Instructor: Suman Ganguli | Fall 2023

Month: December 2023 (Page 1 of 3)

Class 29 Recap (Mon Dec 18) – Final Exam review

Class Info

  • Date: Mon Dec 18
  • Meeting Info: 10a-11:40a, N719

Announcements

  • The final exam is in-class on Wednesday Dec 20 – see below for a list of topics/exercises to review
  • I will be in my office (N724) most of the day Tuesday grading your Exam #3’s. You can drop by if you want to pick up your exam, and/or go over any topics/examples for Wednesday’s final.  I plan to be there 10a-3p.
  • I may also have some office hours Tuesday evening over Blackboard Collaborate (time TBA)
  • I reopened a few WebWork sets, as people were asking me for a chance to complete exercises they didn’t get to earlier in the semester–I reopened sets that may be useful as review for the final exam: “Substitution”, “Area Between Curves”, “Volumes of Revolution”

Topics

I outlined the topics/exercises from the Final Exam Review sheet you should review for the final:

See below for a text version of this outline, which includes a list of relevant midterm and quiz exercises to review for each topic.

We went through examples of integration by parts and improper integrals:

We also introduced power series, for which we use the Ratio Test to find “the interval of convergence” (the x-values for which the given series converges). There will be an exercise on power series on the final exam, but I will count it as extra credit.

Final Exam Review:

Here is the annotated list of topics for the final, listed according to the exercises on the Final Exam Review sheet:

#1(a)-(c): Integration by substitution (see also Exam #1, exercise #4; Exam #2, exercise #1(a); Quiz #2)

#2(a)-(c): Integration by parts (Exam #1, exercise #5, Exam #2, exercise #1(b))

#3(a)(b): Area between two curves (Exam #2, exercise #3)

#4(a)(b): Volume of solids of revolution (Exam #2, exercises #4 & #5)

Skip #5 (Trig substitutions)

#6(a)-(c): Method of partial fractions (Exam #2, exercise #2; Quiz #3)

#7(a)(b): Improper integrals (see examples and video in Class 20 https://openlab.citytech.cuny.edu/mat1575-ganguli-fall2023/2023/11/10/class-20-recap-wed-nov-8/)

#8: Infinite series with positive terms (Exam #3, exercises #1-5)

#9: Alternating series (Exam #3, exercise #6)

#10: Power series (example from today’s class — see above)

Skip #10-11 (Taylor polynomials)

Take-home exam/office hours recording

I recorded office hours yesterday morning with a handful of people from class who logged on. We went through a number of examples from the textbook, the WebWork, and the Final Exam Review which are relevant for the take-home exam exercises. You can view the recording on Blackboard (toggle on the menu to “Recordings”):

https://bbhosted.cuny.edu/webapps/collab-ultra/tool/collabultra?course_id=_2317409_1&mode=cpview

I also scanned my notes from the office hours as a pdf and uploaded it to OpenLab Files.

I opened a WebWork set on “Alternating Series”, which consists of a handful of multiple-choice exercises. These exercises should help you think through how to approach alternating series (#6 on the take-home exam). I went through a number of those WebWork exercises during office hours.

Here is a summary of some relevant examples for each of the take-home exam exercises:

#1 (geometric series): See Final Exam Review #8(b). You can also see the examples of this Class Recap (including a screenshot from Sec 5.2 of the textbook, where geometric series are discussed)

#2 (divergence test): We went through examples in Class 25; I went through Example 5.13(a) from the textbook (Sec 5.3) in office hours; see also FER #8(a)

#3 (p-series): Just apply the p-series test, which we discussed in Class 26; see Example 5.15 (also in Sec 5.3)

#4 (limit-comparison test): See the examples in Class 26 and 27, the example in my “Convergence/Divergence Tests” pdf, and the examples from the WebWork “Comparison Tests”

#5 (ratio test): See the examples we did in Class 27 and 28 (which are also written out in the “Convergence/Divergence Tests” pdf), and FER #8(c)(d) — we went through #8(c) in office hours.

#6 (alternating series): See the explanation and examples in Class 28 and the “Convergence/Divergence Tests” pdf, the exercises in FER #9, and the office hours recording.

Class 28 Recap (Mon Dec 11)

Class Info

  • Date: Mon Dec 11
  • Meeting Info: 10a-11:40a, N719

Announcements

  • The take-home exercises for Exam #3 were distributed in class, which will be due next Monday (Dec 18)
    • the pdf of the take-home exam exercises are available on OpenLab Files, in case you were absent yesterday or need to print out another copy
    • you can also find the pdf of the “Convergence/Divergence Tests” document I wrote up–that should be helpful for completing the take-home exercises
  • We do not meet tomorrow (Wed Dec 13 – reading day) — I will have office hours during class-time over Blackboard Collaborate instead
  • There will also be a short in-class component for Exam #3 which we will take next Monday Dec 18 (the take-home exercises will serve as preparation for the in-class exam); we will also review for the final exam that day
  • The final exam is in-class on Wednesday Dec 20

Topics

We did a couple more examples of the Ratio Test. We analyzed a series which converges, because the ratio “rho” is turns out to be 1/2 (recall that a series converges by the Ratio Test if rho < 1):

And another series which diverges, because rho turns out to be infinity:

See also #8(c) and (d) from the Final Exam Review for applications of the Ratio Test, including another one involved n! (“n factorial”):

We then discussed alternating series, starting with the definitions of absolute vs conditional convergence. We then went through couple introductory examples, and then discussed the alternating series in #9 from the Final Exam Review.

(Alternating series are discussed in Sec 5.5 of the OpenStax textbook.)

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