McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH CALCULUS II

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1 McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH 4 4 CALCULUS II Information for Students (Winter Term, 3/4) Pages - 8 of these notes may be considered the Course Outline for this course. W. G. Brown and N. Sancho April 8, 4

2 Information for Students in MATH 4 4 Contents General Information. Instructors and Times Calendar Description Calendar Description.... Late transfer from MATH Tutorials; Tutors Coordinates..4 Evaluation of Your Progress Your final grade WeBWorK Regular WeBWorK Assignments Written Assignments Quizzes at the Tutorials; Submission of Written Assignments at the Quizzes Final Examination Supplemental Assessments Machine Scoring Plagiarism and Fraud Corrections to grades. 9.5 Published Materials Required Text-Book Optional Reference Books.5.3 Recommended Video Materials Other Calculus Textbooks Website Syllabus Preparation and Workload Prerequisites Calculators Self-Supervision Escape Routes Terminology High and Low Technology and MATH Keep your address up to date Use of Calculators and Computer Algebra Systems Use of the Internet Special Review Sessions A Information Specifically for Students in Lecture Section A. Timetable for Lecture Section of MATH B Information Specifically for Students in Lecture Section B. Timetable for Lecture Section of MATH C Supplementary Notes for Lectures in Lecture Section 3 C. Supplementary Notes for the Lecture of January 5th, C.. 5. Areas and Distances 4 C.. 5. The Definite Integral 5 C. Supplementary Notes for the Lecture of January 7th, 4... C The Fundamental Theorem of Calculus.... C.3 Supplementary Notes for the Lecture of January th, C Indefinite Integrals and the Net Change Theorem C The Substitution Rule9 C.4 Supplementary Notes for the Lecture of January 4th, 4... C The Substitution Rule (continued) C The Logarithm Defined as an Integral... 5 C.5 Supplementary Notes for the Lecture of January 9th, C Areas between Curves7 C Volumes

3 Information for Students in MATH 4 4 C.6 Supplementary Notes for the Lecture of January st, C Volumes (continued) 33 C Volumes by Cylindrical Shells C.7 Supplementary Notes for the Lecture of January 6th, C Work C Average value of a function C.8 Supplementary Notes for the Lecture of January 8th, C Integration by Parts 43 C.9 Supplementary Notes for the Lecture of February nd, C.9. C Trigonometric Integrals Trigonometric Substitution C. Supplementary Notes for the Lecture of February 4th, C Integration of Rational Functions by Partial Fractions C. Supplementary Notes for the Lecture of February 9th, C Integration of Rational Functions by Partial Fractions (continued) 66 C Strategy for Integration C. Supplementary Notes for the Lecture of February th, C Strategy for Integration (continued) C Integration Using Tables and Computer Algebra Systems C Approximate Integration C Improper Integrals 76 C.3 Supplementary Notes for the Lecture of February 6th, C Arc Length C Area of a Surface of Revolution C.4 Supplementary Notes for the Lecture of February 8th, C Applications to Physics and Engineering C Applications to Economics and Biology.. 89 C Probability C.4.4. Curves Defined by Parametric Equations. 89 C.4.5. Calculus with Parametric Curves C.5 Supplementary Notes for the Lecture of March st, C.5.. Calculus with Parametric Curves (continued) 96 C.5..3 Polar Coordinates 99 C.6 Supplementary Notes for the Lecture of March 3rd, C.6..3 Polar Coordinates (continued) C.6..4 Areas and Lengths in Polar Coordinates.. 5 C.7 Supplementary Notes for the Lecture of March 8th, C.7..4 Areas and lengths in polar coordinates (continued) C.7..5 Conic Sections.. 5 C.8 Supplementary Notes for the Lecture of March th, C.8.. Sequences C.9 Supplementary Notes for the Lecture of March 5th, C.9.. Series C. Supplementary Notes for the Lecture of March 7th,

4 Information for Students in MATH 4 4 C...3 The Integral Test and Estimates of Sums. 5 C. Supplementary Notes for the Lecture of March nd, C...4 The Comparison Tests9 C. Supplementary Notes for the Lecture of March 4th, C...5 Alternating Series 33 C.3 Supplementary Notes for the Lecture of March 9th, C.3..6 Absolute Convergence and the Ratio and Root Tests C.4 Supplementary Notes for the Lecture of March 3st, C.4..7 Strategy for Testing Series C.5 Supplementary Notes for the Lecture of April 5th, D Problem Assignments from Previous Years 3 D. 998/ D.. Assignment D.. Assignment D..3 Assignment D..4 Assignment D..5 Assignment D. 999/ D.. Assignment D.. Assignment D..3 Assignment D..4 Assignment D..5 Assignment D..6 Assignment D.3 / D.4 / D.5 MATH E Final Examinations from Previous Years 34 E. Final Examination in Mathematics 89-B (996/997) E. Final Examination in Mathematics 89-4B (997/998) E.3 Supplemental/Deferred Examination in Mathematics 89-4B (997/998) E.4 Final Examination in Mathematics 89-4B (998/999) E.5 Supplemental/Deferred Examination in Mathematics 89-4B (998/999) E.6 Final Examination in Mathematics 89-4B (999/)... 3 E.7 Supplemental/Deferred Examination in Mathematics 89-4B (999/) E.8 Final Examination in Mathematics 89-4B (/) E.9 Supplemental/Deferred Examination in Mathematics 89-4B (/) E. Final Examination in Mathematics 89-4B (/) E. Supplemental/Deferred Examination in Mathematics 89-4B (/) E. Final Examination in MATH E.3 Supplemental/Deferred Examination in MATH F WeBWorK 4 F. Frequently Asked Questions (FAQ)4 F.. Where is WeBWorK? 4 F.. Do I need a password to F..3 F..4 use WeBWorK? Do I have to pay an additional fee to use WeB- WorK? When will assignments be available on WeBWorK? 4

5 Information for Students in MATH 4 4 F..5 F..6 F..7 F..8 F..9 Do WeBWorK assignments cover the full range of problems that I should be able to solve in this course? May I assume that the distribution of topics on quizzes and final examinations will parallel the distribution of topics in the WeBWorK assignments? WeBWorK provides for different kinds of Display Mode. Which should I use? WeBWorK provides for printing assignments in Portable Document Format (.pdf) or PostScript (.ps) form. Which should I use?43 What is the relation between WeBWorK and WebCT? F.. Which browser should I use for WeBWorK?.. 43 F.. What do I have to do on WeBWorK? F.. How can I learn how to use WeBWorK? F..3 Where should I go if I have difficulties with WeB- WorK? F..4 Can the WeBWorK system ever break down or degrade? F..5 How many attempts may I make to solve a particular problem on WeB- WorK? F..6 Will all Regular WeB- WorK assignments have the same length? the same value? F..7 Is WeBWorK a good indicator of examination performance? G Contents of the DVD disks for Larson/Hostetler/Edwards 5 H References 5 H. Stewart Calculus Series H. Other Calculus Textbooks... 5 H.. R. A. Adams H.. Larson, Hostetler, et al. 5 H..3 Edwards and Penney.. 5 H..4 Others, not Early Transcendentals H.3 Other References

6 Information for Students in MATH 4 4 : General Information Distribution Date: This version was mounted on the Web on 6 January, 4 (all information is subject to change) Pages - 8 of these notes may be considered the Course Outline for this course. These notes may undergo minor corrections or updates during the term: the definitive version will be the version accessible at or on WebCT, at or Instructors and Times INSTRUCTOR: Prof. N. Sancho Prof. W. G. Brown (Course Coordinator) LECTURE SECTION: CRN: OFFICE: BURN 3 BURN 4 OFFICE HOURS: MW 9:45 - :5 W 3: - 4:5 (subject to or by appointment F : - : change) or by appointment TELEPHONE: SANCHO@ BROWN@ MATH.MCGILL.CA MATH.MCGILL.CA CLASSROOM: ADAMS AUD ADAMS AUD CLASS HOURS: MWF 8:35 9:5 h. MW 6:35 7:55 h. Table : Instructors and Times Please note that the University considers Tuesday, April 3th, 3, to be a Wednesday for timetable purposes ; this will be the date of the last lectures in both sections of MATH 4 4.

7 Information for Students in MATH 4 4 :. Calendar Description.. Calendar Description MATH 4 CALCULUS II. (4 credits; 3 hours lecture; hours tutorial) (Not open to students who have taken MATH or CEGEP objective UP or equivalent; not open to students who have taken or are taking MATH 3 or MATH 3, except by permission of the Department of Mathematics and Statistics. Prerequisites: MATH 39 or MATH 4 or MATH 5. Each Tutorial section is enrolment limited.) The definite integral. Techniques of integration. Applications. Introduction to sequences and series... Late transfer from MATH 5 Some students from MATH 5 may be permitted to transfer into MATH 4 after the end of the Change of Course Period. If you are in this category, please send an message to Professor Brown as soon as your transfer has been approved..3 Tutorials; Tutors Coordinates Every student must be registered in one lecture session and one tutorial for this course. Tutorials begin in the week of January th, 4. The last tutorial in all Tutorial sections will be in the week beginning Monday, April 5th, 4; Table below gives times, locations, and the tutor s name for each of the tutorials; Table 3 gives the tutors coordinates. The information in these tables is subject to change. We try to publicize changes but sometimes we are not informed in advance. 3 Tutors The tutors in MATH 4 4 are graduate students in Mathematics and Statistics. Like you, they are students, albeit at the graduate level; they have deadlines and commitments and personal lives, and the time they have available for MATH 4 is limited and controlled by a collective agreement (union contract). Please respect the important functions that our tutors provide, and do not ask them to perform services they are not expected to perform: Tutors are not expected, nor authorized, to administer a special quiz or a quiz that has already been administered to others outside of the normal quiz times in their tutorials. The previous designation for this course was 89-4, and the version given in the winter was labelled 89-4B; an earlier number for a similar course was 89-. This is to ensure that your WeBWorK account is opened, and that your date of entry to the course is recorded. 3 The current room for your tutorial should always be available by clicking on Class Schedule on MINERVA FOR STUDENTS,

8 Information for Students in MATH 4 4 : 3 Tutors are not expected, nor authorized, to accept a Written Assignment after the week when it was due. Tutors in MATH 4 4 are not permitted to offer paid, private tuition to students in any tutorial section of this course. If you miss a quiz or assignment for a valid reason (medical or otherwise), please communicate with Professor Brown, providing a copy of the medical or other supporting documents..4 Evaluation of Your Progress.4. Your final grade (See Table 5, p. ) Your grade in this course will be a letter grade, based on a percentage grade computed from the following components:. Assignments submitted over the Web: Ten Regular WeBWorK homework assignments (cf..4.3) counting together for % The Regular Assignments will be numbered #R,... #R.. Materials graded by your Tutor: Five Written Assignments counting together for 5%. The Written Assignments will be numbered #W,..., #W5. Five Quizzes given at the tutorials counting together for 5%. The Quizzes which count will be numbered #Q,..., #Q5. 3. The final examination counting for 7%. Where a student s performance on the final examination is superior to her performance on the tutorial quizzes, the final examination grade will replace the quiz grades in the calculations. It is not planned to permit the examination grade to replace the grades on WeBWorK assignments or on written assignments..4. WeBWorK We will be using the WeBWorK system, developed at the University of Rochester which is designed to expose you to a large number of drill problems, and where plagiarism is discouraged. WeBWorK is accessible only over the Web. Details on how to sign on to WeBWorK are contained in Appendix F to these notes, page 4. WeBWorK assignments carry a due date and time; only answers submitted by this date and time will count. UPDATED TO April 8, 4

9 Information for Students in MATH 4 4 : 4 # CRN Day Begins Ends Room Tutor T3 858 Tue 8:5 9:55 BURN 4 S. Nashaat T4 859 Tue :5 :55 BURN 4 M. Al-Khaleel T5 86 Tue 4:5 5:55 BURN 4 P.-J. Bergeron T6 86 Tue 6:5 7:55 BURN 4 J. Cao T7 86 Tue 6:5 7:55 BURN B39 S. Shahabi T8 863 Thurs 4:35 6:5 BURN 9 Y. Han T9 864 Thurs 6:5 7:55 BURN 4 P. Poulin T 865 Thurs 6:5 7:55 BURN B4 R. Kolhatkar T 866 Mon 3:35 5:5 BURN 4 M. Fortin-Boisvert T 867 Mon 4:35 6:5 ARTS W- X. Liu T3 868 Mon 4:35 6:5 ADAMS 5 C. Foley T4 869 Wed 3:35 5:5 BURN 4 C. Fortin T5 87 Wed 4:35 6:5 BURN B3 M. Hosseini T6 87 Wed 4:35 6:5 ARTS W- G. Paek T Mon 3:35 5:5 ENGMD 8 M. Sohrabi T Wed 3:35 5:5 ENGMD 8 M. Sababheh Table : Schedule and Locations of Tutorials, as of April 8, 4 (subject to change) Tutor address Office Office Hours Day Begins Ends Day Begins Ends Al-Khaleel, M. khaleel@math.mcgill.ca 9 Th 4: 7: Bergeron, P.-J. bergeron@math.mcgill.ca 3 M 9:3 :3 Cao, J. cao@math.mcgill.ca 8 T 5: 6: T 8: : Foley, C. foley@math.mcgill.ca 34 M 9:3 :3 T 9: : Fortin, C. fortin@math.mcgill.ca 3 Th 3: 6: Fortin-Boisvert, M. boisvert@math.mcgill.ca 8 M :3 3:3 M 5:3 7:3 Han, Y. ying@math.mcgill.ca 7 T : :3 Th : :3 Hosseini, M. hosseini@math.mcgill.ca P+P Th 8: : Kolhatkar, R. ratnadha@math.mcgill.ca 7 T 8:3 :3 Liu, X. liu@math.mcgill.ca M :3 4:3 Nashaat, S. nashaat@math.mcgill.ca 5 W 3: 6: Paek, G. paek@math.mcgill.ca 7 W 9:3 : F 9:3 : Poulin, Ph., ppoulin@math.mcgill.ca 3 T : 5: Sababheh, M. sababheh@yahoo.com W :3 3:3 Shahabi, S. shahabi@math.mcgill.ca 85 T 5: 6: F 6: 8: Sohrabi, M. sohrabi@math.mcgill.ca 85 T 4:3 5:3 F 3:3 5:3 Table 3: Tutors Coordinates, as of April 8, 4

10 Information for Students in MATH 4 4 : 5 Some of the conditions described below may appear to be strict. They are designed to reduce the chance that you fail the course, not to impose unreasonably bureaucratic rules..4.3 Regular WeBWorK Assignments There will be ten Regular WeBWorK assignments. These will be paired each even numbered assignment (##R, R4, R6, R8, R) will contain the same types of problems as on the preceding odd numbered assignment (##R, R3, R5, R7, R9). You will have an unlimited number of tries on the problems on the odd numbered assignments; but there will be restricted numbers of tries on the even numbered assignments. The intention is that you should use each odd numbered assignment to thoroughly learn how to solve the problems, and then attempt the following even numbered assignment. The data on the even numbered assignment will probably be different different numbers and/or functions, but the same concepts. 4 It is expected that the due date for Regular assignments will be on specified Mondays, at midnight. As mentioned in the WeBWorK FAQ (cf. Appendix F) if you leave your WeBWorK assignment until the hours close to the due time on the due date, you should not be surprised if the system is slow to respond. This is not a malfunction, but is simply a reflection of the fact that other students have also been procrastinating! To benefit from the speed that the system can deliver under normal conditions, do not delay your WeBWorK until the last possible day! If a systems failure interferes with the due date of an assignment, arrangements may be made to change that date, and an message may be broadcast to all users (to the addresses on record), or a note posted in the course announcements on WebCT; but slowness in the system just before the due time will not normally be considered a systems failure. 5 BONUS assignment It is hoped that there will be one additional WeBWorK assignment, which will be optional. The grade on this assignment would replace the grade on any other WeBWorK assignment with a lower grade, or replace a missed assignment. This assignment will have limits on the numbers of attempts. Further details will be announced in the lectures or on the WebCT or WeBWorK sites. 4 The assignments will count equally in your WeBWorK grade, provided you have received a grade of at least 75% on one Precalculus Assignment; if you have not received that grade on any Precalculus Assignment by the end of the term, your WeBWorK grade will be. 5 Should you find that the system is responding slowly, do not submit your solutions more than once; you may deplete the number of attempts that have been allowed to you for a problem: this will not be considered a systems failure.

11 Information for Students in MATH 4 4 : Written Assignments Written assignments will be posted at about the same time as regular WeBWorK assignments ##R, R4, R6, R8, R. These assignments will contain one or more problems for which you will be expected to write full solutions, modelled on similar types of solution in worked examples in the textbook, or solved problems in the Student Solutions Manual. You will be assigned an individualized version of the problems. You should prepare these solutions and carry them to the Tutorial Quiz the following week; you will be expected to hand in your solutions with your quiz paper, and they will be graded by your tutor and returned to you with that graded quiz paper. Please do not attempt any other methods for submitting your written assignments, to minimize the risk of loss. Written Where to Find How to Submit Date of Assignment Your Version of the Questions Your Solutions Quiz #W on WeBWorK site with Quiz #Q Jan. 9- #W on WeBWorK site with Quiz #Q Feb. -5 #W3 on WeBWorK site with Quiz #Q3 Feb. 6-9 #W4 on WeBWorK site with Quiz #Q4 Mar. 8- #W5 on WeBWorK site with Quiz #Q5 Mar. -5 Table 4: Timetable for Written Assignments.4.5 Quizzes at the Tutorials; Submission of Written Assignments at the Quizzes. There will be 5 short quizzes, numbered #Q, #Q, #Q3, #Q4, #Q5, administered at the tutorials. These quizzes will be graded, and returned (together with the written assignment that will be handed in with the quiz paper). The primary purpose of a quiz is to diagnose possible gaps in your understanding, not to drill on examination skills. No provision is being made for students who miss a quiz. The grading formula permits the quiz component of the final grade to be replaced by the final examination grade, if this is to a student s advantage.. The quizzes will be based on current topics in the syllabus of the course, most 6 of which topics will have been discussed in the lectures before the quiz; the quizzes are not based directly on WeBWorK assignments. To prepare for a quiz you 6 but possibly not all

12 Information for Students in MATH 4 4 : 7 should be working exercises in the textbook based on the material currently under discussion at the lectures, and you should have attempted any open WeBWorK assignments. But, unlike the WeBWorK assignments where the emphasis is on correct answers alone students may be expected to provide full solutions to some or all problems on quizzes. 7 The quiz time available in MATH 4 is not sufficient to examine on all topics; consequently the quizzes will examine on only a sampling of the topics. Students should not assume that topics not examined are in any subsidiary parts of the syllabus. Note also that the last quiz will be administered before the full syllabus has been discussed in lectures and/or tutorials, so a portion of the syllabus will not be examined in quizzes or assignments. 3. You are expected to write the quiz in the tutorial section in which you are registered. Should the classroom become filled while some registered students still cannot be seated, the tutor may insist that any students who are not registered in that tutorial to leave the room Submission of Written Assignments at Quizzes Individualized written assignments ##W, W, W3, W4, W5 will be mounted on the WeBWorK site about a week before they are to be handed in at a tutorial quiz; you should download or copy each written assignment from the web site, write out your solutions at home, bring the completed assignment with you to the tutorial, and hand it in with your quiz paper no other submission method is acceptable. The assignment must be ready for submission when you arrive, as you will not be provided time in the tutorial room to complete it. You will enclose the written assignment in your folded answer paper. 5. Your tutors will normally bring graded quizzes and graded assignments submitted with them to the tutorial to be returned to you. University regulations do not permit us to leave unclaimed materials bearing names and student numbers in unsupervised locations; you may be able to recover an unclaimed quiz from the tutor who graded it, during her/his regular office hours. 7 In Math 4 the general rule for quizzes is that full solutions are expected to all problems, unless you receive explicit instructions to the contrary: ALWAYS SHOW YOUR WORK! The solutions in the Student Solution Manual [3] to the textbook can serve as a guide to what should be included in a full solution. 8 Anyone who is not registered, and who does not leave the room when so requested may forfeit the right to write quizzes at any tutorial, and could be subject to disciplinary action through their Faculty. We ask you to respect the prior right of students who have registered for each tutorial.

13 Information for Students in MATH 4 4 : Final Examination A 3-hour-long final examination will be scheduled during the regular examination period for the winter term (April 5th, 4 through April 3th, 4). You are advised not to make any travel arrangements that would prevent you from being present on campus at any time during this period..4.7 Supplemental Assessments Supplemental Examination. There will be a supplemental examination in this course. (For information about Supplemental Examinations, see the McGill Calendar, [7, 8., p. 5-5; or 8., p. 5-53].) There is No Additional Work Option. Will students with marks of D, F, or J have the option of doing additional work to upgrade their mark? No. ( Additional Work refers to an option available in certain Arts and Science courses, but not available in this course.).4.8 Machine Scoring Will the final examination be machine scored? It is possible that the final examination will contain some questions that will be machine scored. Not more than % of the marks available on the examination possibly less will derive from problems that are machine scored. It is likely that some parts of the final examination will consist of questions where only you need not show your work, but where you should not expect part marks for an answer that is not perfectly correct..4.9 Plagiarism and Fraud Plagiarism While students are not discouraged from discussing methods for solving WeBWorK assignment problems with their colleagues, all the work that you submit whether through WeBWorK or written assignments, or on tutorial quizzes or the final examination must be your own. It is a violation of University regulations to permit others to solve your WeBWorK problems, or to extend such assistance to others; you could be asked to sign a statement attesting to the originality of your work. The Handbook on Student Rights and Responsibilities states in 5(a) 9 that No student shall, with intent to deceive, represent the work of another person as his or her own in any academic writing, essay, thesis, research report, project or assignment submitted in a course or program of study or represent 9

14 Information for Students in MATH 4 4 : 9 as his or her own an entire essay or work of another, whether the material so represented constitutes a part or the entirety of the work submitted. You are also referred to the following URL: The preceding paragraph was prepared before all faculty were advised that the Senate of the University requires the following message in all course outlines: McGill University values academic integrity. Therefore all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures. (See for more information). L université McGill attache une haute importance à l honnêteté académique. Il incombe par conséquent à tous les étudiants de comprendre ce que l on entend par tricherie, plagiat et autres infractions académiques, ainsi que les conséquences que peuvent avoir de telles actions, selon le Code de conduite de l étudiant et des procédures disciplinaires. (Pour de plus amples renseignements, veuillez consulter le site Other Fraud It is a serious offence to alter a graded quiz paper and return it to the tutor under the pretense that the work was not graded properly..4. Corrections to grades Posted Grades Grades will be eventually posted on WebCT. If you believe a grade has been recorded incorrectly, you must advise your tutor not later than 4 weeks after the grade has been posted, and not later than the day before of the final examination whichever of these dates is earlier. It is hoped that grades will be posted within weeks of the due date. You will have to present the graded quiz or assignment to support your claim, which must be submitted to the tutor that graded the quiz or assignment. If he/she believes there has been an error, the tutor will advise Professor Brown. New corrections to the WebCT posting will appear the next time grades are uploaded to WebCT..5 Published Materials.5. Required Text-Book The textbook for the course is J. Stewart, SINGLE VARIABLE CALCULUS: Early Transcendentals, Fifth Edition, Brooks/Cole (3), ISBN , [].

15 Information for Students in MATH 4 4 : Item # Due Date Details Regular WeBWorK Assignments (cf..4.3) R 6 Jan 4 No limit to number of attempts at WeBWorK R problems R Feb 4 Same scope as R, but with limited numbers of attempts R3 9 Feb 4 R4 6 Feb 4 Same scope as R3, but with limited numbers of attempts %. R9 9 Mar 4 R 5 Apr 4 Same scope as R9, but with limited numbers of attempts BONUS 3 Apr 4 see page 5, limited numbers of attempts Written W with Q Download; complete at home; hand in with Q Assignments W with Q Download; complete at home; hand in with Q (cf..4.4) W3 with Q3 Download; complete at home; hand in with Q3 W4 with Q4 Download; complete at home; hand in with Q4 5% W5 with Q5 Download; complete at home; hand in with Q5 Quizzes (cf..4.5) Q 9 Jan 4 5% or %. Q5 5 Mar 4 Final Exam 5 3 Apr 4 Date of exam to be announced by Faculty 7% or 85% Supplemental Exam 5 6 August 4 Only for students who do not obtain standing at the final. Supplemental exams count in your average like taking the course again; exam counts for %. Table 5: Summary of Course Requirements, as of April 8, 4 (all dates are subject to change) This book is the first half of J. Stewart, CALCULUS: Early Transcendentals, Fifth Edition, Brooks/Cole (3), ISBN , []; this edition covers the material for Calculus III (MATH ) as well, but is not the text-book for that course at the present time..5. Optional Reference Books It is recommended that students make use of the student solution manual: D. Anderson, J. A. Cole, D. Drucker, STUDENT SOLUTIONS MANUAL

16 Information for Students in MATH 4 4 : FOR STEWART S SINGLE VARIABLE CALCULUS: Early Transcendentals, Fifth Edition, Brooks/Cole (3), ISBN , [3]. This book is also sold bundled with either version of the text book; the bundles are numbered ISBN [4] and ISBN [5]. The publishers of the textbook and Solutions Manual also produce a Study Guide, designed to provide additional help for students who believe they require it: R. St. Andre, STUDY GUIDE FOR STEWART S SIN- GLE VARIABLE CALCULUS: Early Transcendentals, Fifth Edition, Brooks/Cole (3), ISBN , [6]. (The Study Guide resembles the Student Solution Manual in appearance: be sure you know what you are buying.) a Companion which integrates a review of pre-calculus concepts with the contents of Math 4, including exercises with solutions: D. Ebersole, D. Schattschneider, A. Sevilla, K. Somers, A COMPANION TO CALCULUS. Brooks/Cole (995), ISBN [6]..5.3 Recommended Video Materials Use of the following materials is recommended, but is not mandatory. Larson/Hostetler/Edwards DVD Disks A set of video DVD disks produced for another calculus book, [8] Calculus Instructional DVD Program, for use with (inter alia) Larson/Hostetler/Edwards, Calculus of a Single Variable: Early Transcendental Functions, Third Edition [9] is produced by the Houghton Mifflin Company. A copy has been requested to be placed on reserve in the Schulich Library, and the set of 4 disks is for sale in the bookstore among the reference materials for this course. In Appendix G of these notes there are charts that indicate the contents of these disks that pertain to MATH 4. Videotapes for Stewart s Calculus The publisher of Stewart s Calculus has produced a series of videotapes, [7] Video Outline for Stewart s Calculus (Early Transcendentals), Fifth Edition. These will initially be available for reserve loan at the Schulich Library. There may not be VCR viewing equipment in the library; the intention is that interested students borrow a tape for viewing on their own equipment at home. No one will check whether you have used any of these aids; a student can obtain a perfect grade in the course without ever consulting any of them. No audio-visual or calculator aid can replace the systematic use of paper and pencil as you work your way through problems. But the intelligent use of some of these aids can deepen your understanding of the subject.

17 Information for Students in MATH 4 4 : Tools for Enriching Calculus This is a CD-ROM included with new copies of Stewart s Calculus. From [9, Introduction]: Tools for Enriching Calculus (TEC) enhances a topic that is covered in the textbook by providing both broader and deeper coverage of those aspects for which technology is particularly useful. The basic format of most modules is a point-andclick laboratory environment in which you can easily visualize functions and their derivatives, experiment with suggested examples and exercises, explore your own choices of examples, and perhaps even test some of your own conjectures. You need to be a well-prepared and active player to reap the benefits from these approaches. First, you need to read the textbook materials carefully to gain an understanding of the essential ideas. Next, you need to read the introductory material for each TEC module, which explains the basic mathematical approaches and describes how to use the module. Each module has several examples which will familiarize you with its basic features. When you have finished reviewing this material, and have some paper and pencil in hand, you are ready to get the most benefit from using the module. You can improve your understanding of the topic by exploring mathematical questions that you find puzzling, and checking your ideas for solutions using the module. Try to work through some of the exercises in the module to gauge your understanding of the topic. Be willing to use pencil and paper to first guess what the answer might be before seeing an electronic graph... Another important TEC feature is the homework hints. Hints have been created for several selected exercises in each section of your textbook to help you understand some key points in finding solutions for these exercises. Similar to a good instructor or teaching assistant, these hints ask you questions that will allow you to make progress toward a solution without giving you the actual answer. You need to actively pursue each hint with pencil and paper and fill in many of the computations and details. If you can complete the solution after reading only one or two hints, you can feel proud of your achievements. If you still have questions after completing all of the hints for a problem, your work should help you to better understand the solution presented in the Student Solutions Manual. Interactive Video Skillbuilder CD for Stewart s Calculus: Early Transcendentals, 5th Edition [8] This CD-ROM is included with new copies of the textbook. It contains, after an enlightening pep-talk by the author, a discussion of some of the worked examples in the text-book, followed by a quiz for each section in the book. Some students may find the animations of the examples helpful, although the examples are all worked in the book. You might wish to try some of the quiz questions using paper and pencil, and then check your answers with those given on the CD. It is not recommended that you attempt to enter your answers digitally, as this is a time-consuming process, and uses a different input method from your WeBWorK assignments, which serve the same purpose. Note that homework refers to odd-numbered problems in the exercises in [], not to problems on WeBWorK.

18 Information for Students in MATH 4 4 : Other Calculus Textbooks While students may wish to consult other textbooks, instructors and teaching assistants in Math 4 will normally refer only to the prescribed edition of the prescribed textbook for the course. Other books can be very useful, but the onus is on you to ensure that your book covers the syllabus to at least the required depth; where there are differences of terminology, you are expected to be familiar with the terminology of the textbook. In your previous calculus course(s) you may have learned methods of solving problems that appear to differ from those you find in the current textbook. Your instructors will be pleased to discuss any such methods with you personally, to ascertain whether they are appropriate to the present course. In particular, any methods that depend upon the use of a calculator, or the plotting of multiple points, or the tabulation of function values, or the inference of a trend from a graph should be regarded with scepticism..5.5 Website These notes, and other materials distributed to students in this course, will be accessible at the following URL: The notes will be in pdf (.pdf) form, and can be read using the Adobe Acrobat reader, which many users have on their computers. This free software may be downloaded from the following URL: 3 The questions on some old examinations will also be available as an appendix to these notes on the Web. 4 It is expected that most computers in campus labs should have the necessary software to read the posted materials. Where revisions are made to distributed printed materials for example these information sheets we expect that the last version will be posted on the Web. The notes and WeBWorK will also be available via a link from the WebCT URL: Your grades on assignments and quizzes will be posted on WebCT within about weeks after they become available. Some other features of WebCT 5 have not yet been implemented. There should be multiple copies of the textbook on reserve in the Schulich library. 3 At the time of this writing the current version appears to be There is no reason to expect the distribution of problems on quizzes or in assignments and examinations from previous years be related to the frequencies of any types of problems on the examination that you will be writing at the end of the term. 5 cf. Appendix F to these notes, p. 4

19 Information for Students in MATH 4 4 : 4.6 Syllabus In the following list section numbers refer to the text-book []. The syllabus will include all of Chapters 5, 6, 7, 8,, with omissions, as listed below. 6 Chapter 5: Integrals The derivation 7, in 5.6, of properties of the logarithm and integral is not examination material. The Midpoint Rule, defined in 5., and appearing from time to time subsequently, is not examination material. Chapter 6: Applications of Integration ; 6.5. ( 6.4 is not examination material, but Science students are urged to read it.) Chapter 7: Techniques of Integration ; 7.8. ( 7.6, intended for use in conjunction with integral tables and/or computer algebra systems, is not examination material, but students are to try to solve the problems manually; 7.7 is not examination material.) Chapter 8: Further Applications of Integration. 8., 8. only. ( 8.3, 8.4 are not examination material, but students are urged to read the applications relevant to their course of study; 8.5 is not examination material.) Chapter 9: Differential Equations. (No part of this chapter is examination material; however, students are urged to read 9.4 Exponential Growth and Decay). Chapter : Parametric Equations and Polar Coordinates...4. (.5,.6,.7 are not examination material.) Chapter. Infinite Sequences and Series...7. (.8. are not examination material; however, students are urged to peruse these sections.) Please do not ask the tutors to provide information as to which textbook sections should be emphasized. Unless you are informed otherwise by the instructors in the lecture sections or published notes printed, or mounted on the Web you should assume that all materials listed are included in the syllabus. You are not expected to be able to reproduce proofs of the theorems in the textbook. 6 If a textbook section is listed below, you should assume that all material in that section is examination material even if the instructor has not discussed every topic in his lectures; however, the instructors may give you information during the term concerning topics that may be considered subsidiary. Do not assume that a topic is omitted from the syllabus if it has not been tested in a WeBWorK assignment or a quiz, or if it has not appeared on any of the old examinations in the course! Some topics to not lend themselves to this type of testing; others may have been omitted simply because of lack of space, or oversight. By the same token, you need not expect every topic in the course to be examined on the final examination. 7 But N.B., students are expected to know the properties themselves, which were discussed in MATH 39 and MATH 4.

20 Information for Students in MATH 4 4 : 5.7 Preparation and Workload.7. Prerequisites. It is your responsibility as a student to verify that you have the necessary prerequisite. It would be foolish 8 to attempt to take the course without it. Students who obtained only a grade of C in MATH 4 would be advised to make a special effort to reinforce their foundations in differential calculus; if weakness in MATH 4 or MATH 39 was a consequence of poor preparation for that course, it is not too late to strengthen those foundations as well. In the past students with a grade of D in MATH 4 or MATH 39 have been permitted to take MATH 4 at their own risk, provided they have also registered for MATH 4 or have applied to write the supplemental examination in the course in which they obtained the D. Students with only a D, or students whose examination in MATH 4 has been deferred, who contemplate registering for MATH 4 should recognize that it places a large number of credits at risk, and could have a substantial negative effect on their GPA if they have overestimated their ability; it is safer to obtain standing (grade C or better) in MATH 39 or MATH 4 before registering for MATH 4. Students with a grade of F in MATH 39 or MATH 4 are not authorized to register in MATH 4 4, even if they plan to write a supplemental examination in MATH 4 or are taking MATH 4 concurrently..7. Calculators The use of calculators is not permitted in either quizzes or the examination in this course. Students whose previous mathematics courses have been calculator-oriented would be advised to make particular efforts to avoid the use of a calculator in solving problems in this course, in order to develop a minimal facility in manual calculation. This means that you are urged to do all arithmetic by hand..7.3 Self-Supervision This is not a high-school course, and McGill is not a high school. The monitoring of your progress before the final examination is largely your own responsibility. While the tutors and instructors are available to help you, they cannot do so unless and until you identify the need for help. WeBWorK and quizzes are designed to assist you in doing this. Time Demands of your Other Courses. Be sure to budget enough time to attend lectures and tutorials, for private study, and for the solution of many problems. Don t be 8 and contrary to McGill regulations

21 Information for Students in MATH 4 4 : 6 tempted to divert calculus study time to courses which offer instant gratification. While the significance of the tutorial quizzes in the computation of your grade is minimal, these are important learning experiences, and can assist you in gauging your progress in the course. This is not a course that can be crammed for: you must work steadily through the term if you wish to develop the facilities needed for a strong performance on the final examination. Working Problems on Your Own. An effective way to master the calculus is through working large numbers of problems from the textbook. Your textbook was selected partly because of the availability of an excellent Student Solutions Manual [3]; this manual has brief but complete solutions to most of the odd-numbered exercises in the textbook. The skills you acquire in solving textbook problems could have much more influence on your final grade than either WeBWorK or the quizzes. The real uses of WeBWorK and the quizzes. Students often misunderstand the true significance of WeBWorK assignments and the quizzes. While both contribute to your grade, they help you estimate the quality of your progress in the course. Take proper remedial action if you are obtaining low grades on quizzes 9, or if you require many attempts before being able to solve a problem on WeBWorK. However, while both WeBWorK and the quizzes have a role to play in learning the calculus, neither is as important as reading your textbook, working problems yourself, and attending and listening at lectures and tutorials..7.4 Escape Routes At any time, even after the last date for dropping the course, students who are experiencing medical or personal difficulties should not hesitate to consult their advisors or the Student Affairs office of their faculty. Don t allow yourself to be overwhelmed by such problems; the University has resource persons who may be able to help you..7.5 Terminology Do not be surprised if your instructors and tutors use different terminology from what you have heard in your previous calculus course, particularly if that course was at a high school. Sometimes the differences are purely due to different traditions in the professions. Negative x Your instructors and tutors will often read a formula x as minus x, not as negative x. To a mathematician the term negative refers to real numbers which are not squares, i.e. which are less than, and x can be positive if x itself is negative. 9 The worst action is to miss the quizzes, and thereby block out an unwelcome message.

22 Information for Students in MATH 4 4 : 7 However, mathematicians will sometimes refer to the operation of changing a sign as the replacement of x by its negative ; this is not entirely consistent with the usual practice, but is an abuse of language that has crept into the professional jargon. Inverse trigonometric functions A formula like sin x will be read as the inverse sine of x never as sine to the minus or sine to the negative. However, if we write sin n x, where n is a positive integer, it will always mean (sin x) n. These conventions apply to any of the functions sin, cos, tan, cot, sec, csc; they also apply to the hyperbolic functions, which we have met in [, 3.9]: sinh, cosh, tanh, coth... We will usually not write exponents on general functions, so a formula like f (x) does not have an obvious meaning, and we will avoid writing it when f is other than a trigonometric or hyperbolic function. Logarithms These days mathematicians rarely use logarithms to the base. If you were taught to interpret log x as being the logarithm to base, you should now forget that although it could be the labelling convention of your calculator. Most often, if your instructor speaks of a logarithm, and writes log x, he will be referring to the base e, i.e. to log e ; that is, he is referring to the function that calculus books call ln. When a logarithm to some other base is intended, it will either be denoted by an explicit subscript, as log, or some comment will be made at the beginning of the discussion, as all logarithms in this discussion are to the base. Your instructors try to think like mathematicians even when lecturing to their classes, and so we use the language and terminology we use when talking to each other..8 High and Low Technology and MATH 4.8. Keep your address up to date Both WebCT and WeBWorK contain an address where we may assume you can be reached. If you prefer to use another address, the most convenient way is to forward your mail from your student mailbox, leaving the recorded addresses in these two systems unchanged. You can enter or change a forwarding address by going to and logging in to your student mailbox at po-box.mcgill.ca..8. Use of Calculators and Computer Algebra Systems Insofar as course content is concerned, we will emphasize the lowest possible use of technology: we will avoid the use of calculators and computers. You are urged to do all calculations manually, and to avoid the use of computer algebra systems, until you have completed MATH 4. You should not use a calculator or computer in the solution of

23 Information for Students in MATH 4 4 : 8 WeBWorK problems, as it prevents you from developing skill for detecting errors in manual calculations a skill that you will need for the quizzes and final examination..8.3 Use of the Internet However, insofar as course delivery is concerned, students are expected to be able to access materials through the Internet, whenever required. Here are some of the uses that are expected: To access and submit Regular WeBWorK (.4.3) assignments To access the individualized problems for a student s Written Assignments To access WebCT, where course grades and announcements will be mounted To access the web page for the course also available through WebCT where these and other notes will be available in.pdf form; the site also contains notes and examinations from previous years..9 Special Review Sessions All tutorials hold their last sessions during the week 5-8 April, 3. The following review sessions are planned at the end of the term: (subject to revision) Date Instructor Location Starts Ends UPDATED TO April 8, 4

24 Information for Students in Lecture Section of MATH 4 4 A Information Specifically for Students in Lecture Section A. Timetable for Lecture Section of MATH 4 4 Distribution Date: th version: Monday, January 5th, 4 (Subject to correction and change.) Section numbers refer to the text-book. MONDAY WEDNESDAY FRIDAY JANUARY Tutorials begin week of January th, Course changes must be completed on MINERVA by Jan. 8, Q W 6., 6.3 Q W 3 6., 6.3 Q W Deadline for withdrawal with fee refund = Jan. 5, , 6.3, 6.5 R FEBRUARY Verification Period: February 6, 4 7. Q W R Q W Q W Deadline for withdrawal (with W) from course via MINERVA = Feb. 5, R Q 3 W 3 R Q 3 W 3 X Q 3 W 3 Study Break: February 3 7, 4 No lectures, no regular office hours, no regular tutorials! 3 NO LECTURE 5 NO LECTURE 7 NO LECTURE (Page of the timetable will not be circulated; however, a version is available in the online version of these notes.) Notation: R n = Regular WeBWorK Assignment #Rn due at midnight on Monday this week R = Read Only Q n W n = Quiz #Qn planned for the tutorials this week Hand in Written Assignment #Wn with your Quiz paper X = reserved for expansion or review

25 Information for Students in Lecture Section of MATH 4 4 MONDAY WEDNESDAY FRIDAY MARCH. R Q 4 W 4 R 6.4 Q 4 W 4 X,. Q 4 W 4 5. R Q 5 W 5 R 8 4.,.3 Q 5 W Q 5 W R 9 3.5,.6 APRIL.5, R 7 X 9 NO LECTURE NO LECTURE 3 (TUESDAY) X R BONUS

26 Information for Students in Lecture Section of MATH 4 4 B Information Specifically for Students in Lecture Section B. Timetable for Lecture Section of MATH 4 4 Distribution Date: th version: Monday, January 5th, 4 (Subject to correction and change.) Section numbers refer to the text-book. MONDAY WEDNESDAY JANUARY 5 5., Tutorials begin week of January th, 4 5.4, , 5.6 Course changes must be completed on MINERVA by Jan. 8, , 6. Q W 6.3 Q W Deadline for withdrawal with fee refund = Jan. 5, R 8 7. FEBRUARY Verification Period: February 6, 4 7., 7.3 Q W R Q W 9 7.4, 7.5 R 3 7.8, X Deadline for withdrawal (with W) from course via MINERVA = Feb. 5, , 8. Q 3 W 3 R ,.,. Q 3 W 3 Study Break: February3 7, 4 No lectures, no regular office hours, no regular tutorials! 3 NO LECTURE 5 NO LECTURE (Page of the timetable will not be circulated; however, a version is available in the online version of these notes.) Notation: R n = Regular WeBWorK Assignment #Rn due at midnight on Monday this week R = Read Only Q n W n = Quiz #Qn planned for the tutorials this week Hand in Written Assignment #Wn with your Quiz paper X = reserved for expansion or review

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