Murphy's Laws and Corollaries - Richard's Law: If you keep anything long enough, you can throw it away. If you throw it away, you will need it the next day.

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Complete these tasks before our first class:

* Watch this video: Welcome! I recorded this video last spring, and our first day of class is February 8, but the rest is still accurate.

* Read the Carthage Mask Policy

* Watch this video on mathematical word processing.

2/8: In-person instruction today! - FINALIZED

Written exercises should be typed and turned in via email or in person before the beginning of class on the indicated date. Of course, there is nothing to turn in for reading assignments, but there will be a quiz or writing assignment over that material. **Read the notes for insights into style
and word choice. Exercises with a (T) to the left of the number may be written in two-column format as described in class.** Exercise without a (T) need to be written in paragraph format as demonstrated in the notes.

You are welcome and encouraged to stop by my office for help on the homework, either in person or via Zoom. However, you may not ask questions over homework on the day that the assignment is due. I also will not "pre-grade" your homework to see if your proofs are correct.

Do NOT assume that any assignment beyond today's is correct! I often change them the day of class. Look for the word next to an assignment to know when you can begin watching the videos. If you take a quiz before the lesson is , you may need to take it again if I change the questions.

The Homework assigned today is due 2/13.

Homework

* No written homework assigned today.

*
Check the blog for new entries. As I think about how class went, I will post insights and hints in the blog.

Required Reading Assignment to be Done Before Class on 2/10.

(1) Read the Vital Information section of the notes, pages 1-14.
Browse the Index too so you get an idea of what's there. This information is also available on Schoology as the Course Syllabus.

(2) Read this comment on analysis homework - please read this carefully.

(3) Read this page on homework expectations.

(4)
Read the Carthage Academic Honesty Guidelines. There is a link in the header above.

(5) Read Section 1.1 in the notes.

(6) Read ALL of the exercises in section 1.1! In section 1.2 and beyond, you may need those results, even those problems that werre not assigned as homework, and you can use them!

(7) Read this article on writing mathematics. There are some very good points that extend well beyond mathematics!

Required Videos to be Viewed Before Class on 2/10.

* Office hours - Zoom and in-person (recorded last fall - still mostly accurate, except that office hours are actually an hour this spring)

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook. I think taking notes with a highlighter is a great idea!

(1) The Algebraic Properties of the Real Number System

(2) Our First Proof

(3) Uniquemess Theorems

(4) Multiplication Meets the Additive Identity

(5) Using the Uniqueness Theorems

(6) Cancellation, Subtraction, and Division

(7) Here is a link to a video produced by Arizona State University, which I recommend very highly. It could increase your grade significantly! You could also listen to this radio program. You can read the article, or click on the blue "Listen" button on the left side of the page. You can read the text as well, but listening might be more effective.

* After you have watched the videos and read the notes, take this online quiz. This will count as a quiz. After you take the quiz, you should receive an email confirmation so you can check your answers.

Optional Material

* If you wish, you can download and install Mathematica on your personal computer, and have access to Mathematica Online. Links to the downloads are on this page.

* If you wish, you can sign up for your free MAA membership at this link.

2/10: In-person instruction today! We will discuss Section 1.1. Be ready with questions! - FINALIZED

The Homework assigned today is due 2/15.

Homework

* Re-read Section 1.1 in the notes, concentrating on parts that werre not clear on the first reading.

* Do Exercises 1abd, 5bf from Section 1.1 in the notes. You watched the videos over this material to prepare for today's class. Remember that your proofs need to include complete sentences that use words, and every statement you make needs justification!

Required Reading and Video Assignment to be Done Before Class on 2/13.

(1) Read Section 1.2 in the notes. Take a good look at 1.2.12(c) and 1.2.13(c). You'll need those a lot in the homework; if not in this section, certainly in the future!

(2) Read all of the Exercises in Section 1.1 - they can be used in this section!

Required Videos and Reading to be Completed Before Class on 2/13.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Important Subsets of the Real Numbers

(2) There Is No Rational Number That When Squared Equals 2

(3) The Order Properties

(4) Positivity

(5) Introducing \(\varepsilon\)

(6) Useful Inequalities

* After you have watched the videos and read the notes, take this quiz. This will count as a quiz. After you take the quiz, you should receive an email confirmation so you can check your answers.

Optional Material

* I didn't see any good additional material online.

Today's VQ for Class - you can't take the quiz here! This is for me to use in class. Take the quiz at the link above.

2/13: In-person instruction. - FINALIZED

The Homework assigned today is due 2/17.

Homework

* Re-read Section 1.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from Section 1.1. You may need them!

*
Check the blog for new entries.

* Do Exercises 1cdik, 7, 12, 17 from Section 1.2 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 2/15.

(1) Read Section 1.3 in the notes.

(2) Read all of the exercises in Section 1.3.

Required Videos to be Viewed Before Class on 2/15.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Propositional Functions

(2) The Well-Ordering Principle

(3) Two Example Proofs

(4) A Proof of an Equality

(5) A Proof of an Inequality

(6) The Strong Form of Induction

* After you have watched the videos and read the notes, take this quiz. This will count as a quiz. After you take the quiz, you should receive an email confirmation so you can check your answers.

Optional Material

* There are interactive induction proofs on this page. I recommend working through the proofs, filling in each step and checking your work.

2/15: In-Person Instruction. Last day to add or drop a course. Today's Mathematica File. - FINALIZED

The Homework assigned today is due 2/20.

Homework

* Re-read Section 1.3 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do exercises 1ae, 3i, 6a, 17, 18 from section 1.3 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 2/17.

(1) Read Section 1.4 in the notes.

(2) Read all of the exercises in section 1.4, especially exercises 21 and 22. We will use these in a future section, and you may need to use them in your homework.

(3) Re-read exercises 5, 6, 7, 8, 9, and 14-19 in Section 1.3. We will use some of those inequalities. You may even want to write them down somewhere in your notes.

Required Videos to be Viewed Before Class on 2/17.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Set Proofs and Cartesian Products

(2) Functions, Restrictions, and Extensions

(3) Injective Functions

(4) Surjective Functions

(5) Direct and Inverse Images

(6) Direct Image Proof

(7) Inverse Image Proof

* After you have watched the videos and read the notes, take this quiz. This will count as a quiz. After you take the quiz, you should receive an email confirmation so you can check your answers.

Optional Material

* Without looking back at the notes, work through this proof of one part of DeMorgan's Laws.

2/17: In-Person Instruction. - FINALIZED

The Homework assigned today is due 2/22.

Homework

* Re-read Section 1.4 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 10, 13, 15 from Section 1.4 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 2/20.

(1) Read Section 1.5 in the notes.

(2) Read all of the exercises in Section 1.5.

Required Videos to be Viewed Before Class on 2/20.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Definition of Absolute Value

(2) The Triangle Inequality

(3) Extending the Triangle Inequality

(4) \(\varepsilon\) Neighborhoods

(5) An Important Proof

(6) Bounded Away From a Point

Optional Material

* I've been told that this comic tells some math-related jokes, and that you might enjoy this show. He performs on campus on February 25. Here is a sample of his show.

* A friendly reminder: Although it's early in the semester, you should all know that your proofs must contain complete sentences, not just phrases and random thoughts. No scratch work should be on the page. Sentences start with words (and capital letters), not mathematical symbols, variables, or numbers. Sentences also end with periods, question marks, or exclamation points. It's not appropriate to use mathematical symbols as verbs (although an occasional \(\Rightarrow\) in a sequence of computations is OK), and abbreviations really aren't appropriate either. I expect good, polished writing. Yes, this may mean re-copying your work. Use pencil so you can erase; scribbling out words you don't like is not appropriate. Missing punctuation really does matter!

2/20: In-Person Instruction. - FINALIZED

The Homework assigned today is due 2/24.

Homework

* Re-read Section 1.5 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 5, 6 from Section 1.5 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 2/22.

(1) Read Section 1.6 in the notes.

(2) Read all of the exercises in Section 1.6.

Required Videos to be Viewed Before Class on 2/22.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Definition of Bounded

(2) Definition of Supremum and Infimum

(3) Uniqueness of the Supremum

(4) An Important Lemma and its Proof

(5) The Completeness Property

(6) The Supremum of a Union

Optional Material

* Read the interactive proof involving sups and infs on the website.

2/22: In Person Instruction Today. - FINALIZED

The Homework assigned today is due 2/27.

Homework

* Re-read Section 1.6 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 5, 6, 8, and 10 from Section 1.6 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 2/24.

(1) Read Section 1.7 in the notes.

(2) Read all of the exercises in Section 1.7.

Required Videos to be Viewed Before Class on 2/24.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) An Example Involving a New Definition

(2) Another Example Involving Another New Definition

(3) An Example Involving Direct Images

(4) The Archimedean Property

(5) The Existence of Irrational Numbers

(6) The Density Theorem

Optional Material

* None at this time.

2/24: In-Person Instruction Today. - FINALIZED

The Homework assigned today is due 3/1.

Homework

* Before you start this homework, study the definition of supremum and the definition of infimum again.

* Re-read Section 1.7 in the notes, concentrating on parts that were not clear on the first reading. Re-reading section 1.6 would not be a bad idea either.

* Review the exercises from previous sections. You may need them!

* Do Exercises 1bd, 3, 14ad from Section 1.7 in the notes. You watched the videos over this material to prepare for today's class.

*
Check the blog for new entries.

Required Reading Assignment to be Completed Before Class on 2/27.

(1) Read Section 1.8 in the notes.

(2) Read all of the exercises in Section 1.8.

Required Videos to be Viewed Before Class on 2/27.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The Definition of an Interval

(2) Nested Intervals

(3) Infinite Intersections

(4) The Nested Intervals Property

(5) Countably Infinite Sets

(6) \(\mathbb R\) Is Not Countably Infinite

(7) Cantor's Theorem

Optional Material

* I asked ChatGPT to prove the result we worked on in class. I put its response here. Can you find anything wrong with the proof?

* A somewhat-less-friendly reminder: You should all know that your proofs must contain complete sentences, not just phrases and random thoughts. No scratch work should be on the page. Sentences start with words (and capital letters), not mathematical symbols, variables, or numbers. Sentences also end with periods, question marks, or exclamation points. It's not appropriate to use mathematical symbols as verbs (although an occasional \(\Rightarrow\) in a sequence of computations is OK, as long as you could replace the symbol with the phrase "which implies" and the sentence makes sense), and abbreviations really aren't appropriate either. I expect good, polished writing. Yes, this may mean re-copying your work. Also, you really don't want to assume what you're trying to prove! That can result in an immediate 1 out of 5. **If you have any missing homework assignments or numerous blank problems on assignments you did turn in **, I recommend re-reading the bold statement on page xi of the notes.

2/27: In-Person Instruction Today - FINALIZED

The Homework assigned today is due 3/3.

Homework

* Re-read Section 1.8 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bdh, 3, 8 12 (only the supremum part) from Section 1.8 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 3/1.

(1) Read Section 2.1 in the notes.

(2) Read all of the exercises in Section 2.1.

Required Videos to be Viewed Before Class on 3/1.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The Definition of a Sequence

(2) The Definition of Sequence Convergence

(3) The Uniqueness of a Limit

(4) Equivalent Forms of the Definition of Convergence

(5) Using the Definition of Convergence

(6) Using the Template (note that I only did the \(k\neq0\) case in this video)

(7) Making \(K\) Larger or \(\varepsilon\) Smaller

(8) Important Writing Guidelines

Here is the sequence proof template.

Optional Material

* Work through these tutorials on sequence convergence proofs: a tutorial, example 1, example 2, example 3.

3/1: In-Person Instruction Today. - FINALIZED

Here is the sequence proof template.

The Homework assigned today is due 3/6.

Homework

* Re-read Section 2.1 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1be, 2, 5, 10, and 14 from Section 2.1 in the notes. You watched the videos over this material to prepare for today's class. As indicated in the exercises, you must use the template provided!

Required Reading Assignment to be Completed Before Class on 3/3.

(1) Read Section 2.2 in the notes.

(2) Read all of the exercises in section 2.2, paying particular attention to 5, 6, 7, 9, and 10. We will use those often!

Required Videos to be Viewed Before Class on 3/3.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Showing Sequence Elements Get Close to the Limit

(2) A VERY Important Theorem

(3) Tails of Sequences

(4) Using Theorem 2.2.2

(5) Constructing a Sequence

Optional Material

* None at this time.

3/3: REMOTE Instruction Today - FINALIZED - View today's class on the Live Stream

Today's Demo (you will need to log in to your wolfram account to view this file)

The Homework assigned today is due 3/8.

Homework

* Re-read Section 2.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1b, 2ab, 5, 8b from Section 2.2 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 3/6.

(1) Read Section 2.3 in the notes.

(2) Read all of the exercises in section 2.3. You might even write out a list of exercises that seem like they could come in handy in the future; that list might include 3-8, 12, 14, 17, 22-27. Pay particular attention to 7-8, which examine what can happen when we add or multiply divergent sequences. We will need 14 and 26 in future computations as well.

Required Videos to be Viewed Before Class on 3/6.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Bounded Sequences

(2) CoS

(3) \(\varepsilon/2\) Arguments Again

(4) Using CoS

(5) Corollaries of CoS

(6) Passing to the Limit

(7) The Squeeze Theorem

(8) Unbounded Sequences

Optional Material

* Work through this interactive example to prepare you for the Section 2.3 homework.

3/6: In-Person Instruction Today - FINALIZED

The Homework assigned today is due 3/10.

Homework

* Re-read Section 2.3 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1aeg, 2de, 4, 19 from Section 2.3 in the notes. You watched the videos over this material to prepare for today's class. Make sure that you evaluate every limit carefully.

You can download the Mathematica code for generating recursively defined sequences here.

Required Reading Assignment to be Completed Before Class on 3/8.

(1) Read Section 2.4 in the notes.

(2) Read all of the exercises in Section 2.4, especially exercise 12. The rest are computational.

Required Videos to be Viewed Before Class on 3/8.

**Many of the videos you will view from now on were recorded in the spring of 2020 during the lockdown.** Ignore any references to homework problems that aren't assigned, or to things that just don't seem relevant to us this semester. Special thanks to Wildwood Presbyterian Church in Grayslake, Illinois, for allowing me to use a classroom to record these videos.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Introduction video, definition of monotone

(2) The Monotone Convergence Theorem

(3) Example 2.4.8

(4) Example 2.4.11

(5) Example 2.4.15

Optional Material

* Read and work this interactive example.

* You can download Mathematica code for generating recursively defined sequences here.

3/8: In Person Instruction Today - FINALIZED - Need a little smile today?

The Homework assigned today is due 3/20.

Homework

* Re-read Section 2.4 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 3, 4, 11, 13bc from Section 2.4 in the notes. You watched the videos over this material to prepare for today's class. Model your proofs after the examples, but do not include your scratch work even though I have done that in a few examples.

Required Reading Assignment to be Completed Before Class on 3/10.

(1) Read Section 2.5 in the notes.

(2) Read all of the exercises in Section 2.5, especially numbers 3, 6-8, and 12-19.

Required Videos to be Viewed Before Class on 3/10.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The definition of a subsequence

(2) Subsequences explained (Remember that subsequences must be infinite.)

(3) The major theorems involving subsequences

(4) The limit superior

(5) Four Alternate Definitions of the Limit Superior

(6) A Proof Using the Limit Superior

(7) Another way to think about the BW Theorem and the limit superior

Optional Material

* There are many videos on this material on YouTube, but I don't recommend any one specifically. If you find one you like, let me know!

3/10: In Person Instruction Today - FINALIZED

The Homework assigned today is due 3/24.

Homework

* Re-read Section 2.5 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1be, 2a, 4 from Section 2.5 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 3/20.

(1) Read Section 2.6 in the notes.

(2) Read exercises 4-8 from Section 2.6. Also, read Example 2.6.16 carefully. That relates to Exercise 11 in Section 2.3.

Required Videos to be Viewed Before Class on 3/20.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The need for Cauchy sequences

(2) The definition of a Cauchy sequence

(3) Two important theorems

(4) The Cauchy Criterion

(5) Contrative sequences

(6) An important example

Optional Material

* None at this time.

March 13-17: Spring Break!

3/20: In Person Instruction Today - FINALIZED

After discussing today's material, I will take questions over the exam.

The Homework assigned today is due 3/27.

Homework

* Re-read Section 2.6 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 2, 5, 9 from Section 2.6 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 3/24.

(1) Read Section 2.7 in the notes.

(2) Read all exercises from Section 2.7 except for 8 and 21. There re many useful results in there, and I just can't assign them all!

Required Videos to be Viewed Before Class on 3/24.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The Definition of Properly Divergent

(2) Examples of Properly Divergent Sequences

(3) More Useful Examples

(4) Comparison Tests for Proper Divergence

(5) An Example Involving the Limit Comparison Test

Optional Material

* None at this time.

3/22: Exam 1 Today! - You may use a hand-held calculator on this exam if you wish. You may not use your phone. - FINALIZED

You will be typing your proofs into aGoogle Doc that will be available through Schoology. I will NOT be taking points off for formatting mathematical expressions.

Everyone who is not registered as fully remote through the Registrar's Office or is quarantined because of the COVID are expected to be in class to take the exam, unless I have granted you permission to take the exam remotely. The exam will cover the material we discussed in class from 2/8-3/8; that is, all of Chapter 1, and sections 2.1-2.4. The exam will be distributed at 9:15 am and I need to collect the exam by 10:20.

**You may use anything on my website www.marksnavely.us, including the text and the homework solutions. However, do not stray from my website!**

**You are not permitted to communicate with any human being by any means, including but not limited to, talking, phone conversations, text messages, email messages, hand signals, smoke signals, semaphore, Morse code, Norse code (used to communicate with vikings), social media, antisocial media, or any other means that I've omitted. You are not permitted to use any internet resource whatsoever other than my the contents of my website. You are not permitted to use any text other than mine, any app on your phone, any app that's not on your phone, Google, WolframAlpha, or any other print or internet resource that I've omitted.**

3/24: In Person Instruction Today - FINALIZED

The Homework assigned today is due 3/29.

Homework

* Re-read Section 2.7 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 3, 7a, 9, 21ad from Section 2.7 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 3/27.

(1) Read Section 2.8 in the notes.

(2) Read all of the exercises in Section 2.8.

Required Videos to be Viewed Before Class on 3/27.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) An Intuitive Introduction to Series

(2) The Formal Definition of a Series

(3) Two Very Important Series

(4) Telescoping Series

(5) The nth Term Conditions

(6) Convergence Theorems

(7) A Very Powerful Test

Optional Material

* None at this time.

3/27: In Person Instruction Today - FINALIZED

The Homework assigned today is due 3/31.

Homework

* Re-read Section 2.8 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bei, 2fj, 5c, 14a from Section 2.8 in the notes. You watched the videos over this material to prepare for today's class. Remember that these are still proofs, even when the result is computational! Cite theorems, use complete sentences - the works!

Required Reading Assignment to be Completed Before Class on 3/29.

(1) Read Section 3.1 in the notes.

(2) Read exercises 6-9 in Section 3.1. These are all very usefulu!

Required Videos to be Viewed Before Class on 3/29.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

The proof template (Don't forget to include the number of the definition of a limit in your proofs!)

(1) Introduction to Limits of Functions at a Point

(2) The Definition of a Limit

(3) Important Initial Theorems About Limits

(4) A Template Proof Example

(5) Bonus! An Example That Is Not In the Notes

(6) Very Useful Techniques

(7) How to Show that a Limit Doesn't Exist

Optional Material

* None at this time.

3/29: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/3.

Homework

* Re-read Section 3.1 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bc, 2deg, 12ac from Section 3.1 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 3/31.

(1) Read Section 3.2 in the notes.

(2) Read all of the exercises in Section 3.2, especially 6, 7, and 8.

(3) Re-read the examples from Sections 2.2-2.3 to remind yourselves how to apply a "Co_" theorem, like CoS or CoF.

* Read exercises

Required Videos to be Viewed Before Class on 3/31.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Bounded vs. Bounded at a Point

(2) Combinations of Functions

(3) The Squeeze Theorem

(4) Examples Involving Trig Functions

(5) A Very Important Example

Optional Material

* The geometric proof of Example 3.2.13(c).

*
Graphs of functions relevant to the material covered in today's class.

*
Don't forget the interactive web pages - they can be very useful tools!

3/31: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/5.

Homework

* Re-read Section 3.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 2c, 3b, 9c from Section 3.2 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/3.

(1) Read Section 3.3 in the notes.

(2) Read exercises 3-6 from Section 3.3.

Required Videos to be Viewed Before Class on 4/3.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) One-Sided Limits

(2) Sequential Criterion

(3) As x tends to Infinity

(4) Examples

(5) Divergence

Optional Material

* None at this time.

4/3: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/12.

Homework

* Re-read Section 3.3 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ce, 2ad, 5 from Section 3.3 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/5.

(1) Read Section 4.1 in the notes.

(2) Read exercises 11-19 from Section 4.1 in the notes. Exercises 12 and 13 are particularly useful.

Required Videos to be Viewed Before Class on 4/5.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Introduction to Continuity

(2) Continuity and Limits

(3) Template Proofs

(4) Discontinuity

(5) Dirichlet's Function

(6) The Thomae Function

Optional Material

* None at this time.

4/5: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/14.

Homework

* Re-read Section 4.1 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bce, 3, 4, 15, 19 from Section 4.1 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/12.

(1) Read Section 4.2 in the notes.

(2) Read exercises 2-6, 8, 9, 14, 15, 16, 21, 22, 23 from Section 4.2. Exercise 22 is particularly important!

Required Videos to be Viewed Before Class on 4/12.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) CoCF - Combinations of Continuous Functions

(2) Why We Need To Improve Continuity

(3) Closed Bounded Sets Are Awesome

(4) Extreme Values

(5) The Intermediate Value Theorem

Optional Material

* Don't forget the interactive web pages - there is one just for studying for the third exam!

4/7: Easter Break

4/10: Easter Break

4/12: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/17.

Homework

* Re-read Section 4.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 3b, 17, 22a from Section 4.2 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/14.

(1) Read Section 4.3 in the notes.

(2) Read 8, 9, 11, 12, and 17 from Section 4.3.

Required Videos to be Viewed Before Class on 4/14.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Uniform Continuity Introduced

(2) Continuity vs. UNIFORM Continuity

(3) Showing a Function is Not Uniformly Continuous

(4) Conditions That Imply Uniform Continuity

(5) Uniform Continuity and Cauchy Sequences

(6) Piecewise Continuous Functions

(7) Piecewise Linear Functions

Optional Material

* Here is an example showing uniform continuity, and here is one showing a function is not uniformily continuous.

4/14: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/21.

Homework

* Re-read Section 4.3 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1cd, 5, 17, 19 from Section 4.3 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/17.

(1) Read Section 4.4 in the notes.

(2) Read all of the exercises in Section 4.4.

Required Videos to be Viewed Before Class on 4/17.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Monotone Functions

(2) Monotone Functions Can Jump!

(3) Inverse Functions

Optional Material

* None at this time.

4/17: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/24.

Homework

* Re-read Section 4.4 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 3, 7 from Section 4.4 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/21.

(1) Read Section 4.5 in the notes.

(2) Read all of the exercises in Section 4.5.

Required Videos to be Viewed Before Class on 4/21.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Learning About Gauges

(2) The Definition of a Gauge

(3) Forcing a Specific Tag

(4) Cousin's Lemma

(5) Using Cousin's Lemma

(6) Thinking About Gauges

Optional Material

* None at this time.

4/19: Exam 2 You may use a hand-held calculator on this exam if you wish. You may not use your phone.

Everyone is expected to be in class to take the exam, unless I have granted you permission to take the exam remotely. The exam will cover the material we discussed in class from 3/10-4/12; that is, sections 2.5-2.8, Chapter 3, and Sections 4.1-4.2.

**This is an exam, and whether you are taking the exam in class or elsewhere, you should not do anything that you would not do if you were taking the exam in class in person. You may use anything on my website www.marksnavely.us, including the text and the homework solutions. However, do not stray from my website!**

**You are not permitted to communicate with any human being by any means, including but not limited to, talking, phone conversations, text messages, email messages, hand signals, smoke signals, semaphore, Morse code, Force code (used to communicate with computer Jedi knights), social media, antisocial media, or any other means that I've omitted. You are not permitted to use any internet resource whatsoever. You are not permitted to use any text other than mine, any app on your phone, any app that's not on your phone, Google, WolframAlpha, or any other print or internet resource that I've omitted.**

4/21: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/26.

Homework

* Re-read Section 4.5 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1c, 4, 7, and 9 from Section 4.5 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/24.

(1) Read Section 4.6 in the notes.

(2) Read all of the exercises in Section 4.6.

Required Videos to be Viewed Before Class on 4/24.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Coverings

(2) Open Covers and Compactness

(3) Compact Sets

(4) The Heine-Borel Theorem

(5) Full Covers

Optional Material

* None at this time.

4/24: In Person Instruction Today - FINALIZED

The Homework assigned today is due 4/28.

Homework

* Re-read Section 4.6 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 3, 11 from Section 4.6 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/26.

(1) Read Section 5.1 in the notes.

(2) Read all of the exercises in Section 5.1, especially exercise 11.

Required Videos to be Viewed Before Class on 4/26.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(0) I taught this material last spring on the fourth day of May. This page explains a few things.

(1) The Definition of the Derivative

(2) Examples Written above the examples is the definition of the derivative: \(\displaystyle{\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}}\). Sorry - that got cut off for some reason.

(3) Continuity and Differentiability

(4) Some Familiar Rules

(5) The Chain Rule

(6) An Important Example

Optional Material

* None at this time.

4/26: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/1.

Homework

* Re-read Section 5.1 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 2d, 13b, 17, and 22 from Section 5.1 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 4/28.

(1) Read Section 5.2 in the notes.

(2) Read all of the exercises in Section 5.2.

Required Videos to be Viewed Before Class on 4/28.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

Yes, one of these was actually recorded on a Wednesday. And no, I'm not sick.

(1) Rolle's Theorem

(2) The Mean Value Theorem

(3) Using the MVT

(4) Two Cool Examples

(5) The Most Amazing Theorem (The handwritten text should be "If \(f\) is differentiable on \([a,b]\), that does not imply that \(f'\) is continuous on \([a,b]\). For example, consider \(f(x):= x^2 \sin (1/x)\) on \([-1,1]\).")

Optional Material

* The Khan Academy describes the MVT intuitively.

4/28: In-Person Instruction Today - FINALIZED

The Homework assigned today is due 5/3.

Homework

* Re-read Section 5.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1b, 3, 8, and 11 from Section 5.2 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/1.

(1) Skim Section 5.3.

(2) Read Section 5.4 in the notes.

(3) Read all of the exercises in Section 5.4.

Required Videos to be Viewed Before Class on 5/1.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(0) L'Hopital's Rules

(1) Taylor's Theorem

(2) Inequalities

(3) Taylor's Theorem and the Second Derivative Test

Optional Material

* None at this time.

5/1: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/5.

Homework

* Re-read Section 5.4 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1cd, 3, 5 9bf from Section 5.4 in the notes. You watched the videos over this material to prepare for today's class. Note that I had a paging problem so problem 9 is on page 286.

Required Reading Assignment to be Completed Before Class on 5/3.

(1) Read Section 6.1 in the notes.

(2) Read all of the exercises in Section 6.1.

Required Videos to be Viewed Before Class on 5/3.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The Definition of Integrable

(2) Integrating a Few Functions

(3) Theorems About Integrable Functions

(4) Integrating an Unusual Function

(5) The Dirichlet Function and Thomae Function Revisited

(6) How to Show That a Function is Integrable

Optional Material

* None at this time.

5/3: In-Person Instruction Today! - FINALIZED

The Homework assigned today is due 5/8.

Homework

* Re-read Section 6.1 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1bd, 2, 3, 7 from Section 6.1 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/5.

(1) Read Section 6.2 in the notes.

(2) Read all of the exercises in Section 6.2.

Required Videos to be Viewed Before Class on 5/5.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Antiderivatives

(2) Proving the FTC - A Video for Context

(3) Examples

(4) The Second Form of the FTC

(5) A Few Final Theorems

Optional Material

* None at this time.

5/5: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/10.

Homework

* Re-read Section 6.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1abc, 2c, 4, 7 from Section 6.2 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/8.

(1) Read Section 6.3 in the notes.

(2) Read all of the exercises in Section 6.3.

Required Videos to be Viewed Before Class on 5/8.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Why Improve the Riemann Integral?

(2) Getting to Know the Gauge Integral

(3) The Fundamental Theorem of Integral Calculus for Gauge Integrals

(4) Improper Gauge Integrals Aren't Improper

Optional Material

* None at this time.

5/8: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/12.

Homework

* Re-read Section 6.3 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1abc, 4a, 5, 7b from Section 6.3 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/10.

(1) Read Section 6.4 in the notes.

(2) Read all of the exercises in Section 6.4.

Required Videos to be Viewed Before Class on 5/10.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) The Darboux Integral

(2) Upper and Lower Integrals

(3) Examples

(4) Riemann and Darboux

(5) The Riemann-Stieltjes Integral

(6) How the Riemann and Riemann-Stieltjes Integrals Are Related

Optional Material

* None at this time.

5/10: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/15.

Homework

* Re-read Section 6.4 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 5, 6a from Section 6.4 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/12.

(1) Review Section 2.8 in the notes.

(2) Read Section 7.1 in the notes.

(3) Read all of the exercises in Section 7.1.

Required Videos to be Viewed Before Class on 5/12.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Absolute Convergence

(2) Groupings - Infinite Associativity

(3) Rearrangements of Absolutely Convergent Series - Infinite Commutivity I

(4) Rearrangements of Conditionally Convergent Series - Infinite Commutivity II

(5) Strengthening the Limit Comparison Test

Optional Material

* None at this time.

5/12: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/17.

Homework

* Review Section 2.8 in the notes. You may need some of the theorems and definitions from that section, and you may use the results in that section, including the exercises.

* Re-read Section 7.1 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 3cef from Section 7.1 in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/15.

(1) Read Section 7.2 in the notes.

(2) Read all of the exercises in Section 7.2.

Required Videos to be Viewed Before Class on 5/15.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Familiar Convergence Tests

(2) Raabe's Test

(3) Generalizing the Condensation Test

(4) Two Product-Related Tests

Optional Material

* None at this time.

5/15: In Person Instruction Today - FINALIZED

The Homework assigned today is due 5/19.

Homework

* Re-read Section 7.2 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises 1ab, 3ad, 4ag from Section 7.2 in the notes. You watched the videos over this material to prepare for today's class. NOTE THE CORRECTION - I ORIGINALLY TYPED 2AD INSTEAD OF 3AD.

Required Reading Assignment to be Completed Before Class on 5/17.

(1) Read Section 7.3 in the notes.

(2) Read all of the exercises in Section 7.3.

Required Videos to be Viewed Before Class on 5/17.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) Definition of Pointwise Convergence

(2) Examples

(3) Uniform Convergence

(4) Proving Uniform Convergence

(5) The Uniform Limit of Continuous Functions

(6) Power Series and Dini's Theorem

Optional Material

* Want to stay in touch with Mathematics at Carthage? Join our facebook group!

5/17: In Person Instruction Today - FINALIZED

The Homework assigned today will not be collected. Solutions are posted already.

Homework

* Re-read Section 7.3 in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Read the solutions to Exercises 1ab, 2, 3, 6a, 11 from Section 7.3 in the notes. You watched the videos over this material to prepare for today's class.

5/19: In Person Instruction Today - Nothing New Today! - FINALIZED

5/22: FINAL EXAMS

5/23: FINAL EXAMS - Our final exam period is today from 8:00-10:00. The exam will cover the material from Section 4/14 through the end of the course; that is, Sections 4.3-4.6, Chapters 5, 6, and 7, together with the material described on this sheet. As before, the exam will be open marksnavely.us, including the text and homework solutions.

**This is an exam, and whether you are taking the exam in class or elsewhere, you should not do anything that you would not do if you were taking the exam in class in person. You may use anything on my website www.marksnavely.us, including the text and the homework solutions. However, do not stray from my website!**

**You are not permitted to communicate with any human being by any means, including but not limited to, talking, phone conversations, text messages, email messages, hand signals, smoke signals, semaphore, Morse code, Source**** code (used to communicate with computer computer programmers), social media, antisocial media, or any other means that I've omitted. You are not permitted to use any internet resource whatsoever other than marksnavely.us. You are not permitted to use any text other than mine, any app on your phone, any app that's not on your phone, Google, WolframAlpha, or any other print or internet resource that I've omitted.**

5/24: FINAL EXAMS -

Template:

The Homework assigned today is due 5/28.

Homework

* Re-read Section ?? in the notes, concentrating on parts that were not clear on the first reading.

* Review the exercises from previous sections. You may need them!

*
Check the blog for new entries.

* Do Exercises ??? from Section ?? in the notes. You watched the videos over this material to prepare for today's class.

Required Reading Assignment to be Completed Before Class on 5/26.

(1) Read Section ??? in the notes.

(2) Read all of the exercises in Section ???.

Required Videos to be Viewed Before Class on 5/26.

* With the notes available either printed or online, watch these videos. You should take notes in your printed manuscipt or in a notebook.

(1) ??

(2) ??

(3) ??

(4) ??

(5) ??

(6) ??

Optional Material

* None at this time.

Today's VQ For Class