + 1) , xdx x b 6. = - - -2 log(ax + b). I ax+ b a a 7 I xdx = b 1 I ( b) • (ax + b)2 a2(ax + b) + -;}i og ax + . 369 370 • Appendix: Table of Integrals 8 I xdx = b _ 1 • (ax + b)3 2a2(ax + b)2 a2(ax + b) . 1 (ax+ b)n+3 (ax+ b)n+2 9. x2(ax + b)n dx = - - 2b--'------'-- I a2 n + 3 n + 2 2 b (ax + b) n+ 1 ) + , ni=-I,-2,-3. n+l 2 10. I x dx = ~ (.!..(ax + b)2 - 2b(ax + b) + b2 10g(ax + b)). ax+ b a 2 2 2 11. I x dx 2 =~(ax+ b) - 2blog(ax+ b) _ b ). (ax + b) a ax + b 2 2 12 I x dx = _1(10 (ax + b) + 2b _ b ) • (ax + b) 3 a3 g ax + b 2(ax + b) 2 . -
(Hons.) I Year Subject: Business Mathematics
B.Com. (Hons.) I Year Subject: Business Mathematics SYLLABUS Class: - B.Com. (Hons.) I Year Subject: - Business Organization and Communication Unit-1 Average, Ratio and Proportion, Percentage Unit-2 Profit and Loss, Simple Interest, Compound Interest 45, Anurag Nagar, Behind Press Complex, Indore (M.P.) Ph.: 4262100, www.rccmindore.com 1 B.Com. (Hons.) I Year Subject: Business Mathematics UNIT-I AVERAGE The average of the number of quantities of observations of the same kind is their sum divided by their number. The average is also called average value or mean value or arithmetic mean. Average = For observations x1 ,x2, x3, ……………xn Average = Functions of Average a) To present the salient features of data in simple and summarized form b) To compare and draw conclusion c) To get a simple value that describes the characteristics of the entire group d) To help in statistical analysis Merits of Average a) Simple and rigid definition b) Based on all observations c) Use in analysis because of further algebraic treatment Demerits of Average a) Sometimes it is not a true representation of data b) Sometimes it may give absurd results c) It cannot be obtained by observation d) It can be affected by extreme values RATIO A ratio can exist only between two quantities of the same type. If x and y are any two numbers and y 0 then the fraction is called the ratio of x and y is written as x:y. Characteristics of Ratio – The following characteristics are attributed to ratio relationship: i) Ratio is a cross relation found between two or more quantities of same type. -
International Business
International Business ABOUT THE PROGRAM The International Business PAY College Certificate program The average salary for people provides students with the with a bachelor's degree technical skills for entry-level in international business range positions as specialists in between about $44,000 and exporting and importing for $75,000 starting out. the significant and growing international trade community. JOB OUTLOOK Most students pursue a career According to the Bureau of Labor in import-export trading, Statistics, careers in administrative international transportation services are expected to be some and logistics, global supply of the fastest growing between chain management, international 2006 and 2016. Careers for marketing, or various international financial analysts and marketing business support services. consultants should also grow. The program offers courses that Although career opportunity can prepare students to take the should grow, competition for the National Association of Small positions will still be high as more Business International Trade people are expected to enter the Educators Certified Global field of Business Administration. Business Professional Exam. WHAT DO PERSONS IN INTERNATIONAL BUSINESS DO? International business refers to business activities which involve cross- border transactions of goods, services or resources between two or more Daniels, J., Radebaugh, L., Sullivan, D. (2007). nations. Transaction of economic resources include capital, skills, people International Business: environment and operations, -
FUNDAMENTALS of BUSINESS MATHEMATICS and STATISTICS Foundation Study Notes
FOUNDATION : PAPER - 4 US - 2016 B A LL SY FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS FOUNDATION STUDY NOTES The Institute of Cost Accountants of India CMA Bhawan, 12, Sudder Street, Kolkata - 700 016 First Edition : August 2016 Reprint : April 2017 Reprint : January 2018 Published by : Directorate of Studies The Institute of Cost Accountants of India (ICAI) CMA Bhawan, 12, Sudder Street, Kolkata - 700 016 www.icmai.in Printed at : Mega Calibre Enterprises Pvt. Ltd. 06/315 Action Area 3, New Town, Rajarhat, Kolkata 700 160. Copyright of these Study Notes is reserved by the Institute of Cost Accountants of India and prior permission from the Institute is necessary for reproduction of the whole or any part thereof. Syllabus - 2016 PAPER 4: FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS (FBMS) Syllabus Structure A Fundamentals of Business Mathematics 40% B Fundamentals of Business Statistics 60% A 40% B 60% ASSESSMENT STRATEGY There will be written examination paper of three hours. OBJECTIVES To gain understanding on the fundamental concepts of mathematics and statistics and its application in business decision- making Learning Aims The syllabus aims to test the student’s ability to: Understand the basic concepts of basic mathematics and statistics Identify reasonableness in the calculation Apply the basic concepts as an effective quantitative tool Explain and apply mathematical techniques Demonstrate to explain the relevance and use of statistical tools for analysis and forecasting Skill sets required Level A: Requiring the skill levels of knowledge and comprehension CONTENTS Section A: Fundamentals of Business Mathematics 1. Arithmetic 20% 2. Algebra 20% Section B: Fundamentals of Business Statistics 3.