This week I’ll be in my office (probably!) around 11:30 or 12:00 noon; check in at N707.
Tomorrow (Thursday) I’ll be in my office N707 from 12:00 noon to about 12:30.
Author: Kate Poirier (Page 6 of 6)
Test #1
Test #1 will be given in class on Monday, March 2 and will cover material/homework from Sessions 1-7 (in pink on the schedule). The test will take the full class period and will include 10-15 questions similar to questions from the textbook, Webwork, and the final exam review sheet.
Test #1 Review Assignment – short version
To prepare for the test, you as a class will construct a crowd-sourced review sheet. Each of you is responsible for choosing one question and posting the question and its solution on the OpenLab. Try to pick questions that you think make good test questions (not too easy); try to pick topics that haven’t been posted by your classmates. Select the category Test #1 Review from the right-side of the screen before posting your question/solution. Your post is due on the OpenLab by Friday, February 28 at 11:59pm.
Test #1 Review Assignment – detailed instructions
Choose one question from
- the textbook homework on the course outline,
- Webwork, or
- the final exam review sheet
to complete and post your full solution on the OpenLab. Title your post “Test #1 Review” and add the section and problem number. Add the category Test #1 Review before submitting your post. You may type out the solution or upload a photo of hand-written work.
Make sure nobody has submitted your problem already; try to make sure every section that will be on Test #1 is represented.
It is up to you as a class to ensure that all solutions are correct. If you have a question about someone else’s post, if you would like more detail, or if you think the solution contains an error, post a comment asking the question or correcting the error.
The idea here is that you as a class are creating a review sheet for everyone to study from for the first test. You will be given participation credit for this post.
Due date: Friday, February 28 at 11:59pm
I wanted to return to the example we were considering in class today, before too much time passes and we forget where we were.
Recall, we were trying to evaluate the integral $\int \frac{\sqrt{x^2-9}}{x}dx$. To do so, we tried making a substitution where $x = 3 \sec(\theta)$. In order to make this substitution, we used $x = 3 \sec(\theta)$ to create a dictionary that would take us from an integral in terms of $x$ to an integral in terms of $\theta$.
We saw that if $x = 3 \sec(\theta)$, then (after a little work) $\sqrt{x^2-9} = 3 \tan(\theta)$ and $dx = 3 \sec(\theta) \tan(\theta)d \theta.$
This means that $\int \frac{\sqrt{x^2-9}}{x}dx = \int \frac{3 \tan(\theta)}{3\sec(\theta)}3\sec(\theta)\tan(\theta)d\theta.$
After simplifying, this is equal to $3\int \tan^2(\theta)d\theta$.
This is how far we got in class. The point is that this is an integral we know how to evaluate.
To continue, $3\int \tan^2(\theta)\d\theta = 3 \int(\sec^2(\theta) – 1)d\theta = 3(\tan(\theta) – \theta) + C.$
To finish, we have to go back to our dictionary to write our answer in terms of $x$ instead of $\theta$. Remember, $x = 3 \sec(\theta)$, which means that $\frac{x}{3} = \sec(\theta)$. We have a $\theta$ to replace AND a $\tan(\theta)$ to replace.
To replace the $\tan(\theta)$ we don’t have to get $\theta$ by itself. We use $\frac{x}{3} = \sec(\theta)$ to label the sides of a right triangle. Draw a right triangle and label one of the acute angles by $\theta$. Since $\sec(\theta)$ is defined as the length of the hypotenuse over the length of the adjacent side, we can label the hypotenuse by $x$ and the side adjacent to $\theta$ by $3$. We can use the Pythagorean theorem to find the length of the side opposite to $\theta$: it’s $\sqrt{x^2-3}$. Then the $\tan(\theta)$ term in our answer is opposite over adjacent, which is $\frac{\sqrt{x^2-3}}{3}$.
To replace $\theta$ we do have to get $\theta$ by itself. We apply $\sec^{-1}$ to both sides of the equation $\frac{x}{3} = \sec(\theta)$ to get $\sec^{-1}(\frac{x}{3}) = \theta$.
This means our final answer is:
$\int \frac{\sqrt{x^2-9}}{x}dx = 3(\tan(\theta) – \theta) + C = 3(\frac{\sqrt{x^2-3}}{3} –Â \sec^{-1}(\frac{x}{3})) + C
= \sqrt{x^2-3} – 3\sec^{-1}(\frac{x}{3}) + C$.
A few people asked during class, “Where does $x = 3 \sec(\theta)$ come from?” That’s a tricky question to answer, because it doesn’t really *come from* anywhere. But we see from this example, that by making this substitution, we’re able to evaluate the integral. In some sense it *comes from* the Pythagorean identity $\tan^2(\theta) = \sec^2(\theta) – 1$.
In general, to make a trigonometric substitution:
- if you see $x^2 – a^2$ let $x=a\sec(\theta)$,
- if you see $a^2 – x^2$ let $x=a\sin(\theta)$,
- if you see $x^2 + a^2$ let $x=a\tan(\theta)$.
This should be enough for you to get a solid start on the Trigonometric Substitution Webwork set, which is due on March 1.
Webwork
- Integration by Parts due tomorrow
- Trigonometric Integrals due Sunday
- Trigonometric Substitution due March 1
Class Schedule
- No class this Wednesday
- No class next Monday
Quiz #2
- Quiz #2 will be given in class at the beginning of class on Wednesday, February 19. It will cover material from Sessions 3, 4, and 5.
Trigonometric Substitution
- After next Wednesday’s quiz, we’ll start the next topic: trigonometric substitution. You’ll want to make sure that you remember your trig identities and derivatives to prepare for this topic.
Test #1
- Trigonometric substitution is the last topic that will appear on Test #1, which will be given in class on Monday, March 3. This might sound far away, but you’ll want to start preparing now. You won’t be told which integration technique to use, like you are in Webwork; you’ll have to decide on the best technique yourself (there may be more than one appropriate technique to choose for a given problem). Make sure to practice exercises where you are not told which technique to use. See the final exam review sheet and the end-of-chapter review exercises in the textbook.
Test #1 Review – OpenLab Assignment
Stay tuned for your next OpenLab assignment (spoiler: you’ll choose and solve one review question for Test #1 and post it to the OpenLab… details TBA).
I mentioned during office hours today that I work closely with the director of the Atrium Learning Center and with the math tutors who work there. One of the things I do each semester is create a flyer to encourage students to attend tutoring. For the past few semesters, I’ve used this meme combo:
These memes are getting a little old, so I was thinking of changing them up this semester. I’ve come up with a few ideas of my own, but I thought it’d be fun to ask you to come up with your own too. This assignment is optional and purely for fun but anyone who creates their own meme will earn one participation point (you can enter as many times as you want, but you’ll still get only one point for entering). If one of your memes is ultimately chosen for the official poster, you’ll be declared the winner and will earn two extra participation points. Continue reading
- A new Webwork feature has just been enabled. When you are logged in and looking at one of your questions, you’ll see a button at the bottom labelled “Ask for Help.” Clicking on this button will take you to an OpenLab page which is a question-answer forum for that question specifically (each student has the same template for each question; the numbers may be different but the procedures for answering the questions will be almost the same). If there’s a Webwork question that’s troubling you, please submit the details on the “Ask for Help” page and/or scroll down to check if your question has been submitted previously by someone else. You’ll get participation credit for asking your own question or answering someone else’s on this page.
- Tutoring is available in different places around campus. The schedule for the Atrium Learning Center is available here: PERKINS MATH TUTOR SCHEDULE FOR SPRING 2020. You’ll get participation credit for attending tutoring; ask the tutor for a verification slip and give it to me.
The Webwork set “Substitution Method” is due next Sunday night. Many of the exercises are similar to the example we saw in class today. We’ll see another example or two after Thursday’s quiz, before we start the next section: Integration by Parts.
The following sets are due next Tuesday at 11:59pm:
- Definite Integrals
- FTC Part I
- FTC Part II
Remember to use the definition of a definite integral from today’s class and not the one from the text (which uses a section of the text we haven’t covered yet).
Remember that the text and Webwork disagree about which is the first and which is the second fundamental theorem. We used the text’s ordering in today’s class.
The exercises in the FTC Part I set are straightforward applications of the (second) fundamental theorem. We’ll look at an example in class on Monday, but you should get started on this set before then.
Most of the exercises in the FTC Part II set are straightforward applications of the (first) fundamental theorem and similar to the example from today’s class. On Monday, we’ll look at a less straightforward example (like problem #10) in class, so you should make sure you’re familiar with the straightforward ones at the beginning of the set.
Welcome to your MAT 1575 OpenLab site! All course information is posted here. Please take a look around.
OpenLab Homework #1
Your first assignment is to add yourself to our OpenLab course and to submit a post according to the instructions below. Your post is due by 8 am on Monday, February 3. Your post will count toward your participation grade.
- Sign up for an OpenLab account using your CityTech email address.
- After you’ve logged in, add yourself to our MAT 1575 class.
Submit your post:
- From the class OpenLab site you can create a new post by clicking the symbol that looks like a + sign inside a circle at the top of the screen.
- Title your post “Introducing [your first name] [your last name].” (You may use just initials if you like, as long as there’s enough information for me to locate you on my official roster.)
- From the categories on the right-hand side of the screen, make sure to select “Introduce yourself” so that the rest of us can locate your post later.
- Tell us anything you’d like us to know about you! Write one paragraph with at least five sentences. Some suggestions are…
- Your pronouns (examples: she/her/hers or he/him/his or they/them/theirs)
- Your major
- How you became interested in your major
- Which math classes you’ll take after MAT 1575 (if you don’t know, you can check your program requirements in the college catalog here)
- What you hope to do after you graduate from CityTech; short-term goals, long-term goals, whatever…
- Other interests/hobbies you have
- Something you’re good at
- Something you’re proud of
- A time when you overcame something difficult
- Your favorite book
- Your favorite animal
- Read and comment on at least one your classmates’ posts. (Note: the administrator will have to approve your comments, so don’t worry if they don’t appear right away.)
- If you have anything private that you’d like to share with me but not with the whole class (for example, if you are to receive a testing accommodation through Student Support Services), please email me directly.
Webwork #1:
Your first Webwork assignment consists of two short sets. Answers are due by Sunday, February 2, 2020 at 11:59pm
- Review-ChainRule
- Review-PowerRule
- Review-ProductRule
- Review-QuotientRule
- Antiderivatives
To log into Webwork, follow this link.
- Your username is your first initial +Â your last name. For example, if your name is Barack Obama, your login name is bobama. (If your last name has a space or a hyphen, delete it.) All letters are lower case. Both username and password are case sensitive.
- Your password is your CUNY EMPLID number.
- If you can’t log in, let me know after class.
Click on the homework set’s name to view the problems. Click on individual problems and enter your answer. After you have entered an answer, the system will tell you if it is correct. Make sure you have entered all your answers before the deadline. There is nothing else for you to do after that.
Here are some tips:Â WeBWorK Guide for Students.
Quiz #1
Your first quiz will be given at the beginning of class on Wednesday, February 5. It will be based on material/exercises from Sessions 1 and 2. More details will be announced in class.
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