Category: Student Work (Page 1 of 30)

Calculus 2 (MAT1575) has been a wild journey for me, especially with having an partially asynchronous format. This semester, we have been learning most of the material on our own, while taking an advantage to explore many different options. At first, I had an very hard start as I am disorientated with the new structure, but I was able to finish this class with a blast. The last thing I am going to do before I leave this class is to answer my own prompt from Week 16.

Prompt #1

Integration – I only count integration techniques on this list.

• Easiest (1/10) – Improper Integrals – I know that improper integrals can be hard for some students due to limits. However, I had a blast doing this topic, because it is a fun exploration when integrals go towards infinity or through asymptotes. While evaluating improper integrals, I like to imagine that infinity would actually be a number. In these WeBWorK sets, each problem took me less than one minute to do them (if they were divergent)
• (2.5/10) – U-substitution – Objectively, u-substitution is the easiest integration technique to apply. However, I had a hard time applying this particular technique, because I didn’t knew what to do if the u and du didn’t perfectly fit the forms. After Week 5, I was able to apply this technique almost seamlessly. To ensure my understanding of u-substitution, I tried a problem with u-substitution and posted the step-by-step solution on Week 6.
• (4/10) – Integration by Parts – I was able to partially understand this concept from the start. However, I made a conceptual error in Test #1, in which I mixed up my u and dv, resulting in far more complicated answers. After I realized my mistakes, I have used the mnemonic (LIATE) to help remember the order in which I used my u and dv.
• (5/10) – Partial Fraction Decomposition – I didn’t know how to do partial fraction decomposition, the easy way… until Week 15! Sure, I have used the methods from Mathispower4u to solve these problems, but it is not exactly the easiest way. With his way, I can only do these problems involving A and B, but not more than that. However, with the easy way (substituting values), I am able to do more complicated problems, and maybe the ones using long division.
• Hardest (8.5/10) – Trigonometric Substitution – What I can say about trigonometric substitution?! This is one of the hardest integration techniques to apply. I mean, we have a rational function, but it is replaced with sin(), cos(), tan(), and my head keeps on spinning! And, I have to know all these trigonometric integrals and identities in order to do these seamlessly! Without any knowledge of trigonometric integrals and identities, you will definitely get stuck! However, I’m so genuinely blessed that I able to follow specific examples used on class (since their randomization is so limited) to obtain the procedure. Otherwise, I will get stumped.

Series– I don’t count sequences and infinite series on this list.

• Easiest (1/10) – Divergence Test – One of the easiest tests to apply. It involves p-values, and that’s it… It’s simple as that.
• (1.5/10) – Comparison Test – It is also easy to apply the Comparison Test. When I use this test, I remove other non-involved terms, and keep the ones with the highest degree (or growth) in the numerator and denominator. Note this test works best when it only involves powers.
• (3/10) – Taylor/Maclaurin Polynomials – The topic is quite the breeze, because I just need to repeatedly take derivatives to certain degrees. The easiest I know is taking the x degree of Maclaurin polynomial of e^x. Yes, there are hard problems, but in this class, it is focused on the easy ones.
• (3.5/10) – Integral Test – This test is quite intuitive to apply. Since I like to integrate, I like to use this test most of the time to determine convergence. Yes, I know that the integral test may cause problems at times, but I know when to use integral test or not.
• (4.5/10) – Ratio/Root Test – This test is considered “not bad”. It involves multiplying, dividing, and the cancellation of terms. I know that I am not able to cancel out all of the terms like (n+1)², and there are several problems where I get stumped using the Ratio/Root Test, but I do understand the overall concept.
• (5.5/10) – Alternating Series Test – This test gives us clear conditions to whether it converges. It may quite be easy to tell by just testing these conditions, but I would likely have to use another test to prove absolute convergence, conditional convergence, or divergence. Applying this test by itself isn’t bad at all, but when I have to apply another test, it bumps up the difficulty slightly.
• Hardest (9/10) – Power Series – Not only I have to apply different tests, but also I have to find the radius of convergence!!! This has to be one of the most complicated series topics we have learned. Enough said!

Applications of Integration – I don’t count Riemann sums here.

• Easiest (2.5/10) – Areas between Curves – With the exception of having to set up integrals and having the correct bounds, it is just solving a definite integral and using the systems of equations. To make this topic easier, I used my handy-dandy tool, the TI-84.
• Hardest (6/10) – Volumes – While it may be the hardest topic for many students, I find this topic not-so-difficult. Yes, I get that there is considerably more workload than Areas between Curves, it is still just definite integrals… no problems! However, the thing is that I may be stumped with some problems, but understood the general procedure for finding the volume.

Prof. Poirier, if you are reading this, again, it has been a pleasure having you as my Calc 2 professor. I will definitely miss you… so much

(And I may update this blog post as necessary)

12/14/2020 at 5:16pm-6:49pm

Dear my past self,

Anthony, it has been almost 4 months since I have learned tons of calculus, focusing on integrals and series. My Calculus 2 journey has been rough, especially at the beginning. You know, when I began Calculus 2, I was disorientated for the first few weeks. I thought everything was new especially the format of the homework, how much participation credit I needed, and the heavy truckload of (450?!) questions I was assigned for homework. Especially with the integration part of the class, I felt like these integration techniques were forced upon me! Due to so much work I have to do, I felt confused about the new integration techniques and didn’t make progress like I hoped to. I didn’t even begin my official HOT Topic presentations until the start of Week 3, where I presented integration by parts, and I managed to get my first H there.

Starting Week 3, the WeBWorK workload was slightly more manageable, but still tedious. Especially when working with trigonometric substitution, I was very confused about this topic especially I had little to no background knowledge about trigonometric integrals and identities. However, I took the advantage to watch videos on the course hub to understand the procedure for certain problem formats. Because of these videos, I was able to get my second H, and solve the group trigonometric substitution problem presented in Week 4.

In Week 4, I am relieved by partial fraction decomposition, but not too much. Partial fraction decomposition is an algebraic technique, but it can be complex at times. I was able to do simple problems involving splitting fractions into A and B, however, I remain confused when there are more complex problems involving this topic. But again, I was also to successfully get my third H.

In Week 5, I am starting to jump for joy as we are learning improper integrals. Since we are learning about improper integrals, they somewhat remind me of basic u-substitution problems. U-substitution was an easy topic, but couldn’t understand it until we learned improper integrals. But, because this topic was easy, I have blasted through most of the problems in this topic. This is also the first time that my HOT topics are up to par.

After Week 5, I have learned to manage my time and learn these topics early enough. The Series topics were harder, but I was able to progressively understand it over time. Even though that we have to learn a handful of convergent tests, it is going to be manageable as long as we look carefully at the problem and read the instructions. (By the way, I love the topic sequences, because they remind me of Frames-and-Arrows problems back in 1^st grade).

I am now going to give you advice:

1) Manage your time to do the assignments in this class. For instance, if you usually squeeze some time for Friday to do the WeBWorK sets and during the class time to watch videos and do assignments, try to keep that regular schedule unless plans change. Avoid doing homework spontaneously, because chances are, you may somehow fall behind. I also have been hearing some students doing homework in the last minute, so I ask you not to do it.

2) Watch videos (and take notes) before doing any math work yourself. As you watch the videos, it is important that you take notes of the entire procedure of math problems. If you don’t understand anything from the video, it is best to ask questions during your COLD session. I know that during in-person classes a while ago, videos do help, but they aren’t a replacement for attending class and taking notes. However, now with online class, it is important that you watch videos in order to understand the material (especially with asynchronous classes!).

3) Have your integration and trig identity tables ready. I have seen some integration tables appear on the Web. These tables are very helpful, but you must choose the appropriate table(s) you will use during your time on Calculus 2. One thing I have found integration tables helpful is when I have to find the integrals resulting in inverse trigonometric functions. I believe that trigonometric substitution is one of the hardest integration techniques to apply, but the use of trig identity tables will make this topic more bearable.

4) I will admit that fact that study groups in this class didn’t work like the way I wanted. Therefore, I rarely join study group sessions. However, I will say this: Try to do weekly study groups. I really want you to join study groups for few reasons. You may think that you can do all the studying alone, however, study groups allow you to check each other’s progress, teach others what you know, and to ask other questions. To some degree, study groups may enhance your social skills.

5) Since you will be taking this class for the first time, make sure you orientate yourself and familiarize yourself will all of the topic you will learn this semester. Practice the WeBWorK problems before trying to solve HOT Topic problems. These HOT Topic problems will demonstrate your understanding of that particular topic. Make sure you prepare your HOT Topic solution ahead of time and write down the numbered steps to help you walk through your solution. If your ever get an O in your HOT Topic presentation, don’t panic! Analyze what you have done wrong in your solution and fix them. You can always try again. And, I know that you can get 10 H’s this semester. You can do it, Anthony!

Have a great semester! One more thing, don’t cause a time paradox!

Best,

My present self (Anthony John Regner)