:::info Welcome to MTH 201! I'm Dr. Robert Talbert, Professor of Mathematics, and I am grateful that you are signed up for the course and am looking forward to working with you this semester. ::: What's MTH 201 all about? MTH 201 is a first course in Calculus, which is all about modeling and understanding change. Change is maybe the most important facet of the world around us, and we care about it more than we realize. For example, we care a lot about the number of Covid-19 cases in our community, but we might care even more about how fast the number of cases is changing (either up or down). In MTH 201, you'll learn the mathematical language of change and apply it to models that you build to draw conclusions, make predictions, and give meaningful answers to real problems. MTH 201 goes beyond just computation. In MTH 201, you'll build skills with understanding complex concepts, communicating those concepts and the meaning of your results to appropriate audiences, using professional tools to help you in your work, and practice working with others to improve your learning (and theirs). These are valuable skills no matter where you go next. Success in this course doesn't come easy, and you can expect to be pushed and stretched intellectually. But the struggle you experience is normal and healthy, a sign of growth and that you are doing things the right way. And you will receive tireless support from me and your classmates in the process. Above all, my top priority is to support you in your work and help you succeed.

Initial due date: Sunday, April 9 at 11:59pm ET Overview Our final miniproject reaches back into linear algebra to look at diagonalizable matrices and their uses in solving systems of differential equations. Prerequisites: You'll need to be able to solve basic systems of differential equations and find the eigenvalues and eigenvectors for a small matrix. You'll also need a basic comfort level with concepts of linear independence and matrix arithmetic from earlier in the course. Background This entire problem comes from Section 3.9.1 in your textbook. Here is a rephrased version of the introduction to that section.

Initial due date: Sunday, April 9 at 11:59pm ET Overview This miniproject will teach you about the Runge-Kutta method, a standard numerical solution technique for differential equations. Prerequisites: A strong grasp of Euler's Method for single DE's is needed. You will also need to be comfortable using a spreadsheet. Miniproject 6 (Euler's Method for systems) is also recommended. Background A description of the Runge-Kutta method along with an example is given in this tutorial. Read it carefully and make sure you can work along with the example before proceeding.

Initial due date: Sunday, April 9 at 11:59pm ET Overview This miniproject introduces a version of Euler's Method as a numerical solution technique for systems. Prerequisites: You will need to be comfortable with using Euler's method for single differential equations. You'll also benefit from some familiarity with spreadsheets or Python in order to automate the calculations. Background This tutorial gives you the background you need for this assignment. Please read it and make sure you understand the concepts and the example: https://github.com/RobertTalbert/linalg-diffeq/blob/main/assignments/Euler's_Method_for_Systems.ipynb

Initial due date: Sunday, March 26 at 11:59pm ET Overview Eigenvalues of a matrix are incredibly useful and important for many applications. (Some of these applications are in Miniprojects 1-3.) But computing eigenvalues of a matrix, even of relatively small size, can be difficult or impossible to do exactly. So we need numerical approximation methods for most practical uses of eigenvalues. This miniproject will teach you one such method. Prerequisites: You'll need to know what an eigenvalue and eigenvector for a matrix are, and how to find these using SymPy. You'll also need to know how to multiply matrices and vectors. Background Complete the following warmup exercises first. These don't go in your writeup. They are just here to teach you some terminology you'll need in the main assignment.

Initial due date: Sunday, March 12 at 11:59pm ET Overview In this miniproject, you'll apply concepts from linear algebra to study discrete dynamical systems. These are related to the idea of Markov chains that was the subject of Miniproject 1 and are the linear algebra analogue of systems of differential equations which we will study later. Prerequisites: You'll need to know what an eigenvalue and eigenvector for a matrix are, and how to find these using SymPy. This miniproject also requires some knowledge of matrix-vector multiplication. Background Complete the following before beginning this miniproject. These are not part of your writeup, but you'll need the knowledge before you can understand the tasks in the assignment.

Initial due date: Sunday, February 26 at 11:59pm ET Overview In this miniproject, you'll explore the concept of matrix transformations and use matrix transformations in $\mathbb{R}^3$ to create a very simple computer animation --- and see how linear algebra is used to make more complex animations happen. Prerequisites: You'll need to know how to multiply a $3 \times 3$ matrix to a $3 \times 1$ vector using SymPy, as well as what some of our recent theorems about matrices and invertibility say. Also, you'll need to do a bit of background learning using the video mentioned below. Background Complete the following before beginning this miniproject. These are not part of your writeup, but you'll need the knowledge before you can understand the tasks in the assignment.

This quiz contains the second version of Foundational Skill LA.1 and the first version of Skill LA.2. Instructions: If you had a "Success" mark on Skill LA.1 from the first quiz, do not do the problem for that skill again. Make sure to consult the Standards for Student Work in MTH 302 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also check the Success criteria below each problem. This week's quiz is done entirely asynchronously since we are not meeting as a class. Please submit your work on Blackboard by 11:59pm ET Monday, January 30. As before, you may hand-write your work on paper, hand-write it in a notes app, or type it up. But for this and all subsequent quizzes, put the work for each Foundational Skill on its own page. So, if you are doing both LA.1 and LA.2, put the work for LA.1 on a different page than the work for LA.2. Do not put both problems on the same page. When you are ready to submit your work: Scan your handwritten work to a clear, legible, black-and-white PDF using a scanner or scanning app -- one PDF per problem. So if you are doing both problems, you will have two PDFs: one for Skill LA.1 and another for Skill LA.2 (all parts). Then, upload each PDF to its designated folder on Blackboard: The PDF for Skill LA.1 goes into the folder for Skill LA.1, and the PDF for Skill LA.2 goes into the folder for Skill LA.2. Make sure to click "Submit" on each before exiting. Your work will be graded on Blackboard, receiving a mark of either Success or Revise along with feedback. If you need to Revise, a new version of each of these skills will appear on Skill Quiz 3.

Overview In this miniproject you will learn about Markov chains and use them to make predictions about long term trends in population growth and US voter activity. Prerequisites: You'll need to know how to multiply a matrix times a vector, both by hand and using SymPy. Initial deadline: Sunday, February 5 at 11:59pm ET.. This is the deadline for all first drafts of your solution. If you turn in a good-faith effort at a complete and correct solution by this date, you may continue to revise and resubmit as needed with no additional deadlines, until the final deadline of 11:59pm ET Sunday April 16. However, no first drafts will be accepted after the initial deadline. Background Complete the following before beginning this miniproject. These are not to be turned in! But you won't get far on the assignment without doing them.

Grading on your work from Quiz 10 is now complete! As we head into the final last-chance quiz on Tuesday and the mini-quiz on Friday, here are some notes for you that might be useful if you will be participating in either of those. Where we are Here is a chart that shows your current status (as of Thursday afternoon) on each of the learning targets. This is a combined chart with both Sections 03 and 04: Green means the standard has been met. Blue means that one successful demonstration of skill has happened, but not a second one. Yellow means that there have been attempts at the standard but none have been successful so far. And gray means that the standard has no attempts so far. Two takeaways from this chart: Standards P.1 through DR.1 are in good shape. All students have now met standards P.1, G.1, and G.2 (so you will not be seeing those on any more quizzes).

:::info This quiz contains new versions of all content standards except the ones that all students have met. Instructions Work only the problems that you need to work and feel ready to work. Do your work on separate pages with each Content Skill Standard on its own separate page. Please do not put work for multiple Standards on the same page. Make sure to consult the Specifications for Student Work in MTH 325 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also, make sure to carefully read the Success Criteria below each problem to know exactly what is expected from that problem. When you are done: Scan each Learning Target to a clear, legible, black-and-white PDF and upload the PDF to the designated folder on Blackboard. Please do not just take a picture with your camera --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.

Instructions First of all please note: You do not have to do all of the problems below. The Applications badge instructions in the syllabus only require that you complete three different Application problems at least one of which is a "Programming Focused" problem. We will add more problems through the semester, eventually totalling between 4 and 10 (probably 6-8). Some of these problems focus on writing Python code to accomplish some task. These are specifically labelled as Programming Focused and have a computer emoji :desktop_computer: under the problem number. Your job is to find a problem that looks interesting to you and then solve it. "Solve" here can look like different things, depending on the problem. However, all Application Problem solutions must be complete, correct, and professionally presented solutions that explain why the work you did actually solves the problem. Please see the Specifications for Students Work in MTH 325 document for precise details of what this involves. Among the criteria for acceptable work are these important points: Your work must constitute a good-faith effort at a completed solution. You may not write up part of a solution and leave other parts of it incomplete or blank.

:::info This quiz contains new versions of Content Skill Standards P.1-P.3, G.1-G.8, and DR.1-DR.6, and T.1 and introduces Content Skill Standards T.2 and T.3. Instructions Work only the problems that you need to work and feel ready to work. Do your work on separate pages with each Content Skill Standard on its own separate page. Please do not put work for multiple Standards on the same page. Make sure to consult the Specifications for Student Work in MTH 325 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also, make sure to carefully read the Success Criteria below each problem to know exactly what is expected from that problem. When you are done: Scan each Learning Target to a clear, legible, black-and-white PDF and upload the PDF to the designated folder on Blackboard. Please do not just take a picture with your camera --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.

:::info This quiz contains new versions of Content Skill Standards P.1-P.3, G.1-G.8, and DR.1-DR.6 and introduces Content Skill Standard T.1. Instructions Work only the problems that you need to work and feel ready to work. Do your work on separate pages with each Content Skill Standard on its own separate page. Please do not put work for multiple Standards on the same page. Make sure to consult the Specifications for Student Work in MTH 325 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also, make sure to carefully read the Success Criteria below each problem to know exactly what is expected from that problem. When you are done: Scan each Learning Target to a clear, legible, black-and-white PDF and upload the PDF to the designated folder on Blackboard. Please do not just take a picture with your camera --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.

:::info Selected solutions for this Proof Problems can be found here: https://hackmd.io/@rtalbert235/H1HOeX1Bi ::: Instructions for all Proof Problems First of all please note: You do not have to do all of the problems below. The Proofs badge instructions in the syllabus only requires that you complete three different Proof problems --- one that uses mathematical induction in the proof, one that does not use induction, and a third one that you can prove any way you want. Proof problems will be added continuously to this list through the semester, and there may be dozens of problems from which to choose by the end. Please note that problems can be solved in many different ways, and it's possible that some Proof problems could be done both with induction and without it. Your job is to find a problem that looks interesting to you and then solve it. "Solve" here means write a complete, correct, and polished proof for the statement that's given in the problem. Please see the Specifications for Students Work in MTH 325 document for precise details of what this involves. Among the criteria for acceptable work are these important points: Your work must constitute a good-faith effort at a completed proof. You may not write up part of a proof and leave other parts of it incomplete or blank.

Problem 2 Click here for the problem itself Proof: We prove this by induction. The base case is $n=1$. In this case, the function would return 0 because of the first part of the if statement. And, we can calculate that $\lfloor \log_2(1) \rfloor = \lfloor 0 \rfloor = 0$. So the base case holds. Now suppose that for some positive integer $k$, the function returns $\lfloor \log_2(k) \rfloor$ when we input k. Note: This is equivalent to saying that the function returns $\lfloor \log_2(m) \rfloor$ when we input m for any value of $m$ less than or equal to $k$. We want to show that the function returns $\lfloor \log_2(k+1) \rfloor$ when we input k+1. We may assume that $k > 1$ since we already handled the case where $k = 1$. Therefore when given an input of k+1, the function will default to the else branch of the if-then statement and recursively compute L = r((k+1)//2). Now the value of (k+1) // 2 is an integer that is less than or equal to k. Therefore, by the induction hypothesis, r((k+1)//2) evaluates to $$\left \lfloor \log_2\left( \frac{k+1}{2} \right) \right \rfloor$$

:::info This quiz contains new versions of Content Skill Standards P.1-P.3, G.1-G.8, ad DR.1 and introduces Content Skill Standards G.8 and DR.2-DR.6. Instructions Work only the problems that you need to work and feel ready to work. Do your work on separate pages with each Content Skill Standard on its own separate page. Please do not put work for multiple Standards on the same page. Make sure to consult the Specifications for Student Work in MTH 325 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also, make sure to carefully read the Success Criteria below each problem to know exactly what is expected from that problem. When you are done: Scan each Learning Target to a clear, legible, black-and-white PDF and upload the PDF to the designated folder on Blackboard. Please do not just take a picture with your camera --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.

:::info This quiz contains new versions of Content Skill Standards P.1-P.3 and G.1-G.7 and introduces Content Skill Standards G.8 and DR.1. Instructions Work only the problems that you need to work and feel ready to work. Do your work on separate pages with each Content Skill Standard on its own separate page. Please do not put work for multiple Standards on the same page. Make sure to consult the Specifications for Student Work in MTH 325 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also, make sure to carefully read the Success Criteria below each problem to know exactly what is expected from that problem. When you are done: Scan each Learning Target to a clear, legible, black-and-white PDF and upload the PDF to the designated folder on Blackboard. Please do not just take a picture with your camera --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.

Let us change our traditional attitude to the construction of programs: Instead of imagining that our main task is to instruct a computer what to do, let us concentrate rather on explaining to human beings what we want a computer to do. Donald Knuth, Literate Programming About the course and this syllabus Welcome to MTH 325! I'm Dr. Robert Talbert, the professor of this course. I'm grateful you're here, and I think you're going to love what you learn this semester. MTH 325 really gets into the structures part of Discrete Structures by looking at three fundamental constructs that are everywhere in math and CS: graphs, relations, and trees. We'll be learning all about these, and also learning how to think rigorously about these structures through the use of proof. This syllabus contains all the information you need to navigate the course. It is available in an online format as well as a PDF. Both versions will be updated weekly. When you see blue- or purple-underlined text in the syllabus or any other document, it's a clickable link. For example, click here for a cat video.

:::info This quiz contains new versions of Content Skill Standards P.1-P.3 and G.1-G.6 and introduces Content Skill Standards G.7. Instructions Work only the problems that you need to work and feel ready to work. Do your work on separate pages with each Content Skill Standard on its own separate page. Please do not put work for multiple Standards on the same page. Make sure to consult the Specifications for Student Work in MTH 325 document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also, make sure to carefully read the Success Criteria below each problem to know exactly what is expected from that problem. When you are done: Scan each Learning Target to a clear, legible, black-and-white PDF and upload the PDF to the designated folder on Blackboard. Please do not just take a picture with your camera --- use a scanning app to create a PDF, and upload the PDF. You can also type up your work and save to a PDF if you want; or use a notes app on a tablet to handwrite your work and save that to a PDF.