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Date Created: 09/07/15
CONTINUING EDUCATION The University of Kansas MATH I I6 Calculus 11 Written by Marian K Hukle PhD KU Independent Study ContinuingEdkuedu Mathematics 116 Calculus II 3 hours credit Written by Marian K Hukle PhD The University of Kansas Independent Study Continuing Education The University of Kansas Lawrence Kansas Contents Preface Summary of Course Components 1 Antiderivatives and the Inde nite Integral De nite Integrals Applications of the De nite Integral Trigonometric Functions Topics in Integration Introduction to Functions of Several Variables Applications of Functions of Several Variables Double Integrals and Their Applications Taylor Polynomials and In nite Series Appendix A Review of Exponential and Logarithmic Functions Appendix B First Steps with Your Graphing Calculator iii 11 17 23 29 35 43 49 55 59 Preface The origins of calculus extend back to the time of the great Greek mathematician Archimedes 287 212 BC who first considered the problem of the area under a curve It was not until the seventeenth century that Isaac Newton and Gottfried Leibnitz perfected the calculus we know today Calculus addresses two main questions 0 How do you find the area of a region in the plane 0 How do you calculate the tangent to a given curve The differential calculus addresses the tangent problem and the integral calculus addresses the area problem Amazingly it turns out that these two questions that seem to be completely independent of each other turn out to be ip sides of the same coin In fact both Newton and Leibnitz reached this conclusion independently This concept is so important that it is called the Fundamental Theorem of Calculus In a two semester calculus sequence it is customary to introduce the differential calculus in the first semester and integral calculus in the second The first course in this sequence Math 115 introduced the general concepts of differential and integral calculus looked at the major applications of both branches of calculus and stated and used the Fundamental Theorem of Calculus which ties the two branches together We will begin this course with a short review of integration techniques Sections 61 65 of the text before beginning the main part of this second course on calculus We will explore further techniques for finding indefinite integrals and evaluating definite integrals Following this we will review another set of functions of a single variable the trigonometric functions then learn to differentiate and integrate these functions Then we begin the study of functions of more than one variable and will extend the concepts of single variable calculus to functions of several variables Finally we will introduce Taylor polynomials infinite sequences and infinite series By the end of this course all the primary and most of the secondary ideas of calculus will have been introduced After Chapter 6 the course material consists basically of expanding the application of calculus to include new functions For students who have not studied calculus for some time you should review the differentiation and integration formulas from Calculus l and memorize them again In particular you must know the Summary of Principal Formulas and Terms for Chapter 3 on pages 246 247 There is also a review of exponential and logarithmic functions included in Appendix A of this study guide Text The text for this course is Applied Calculus for the Managerial Life and Social Sciences 6th ed by ST Tan Brooks Cole 2005 Graphing Calculator This course requires the use of a graphing calculator also referred to as a graphing utility Instructions on various techniques of using your graphing calculator are fully integrated into the text with step by step instructions If you are unfamiliar with the calculator you should start by reading Appendix B in this study guide which provides beginning and intermediate calculator instruction Also be sure to try the examples in the text that are geared toward calculator techniques Acceptable calculator models include the Tl 83 Tl 83 Plus Tl 84 Tl 84 Plus HP 38C and others the author of this course uses a Tl 84 Plus The Tl 89 is not suitable and therefore is not recommended Graphing calculators are now available at most discount stores office supply stores as well as the Kansas Union Bookstore To be suitable for this course your calculator must have the following capabilities H It must be able to graph at least four functions si1nultaneously N It must have a key that allows you to trace along each graph showing both the x value as well as the function value y value and allows you to switch from one graph to another 03 It must have zooming capabilities 4 It must allow the user to specify or change the graphing window 5 It must be programmable The calculator is an integral part of this course You are required to complete problems that can only be solved with the calculator The calculator helps you visualize some of the concepts being taught in the course It also allows you to check some of your answers on problems that you have solved symbolically Course Structure The course is broken up into nine lessons Each lesson has a list of objectives a reading assignment and instructor notes and comments written to supplement the text There is also a set of practice assignments for each assigned section in the text These are odd numbered problems which have the answers provided You should try these problems after studying each section They are for practice only and are not to be submitted for grading Mathematics is a subject that is reinforced by doing problems the more you practice the more proficient you will become At the end of every lesson is a list of problems that you will submit to the instructor Each written assignment consists of 15 to 30 problems worth 2 points each These problems are selected from the even numbered problems for which the answers have not been provided in the text If you are not sure you are doing an assigned problem correctly please note that problems are usually written in pairs You should be able to find a similar odd numbered problem with an answer provided try that problem first then apply the method to the even numbered problem Course Mechanics You will be expected to observe the following rules 1 Use only the paper provided to you along with your study guide You must include an Independent Study cover sheet with each lesson as it is submitted Your assignment will not be graded without this cover sheet 2 Keep your papers organized neat and legible Leave room between problems for instructor comments 03 Write on one side of the paper only with problems submitted in the order they are assigned You must also include the page number problem number and the problem itself 1 Give exact answers for your problems whenever possible For example if your answer is 2 or 3739 leave your answer in this form Do not give a decimal approximation for these numbers U I You must show all your work to receive credit for your answer An answer with no justification is worth 0 points If you use your graphing calculator to solve a problem you must include a detailed description of the steps you followed 9 On problems that require drawing a graph be sure to include a scale on both the x and y axes and label at least three points It is permissible to copy a graph from your graphing calculator but you must include the scale and labeled points 1 Do not submit Assignment 2 until Assignment 1 has been returned to you 00 After Assignment 1 has been returned to you you may submit up to two assignments per seven day period It is better to wait to receive a corrected assignment before getting too far ahead This will help you avoid learning bad habits gt9 Do not submit assignment 5 until you have taken the midterm exam It is very important that you communicate your ideas and your work clearly and concisely This will indicate to the instructor that you understand the lesson Answers that are ambiguous messy or hard to read will be marked down Examinations There are two written exams for this course a midterm and a final exam The best way to study for these exams is to re do the assigned problems without looking at your corrected written assignment You should also try the odd problems from each section that are similar to those in the written assignments The Chapter Review problems at the end of each chapter are a good source of practice problems Another good source of study problems are the examples in the book Cover the solutions to the examples try to work them out then check your answer and method compared to that presented in the book This will allow you to check not only your answer but also the method you used Learning mathematics vi requires doing lots of practice problems and you should do as many as you can to prepare for the exam It is important that you show detailed work on your exams in order to receive partial credit when possible You will need your graphing calculator for both examinations Midterm Examination You may take the midterm examination after you have completed all assignments up to and including Written Assignment 4 and they have been graded and returned to you The exam is worth 200 points and covers material from the first four lessons Problems on the exam will be similar to those you have done on your practice and written assignments You will be allowed up to two hours to complete the midterm exam Final Examination You may take the final exam after you have received your grade for the midterm exam and you have completed all assignments and they have been graded and returned to you The exam is comprehensive It covers all nine lessons and consists of problems similar to those you have done on your practice and written assignments The exam is worth 300 points and you must score at least 180 points to pass the course You will be allowed up to three hours to complete the exam Plagiarism Plagiarism is presenting someone elses words or work as your own Plagiarism applies to material taken from a book article or the Internet or to material taken from another person without properly citing your sources Paraphrasing another writer substituting words or rearranging sentences from the work of another also constitutes plagiarism Plagiarism is easily detected with databases and search engines Plagiarism is academic misconduct and is a violation of rules and regulations of the University of Kansas Penalties for academic misconduct range from failure of the assignment to expulsion from the University In this course plagiarism on an assignment will result in an F for that assignment and any additional plagiarism will result in failure of the course and possible further penalties Plagiarism on an examination will result in failure vii of the course regardless of the current status of your grade lf plagiarism is discovered after you complete the course your instructor may reexamine your work and will notify you of the proposed penalty If you disagree with a charge of academic misconduct you may request a review by Continuing Education The KU Writing Center wwwwritingkuedu students guidesshtml2 provides guidance on academic integrity incorporating and properly citing reference sources and how to avoid plagiarism Grading The total number of points for the course is 900 points The nine written assignments are worth a total of 400 points the midterm examination is worth 200 points and the final exam is worth 300 points You must receive a score of at least 180 points on the final to pass the course The grading scale is as follows 0 810 900 pointsA 0 720 809 pointsB 0 630 719 pointsC 0 540 629 pointsD About the Author Marian Hukle earned her PhD in Mathematics from the University of Kansas where she has taught mathematics since 1994 Her area of research is in functional analysis the study of spaces of functions and the operators that act upon them For the past five years she has been involved in developing teaching techniques to increase the success of students in college algebra and calculus courses Acknowledgements The author would like to thank Phil Montgomery for his contributions to the text and Lon Mitchell for his assistance in formatting and designing the layout for this study guide viii Summary of Course Components 0 ST Tan Applied Calculus for the Managerial Life and Social Sciences 6th ed Brooks Cole 2005 0 Required equipment A graphing calculator 0 Number of lessons 9 0 Number of writing assignments 9 0 Maximum number of writing assignment submissions No more than two in a seven day period You may not submit your second writing assignment until writing assignment one has been graded and returned to you Thereafter you may submit assignments without waiting for previous ones to be returned 0 Midterm Exam Time allowed for midterm 2 hours Aids allowed graphing calculator Lessons covered on midterm 1 4 Restrictions You must have received a grade for written assignment 4 before you apply for the midterm exam Do not submit written assignment 5 until you have taken the midterm exam 0 Final Exam Time allowed for final 3 hours Aids allowed graphing calculator Lessons covered on final 1 9 Restrictions You must have received a grade for your midterm exam and all assignments before you apply for the final exam 0 Course Grading You must score at least 180 points on the final exam to pass the course The following formula applies only if you meet this requirement Written assignments 8 400 points Midterm exam 200 points Final exam 300 points Total 900 points 810 900 pointsA 720 809 pointsB 630 719 pointsC 540 629 pointsD Antiderivatives and the Inde nite Integral Reading Assignment Chapter 6 Sections 1 2 Pages 398 421 Practice Assignments 0 Section 61 Pages 407 411 3 7 9 13 17 19 29 31 37 41 45 47 53 61 71 75 79 0 Section 62 Pages 419 421 All odd numbered problems 1 41 51 55 59 63 Objectives In this lesson we will first look at the problem of trying to reverse the derivative process That is we will ask the questions given a function y f x is f x the derivative of some other function y Rx If so how can we find this new function Rx You will learn to find an antiderivative and the indefinite integral of functions you have previously studied You will learn to find the antiderivative using the power rule and then proceed to functions that do not fall into the category of the power rule but instead require the use of the chain rule The material in Section 61 of the text is a review of material normally covered in a typical Calculus 1 course If this material is unfamiliar to you be sure to work through all the practice problems Comments 1 P3 5 Let y f x be a function In Calculus l we were interested in finding the rate of change or slope of the tangent line given by the derivative of the function which itself is another function In other words when given a function f x we were asked to find gx f In this lesson we will learn to do the reverse Now given the function fx we would like to find a function gx such that gx This turns out to be a much more difficult problem to solve To begin we will use the derivative rules from Chapter 3 and reverse them to solve our problem For example to find the derivative of the function y f x xquot we multiply the variable x by the exponent n then subtract one from the exponent We do the reverse operations to solve the present problem We add 1 to the exponent of the function to get 11 1 and then divide by n 1 We call the function we are seeking the antiderivative The definition is given on page 398 but to paraphrase a function F is an antiderivative of f if P x f One problem arises Because the derivative of any constant C is equal to zero then the derivative of Fx C is also f Thus there can be an infinite number of antiderivatives for a given function f The set of all antiderivatives of a given function y f x is called the indefinite integral denoted fx dx Fx C Be sure to learn the six rules of the indefinite integral in the shaded boxes on pages 712 1 i x 2 does apply since 72 1 71 in the first example and 71 1 Jr in the second 401 404 For the functions y x z and y x the power rule example However you must be very careful when using the power rule as it does not work for n 71 Why This case is covered in Rule 6 on page 404 Note that the 1 indefinite integral of the function y E is the function y ln lxl not y ln x because 1 both In x and ln 7x have the same derivative y E 5 Q 1 9 Sometimes we need to find a specific antiderivative from among the infinite number of them given by the indefinite integral Geometrically you can see that the set of all antiderivatives is given by graphs that are similar in shape as shown in Figure 64 on page 405 In fact these graphs are just vertical displacements of each other In order to find a specific antiderivative we need at least one other piece of information This information is called an initial condition and is usually in the form of at least one point on the graph of the antiderivative or a specific value that the function takes for a specific value of x This point on the graph allows us to solve for the constant C and thus find the specific antiderivative that we are seeking In Section 62 we are introduced to a method that will allow us to find the antiderivative for functions that do not fit any of the rules we learned above For example if f x x 32 then the only way to use the power rule is to multiply x 3 by itself to get the polynomial x2 6x 9 for which we can easily find the antiderivative However in the case of y x 38 this would be more difficult and impossible in the case of x 312 Note that if we were to find the derivative of such functions we would need the chain rule It turns out that we can do the chain rule in reverse to find the antiderivative We can write x 32 as hgx where gx x 3 appears inside the power function hx x2 Let u gx Then the differential du dx Substitute u for gx and tin for dx and rewrite the integral as x32dx u2du Now use the power rule to find the indefinite integral 3 2 7 LL M iii 7 3 C Last make the reverse substitution to get x32dx x33 C The method described above is called integration by substitution because we substitute the function u gx into the expression f gxquotgx dx to get f uquot du n1 which we have seen has the solution y C To complete the process we gx 1 n 1 C 7 n 1 substitute gx back into the solution to get f gxquotgx dx 9 This method can be used for other types of functions requiring the chain rule in 3 reverse For example if ln3x i 7 then fx Thus we can find 3 the indefinite integral of y m using the process shown above If we substitute u 3x 7 7 and du 3 dx into f M dx we end up with the indefinite integral f idu which has Valueln M C Thus f 3 3x77dxilnl3xi7l C 10 The general process of substitution is described on pages 413 414 Note that there are some circumstances in which you do not quite have the differential du g x dx that you need for the u gx you have chosen See Example 3 on page 415 for how to handle this situation 11 There are three steps to integrate by substitution First we identify the function u gx Second we must determine if the integrand is in the form gxg x dx Third the new integral formed by the substitution must be one that can be successfully integrated In the exercises in this text none of the above problems will be difficult to overcome You may find it difficult at first to identify the function u gx Just make a guess at what u needs to be try the integral and if it doesn39t work guess again Your guesses will get better with practice Written Assignment 1 Section Pages Problems 61 407 411 20 26 30 36 40 46 58 68 72 62 419 421 2 4 6 8 10 12 14 20 22 28 32 36 38 54 56 60 Please reView the Course Mechanics section in the preface to this study guide Reminder Be sure to include an Independent Study Cover Sheet labeled Written Assignment 1
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