Math 212: Calculus II with Introduction to Multivariable Functions
Fall 2021
Instructor: Prof. Cleary
Email: scleary (at sign) ccny.cuny.edu
Office: Marshak 301C
Phone: 6505122
Office Hours: see webpage here for uptodate office hours.
Text: Thomas' Calculus: Early Transcendentals (14th ed.), Hass, Heil, and Weir (Pearson), ISBN13: 9780134439020
Course documents:
Announcements:
 Final exams graded, congratulations to those who did well and good to everyone on their remaining exams! Scores and grades are on Gradescope and Blackboard.
 The CUNY Tutor Corps is hiring qualified students. This may be a good opportunity to help students:
We are recruiting CUNY students (at any campus) with a GPA of 3.0 or better who have 12+ credits in any STEM subject and/or education. We want to hire them to work as tutors in NYC Public Schools. We pay our tutors $17 per hour and they work anywhere from 619 hours per week. They will be tutoring math or science or computer science during the school day in person. We will place them in a school that is in the most convenient location possible considering where they live and the campus they attend. We offer an intensive, paid week of training as well as ongoing professional development. It's a great job for students who need something with a flexible schedule to fit in around their coursework.
 Final exams graded, scores on Gradescope and Blackboard.
 There were a few typos on the early seats page and it has been updated please check the updated seating page.
 Seats for final exam are at this page. As usual, no notes, books, calculators, or electronic devices for the exam.
 Seats for the final exam in NAC 0/201 Thurs Dec 16th 3:30pm  5:45pm will be assigned here on this webpage but have not been finalized yet.
 Office hours for Monday, Dec 13th are postponed to 5:45pm as I am covering another instructor's class during my usual office hours.
 The last day to drop the course is Monday, Dec 13th. This is also the last day of classes. We will meet as usual on Monday, Dec 13th. The final exam is Thursday, Dec 16th.
 Note that campus access has gotten more difficult with recent policy changes. No longer are the V22 ID stickers or Excelsior passes enough to enter any campus buildings. You must use the Cleared4 pass procedure that you recently got email about. You must be able to display the pass when entering buildings or approached by security within the buildings. Campus administrators want instructors to know that there may be large backups to enter campus buildings and to move about inside campus buildings during final exams, so please allow extra time and arrive very early for the final exam on Thursday, Dec 16th.
 Check your email for information about campus access information. There should be email from noreply@Cleared4Work.com about using the Cleared4Work pass procedure to enter campus buildings. Also check your email as some students still have not satisfied the vaccine documentation procedures and will be administratively dropped from their courses.
 The final exams are long (usually ten questions, but typically with multiple parts, often not related to each other) and it can be difficult to get started on the problems when they are presented out of context. Looking at old final exams is good preparation for the important skill of classifying what approaches to use for problems.
 As mentioned in class, the grading criteria are on the syllabus and you can estimate your prospects and evaluate "whatif" scenarios using that. There is no extra credit and it is your obligation to show mastery of the material by exam work in the course.
 If you are interested in the "twopart" exam option for the final exam to be able to take a break during the exam, let me know by 10am on Monday Dec 13th via email or posting to the Blackboard Discussion thread so that I can arrange correct duplication.
 The last homework deadline has been pushed to Monday, Dec 13th, the last day of class but since the final is not far away, I suggest finishing that sooner and focusing on final exam preparation.
 I extended the deadlines on quizzes 2 through 5 for additional time for those who have not yet completed them.
 Exam 3 graded, scores available on Gradescope.
 For the final exam, there are multiple resources to prepare but they are not all in the same place. Some resources:
 The Math 212 page has a few sample final exams covering the material of the course. Some of those were from the times when the exams were online and not representative as far as the expected format.
 The Math 202 page has many old exams, but covers slightly different topics. Pretty much all of the questions are appropriate aside from some that refer to volumes, work, and some of the conic section questions.
 The Math 203 page has many old exams, covering some of the topics we covered. There are also many solutions there but it is generally more helpful to make a good effort on the exams before consulting the solutions. The questions about series, power series, lines, planes, vectors, quadrics, partial derivatives, and limits of functions of several variables that should all be doable for 212 students. There are some questions about double and triple integrals, volumes, areas, centers of mass, rates of change, differentials, tangent planes, and some other topics that we did not cover in 212.
 Pretty much every topic that we covered is in one of Math 202 or 203, but we did not cover all of the topics in either course. So there are some questions that you may ignore on those sample final exams.
 Note: our final exam takes place Thursday, Dec. 16, 2021 3:30pm  5:45pm, because it is a group final. All 212 students will be taking the final then. The final exam time for Math 212 is not based on the class meeting time, unlike many courses at CCNY.
 For the students who were asking about my Klein bottle hat, here is a video on how to make one. It's a onesided surface, similar to a Mobius band. There are nice glass Klein bottles described here.
 The 203 video lessons page has Lessons 19 on vector and spatial topics we cover in 212, and 1114 as well on functions of several variables that we are doing until the end of the term.
 Final exam information:
 Final exam will be Thursday, Dec. 16, 2021 3:30pm  5:45pm.
 Room is planned to be NAC 0/201 in the North Academic Center (huge hulking grey building across from Marshak and diagonal from Shephard)
 Seat assignments to be announced once room is confirmed
 If you are interested in the "twopart" exam option for Exam 3, let me know by 11am on Monday Nov 29th via email or posting to the Blackboard Discussion thread so that I can arrange correct duplication.
 Exam 3 covers material up to the homework due on Dec 1st, the day of the exam.
 Some homework due dates postponed near the end of the term.
 Homework on 12.5 duedate postponed to Nov 24th.
 Exam 3 info for Wed, Dec 1st:
 212CD: 12pm1:40pm in SH 210 (same room as first exam, in Shephard Hall)
 212EF: 2pm3:40pm in MR117 (same room as the first two exams, in Marshak)
 Make sure that you have a photo ID (CCNY preferred but not required) and that you know your student EMPLID.
 Students who are interested in having the "two part" exam should either email me or post in the Blackboard discussion thread under "Administration" before the Thanksgiving break so that I can duplicate enough exams in that format.
 Topics are all material covering up to the section with homework due on the day of the exam.
 Blackboard Quiz 2 was inadvertently set to only allow one attempt in some cases. That should be sorted out now so you can try again if needed.
 There are some quizzes available on Blackboard. They are not due until Monday morning after the Thanksgiving break and only the most recent score for each quiz is counted.
 For the attendance scores reported on Blackboard, it does not always record properly, particularly in instances of disconnection and reconnection. I have my own records and if your Blackboard attendance is not being recorded correctly, let me know and I can update those. Furthermore, to allow for the inevitable difficulties that arise over the course of a semester, the lowest 3 attendance scores are not included in scoring.
 Fall 2007, Q10 and Spring 2009, Q11 from the old Math 203 final exams at this link, about the series for arctan, was related to a question on some versions of Exam 2
 We will return to Section 11.3 on Polar Coordinates at some point. You may find Lessons 24 to 26 in the Math 202 videos page helpful preparation for that, particularly if you haven't seen polar coordinates in a while. Also there is Lesson 27 on Conic Sections, which some students will find helpful when we get to quadric surfaces later in the term.
 Exam 2 graded and available on Gradescope.
 Topics on space, vectors. vector operations, lines, and planes are covered in lectures 1 through 6 from the 203 lectures page of Dr. Marchese and me.
 Exam 2 info for Wed, Nov 3rd
 212CD: 12pm1:40pm in NAC 5101 (different room and building)
 212EF: 2pm3:40pm in MR117 (same room as the first exam)
 Make sure that you have a photo ID (CCNY preferred but not required) and that you know your student EMPLID.
 Students who are interested in having the "two part" exam should either email me or post in the Blackboard discussion thread under "Administration" by 9am on Monday so that I can duplicate enough exams in that format.
 Topics are all material covering up to the section with homework due on the day of the exam.
 The planned final exam date and time have been announced: Thursday, Dec. 16, 2021 3:30pm  5:45pm.
 In the Pearson system, when they ask for "Cartesian equation", they generally mean that they would like y as an explicit function of x. This can often be accomplished by eliminating the parameter.
 The due dates for the assignments for 10.9 and 10.10 have been postponed to Monday, Nov 1st.
 Parametric curves are in Lessons 22 and 23 in lectures from Dr. Marchese, available at this page.
 The topics from Taylor series are in Lesson 33, 34, and 35 from the the 203 video lectures page.
 For some practice on series problems, a good place to look is at the collection of many old Math 203 final exams at this link.
 I recommend all students who did not pass the first exam come to office hours to discuss strategies for doing better on the upcoming work. In many instances, mastering prerequisite material will be important to making good progress in 212.
 The first exam was very calculational there were no conceptual problems or proofs on it. That does not mean that those types of problems will not be on future exams.
 Exam 2 is coming up on Wednesday, Nov 3rd. I have obtained room assignments for those oncampus exams:
 212CD: 12pm1:40pm in NAC 5101 (different room and building)
 212EF: 2pm3:40pm in MR117 (same room as the first exam)
 Power series are in lecture 32 from the 203 video lectures page.
 Participating via useful Blackboard discussions is useful. A number of students have posted exam questions and solutions there.
 Some students have found CCNY instructor Jhevon Smith's video lectures on calculus topics useful. Those are available on Youtube. For example, one on sequences and beginning series is at this link.
 Exam 1 graded, scores on Blackboard. Exams results are available on Gradescope and you should have gotten an email from them if your exam scan went smoothly.
 Students may want to work in a campus library before the exam. Apparently, they just changed the policy so that reservations are no longer needed. I recommend the CCNY Music Library for CD students as it is a beautiful space in Shepard Hall (where the CD exam is) and is open 10am5pm on the day of our exam. For EF students, there is the Science and Engineering Library in the Science Building which is open from 9am to 5pm on the day of Exam 1 and is in the Science Building, not too far from where the EF exam is.
 Homework that was originally due on the day of Exam 1 has been postponed to allow for more time to study for the exam.
 No class meeting on Monday Oct 11th due to CUNY holiday closure.
 Students who are interested in having the "two part" exam should either email me or post in the Blackboard discussion thread under "Administration" by noon on Tuesday so that I can duplicate enough exams in that format
 Make sure that you have a photo ID (CCNY preferred but not required) and that you know your student EMPLID.
 Update: Exam 1 will take place in the following rooms on Wednesday, Oct 6th:
 212CD: 12pm1:40pm in Shepard Hall 210, on the second floor of Shepard Hall
 212EF: 2pm3:40pm in Marshak 117, on floor 1 or Marshak Hall (the Science Building) which is not the floor you enter on, but one flight up from the Science Cafe and Science Library areas, which themselves are one short staircase flight up from the staffed entrance on Convent Avenue. I suggest using the staircase close to the (defunct) Science Cafe to get to Marshak 117.
 Both the Science Building and Shepard Hall are on the east side of Convent Avenue. The Science Building is the very rectangular box across from the hulking grey NAC, and Shepard Hall is the stately building which looks kind of like a castle. Both buildings have guarded entrances facing Convent Avenue.
 For Exam 1, there will be two options:
 Get the whole exam at the beginning of class, and complete it in one sitting.
 Get the first half of the exam at the beginning of class. After 50 minutes, turn that part in and then take a break of up to 5 minutes. Then get the second half of the exam and complete the second half in the remaining time.
If you expect to need to leave during the exam, you should opt for the twopart option. No student who arrives later than 15 minutes after the exam begins may choose the twopart option. No onepart exam student may leave until one hour has elapsed.
 Per CCNY guidelines, students must be masked while in the Science and Shepard Hall Buildings.
 For Exam 1, there are no notes, books, calculators, phones or smartwatches permitted during the exam. Your guide as to completing your answers is what we do in lecture: we want correct exact answers. For example (x ln(2)+15)/13 is in a fine format for full credit. Your belongings should be left in the front of the lecture hall. At your seat, you may have pencils, pens, erasers, and a photo ID. If needed, scratch paper can be provided but remember that to receive full credit, your work must be shown in full on the submitted exam.

Exam 1 will be held in room MR3 of Marshak Hall, the Science Building on the east side of Convent Ave.
 Some sample problems to help prepare for the upcoming exam can be found in old Math 202 final exams. Pretty much any integral or improper integral from the final exams on this Math 202 page are appropriate. For the exams that have posted solutions, make sure to give the problems a proper effort before consulting the solutions.
 Here is a site with some comments about choosing strategies for integration, together with more problems.
 The "Assorted Integrals" practice problems (69139) on p. 535 of the text are good practice for solving integration problems where the method is not specified.
 Some of the homework deadlines on the Pearson homework site have been moved forward as we are falling behind the pace to cover the syllabus and will need to speed up later in the term.
 As we begin Ch. 10 on sequences and series, topics will come from the old 203 lectures page, starting with lectures 29, 30 and 31.
 Topics for lecture 6 are from 8.8 principally. The relevant lecture from Dr. Marchese are Lectures 15 and 16, available at this page.
 Topics for lecture 5 were from 8.5 and 8.7 principally. The relevant lectures from Dr. Marchese is Lectures 13 and 14, available at this page.
 Topics for lecture 4 were from 8.4 principally. The relevant lecture from Dr. Marchese is Lecture 12, available at this page.
 Reminder that Exam 1 is Wednesday, Oct 6th during our normal class meeting time. Other dates on the Course Information sheet
 I'm working on resolving the technical difficulties in class. The notes are posted and deadlines extended for the applicable assignments.
 MyLab homeworks from Chapter 8 now have later deadlines. Check MyLab for the current deadlines.
 Quiz 1 (the secondeasiest quiz of the term) is now open on Blackboard, and Quiz 0 is reopened for those who missed the easiest quiz of the term.
 Class does not meet on Wednesday, Sept 15th so our next class meeting is Monday, Sept 20th. Thank you to students who corrected me on the upcoming holiday schedule.
 Reminder from our Dean of Science Susan Perkins about vaccination affecting enrollment:
Please tell your students that they must upload vaxx info on CUNYfirst by Monday Sept 27 or they will be dropped! I hope most students are vaxxed and only need to upload info but if not they must get J&J vaxx very soon. There is no undo button, no refunds, this can screw up financial aid and leave them with a big bill, there is no appeal for emergency funds. This is a CUNY policy.
 The "Product to sum" trig identies used in 8.3 on integration have some nice demonstrations available at Wikipedia on beat phenonema and at on Wikibooks
 Topics for lecture 3 were from 8.3 and we will be covering material from 8.4. Relevant lectures from Dr. Marchese include Lectures 11 and 12, available at this page
 Topics for lecture 2 were from 8.1 and 8.2, as mentioned in class. Relevant lectures from Dr. Marchese include Lesson 10: integration by parts with notes
 A table of basic integrals from the text is here
 For those interested in applications of the hyperbolic trig functions, see this page about the St. Louis Gateway arch, or the part of this page which is about the caternary curves used by Antonin Gaudi in the Sagrada Familia in Barcelona.
 PDF notes from lecture 1 are posted on Blackboard. A video lecture on hyperbolic functions is here with coordinated notes here.
 The video lessons from the old Math 202 course are here. Good places to start are Lesson 2 and Lesson 8 which correspond to where we are starting.
 There are some resources at the departmental Math 212 page including some review material which may be helpful.
 Please use the appropriate Blackboard discussion boards for general questions that may be of interest to other students as described in lecture.
 Office hours are in the Blackboard Classrooms, but there are two of them and I can only be present in one at a time. If I am not in yours during office hours, send email and I can switch when feasible.
 Well done to those who have gotten started on the MyLab assignments. If you are a student with the free trial "unpaid access" keep an eye on when the expiration date is on your trial and make sure to complete your Pearson purchase by then.
 First Mylab assignments not due until Sept 13th 11am, but there are many due that day so get started soon.
 Lecture notes posted on Blackboard under "Content"
 Class meets on Blackboard Collaborate.
 The CCNY final exam schedule does not yet have confirmed dates for Mathematics Department group finals but those will be confirmed in due course.
 Questions about administrative aspects of the course or specific mathematical topics are generally best posted in the appropriate Blackboard discussion board, as that way others can either answer them or benefit from responses.
 An introductory quiz "Quiz0" is available on Blackboard and should be completed by Sept. 7th.
 First lecture is Wednesday, Aug 25th.
 Instructions for enrolling in the Mylab course have been emailed to enrolled students. If you have already recently purchased the correct version of the text for CCNY Math 201 or elsewhere, you should not need to purchase it again and should be able to complete your Mylab enrollment already.
 A "Welcome" message has been emailed to all enrolled students. If you did not get such an email, make sure that your Blackboard and CUNY First email addresses are set to places that you check.
 We will be using the Pearson MyLab system for the text and online homework assignments. I will send instructions but if you already have purchased an electronic or looseleaf version of the Thomas' Calculus 14th edition Early Transcendentals text, for example for CCNY Math 201, you shouldn't need to purchase anything new. If not, CCNY has arranged for electronic or looseleaf text purchase both of which include electronic access to MyLab for the homework system. There will be instructions sent to enrolled students for those which are about $80 for the electronic version, or about $115 for the looseleaf paper copy, both including required MyLab homework system access.
 The video lessons from the old Math 202 course are here. Good places to start are Lesson 2 and Lesson 8 which correspond to about where we are starting.
 There are some resources at the departmental Math 212 page including some review material which may be helpful.
 First class meets Aug 25th.
 The prerequisite topics are those in Math 201 as taught at CCNY since Fall 2019 see this description of 201 topics.
If you are missing some of those topics from taking a course which is judged equivalent
(generally having about 80% of the topics) it is a good idea to fill those gaps before the term starts as it will be difficult to pass the class without mastery of
topics from 201 including
 Trig and exponential functions and their inverses
 Differentiation, optimization, max/min word problems
 Integration, Fundamental Theorem of Integral Calculus
 Substitution as an integration technique
 There are some helpful videos from Dr. Marchese at
this YouTube channel which cover material from old 201 and 202. These can be very helpful either to fill in gaps in preparation
(for example for those who took calculus elsewhere and may be missing some topics) or as a review or supplement.
 A sample final from Math 201 is available here.
Webpages for this course:
 Blackboard page requires login for registered students
 Pearson My Lab page requires registration via textbook
 Math 212 page at CCNY Math has course documents and sample final exams
 Math 202 at CCNY Math has course documents and resources covering topics from 202
 Math 203 at CCNY Math has course documents and resources covering topics from 203
 Homework Assignments summary page
Other Links