MAT1575: Calculus 2

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)

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.)