Final Exam Study Guide

The final exam is this Wednesday, May 20, at the usual class time. Below is an outline of topics to study, and here are some tips to how to prepare:

  • Start by working through the corresponding exercises on the Final Exam Review sheet (handed out in class, also available here). Many of these were assigned as homework exercises, so you should have already worked through them.
  • Also review the corresponding questions┬áon the midterm exams. I have uploaded┬ásolutions to the exams in the Files section of the course profile.
  • Of course you can also study the┬áexamples and homework exercises from the corresponding sections in the textbook.

Here is a list of topics, according to the questions in the Final Exam Review sheet:

  • Final Exam Review #1: Integrating by substitution (see also Exam #1 & Sec 5.6)
  • Final Exam Review #2: Integration by parts (see also Exam #1 & Sec 7.1)
  • Final Exam Review #3: Using integrals to find the area between two curves (see also Exam #2 & Sec 6.1)
  • Final Exam Review #4: Using integrals to find the volumes of solids of revolution (see also Exam #2 & Sec 6.3)
  • Final Exam Review #5: Integrals using trigonometric Substitutions (see also Exam #2 & Sec 7.3)
  • Final Exam Review #6: Integrals using the Method of Partial Fractions (see also Exam #2 & Sec 7.5)
  • Final Exam Review #7: Improper Integrals (see also Exam #2 & Sec 7.6)
  • Final Exam Review #8 & #9: Infinite Series (Exam #3, Sections 10/2-10.5)
  • Final Exam Review #10: Power Series (Sec 10.6 – Examples 1 & 2)

Summary of Tests for Infinite Series

I wrote up a summary of the various tests for convergence/divergence of infinite series that we discussed from Sections 10.2 and 10.3 of the text. You can use this as a reference as you do exercises, but also look at the examples in the textbook and the examples we did in class for each test:
https://openlab.citytech.cuny.edu/?get_group_doc=15883/1430748441-Math1575-InfiniteSeriesTests.pdf

UPDATE: I have extended the document to include the Alternating Series Test (Section 10.4) and the Ratio Test (Section 10.5). You can download the new version of the document here:

https://openlab.citytech.cuny.edu/?get_group_doc=15883/1431359547-Math1575-InfiniteSeriesTests-Updated.pdf

 

The trickiest of these is probably the limit comparison tests. It may be helpful to look at some video examples:

Exam #2: Review Exercises + Example Videos

As we’ve been discussing in class, our 2nd exam will be this Wednesday, April 15. Below┬áis the outline of topics that will be covered on the exam. The┬áHW#8┬áexercises cover these topics:

  • From the Final Exam Review sheet (handed out in class, also available here):
    • #5 (Sec 7.3: Trigonometric Substitutions)
    • #6 (Sec 7.5, Method of Partial Fractions)
    • #7 (Sec 7.6, Improper Integrals)
  • Sec 6.1, Integrals for the area between two curves: (pp361-362): #1, 3, 4, 27
  • Sec 6.3, Volumes of Revolution (p381): #1, 3, 5, 6

Trigonometric Substitutions:

Do #5(a), (b), (c), and (d) on the Final Exam Review Sheet. Take a look at the outline of how to approach these problems I posted in the discussion forum.

Actually we did #5(a) in class today. If you want to review it again (or missed class today), here is a video I did of a very similar example:

If you want to review this topic further, look at Examples 1, 2 & 4 in Section 7.3. You could also review #53 from Sec 7.5, which was assigned as a HW exercise.

 

Method of Partial Fractions

There are three types of partial fractions integrands we discussed, depending on the factors that appear in the denominator:

  1. “Distinct linear factors”: This is the most basic type, where the partial fraction decomposition has a constant coefficient for each distinct linear factor. For example, in Example 2 in Section 7.5, the partial fraction decomposition of the integrand is of the form:\frac{x^2 + 2}{(x-1)(2x-8)(x+2)} = \frac{A}{x-1} + \frac{B}{2x-8} + \frac{C}{x+2}You should be familiar with how to solve for A, B and C in an example like this. If you need to review, look at Examples 1 and 2 in Section 7.5.#6(b) and #6(c) on the Final Exam Review Sheet are integrands of this type (after you factor the given denominators!)
  2. “Repeated linear factors”: Here the partial fraction decomposition must include terms with constant coefficients for each power of the repeated linear factor.For example, consider #17 from Section 7.5, where the integrand is\frac{x^2 + 11x}{(x-1)(x+1)^2}.Since the (x+1) factor is repeated twice, the partial fraction decomposition must be of the form:\frac{x^2 + 11x}{(x-1)(x+1)^2} = = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{(x+1)^2}I worked out the rest of this exercises here:

    If you need to review further, look at Examples 4 and 5(c) in Section 7.5.

  3. “Irreducible quadratic facors”: If the denominator contains an irreducible quadratic factor, i.e., a quadratic term (such as x^2 + 9) that can’t be factored into linear factors. In that case we include a term in the partial fraction decomposition for such a quadratic factor, but the numerator isn’t just a constant but must be a linear term.This is the case with #6(a) on the Final Exam Review. Review in the example we did in class today, and also read Example 5(a) in Sec 7.5 carefully. You can also view this video by patrickJMT (who has a bunch of integration examples that you may find helpful!)

Improper Integrals

Do the three improper integrals in #7 of the Final Exam Review sheet. For each, first rewrite the integrals as a limit, and then evaluate the definite integral inside the limit–for each of these three, you can find the antiderivative by making a simple u-substitution. ┬á Finally evaluate the limit

Check back here later, I’ll post an example in case you need some guidance with these.

Videos: Integration by Parts, Trigonometric Integrals, Trig Substitutions

In order to learn the various methods of integration, you need to you read, see, and work through as many examples as you can. That’s why I’ve urged you to take notes in class, read the examples in the textbook, try to do all the homework exercises, and ask questions about any examples and exercises you are confused about.

Here are more examples for the various methods of integration we’ve covered in the past few weeks:

Integration by Parts

Khan Academy’s IBP sequence

PatrickJMT’s first IBP video (see the list of all his calculus videos here!):

 

Trigonometric Integrals

Trigonometric Substitutions

 

 

 

 

Visualizing Riemann Sums: Visual Calculus, Wolfram MathWorld, Khan Academy

We won’t spend much time in this course studying Riemann sums and the formal definition of the definite integral, but I encourage you to read the relevant parts of the book (Sec 5.1 and the first part of Sec 5.2) and try to understand the concept.

You can play around with the “Visual Calculus” visualization of Riemann sums I showed in class on Monday. There are a number of such visualizations on the web; for example, take a look at Wolfram MathWorld’s Riemann sum page.

Khan Academy has a sequence of 10 short videos on Riemann sums which you may find helpful. Here is the first one:

Khan Academy’s Integral Calculus Videos: “Antiderivatives and indefinite integrals”

As I mentioned during our first class meeting yesterday, there are numerous resources that you can take advantage of to help you learn the material. Of course there are on-campus resources such as my office hours and the tutoring centers (more information on that will be provided soon!).┬á But there are also numerous online resources–a popular one is Khan Academy.

Here is the link to their Integral Calculus sequence of videos. Try taking a look at the first one, which starts with the same topic we started with yesterday:
Antiderivatives and indefinite integrals

Khan Academy’s videos are also on YouTube, which may be useful in case you want to watch on your phone or on a tablet:

 

Let me know if you find these videos useful (you can always leave a comment below). And of course let me know if you have any questions about the math!