Semester Grades have been submitted to CUNYFirst (they should be available shortly). You can find more detail, including your score in each of the grading categories (WeBWorK, OpenLab, Exams, Quizzes, Final Exam, Overall) on the OpenLab GradeBook.

REMEMBER: Grading this semester is unusual, in the sense that you can CHOOSE whether to receive a letter grade, or convert it to CR/NC. More details can be found in my previous post on the subject, here:

It looks like the link for uploading your final exam written work stopped accepting uploads early this morning – that was a mistake! The exam is still going on, until midnight tonight. The link should be working again now. Full instructions can be found in the ***Final Exam*** post:

Good luck! Feel free to leave a comment here or send me an email if you have questions…

Hi everyone! Today is the last official class day before the final exam. I’ll be holding a final exam review session live on Zoom during our normal class time, 10:00 – 11:40am.

I’ll be using the time to answer your questions – either general questions, or questions from the review sheet, or questions from the practice final exams in WeBWorK. Please bring your questions!

Having trouble joining the meeting? Please comment below, I’ll do my best to check back regularly.

Details about your final exam appear below. Please READ THEM ALL CAREFULLY, and don’t hesitate to leave me a comment or send me an email if you have any questions or run into problems. This will be the only OpenLab post about the final exam – hopefully it contains all the information you need to take the final (starting tomorrow *very* early, Wed 5/20 at 12:00am).

You will take the exam using your “Exam WeBWorK account.” The Exam WeBWorK account is the same account you used for the Practice Final Exam, it is NOT your usual WeBWorK homework account. You should have received an email sometime on or around Friday, 5/8/20, with the title Practice Exams and Final Exam Login Information from the address WeBWorK Administrator <webwork@citytech.cuny.edu>. This email contains your exam WeBWorK account login and password.

You will have a 48-hour window to start the exam, all day Wednesday and Thursday (from 12:00am Wednesday 5/20 through 11:59pm Thursday 5/21), and a 2-hour window to complete the exam once you start it. You can start the exam whenever you like during this period.

You will have 2 hours to complete the exam from the time you start it (WeBWorK will keep track of this for you – you must submit your answers within two hours of starting).

You can complete the exam TWICE if you wish, once on Wednesday and again 24 hours later on Thursday (a new version will be generated for you 24 hours after the first). The exam with the best score will be graded.

Please choose a time to complete the exam that provides you with the best opportunity to work uninterrupted.

You must complete each of the problems on paper and upload them to DropBox within 30 minutes of completing the exam (see Step 2).

When you complete the exam, enter your answers and submit them in WeBWorK.

Step 2: Upload your written work as a single pdf file to the link provided.

You must submit your written work within 30 minutes of completing the exam.

Your exam grade will be primarily based on your written work – so this step is essential!

Please take clear photographs of your written work (one photo per page).

You MUST combine all the photos into a single pdf file. If you need help with this, see the box “Tips for combining photos on various devices” below.

Step 3: You’re done – great work! If you had any trouble carrying out the instructions above, please let me know SOONER rather than later (email is a great way: jreitz@citytech.cuny.edu).

picture_as_pdf

Tips for combining photos in a pdf on various devices

Note 1: Completing the exam will also mark your attendance for the day.

Note 2: Grades are due May 28th, although I hope to get them in sooner. When I post grades on CUNYFirst, I will also post more detailed grades on the OpenLab (so you can see you final score on the exam) – I will also post an announcement on the OpenLab to let you know when grades are available.

Note 3: You all have faced a tough road this semester! I regret we couldn’t spend more time face-to-face, but I’m glad I got to know you as much as I did. I’m proud of you.

Here is today’s Daily Quiz, based on the material covered on Tuesday. As you know, quizzes will count towards your grade, and they will also be used to track your attendance (to be marked PRESENT, you must submit the quiz before midnight).

Final Exam date

Our final exam will take place next week, on Wednesday 5/20 and Thursday 5/21 (the exam will open at 12am on 5/20 and close at 11:59pm on 5/21.

Exam instructions, including a Dropbox link for uploading your work, will be posted here on the OpenLab on Tuesday 5/19.

Final Exam Review Session

Next Tuesday 5/19 I will hold a live final exam review session during our usual class hours 10am-11:40am. Details (including a zoom link) will be posted here one hour before the session, at 9am on Tuesday. I will focus on answering questions from the review sheet or WeBWorK practice exams. I hope to see you there!

I will NOT post a new lecture today – instead, please take this time to review for the final exam. Take a practice exam today (and every day, until the exam!). If you do not yet have access to the WeBWorK Final Exam area, please contact me ASAP. I wish you the best in your studies!

As always, let me know if you have any questions or problems.

Be well, Prof. Reitz

Daily Quiz: 5/14/20

This quiz MUST be taken on 5/14 to record your attendance.

As you may remember, the original syllabus listed 3 in-class exams in addition to the final. Due to coronavirus disruptions, we have held 2 in-class exams and a number of “daily quizzes” – the updated grading policy appears below, and has been reflected on the syllabus:

UPDATE TO EXAM GRADING POLICY: Due to class disruptions caused by coronavirus, the In-Class Exams (45%) portion of your grade for the semester will be computed as follows:

In-Class Exams & Quizzes (45%): There will be two in-class exams during the semester, worth a total of 35% of your grade, and a number of “Daily Quizzes”, worth a total of 10% of your grade.

The original policy appears here, for your reference:

In-Class Exams (45%): There will be 3 exams during the semester (not including the final). No makeup exams will be given. If you miss an exam for a valid reason, your final exam score will take the place of the missing exam.

If you have any questions about this change, please let me know.

Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson 24: The geometric series

Lesson Date: Tuesday, May 12th.

Topic: This lesson covers Chapter 24: The geometric series

WeBWorK: There are two WeBWorK assignments on today’s material, due in one week:

Sequences – Geometric

Series – Geometric

Question of the day: Can we add up infinitely many numbers?

Lesson NOtes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Today we look at a new kind of sequence, called a geometric sequence, and the corresponding series, geometric series.

A geometric sequence is a sequence for which we multiply by a constant number to get from one term to the next, for example:

Definition 24.1. A sequence $\left\{a_{n}\right\}$ is called a geometric sequence, if any two consecutive terms have a common ratio $r$. The geometric sequence is determined by $r$ and the first value $a_{1}$. This can be written recursively as: $$a_{n}=a_{n-1} \cdot r \quad \text { for } n \geq 2$$

Alternatively, we have the general formula for the $n$ th term of the geometric sequence: $$a_{n}=a_{1} \cdot r^{n-1}$$

Example 24.2. Determine if the sequence is a geometric or arithmetic sequence, or neither or both. If it is a geometric or arithmetic sequence, then find the general formula. a) $3,6,12,24,48, \dots$ b) $100,50,25,12.5, \ldots$ c) $700,-70,7,-0.7,0.07, \ldots$ d) $2,4,16,256, \dots$ e) $3,10,17,24, \ldots$ f) $\quad-3,-3,-3,-3,-3, \dots$ g) $a_{n}=\left(\frac{3}{7}\right)^{n}$ h) $a_{n}=n^{2}$

VIDEO: Introduction to geometric sequences, Example

Example 24.3. Find the general formula of a geometric sequence with the given property a) $r=4,$ and $a_{5}=6400$ b) $a_{1}=\frac{2}{5},$ and $a_{4}=-\frac{27}{20}$ c) $a_{5}=216, a_{7}=24,$ and $r$ is positive

VIDEO: Finding the formula of a geometric sequence – Example 24.3

Example 24.4. Consider the geometric sequence $a_{n}=8 \cdot 5^{n-1},$ that is the sequence: $$8,40,200,1000,5000,25000,125000, \ldots$$

Find the sum of the first 6 terms of this sequence $$8+40+200+1000+5000+25000=31248$$

VIDEO: Sum of a geometric series – intro example

Observation 24.5. Let $\left\{a_{n}\right\}$ be a geometric sequence, whose $n$ th term is given by the formula $a_{n}=a_{1} \cdot r^{n-1} .$ We furthermore assume that $r \neq 1 .$ Then, the sum $a_{1}+a_{2}+\dots+a_{k-1}+a_{k}$ is given by $$\sum_{i=1}^{k} a_{i}=a_{1} \cdot \frac{1-r^{k}}{1-r}$$

Example 24.6. Find the value of the geometric series. a) Find the sum $\sum_{n=1}^{6} a_{n}$ for the geometric sequence $a_{n}=10 \cdot 3^{n-1}$ b) Determine the value of the geometric series: $\sum_{k=1}^{5}\left(-\frac{1}{2}\right)^{k}$ c) Find the sum of the first 12 terms of the geometric sequence $$-3,-6,-12,-24, \dots$$

VIDEO: The sum of a finite geometric series, Example 24.6

Infinite geometric series

Sometimes it makes sense to add up not just a finite number of terms in a sequence, but ALL the terms (infinitely many!).

Example 24.7. Consider the geometric sequence $$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$$ What is the initial term? What is the common ratio? Let’s try adding up some of the terms. Try this by hand, and by using the formula for finite geometric series. What happens if we add up ALL the terms?

VIDEO: Infinite geometric series – intro example and formula

Definition 24.8. An infinite series is given by the formula $$\sum_{i=1}^{\infty} a_{i}=a_{1}+a_{2}+a_{3}+\ldots$$

Observation 24.9. Let $\left{a_{n}\right}$ be a geometric sequence with $a_{n}=a_{1} \cdot r^{n-1}$ Then the infinite geometric series is defined whenever $-1<r<1$. In this case, we have: $$\sum_{i=1}^{\infty} a_{i}=a_{1} \cdot \frac{1}{1-r}$$

$\quad$

Example 24.10. Find the value of the infinite geometric series. a) $\sum_{j=1}^{\infty} a_{j},$ for $a_{j}=5 \cdot\left(\frac{1}{3}\right)^{j-1}$ b) $\sum_{n=1}^{\infty} 3 \cdot(0.71)^{n}$ c) $500-100+20-4+\ldots$ d) $3+6+12+24+48+\ldots$

Example 24.11. Consider the real number given by $0.555555\dots$. Rewrite this number as an infinite geometric series. Can you figure out what fraction it is equal to?

Here is today’s Daily Quiz, based on the material covered last Thursday. As you know, quizzes will count towards your grade, and they will also be used to track your attendance (to be marked PRESENT, you must submit the quiz before midnight). Today’s lecture will be posted later this morning.

As always, let me know if you have any questions or problems.

Be well, Prof. Reitz

Daily Quiz: 5/12/20

This quiz MUST be taken on 5/12 to record your attendance.

To give everyone the best chance to improve their grade, I have re-opened all your WeBWorK assignments since the beginning of the semester. They will remain open until next Tuesday, 5/19, at 10:00am. Good luck!

Just a quick update to share two resources for preparing for the final exam:

Take a practice exam (or more than one!) on WeBWorK. The department has prepared a practice final exam – your actual final exam will be similar.

Where can I find the practice exam? The practice exams can be found in the “Final Exam WeBWorK” area (NOT our usual WeBWorK site), the same location where our actual final exam will take place

What is my login to the Final Exam WeBWorK area? You should have received an email sometime Friday, 5/8/20, with the title Practice Exams and Final Exam Login Information from the address WeBWorK Administrator <webwork@citytech.cuny.edu>. This email contains your exam WeBWorK account login and password, and a link to the WeBWorK final exam area for our class.

How many times can I take the Practice Final in WeBWorK? You can one practice final exam per day, from now until the actual finals begin. You will have 2 hours to complete the practice exam once you begin (just like a real exam).

Take a look at the official departmental Final Exam review sheet (follow the link and you will also find videos of the review problems worked out in their entirety).