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# Introductory Mathematical Analysis II MATH 1331

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This 22 page Class Notes was uploaded by Ms. Ally Koelpin on Thursday October 22, 2015. The Class Notes belongs to MATH 1331 at Texas Tech University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/226466/math-1331-texas-tech-university in Mathematics (M) at Texas Tech University.

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Date Created: 10/22/15

MATH 1 331 Introductory Mathematical Analysis Calculus for Business Course Author Heide Mansouri Yourgmder my be di ferentfmm le nulhor MATH 1331 features 0 3 hours credit 0 8 lessons each containing Introduction Lesson Objectives How to Proceed Discussion Review Exercises and Lesson Assignment 1 final examination 0 1 textbook 0 Prerequisites MATH 1330 Student must have access to a graphing calculator that can also perform log at In x and exponential functions 0 All lesson assignments must be submitted via surface mail only MATH 1331 Published by Division of Outreach and Distance Education Texas Tech University BOX 42 191 Lubbock TX 794092191 Outreach 81 Distance Education Course Development Instructional Designer Heide Mansouri Copyright 2006 by the Board of Regents for the College ofArts 81 Sciences acting for and on behalf of Texas Tech University Lubbock Texas 79409 All rights reserved TABLE OF CONTENTS Introduction to MATH 1331 Introductory Mathematical Analysis Calculmfar Easiness V Course Lessons Lesson One The Derivative 1 Lesson Two More Derivative Topics 9 Lesson Three Graphing 15 Lesson Four C u 39 quot 19 Lesson Five Additional Derivative Topics 23 Lesson Six 7 J 1 quot Tntegval 29 Lesson Seven Definite Integral 33 Lesson Eight Additional t o quot Topics 37 Final Examination Dirertinns 43 introduction Introductory Mathematical Analysis Calculus for Business Welcome to MATH 1331 Introductory Mathematical Analysis Calculus for Business MATH 1331 is a three credit hour course for business administration economics life sciences and social sciences students This course is designed for students who have completed MATH 1330 Successful ompletion of this course will enable you to solve many important problems in different disciplines This course comprises eight lessons a sample test answers to the sample test helpful Web sites and helpful hints Each lesson covers up to a chapter in the textbook and consists of an introduction a list of lesson objectives instructions for how to proceed a discussion review exercises and a lesson assignment MATH 1331 covers chapters 9 13 in the textbook College Mntheinnticsfar Business Economics Li e Sciences and Social Sciences The textbook emphasizes mathematical applications and models in business economics life sciences and social sciences Chapter 9 introduces the derivative the power rules and the basic differentiation properties Applications of derivatives in business and economics are also discussed The interplay between graphical numerical and algebraic concepts is emphasized in this chapter and throughout the text Chapter 10 focuses on graphing functions and optimization problems It covers the first and second derivatives and their roles in describing the properties of the graph of functions with application to graphing polynomial and rational functions Application of the derivatives to optimization problems is also covered in this chapter Chapter 1 1 extends the concept of derivatives discussed in Chapters 9 and 10 to exponential and logarithmic functions A general form of the chain rule is also discussed in Chapter 11 Chapter 12 introduces the concept of antiderivatives and integral calculus This chapter also covers the method of integration by substitution MATH 1331 V20 Introduction v Course Objectives Textbook and Materials About the Textbook Texas Tech University differential equations definite integrals and different methods of evaluating definite integrals Chapter 13 covers additional integration topics including integration by parts and application of these methods in business and economics After completing this course you should be able to o recognize some special functions and their applications differentiate functions by applying basic rules of differentiation discuss some applications of derivatives sketch graphs of functions and find their extrema find the antiderivative for a given derivative integrate a wide variety of functions discuss some applications of integrals The required textbook for this course is Barnell Raymond A Zieglar Michael R and Byleen Karl E 2005 College Mnthemnticsfor Business Economics Life Sciences and Social Sciences 10th edition Prentice Hall Inc ISBN 0131432095 Also required is a calculator with at least log x ln x and exponential function ex keys College Mnthemnticsfor Business Economics Li e Sciences and Social Sciences 10th ed is a clearly written textbook for college level mathematics It is rich in examples worked problems and exercises The illustrations are simple and clear This book is organized into three parts A Library of Elementary Functionsquot Finite Mathematicsquot and Calculus and fourteen chapters This course will cover Part Three of the book for MATH 1331 except Section 13 4 and Chapter 14 The chapters have several sections each vi Introduction MATH 1331 v20 Outreach amp Distance Education There are exercise sets marked with capital letters A routine easy mechanics B more difficult mechanics and C difficult mechanics and some theory levels in each chapter and a set of review exercises The review exercises have solutions in the back of the textbook beginning on page A 1 Answers Almost every exercise set contains application problems which are simplified versions of actual real world problems taken from professional journals and books Each example is followed by a similar matched problem to work while reading the material Answers to matched problems are found near the end of each section before the exercise set Answers to odd numbered problems are provided in the Answers section of the textbook Tables of basic geometric formulas and integration formulas are contained in Appendix C Since you want to learn mathematics your textbook s approach is to introduce each mathematical concept with at least one example that shows why it is needed and what question it answers followed by application problems to give you experience in modeling and solving real world problems You will need a graphing calculator that can also perform lag x In x and caICUIator Usage the exponential functions You will find a graphing calculator useful while working on your assignments and during the final examination When you use a calculator you need to be careful to show all of your work for each problem Otherwise it will be difficult to assign partial credit You should always check your work by keying the problem twice to make sure you get the same answer both times Be sure to follow these instructions when preparing your written lessons written to be graded Assignments Do your work in pencil Each lesson assignment is to be started on a clean sheet of notebook paper Work dawn the page not across When you get to the bottom of the page you may start back at the top of the page and make a second column of work MATH 1331 V20 Introduction vii Tutors Grading Policy Texas Tech University Underline your answers Skip at least one line between problems Neatness in this course will save you time and help you avoid mistakes I do not want to have to send work back to you to be redone because I cannot read it I recommend that if you do decide to use any form of tutoring you do so correctly That is you should use a tutor only minimally and always make sure that you can work the problems from each lesson by yourself with no assistance before you go on to the next lessons When you come to a proctored examination you must be prepared to do all the work yourself The course grade is based on eight assignments worth 100 points each and 100 points for the final exam Letter grades will be awarded according to the following percentages 90 10000 A 80 89 B 70 79 C 60 69 D lt 59 F To be successful in this course you must send in your lesson assignments on a regular basis To encourage you to complete the course in a timely manner Iwill add 10 bonus points to your lesson scores if you complete the course successfully within five months of your enrollment date Your final grade for the course will be determined as follows Lesson scores plus 10 you 39m39sh within ve months 30 70 Final examination score For the lesson assignments and the final examination Iwill award partial credit if the work supports it I consider myself generous in this regard Also I won t penalize you if your answer follows from an earlier misstep viii Introduction MATH 1331 v20 Outreach amp Distance Education When you have completed the eight lessons you will be ready for the final examination The final examination will be similar to the assigned About the Final Exam problems and will allow the use of a graphing calculator and a 3quot 5quot formula card which you prepare You will have three hours to complete the final examination Remember that you must pass the final examination in order to pass the course Here are some useful web sites Please type the address exactly the way it is shown Useful Web Sites URL addresses http peopleh0fstraedu faculty S te fan7Waner Re alWorld This is a Finite Mathematics and Applied Calculus resource page for online interactive learning httpvigprenhalloom ltgt Go to the address above Near the top of the page next to Search Our Catalogquot choose By Authorquot type Barnett in the text field and click GO ltgt Scroll down a little and click College Mathematics for Business Economics Life Sciences and Social Sciences10equot This site is related to your textbook Under Resources on the left hand side of the page you will find helpful resources hints and self test questions httparchivesmathutkeduvisualcalculus Learn calculus visually httpwwwmath lesc0m This site offers more than 1200 completely solved mathematical problems httpeducati0nticomeducationportalsitesUSsectionHome studentimainhtml Texas Instruments has created dynamic tutorials to help you learn to use and apply your TI calculator These interactive tutorials walk you through functionality calculations scientific notation and graphing MATH 1331 v20 Introduction ix Study Tips Texas Tech University 1 Familiarize yourself with the course materials Please take a moment to learn about the textbook and the course guide and write down any questions or concerns that you may have 2 Use available resources If you are having difficulty with certain problems and if studying the textbook and the course guide are not helping you see if you have additional resources for getting help Talk to friends other students or the tutoring labs If you need help from me you can contact me via e mail at heidemans0urittuedu or by mailing a letter to me at Outreach 8 Distance Education I check my e mail each working day and that is my preferred method of contact You may also include questions with your assignments and I will provide responses when I return your work 3 Structure your time and work Make a contract with yourself for completing the course by setting a target date and developing a schedule that will help you complete the course by that date Maybe there is a special reward in addition to your grade that you can give yourself upon completing the course I recommend that you make a weekly schedule of all your class hours and other fixed activities Then schedule your study hours Give yourself some free time in your schedule and be realistic If you have a realistic schedule you will follow it As a three credit hour course the day school would expect three hours of in class and six hours of out of class effort for each of ten weeks or 90 hours total effort I have divided the course into eight lessons of up to a chapter each You should expect to spend about 11 12 hours on each lesson including reading the course guide studying the material and working the exercises Some students will be able to work more quickly others will take more time The advantage of independent and distance learning is that you can work at your own pace I recommend that you set your schedule so that you complete the course in an appropriate time frame and that you try to stick to it You will have to pace yourself and you will have to motivate yourself to finish the course The disadvantage of independent and distance learning is that you can rest at your own pace It is my hope that you will find the x Introduction MATH 1331 v20 Outreach amp Distance Education material interesting enough to help motivate you to learn more about it and finish the course in a timely manner In each study period you should try to do the following review the previous material including any returned assignments review and organize your notes read the lesson introduction objectives and discussion material read the assigned text highlighting the key terms and identify where in the text the lesson objectives are addressed make notes as appropriate work on review exercises from the exercise sets to ensure that you understand the material and check your answers in the back of the book work the assigned problems and then submit your answers for grading review the material just covered and then try to summarize it in your own mind The reviews at the beginning and end of each lesson will help reduce the need to cram before the final exam Also many of the review exercises will cover earlier materials without alerting you to that fact This will also help you to stay refreshed on earlier topics 4 Find your place to study Most students do well if they can pick one special spot to study one that they associate with studying Some students cannot study on their beds because they associate this with sleep and hence always fall asleep Others cannot study at a kitchen table because they associate this with eating and become distracted A library is a good place to study because it is quiet and usually does not have distractions such as the phone TV or other people You will want to check the course guide before you begin your lesson and submit your lesson assignments MATH 1331 V20 Introduction xi Texas Tech University 5 Develop a study plan Although you will need to develop your own study plan here are some helpful hints Skim the assigned text readings to get a sense of what the lesson is about Read the material As you come to each key term highlight it It s a good idea to keep a working dictionary of terms You may want to write the definition of each term on a note card and keep those cards as a reference Try to work the examples in the textbook Remember mathematics is not a spectator sport You have to do it to learn it Keep in mind that most students do not fully understand the material the first time they read it You may have to reread a chapter or section several times Work some of the exercises in the review exercise sections Check your answers If you are having trouble with a concept review the previous sections Many times the ideas are introduced earlier and then are fully explained in a subsequent chapter Finally you can send me an e mail at heidemans0urittuedu 6 Do good work and earn a good grade Each lesson has an assignment chosen from the even numbered problems The sooner you submit your assignment the sooner Iwill be able to grade your work and return it to you 7 Prepare for the final examination To prepare for the final examination carefully go over the sample test It illustrates the type of questions that will appear on the final examination I will provide answers to the sample test xii Introduction MATH 1331 V20 Outreach amp Distance Education My name is Heide Mansouri I hold three degrees a BS degree in statistics an MS degree in statistics from the University of Kentucky and an MS degree in operations research from the George Washington University Ihave also completed the core courses for the management information system MIS program and the entire course works for the doctoral program in educational instructional technology at Texas Tech University I have taught mathematics courses at Outreach 8 Distance Education since 1991 at both the freshman and sophomore levels Additionally I have taught resident mathematics courses at Texas Tech University and undergraduate level statistics for three years at the University of Kentucky 1 also authored the course guides for MATH 2300 and MATH 1330 offered by TTU Outreach 8 Distance Education Since 1991 Ihave had the privilege of being the instructor for several Outreach 8 Distance Education math courses including MATH 2300 MATH 1330 and MATH 1331 at Texas Tech University As the reader of this course guide you are my most important critic and commentator I value your opinion and want to know what I am doing right what I could do better and any other words of wisdom you are willing to pass my way If you have an idea that could make this a more useful course guide please let me know by sending me an e mail at heidemans0urittuedu or a note with your lesson assignment I look forward to helping you in this course About the Author MATH 1331 v20 Introduction xiii one The Derivative his lesson begins the study of calculus In calculus we are interested in the effect of change in one variable on another variable Although the main topic of this lesson is derivative of functions a substantial part of the lesson is devoted to the fundamental concepts of limit and continuity After you have completed this lesson you should be able to Lesson Objectives compute the derivative of a function compute the equation of the tangent line to the graph of y f x at a given point determine whether or not a function has a limit at a given point evaluate the limit of a given function determine whether a function is continuous at a given point use some basic rules of differentiation to find the derivative of a polynomial function use the chain rule to find the derivative of composition of two functions differentiate exponential and logarithmic functions MATH 1331 v20 Lesson One 1 How to Proceed Discussion Texas Tech University 1 Make sure you have read the Introduction and Objectives for this lesson Survey read and take notes on Chapter 9 Sections 9 1 through 9 4 and the Chapter 9 Review Important Terms Symbols and Conceptsquot for those sections in College Mnthemnticsfor Business Economics Life Sciences and Social Sciences Read the Discussion in this course guide Work on the matched problems while reading the material You will find the solutions to these problems near the end of each section before the exercise set Distance Education according to the directions given in the Policies 8 Forms Guide Review your notes briefly every day until you complete the course After you have finished this lesson you may proceed to Lesson Two 4 Complete the Lesson One Assignment and submit it to Outreach 8 In calculus we are interested in how a change in one variable affects another variable The derivative of a function y f x depicts how the function is changing at a given point Note It is necessary for the function y f x to be continuous at point x n in order for the function to have a derivative at that point Suppose that f x is a function that is defined for all x in some interval containing c a constant except possibly at x c itself A real number L is called the limit of fXas Kapproaches cif f x can be made as close to L as desired for all x whenever x is close to but not equal to c Then we write fx L Example11 3 z z lim z 11mM 1mmZ 2Z 4 x42 xZ x42 xZ 7542 The concept of limit helps us to describe in a precise way the behavior of a function f x when x is close but not equal to a particular value a 2 Lesson One MATH 1331 v20 Outreach amp Distance Education The Properties of Limits are explained in Theorem 2 on page 548 of your textbook A function f is continuous at the point x cif 1 lim f x exists 2 fc exists i e defined at c and 3 limfx fc xgt All three of these conditions should be satisfied by a function f for it to be continuous at c That is if one or more of the three conditions fail then the function is discontinuous at x cand cis called the point of discontinuity of The average rate of change of y f x over the interval 4 a h is fltahgt fltagt fltahgt fltagt 4T Th defined as a This average rate of change is also called difference quotient Example 12 Let f x 4x2 5 Then the average change of f x over the interval 2 4 is f4 f2 416 5 16 5 59 11224 4 2 4 2 2 The instantaneous rate of change of y f x at x a is THEM if the limit exists see example 3 on page 575 and its unit of measurement is the unit of f x per unit of x Note Rate of changequot always refers to the instantaneous rate of change and not the average rate of change The notion of the limit is one of the most basic concepts in mathematics and is the foundation of calculus MATH 1331 v20 Lesson One 3 Texas Tech University When computing the instantaneous rate of change of a function f x at f4hf4 point a we calculate the difference quotient h a a f4hf4 h and determine the value that it approaches as h assumes values progressively closer to zero This value is called the limit of the difference quotient as 11 approaches zero The derivative of a function y f x at x denoted by f is f x ligw if the limit exists See page 577 of your textbook for the conditions for the existence of a limit and interpretation of the derivative Also see page 582 and Figure 8 for illustration of some common situations where a derivative fails to exist A function which has a derivative is said to be differentiable Example For an object moving in a straight line with position f t at time t the average velocity from time tto time t h is v fthft average h The instantaneous velocity at time tis 7 fthft at 7 ft 7 yggT See applications on pages 291 293 Example 13 Suppose the position of a moving object is given by f t t2 3t 7 miles at time thours Then its velocity at time tis i mthz 3th7 tZ 3t7 ft 7 h mt2hz2th 3t 3h7 tz3t 7 36 h 1 h2 2th 3h 1m h7gt0 h 4 Lesson One MATH 1331 v20 Outreach amp Distance Education hm hh 21 3 hgt0 h limh 21 3 hgt0 2t 3 miles per hour Thus for example its velocity at time t 5 hours is f 5 7 miles per hour Example 14 Revolving credit debt in billions of dollars in the United States can be described approximately by f t 0621 2 t 51 where tis time in years and t 0 corresponds to 1970 What were the debt and the instantaneous rate of change of the debt in 199839 f28 206228Z 1 97116 at f t gig 2 m 062t h2 t h 51 0621 2 t 51 3536 h 062tZ h2 21h t h 51 0621 2 t 51 h hm h062h 2062t 1 hgt0 h 51101062h 2062t l 2062t 1 f 28 206228 1 3372 MATH 1331 v20 Lesson One 5 Texas Tech University Therefore in 1998 the revolving credit debt was 97116 billion and was increasing at the rate of 3372 billion per year The difference quotient and its interpretations Formula or Process Numeric interpretation Geometric interpretation f h f 4 average rate of change or slope of secant line average velocity f a h f a instantaneous rate of slope of graph or tangent Ho h change or velocity line Note f x represents both the instantaneous rate of change of f at x and the slope of the tangent line to the graph of f at the point x f x Fory fx f xy and g are different notations used to represent x the derivative of a function fat x Power rule and basic differentiation properties The derivative of a constant function is zero That is if y f x c a constant then f x 0 The derivative of the product of a constant and a function is equal to the constant times the derivative of the function That is if y f x kux then f x ku x The derivative of a power function is the power of the function times the it function to the power of the function minus one That is if f x x then f x mc H 6 Lesson One MATH 1331 V20 Outreach 8 Distance Education Example 15 If x x396 then f x 7 6x396391 7 6x397 The derivative of the sum or difference of two functions is equal to the sum or difference of the derivatives of the functions That is if x Ux Vx then f x U x V x Example 16 Calculate the derivative of x 3 W 01x3 715x 6 To find f x first we write the function as x 3x 01x3 715x 6 Then f x 3 x 1 01 33 1 715 0 Or f x 3x 032 715 4 03x2 715 435 03xZ 715 Notice that in this example I used a combination of derivation rules mentioned above More rules will be introduced in Lesson Two Work on all review problems if you can If not complete the following ReVieW Exef ie problems found in your textbook and check your answers in the Answers Appendix A56 to A59 1 0 Pages 626629 problems 1 2 3 4 5 6 7 MATH 1331 V 2 0 Lesson One 0 7 Lesson One Assignment Texas Tech University Study Sections 9 1 to 9 4 of Chapter 9 Make sure that you carefully read and understand each example After you have carefully read the material do the assigned problems show all your work and underline your answers Please make sure that your final answers are written neatly and in numerical order 0 Exercise 91 pages 554 557 problems 10 16 22 26 34 42 46 0 Exercise 92 pages 566 569 problems 10 50 58 60 64 78 0 Exercise 93 pages 583 585 problems 4 16 36 64 0 Exercise 94 pages 593 594 problems 8 10 12 44 58 64 72 76 8 Lesson One MATH 1331 v20

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