Intro Modeling Notes - Carroll College

1 Introduction to Mathematical ModelingDifference equations , Differential equations , & linear algebra (The First Course of a Two-Semester Sequence)Dr. Eric R. of MathematicsCarroll College , Helena, MTContent Last Updated: January 8, 20181 Eric Sullivan. Some Rights work is licensed under a Creative Commons International License. You may copy, distribute, display, remix, rework, and per-form this copyrighted work, but only if you give credit to Eric Sullivan, and all deriva-tive works based upon it must be published under the Creative Commons Attribution-NonCommercial-Share Alike United States License. Please attribute this work to EricSullivan, Mathematics Faculty at Carroll College , To view a copyof this license, send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, Cali-fornia, 94105, To the Student and the An Inquiry Based Approach.

2 Online Texts and Other Resources .. To the Instructor ..71 Fundamental Notions from Sections from Active Calculus .. Modeling Explorations with Calculus ..102 An Introduction to linear Why linear algebra ? .. Matrix Operations and Gaussian Elimination .. Gaussian Elimination: A First Look At Solving Systems .. Systems of linear equations .. linear Combinations .. Inverses and Determinants .. Technology For linear algebra .. The Magic Carpet Ride .. Span .. linear Independence, linear Dependence, and Basis .. The Column and Null Spaces of a Matrix .. Modeling Explorations with linear algebra .. Input-Output Economies .. Traffic Networks .. Balancing Chemical equations ..663 First Order Birth, Death, and Immigration Exploration .. Models for Birth, Death, and Immigration.

3 Difference equations and Differential equations .. Stability and Equilibria .. Euler s Method: Numerical Solutions for Differential equations .. Classifying Difference and Differential equations .. Technique: Solving 1stOrder linear Homogeneous equations .. Technique: Solving 1stOrder linear Nonhomogeneous equations .. Technique: Separation of Variables .. Technique: Undetermined Coefficients .. Modeling Explorations with Difference and Differential equations .. 1024 Second Order Modeling Oscillations .. Homogeneous linear 2ndOrder Differential equations .. Forced Oscillations .. Energy in Mass Spring Systems A Lab Exploration .. Modeling Explorations with 2ndOrder Differential equations .. 1295 Systems of Difference and Differential Spread of Disease .. Spreading a Juicy Rumor.

4 The H1N1 Virus .. Writing Systems of Difference equations .. linear Systems .. The Eigenvalue / Eigenvector Problem .. of Eigenvectors and Eigenvalues .. Eigenvalue Eigenvector Problem .. for the Eigenvalue Eigenvector Problem .. Markov Chains .. Analysis of linear Systems .. Systems of linear Difference equations .. of Equilibrium Behavior .. Homogeneous Systems of linear Difference equations .. Modeling Explorations with Systems .. 1696 Infinite Sections from Active Calculus .. 174 Appendices175A MATLAB Vectors and Matrices .. Looping .. For Loops .. The While Loop .. Conditional Statements .. If Statements .. Case-Switch Statements .. Functions .. Plotting .. Animations .. 183 Chapter 0To the Student and the InstructorThis document contains lecture Notes , classroom activities, examples, and challenge prob-lems specifically designed for a first semester of differential equations and linear algebrataught with a focus on mathematical Modeling .

5 The content herein is written and main-tained by Dr. Eric Sullivan of Carroll College . Problems were either created by Dr. Sul-livan, the Carroll Mathematics Department faculty, part of NSF Project Mathquest, partof the Active Calculus text, or come from other sources and are either cited directly orcited in the LATEX source code for the document (and are hence purposefully invisible tothe student). Inquiry Based ApproachProblem (Setting The Stage). Get in groups of size 3-4. Group members should introduce themselves. For each of the questions that follow I will ask you a possible answer on your answers with the rest of the summary of each group s discussionQuestions:Question #1:What are the goals of a university education?Question #2:How does a person learn something new?Question #3:What do you reasonably expect to remember from your courses in 20 years?Question #4:What is the value of making mistakes in the learning process?

6 Question #5:How do we create a safe environment where risk taking is encouraged andproductive failure is valued?N5 CHAPTER 0. TO THE STUDENT AND THE INSTRUCTOR6(The previous problem is inspired by Dana Ernst s first day activity in IBL activitytitled: Setting the Stage.) Any creative endeavor is built in the ash heap of failure. Michael StarbirdThis material is written with an Inquiry-Based Learning (IBL) flavor. In that sense, thisdocument could be used as a stand-alone set of materials for the course but these notesare not atraditional textbookcontaining all of the expected theorems, proofs, examples,and exposition. The students are encouraged to work through problems and homework,present their findings, and work together when appropriate. You will find that this doc-ument contains collections of problems with only minimal interweaving exposition. It isexpected that you do every one of the problems and then use other more traditional textsas a backup when you are stuck.

7 Let me say that again: this is not the only set of materialfor the course. Your brain, your peers, and the books linked in the next section are yourbest resources when you are learn more about IBL go to Thelong and short of it is that the students in the class are the ones that are doing the work;building models, proving theorems, writing code, working problems, leading discus-sions, and pushing the pace. The instructor acts as a guide who only steps in to redirectconversations or to provide necessary insight. If you are a student using this material youhave the following jobs:1. Fight! You will have to fight hard to work through this material. The fight is exactlywhat we re after since it is ultimately what leads to innovative Screw Up! More accurately, don t be afraid to screw up. You should write code,work problems, and prove theorems then be completely unafraid to scrap whatyou ve done and redo it from scratch.

8 Learning this material is most definitely anon- linear path.*Embrace this!3. Collaborate! You should collaborate with your peers with the following caveats: (a)When you are done collaborating you should go your separate ways. When youwrite your solution you should have no written (or digital) record of your collabo-ration. (b) The internet is not a collaborator. Use of the internet to help solve theseproblems robs you of the most important part of this class; the chance for Enjoy! Part of the fun of IBL is that you get to experience what it is like to think likea true mathematician / scientist. It takes hard work but ultimately this should befun!*Pun intended: our goal, after all, is really to understand that linear algebra is the glue that holds mathe-matics 0. TO THE STUDENT AND THE Texts and Other ResourcesIf you are looking for online textbooks for linear algebra and differential equations I canpoint you to a few.

9 Some of the following online resources may be a good place to help youwhen you re stuck but they will definitely say things a bit differently. Use these resourceswisely. The bookDifferential equations with linear algebra , An inquiry based approach tolearningis a nice collection of Notes covering much of the material that we cover inour class. The order is a bit different but the Notes are well (2013 Spring).pdf The ODE Project by Thomas Juson is a nice online text that covers many (but notall) of the topics that we cover in differential Elementary Differential equations by William Trench. This book contains every-thing(!) you would ever want to look up for ordinary differential equations . It is agreat resource to look up ODE A First Course in linear algebra by Robert Beezer. This book is very thorough andcovers everything that we do in linear algebra and much linear algebra Workbook by TJ Hitchman.

10 This is a workbook for Dr. Hitchman sclass at U. Northern Iowa. Even though it is only a workbook it contains somenice explanations and it has embedded executable code for some the InstructorIf you are an instructor wishing to use these materials then I only ask that you adhereto the Creative Commons license. You are welcome to use, distribute, and remix thesematerials for your own purposes. Thanks for considering my materials for your course!My typical use of these materials are to let the students tackle problems in smallgroups during class time and to intervene when more explanation appears to be necessaryor if the students appear to be missing the deeper connections behind problems. Thecourse that I have in mind for these materials is a first semester of differential equationsand linear algebra taught from the standpoint of mathematical Modeling .