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MA103 Mathematical Modeling and Intro to Calculus

Course Guide – AY 19-01


Table of Contents

Lesson Legend

Block I – Modeling with Discrete Dynamical SystemsLesson 1: Mathematical Modeling I - Problem SolvingLesson 2: Mathematical Modeling II - TransformLesson 3: Mathematical Modeling III - InterpretProblem Solving Lab 1: Introduction to Assessment s Lesson 4: Discrete Dynamical Systems ILesson 5: Discrete Dynamical Systems IILesson 6: Modeling with Linear DDS ILesson 7: Modeling with Linear DDS IILesson 8: Writ 1Problem Solving Lab 2: Modeling with DDSLesson 9: Long Term Behavior I Lesson 10: Long Term Behavior IILesson 11: Analytic Solution ILesson 12: Analytic Solution IILesson 13: FCE Lesson 14: Non-Linear DDS Problem Solving Lab 3: Modeling with Linear DDSLesson 15: Written Partial Review 1ALesson 16: Written Partial Review 1B

Block II – Introduction to Vectors and Matrix AlgebraLesson 17: Introduction to Vectors ILesson 18: Introduction to Vectors II Lesson 19: Introduction to Vectors III Lesson 20: Introduction to Vectors IVLesson 21: Introduction to Matrix Algebra ILesson 22: Introduction to Matrix Algebra IILesson 23: Writ 2ALesson 24: Solving Systems of Linear Equations ILesson 25: Solving Systems of Linear Equations IILesson 26: Solving Systems of Linear Equations IIILesson 27: Modeling with Matrix Algebra I Lesson 28: Modeling with Matrix Algebra IILesson 29: Modeling with Matrix Algebra IIILesson 30: Writ 2BLesson 31: Systems of Recursion Equations ILesson 32: Systems of Recursion Equations II Lesson 33: Systems of Recursion Equations IIIProblem Solving Lab 4: Modeling with Systems of RELesson 34: Systems of Recursion Equations IVLesson 35: Systems of Recursion Equations VLesson 36: Eigenvalues and Eigenvectors I Lesson 37: Eigenvalues and Eigenvectors IILesson 38: Analytic Solutions to System of Recursion Equations ILesson 39: Analytic Solutions to System of Recursion Equations II Problem Solving Lab 5: Modeling with Systems of Recursion

Equations Lesson 40: Written Partial Review 2ALesson 41: Written Partial Review 2B

Block III –Modeling with Continuous FunctionsLesson 42: Introduction to Continuous FunctionsLesson 43: Introduction to Linear FunctionsLesson 44: Modeling with Linear FunctionsLesson 45: Introduction to Power FunctionsLesson 46: Modeling with Power FunctionsLesson 47: Writ 3 Problem Solving Lab 6: Modeling with Continuous Functions ILesson 48: Introduction to Exponential FunctionsLesson 49: Modeling with Exponential FunctionsLesson 50: Introduction to Trigonometric FunctionsLesson 51: Modeling with Trigonometric FunctionsProblem Solving Lab 7: Modeling with Continuous Functions IILesson 52: Written Partial Review 3A Lesson 53: Written Partial Review 3BLesson 54: Rates of Change ILesson 55: Rates of Change II Lesson 56: Rates of Change III Problem Solving Lab 8: Course Review


Lesson Legend

OBJECTIVES: Lesson objectives are what students should know or be able to do after completing the ASSIGNMENT tasks. All graded event questions are derived from lesson objectives.

ASSIGNMENT:READ: Students are required to read the READ text with the intent to obtain a full understanding of the material.

READ FOR FURTHER UNDERSTANDING: The content of the READ FOR FURTHER UNDERSTANDING readings is designed to augment the mandatory READ requirement. The READ FOR FURTHER UNDERSTANDING text is not mandatory, but contains information to give students a more complete comprehension of the lesson material.

DO: Students are required to attempt the WebAssign lesson or Do problems prescribed in the DO section.

WATCH: Students are required to watch Videos to explain a concept or help work through a problem are available on the MA103 Video AI Webpage (https://lms.westpoint.edu/math/ma103/SitePages/Video%20AI.aspx).

THINK ABOUT: The THINK ABOUT section offers questions that provide insights into fundamental concepts of the lesson. Instructors have the opportunity to ask students about the THINK ABOUT questions in order to prompt discussion and assess comprehension.

TECH TIPS: The TECH TIPS section offers examples of Excel and Mathematica syntax to assist students.

https://lms.westpoint.edu/math/ma103/SitePages/Video%20AI.aspx



Block IModeling with Discrete

Dynamical Systems

Block I Outcomes

By the end of Block I Students will be able to use the USMA Mathematical Modeling Process to transform, solve, and interpret real world problems involving a single variable.

BLOCK I OUTCOMES:

1. Transform situations involving change at discrete intervals into mathematical models.2. Solve a discrete dynamical system (DDS) numerically (through iteration), graphically, and analytically

(for linear cases).3. Use technology, especially when developing and using math models, to visualize concepts, to explore

potential solutions, and to execute complex and time-consuming computations when solving applied problems.

4. Interpret the results of solving a DDS for a given problem.



Dynamical Systems

Mathematical Modeling I – Problem Solving Lesson 1

OBJECTIVES:1a. Describe what is meant by mathematical modeling.1b. Recognize some of the things you should consider when solving a problem.1c. Explain why military officers must be good problem solvers.

ASSIGNMENT:READ:

1. The Course Introduction in Modeling in a Real and Complex World2. Sections 1.1.1 – 1.1.2 in Modeling in a Real and Complex World

DO: 1. Problems 1.1.1 and 1.1.2 (Section 1.1.8) in Modeling in a Real and Complex World2. Find and bookmark the MA103 webpage: https://lms.westpoint.edu/math/ma1033. Create WebAssign username and register in accordance with instructions attached to your

instructor’s email.4. Collect and bring the following items to the first day of class:

References from Book Issue: a. James Stewart’s Calculus, Early Transcendentals Eighth Edition(referred to as

the “Stewart Text” from this point forward).b. WebAssign booklet (separate package that comes with the Stewart Text).

References from the MA103 Webpage:a. Digital Copy of Modeling in a Real and Complex World (MRCW – MA103 textbook

AY 19-01 edition) located on the MA 103 Course Homepage. Save a copy to your laptop.

b. The MA103 – Course Calendar. Print a copy.c. The MA103 – Formula Reference Card. Print a copy.

Software on your laptop:a. Mathematica – Wolfram 10.4b. Word – Microsoft Office 2013c. Excel – Microsoft Office 2013

https://lms.westpoint.edu/math/ma103/SitePages/default.aspx

https://lms.westpoint.edu/math/ma103

https://lms.westpoint.edu/math/ma103



Dynamical Systems

Mathematical Modeling II - Transform Lesson 2

OBJECTIVES:2a. Explain the benefits of using a problem solving process.2b. Know and be able to apply the three steps in the Mathematical Modeling Process – Transform,

Solve, Interpret.2c. Given an assumption used in the formulation of a mathematical model, explain why the assumption

is reasonable and necessary.

ASSIGNMENT:READ:

1. Sections 1.1.3 – 1.1.4 in Modeling in a Real and Complex World2. Section 1.2 (stop just before Linear Models) in the Stewart Text

DO: Problems 1.1.3, 1.1.5, and 1.1.6 (Section 1.1.8) in Modeling in a Real and Complex World



Dynamical Systems

Mathematical Modeling III - Interpret Lesson 3

OBJECTIVES:3a. Evaluate a model based on the context of the problem and visual inspection (the “eyeball” test).3b. Interpret the mathematical solution to a real world problem and communicate results in non-mathematical terms.3c. Discuss how sensitive a solution is to the assumptions.

ASSIGNMENT:READ: Sections 1.1.6-1.1.7 in Modeling in a Real and Complex World

DO: 1. Introduction to WebAssign2. Problems 1.1.11 and 1.1.12 in Modeling in a Real and Complex World

http://www.webassign.net/



Dynamical Systems

Problem Solving Lab 1: Introduction to Assessments PSL 1

OBJECTIVES:a. Understand the Conduct of Team and Individual Assessments

ASSIGNMENT: READ: Read Ahead for Practice Assessment

THINK ABOUT: Studying in a formal group. Preparation for MA 103 Writs.



Dynamical Systems

Discrete Dynamical Systems I Lesson 4

OBJECTIVES:4a. Explain the difference between discrete and continuous models.4b. Explain the relationship between a difference equation and a recursion equation.4c. Given an initial condition, manually iterate a recursion equation.4d. Create a mathematical model of a recursive relationship using the transform step.

ASSIGNMENT: READ: Section 1.2 in Modeling in a Real and Complex World

DO: WebAssign Lesson 4

THINK ABOUT: When developing a mathematical model of a recursive relationship, it is helpful to include the following elements in the transform step: State what is given. State what you must find. Define variables representing the quantities of interest. Identify reasonable and necessary assumptions. Develop a DDS model

o State the initial condition.o State the recursion equation.




Dynamical Systems

Discrete Dynamical Systems II Lesson 5

OBJECTIVES:5a. Identify when the sequence generated by a recursion equation is arithmetic, geometric, or neither.5b. Classify a sequence as converging or diverging.5c. Find closed-form (analytic) solutions for arithmetic and geometric sequences.5d. Given a sequence of numbers, identify the discrete dynamical system that generated it.

ASSIGNMENT: READ:

1. Section 11.1 (pp. 694 – 696) in the Stewart Text2. Section 1.3 and Section 1.4 in Modeling in a Real and Complex World





Dynamical Systems

Modeling with Linear DDS I Lesson 6

OBJECTIVES:6a. Model situations and solve problems involving linear DDS models.6b. Explain the components of a linear DDS.6c. Explain the difference between linear homogeneous and linear non-homogeneous models.


DO: 1. Problem 1.5.2 in Modeling in a Real and Complex World2. WebAssign Lesson 6

A problem set relating to this block will be assigned at the end of the class and will be due at the beginning of class on 14 September 2018.




Dynamical Systems

Modeling with Linear DDS II Lesson 7

OBJECTIVES:8a. Explain the concept of an equilibrium value as it pertains to recursion equations.8b. Determine equilibrium values algebraically and graphically from a time series graph and fundamental graph. 8c. Classify an equilibrium value using:

- Time series graph.- Fundamental graph.

8d. Create a time series graph for a linear DDS.



Watch: Video AI for Equilibrium Values





Dynamical Systems

Writ 1 Lesson 8

ASSIGNMENT: Prepare for Writ 1.



Dynamical Systems

Modeling with DDS PSL 2

OBJECTIVES:1a. Model Real World Problems with Discrete Dynamical Systems.1b. Recognize some of the things you should consider when solving a problem.1c. Explain why military officers must be good problem solvers.

ASSIGNMENT:TBD by your Instructor



Dynamical Systems

Long Term Behavior I Lesson 9

OBJECTIVES:9a. Explain the concept of long-term behavior as it pertains to sequences and discrete dynamical systems.9b. Determine the long-term behavior (LTB) of a discrete dynamical system or sequence from numerical and graphical representations.9c. Describe the long-term behavior of a discrete dynamical system using:

- Words.- Limit notation.



Watch: Video AI for Long Term Behavior





Dynamical Systems

Long Term Behavior II Lesson 10

OBJECTIVES:9a. Explain the concept of long-term behavior as it pertains to sequences and discrete dynamical systems.9b. Determine the long-term behavior (LTB) of a discrete dynamical system or sequence from numerical and graphical representations.9c. Describe the long-term behavior of a discrete dynamical system using:

- Words.- Limit notation.




Dynamical Systems

Analytic Solutions I Lesson 11

OBJECTIVES:10a. Explain what analytic (closed-form) solutions to discrete dynamical systems are.10b. Determine an analytic solution for homogeneous and non-homogeneous linear Discrete

Dynamical Systems.10c. Determine a specified value in a sequence using an analytic solution.10d. Determine the time (or index value) when a sequence obtains a specified value using a logarithmic

transformation of the analytic solution.

ASSIGNMENT: READ: Sections 1.9.1-1.9.4 in Modeling in a Real and Complex World


Watch: Video AI for the Analytic Solution





Dynamical Systems

Analytic Solution II Lesson 12OBJECTIVES:

11a. Classify the equilibrium value of a linear recursion equation based on the value of the a parameter.

11b. Determine the long-term behavior of linear recursion equations based on the value of the a parameter and the initial condition.

ASSIGNMENT: READ: Section 1.9.6 in Modeling in a Real and Complex World


THINK ABOUT: What does the a in a model of the form pn=a pn−1+d indicate about the long-term behavior of the model?




Dynamical Systems

Fundamental Concept Exam Lesson 13

ASSIGNMENT: Prepare for the Fundamental Concept Exam.



Dynamical Systems

Non-Linear DDS I Lesson 14

***Problem Set 1 is due at the beginning of class***

OBJECTIVES:12a. Model situations and solve problems involving non-linear DDS models using the logistic model.12b. Explain the difference between linear and non-linear models.12c. Find the equilibrium values of non-linear models.

ASSIGNMENT: READ: Section 1.10 (stop prior to Theorem 1.10.1) in Modeling in a Real and Complex World

READ FOR FURTHER UNDERSTANDING: Section 9.1 (pages 586-590) in the Stewart Text

DO: WebAssign Lesson 14 (this was two lessons)

WATCH: Video AI for Problem 1.10.9 (MRCW). The lesson numbers, problem numbers, and page numbers referenced in these videos have changed, but the questions are still the same.

THINK ABOUT:1. Is the logistic difference equation a linear difference equation or non-linear? How can you tell?2. How close do you think the Earth is to reaching limits to growth? What do you think would limit

the growth of humans on the planet?

https://lms.westpoint.edu/math/ma103/SiteAssets/SitePages/Video%20AI/MA103%20Problem%201.10.9.pdf

https://webassign.com/



Dynamical Systems

Modeling with Linear DDS PSL3

ASSIGNMENT: TBD by your instructor.



Dynamical Systems

Written Partial Review I A Lesson 15

ASSIGNMENT: READ: WPR I Memo found under the “Exam Instructions” for WPR I on the MA103 course website.

DO: Attempt previous exams under the “Previous Exams” for WPR I on the MA103 course website.

https://lms.westpoint.edu/math/ma103/SitePages/Previous%20Exams.aspx

https://lms.westpoint.edu/math/ma103/SitePages/Exam%20Instructions.aspx



Dynamical Systems

Written Partial Review 1B Lesson 16

ASSIGNMENT: Read the WPR 1B Read Ahead and prepare for WPR 1B


Block IIIntroduction to Vectors

and Matrix Algebra

Block II Outcomes

By the end of Block II Students will be able to use linear algebra techniques to solve systems of linear equations and will have a base of knowledge in certain fundamental concepts.

BLOCK II OUTCOMES

1. Apply common vector and linear algebra concepts.2. Solve a system of equations manually using linear algebra techniques.3. Represent lines and planes using vectors.



and Matrix Algebra

Introduction to Vectors I Lesson 17

OBJECTIVES:16a. Explain two ways that mathematicians or scientists use the term vector.16b. Algebraically calculate vector addition, vector subtraction, and vector scalar multiplication.16c. Graphically represent vector addition, vector subtraction, and vector scalar multiplication16d. Convert vector notation to and from graphical representation of vectors.16e. Determine the displacement vector algebraically and graphically.16f. Using the 3-dimensional rectangular coordinate system, plot points and vectors.16g. Convert vector notation to and from scalar multiplication of the standard basis vectors.16h. Calculate the magnitude of a vector.16i. Given the magnitudes of two vectors and the angle between them, represent the vectors in

the xy-plane using component notation.

ASSIGNMENT:

READ: 1. Pages 792-803 (stop prior to “Components”) in the Stewart Text2. Chapter 2 Intro through Section 2.2 in Modeling in a Real and Complex World


THINK ABOUT:Note the definition of a vector in the Stewart Text on page 798 and compare it to the vector

definition in the Modeling in a Real and Complex World on page 149. Why are there two different definitions?




and Matrix Algebra

Introduction to Vectors II Lesson 18

OBJECTIVES:18a. Compute the dot (scalar) product of two vectors.18b. Determine the dot product of two vectors, given their magnitudes and the angle between them.18c. Use the dot product to find the angle between two vectors.18d. Use the dot product to determine if two vectors are orthogonal (perpendicular).

ASSIGNMENT:READ: Section 2.3 in Modeling in a Real and Complex World

READ FOR FURTHER UNDERSTADING:1. Section 12.3 (pp 807-810, stop just before Directional Angles and Directional Cosines) in the

Stewart Text2. In physics you will see a problem similar to the one below. Solve it with what you just

learned about vectors.1

The Figure 17 below gives an incomplete map of a road rally. From the starting point at the origin, you must use available roads to go through the following displacements:

a. a⃗ to checkpoint Able (point A), magnitude 36 km, due eastb. b⃗ to checkpoint Baker (point B), due northc. c⃗ to checkpoint Charlie (point C), magnitude 25 km, at angle shownd. Your net displacement d⃗ from the starting point is 62.0 km. What is the

magnitude of b⃗?


WATCH: Video AI for Dot Product

1 Halliday, Resnick, and Walker, Fundamentals of Physics (New York: John Wiley & Sons, 2001), p. 39.

Figure 17.





and Matrix Algebra

Introduction to Vectors III Lesson 19

OBJECTIVES:19a. Find a vector equation of a line given:

- 2 points.- 1 point and a line or vector and its relationship to the line.

19b. Find the angle between two lines.19c. Determine if two lines are parallel or perpendicular.19d. Find the shortest distance between a point and a line.

ASSIGNMENT:READ: Section 2.4 in Modeling in a Real and Complex World

READ FOR FURTHER UNDERSTANDING:1. Section 10.1 (p.640-641, stop prior to Example 2) in Stewart2. Section 12.5 (p.823-825, stop prior to “Planes”) in Stewart


WATCH: Video AI for Vector Equation of a Line

THINK ABOUT:If a vector equation with one parameter represents a line, what would you expect a vector

equation with two parameters to represent?





and Matrix Algebra

Introduction to Vectors IV Lesson 20

OBJECTIVES:20a. Find a vector equation of a plane given:

- 3 non-collinear points.- 2 intersecting lines.

20b. Determine if a point is on a plane.20c. Find the shortest distance between a point and a plane.


READ FOR FURTHER UNDERSTANDING: Section 16.6 (p. 1111, stop prior to Example 1, then read Example 3) in Stewart Text


WATCH: Video AI for Vector Equation of a Plane





and Matrix Algebra

Introduction to Matrix Algebra I Lesson 21

OBJECTIVES:21a. Represent an m×n array of numbers as a matrix using proper mathematical notation.21b. Perform the following operations by hand:

- Find the sum and difference of two matrices.- Multiply a matrix by a scalar.- Multiply a matrix by another matrix.- Find the transpose of a matrix.

21c. Explain when the sum, difference, or product of matrices is not defined.






and Matrix Algebra

Introduction to Matrix Algebra II Lesson 22

OBJECTIVES:21a. Represent an m×n array of numbers as a matrix using Mathematica.21b. Perform the following operations using Mathematica:

- Find the sum and difference of two matrices.- Multiply a matrix by a scalar.- Multiply a matrix by another matrix.- Find the transpose of a matrix.

21c. Explain when the sum, difference, or product of matrices is not defined in Mathematica.



Block IIIntroduction to Vectors and

Matrix Algebra

Team Writ 2A Lesson 23

ASSIGNMENT: Prepare for Writ II A.



and Matrix Algebra

Solving Systems of Linear Equations I Lesson 24

OBJECTIVES:23a. Know the result of multiplying any matrix by an identity matrix, when the matrix

multiplication is properly defined.23b. Find the determinant of a 2x2 matrix without using technology.23c. Describe how the value of the determinant of a matrix can tell you whether the inverse of a

matrix exists.23d. Find the inverse of a 2x2 matrix without using technology23e. Transform a system of linear equations into matrix-vector form.23f. Describe a system of linear equations in two unknowns graphically.23g. Know what the determinant of a coefficient matrix can tell about the number of solutions

to the system.2hd. Solve systems of linear equations in two unknowns graphically and using the inverse of a

matrix (without technology).

ASSIGNMENT: READ: 1. Section 2.9 in Modeling in a Real and Complex World 2. Sections 2.10.1 and 2.10.2 in Modeling in a Real and Complex World3. Section 2.11.1 in Modeling in a Real and Complex World


THINK ABOUT:1. Do all matrices have an inverse? 2. What properties must the matrix have for the inverse to exist?3. Can you describe a situation in which a solution to a system of equations might not exist?4. Is it possible to have exactly two solutions to a system of linear equations?5.




and Matrix Algebra

Solving Systems of Linear Equations II Lesson 25

OBJECTIVES:24a. Transform a system of linear equations into an augmented matrix.24b. Know the three types of elementary row operations used in row reduction and their effects

on the set of solutions to a system of linear equations.24c. Understand what reduced-row echelon form means and be able to use elementary row

operations to transform a matrix with two rows into reduced row echelon form by hand.24d. Solve systems of two linear equations by hand using row reduction.






and Matrix Algebra

Solving Systems of Linear Equations III Lesson 26

OBJECTIVES:25a. Know what it means for a system of equations to be inconsistent or redundant, and what

that implies about the reduced row echelon form and its number of solutions.25b. Know how many consistent, independent linear equations a system must have in order for

there to be a unique solution to the n unknowns.25c. Given that a system has an infinite number of solutions, determine at least two solutions.





Block IIIntroduction to Vectors and Matrix

Algebra

Modeling with Matrix Algebra I Lesson 27

OBJECTIVES:27a. Transform flow network scenarios into mathematical models using systems of linear

equations.27b. Solve systems of linear equations of any size using technology (i.e. using Mathematica’s

“Solve” command, row reduction, or matrix inverse techniques).27c. Interpret the solution of the system of linear equations and be able to explain the results.



TECH TIPS: There are three ways Mathematica can help you solve systems of equations: “Solve” command, row reduction, or inverse techniques. Review Appendix B in Modeling in a Real and Complex World for examples of how to use each of the three techniques.




Matrix Algebra

Modeling with Matrix Algebra II Lesson 28

OBJECTIVES:28a. Determine the adjacency matrix for a social network.28b. Find the degree centrality and the normalized degree centrality of a given network.28c. Find the eigenvector centrality and the normalized eigenvector centrality of a given

network.28d. Interpret the results of centrality analysis to identify nodes that are potentially the most

critical nodes in a social network.



TECH TIPS: The following explains how to use Mathematica to find the eigenvector centrality and

normalized eigenvector centrality of the social network described in Figure 3.13 in the MRCW.

Largest Eigenvalue

Eigenvector Centrality

Largest Value in Eigenvector Centrality

Normalized Eigenvector Centrality




Matrix Algebra

Modeling with Matrix Algebra III Lesson 29




Matrix Algebra

Team Writ 2B Lesson 30

ASSIGNMENT: Prepare for Writ 2B.



Matrix Algebra

Systems of Recursion Equations I Lesson 31

OBJECTIVES:33a. Transform a scenario into a system of linear recursion equations.33b. Given an initial condition, iterate a system of recursion equations and interpret the results.33c. Determine the long-term behavior of a system of recursion equations through iteration.33d. Convert a system of recursion equations into matrix-vector form.



TECH TIPS: The following explains how to use Mathematica to solve the system of recursion

Equations described in Example 3.5.2 in the MRCW.

A problem set relating to this block will be assigned at the end of the class and will be due at the beginning of class on 02 November 2018.




Matrix Algebra

Systems of Recursion Equations II Lesson 32

OBJECTIVES:34a. Explain the concept of an equilibrium as it pertains to systems of recursion equations.34b. Compute the equilibrium vector(s) for a system of linear recursion equations.






Matrix Algebra

Systems of Recursion Equations III Lesson 33




Matrix Algebra

PSL 4: Modeling with Systems of Recursion Equations PSL4




Matrix Algebra

Systems of Recursion Equations IV Lesson 34

OBJECTIVES:36a. Transform information about population dynamics (e.g., reproduction and survival

rates) into a system of linear equations.36b. Use a Leslie matrix, as part of an age dependent population model, to predict the

future state of a population subgroup.



TECH TIPS: By using what you learned about modeling a DDS with more than one variable in

Mathematica, you can use the same technique to model age dependent population models.




Matrix Algebra

Systems of Recursion Equations V Lesson 35

OBJECTIVES:37a. Transform a probabilistic scenario into a system of linear equations using a

transition matrix.37b. Use a transition matrix and an initial state vector to determine the probability of

being in a certain state after a specified number of iterations of a stochastic process.



WATCH: Video AI for Problem 3.7.13 (MRCW). The lesson number, problem number, and page number referenced in the video have changed, but the question is still the same. NOTE: This video can only be viewed while connected to the West Point internet.

TECH TIPS: Modeling in a Real and Complex World shows the Mathematica code used to solve

Example 3.7.2. By using this technique you can quickly solve Markov Chains with a large number of states like the Markov Mouse shown later Example 3.7.3





and Matrix Algebra

Eigenvalues and Eigenvectors I Lesson 36

OBJECTIVES:

26a. Know the definition , whereλ is an eigenvalue and v⃗ is an eigenvector.26b. Recognize when a vector is an eigenvector graphically, and estimate the eigenvalue.26c. Calculate the eigenvectors and eigenvalues of a 2x2 matrix without using technology.



WATCH: Video AI example problem 2.13.7

Video AI for Problem 2.13.4 (MRCW). The lesson number and page number referenced in the video have changed, but the question is still the same. NOTE: This video can only be viewed while connected to the West Point internet.






and Matrix Algebra

Eigenvalues and Eigenvectors II Lesson 37

OBJECTIVES:

26a. Know the definition , whereλ is an eigenvalue and v⃗ is an eigenvector.26b. Interpret wat eigenvectors and eigenvalues represent in a system of recursion equations.26c. Calculate the eigenvectors and eigenvalues of a matrix using technology.


DO: WebAssign Lesson 37 (add a lesson?)

WATCH: Video AI example problem 2.13.7

Video AI for Problem 2.13.4 (MRCW). The lesson number and page number referenced in the video have changed, but the question is still the same. NOTE: This video can only be viewed while connected to the West Point internet.






Matrix Algebra

Analytic Solutions to Systems of Recursion Equations I Lesson 38

OBJECTIVE:39a. Determine the constants that result from vector decomposition of

p⃗0− p⃗¿=c1 v⃗1+c2 v⃗2+…+ck v⃗k .39b. Develop analytic (closed-form) solutions (with eigenvector decomposition) to

homogeneous systems of recursion equations.39c. Use the analytic solution to analyze the long-term behavior of a system of recursion

equations.

ASSIGNMENT: READ: Section 3.8.1 and 3.8.2 in Modeling in a Real and Complex World


TECH TIPS: By combining a number of techniques you have learned in Mathematica you can solve

for the analytic solution to Example 3.8.3 in Modeling in a Real and Complex World as shown below:

This gives you the analytic solution:

p⃗n=[00]+(1 ) (3 )n[11]+(0 ) (−1 )n[−11 ]

Simplified,


p⃗n= (3 )n[11]MA103 Mathematical Modeling and Intro to Calculus


Matrix Algebra

Analytic Solutions to Systems of Recursion Equations II Lesson 39

***Problem Set 2 is due at the beginning of class***

OBJECTIVES:40a. Develop analytic (closed-form) solutions (with eigenvector decomposition) to non-homogeneous systems of recursion equations.40b. Use the analytic solution to analyze the long-term behavior of a system of recursion equations.



TECH TIPS: By combining a number of techniques you have learned in Mathematica you can solve

for the analytic solution to Example 3.8.4 in Modeling in a Real and Complex World as shown below:

Analytic solution:

p⃗n=[−30−10]+(43.7321)(1.1)n[0.857493

0.514496 ]+(−0.707114) (0.3 )n[−0.7071070.707107 ]


Note that this gives you a different analytic solution than the one in MRCW. Why is that? (Hint: think about how many possible eigenvectors each A matrix has) Do both solutions give you the same answer?



Matrix Algebra

PSL5: Modeling with Systems of Recursion Equations PSL 5

REVIEW: Review reading assignments from Block III and come to class prepared to ask questions in preparation for WPR II.

ASSIGNMENT: As per your instructor.

BLOCK III OUTCOMES

1. Transform a situation into a mathematical model using a system of linear equations. 2. Solve a system of linear equations using technology.3. Solve a system of linear recursion equations analytically.4. Evaluate and interpret the results of solving a system of linear equations.



Matrix Algebra

Written Partial Review 2A Lesson 40

ASSIGNMENT: READ: 1. WPR III Memo found under the “Exam Instructions” for WPR II on the MA103 course

website.

DO: Attempt previous exams under the “Previous Exams” for WPR II on the MA103 course website.





Matrix Algebra


ASSIGNMENT: Prepare for WPR 2B.


Block IIIModeling with Continuous

Functions

Block III Outcomes

By the end of Block IV Students will understand the conceptual meaning and be able to use the basic ideas of the single variable derivative.

BLOCK IV OUTCOMES:

1. Distinguish common families of functions (linear, exponential, power, and trigonometric) and how their parameters impact their shape.

2. Transform a given set of data into a model using an appropriate continuous function.3. Solve using a continuous model in order to make predictions.4. Evaluate a continuous model and interpret the results of the model’s predictions.5. Understand the foundational concept to single variable calculus:

- Limits.- Continuity.- Rates of change.



Functions

Introduction to Continuous Functions Lesson 42

OBJECTIVES: 43a. Using the definition of a function, determine which relationships are functions and which are not.43b. Describe the four representations of functions.43c. Determine the domain and range of a function.

ASSIGNMENT: READ: Section 1.1 in the Stewart Text





Functions

Introduction to Linear Models Lesson 43

OBJECTIVES:44a. Explain what a parameter is and how changes in the parameters of a function affect its shape.44b. Given a set of data, develop a linear model.44c. Use a linear model to make predictions.44d. Explain the difference between discrete and continuous models.

ASSIGNMENT: READ:1. Section 4.1 in Modeling in a Real and Complex World2. Section 1.2 (pp 14-17) in Modeling in a Real and Complex World

READ FOR FURTHER UNDERSTANDING: Section 1.2 (pp 24-27, start at Linear Models and stop just before Polynomials) in the Stewart Text





Functions

Modeling with Linear Models Lesson 44

OBJECTIVES:45a. Evaluate a model by considering the following:

- Context of the problem - “Eyeball” test (visual inspection)- SSE- Coefficient of Determination

45b. Describe the advantages and disadvantages of using SSE and r2 .45c. Use Microsoft Excel Solver in order to determine the model of best fit.

ASSIGNMENT: READ:

1. Section 5.1-5.2 in Modeling in a Real and Complex World2. Review Section 1.1.6 in Modeling in a Real and Complex World


WATCH: Video AI titled MA103 Coefficient of Determination and MA103 Microsoft Excel Solver

TECH TIPS: Using Excel to calculate SSE and r2 will be critical in evaluating different models. By

using formulas and cell references in Excel it is possible to perform a large number of calculations in a few seconds, and calculate SSE and r2 for hundreds or even thousands of data points.

Using the data from Example 5.1.1 in Modeling in a Real and Complex World you can use Excel to find SSE and r2 by using the following:

Note that the predicted height calculations are “linked” to cells H3 and H4. This is important when you use “Solver” for the remainder of your modeling classes.




Functions

Introduction to Power Functions Lesson 45

OBJECTIVES:46a. Describe and graph the three cases of power functions given in the text. 46b. Describe how changes in the parameters of power functions affect their shape. 46c. Determine the domain and range of power functions.46d. Recognize when a power function would be appropriate to model a real world situation.

ASSIGNMENT: READ: Section 4.2-4.2.1 in Modeling in a Real and Complex World





Functions

Modeling with Power Functions Lesson 46

OBJECTIVES:47a. Given a data set, develop a best fit power model.47b. Use the power model to make predictions.47c. Discuss the results of your mathematical model so a decision maker can understand your solution and analysis.

ASSIGNMENT: READ: Section 4.2.3-4.2.4 (Do not read Method 2) in Modeling in a Real and Complex World


WATCH: Video AI for Problem 4.2.27 (MRCW).

TECH TIPS: Using the “Solve” command in Mathematica you can solve multiple equations with

multiple unknowns. The following uses Mathematica to estimate parameters as shown in Example 4.2.3.

This gives the initial power model:

f ( x )=−5880 ( x−100 )−1+750

This gives the initial power model:

f ( x )=−23192.6 (x−68.7412 )−1+842.444




Functions

Team Writ 3 Lesson 47

ASSIGNMENT: Prepare for Team Writ 3.



Functions

Problem Solving Lab 6: Modeling with Continuous Functions I PSL 6

ASSIGNMENT: TBD your instructor.



Functions

Introduction to Exponential Functions Lesson 48

OBJECTIVES:49a. Describe the properties of and be able to graph the exponential function.49b. Describe how changes in the parameters of exponential functions affect their shape.49c. Determine the domain and range of an exponential function.49d. Recognize when an exponential function would be appropriate to model a real world situation.

ASSIGNMENT: READ: Section 4.3.1 of Modeling in a Real and Complex World

READ FOR FURTHER UNDERSTANDING:1. Section 1.4 in the Stewart Text2. Power Tools article by The New York Times



http://opinionator.blogs.nytimes.com/2010/03/28/power-tools/?src=me&ref=general



Functions

Modeling with Exponential Functions Lesson 49

OBJECTIVES:50a. Given a data set, develop a best fit exponential model50b. Use the exponential model to make predictions.50c. Discuss the results of your mathematical model so a decision maker can understand your solution and analysis.

ASSIGNMENT: READ: Section 4.3.3-4.3.4 (Do not read Method 2) of Modeling in a Real and Complex World


WATCH: Video AI for Example 4.3.3 (MRCW) titled Parameter Estimation – Exponential Function.

TECH TIPS: Using the “Solve” command in Mathematica you can solve multiple equations with

multiple unknowns. The following uses Mathematica to estimate parameters as shown in Example 4.3.4.

The Mathematica red text is warning the user that it had to use numeric approximation to solve the system of equations. Remember, the b parameter must be greater than zero, so choose the solution with the positive b parameter.This gives the initial exponential model:

f ( x )= (35.9019 )1.42054x+1000

The Mathematica red text is warning the user that it had to use numeric approximation to solve the system of equations. This gives the initial exponential model:

f ( x )= (1338.21 ) 1.03986x−340.28




Functions

Intro to Trigonometric Functions Lesson 50

OBJECTIVES:51a. Describe the properties of and be able to graph the trigonometric functions.51b. Describe how the changes in parameters of trigonometric functions affect their shape.51c. Determine the domain and range of trigonometric functions.51d. Recognize when a trigonometric function would be appropriate to model a real world situation.

ASSIGNMENT: READ: 1. Section 4.4.1 in Modeling in a Real and Complex World2. Section 1.2 (Pages 31-42) in the Stewart Text





Functions

Modeling with Trigonometric Functions Lesson 51

OBJECTIVES:52a. Given a data set, develop a best fit trigonometric model.52b. Use the trigonometric model to make predictions.52c. Discuss the results of your mathematical model so a decision maker can understand your solution and analysis.

ASSIGNMENT: READ:

1. Section 4.4.3 – 4.4.4 of Modeling in a Real and Complex World2. Examples 3 and 4 (pages 39-40) in the Stewart Text

READ FOR FURTHER UNDERSTANDING: Section 3.3 (pages 190-195) in the Stewart Text


TECH TIPS: Excel can calculate trigonometric functions and you can use the techniques for

calculating SSE and r2 from previous lessons to evaluate trigonometric models. The Excel file below shows how to predict the illumination for 1-3 June using the model from Example 4.4.3 and the first few data points from Table 4.30 in Modeling in a Real and Complex World.

Note that you need to create a column B for the days starting at 1, otherwise you will be asking Excel to take the sine of 0.2167*(1-June – 12.5).




Functions

Problem Solving Lab 7: Modeling with Continuous Functions II PSL 7




Functions

Written Partial Review 3A Lesson 52

ASSIGNMENT: READ: 2. WPR IV Memo found under the “Exam Instructions” for WPR III on the MA103

course website. 3. WPR IV Room Assignments under the “Exam Instructions” for WPR III on the MA103

course website.

DO: Attempt previous exams under the “Previous Exams” for WPR III on the MA103 course website.






Functions


ASSIGNMENT: Prepare for WPR 3B.



Functions

Rates of Change I Lesson 54

OBJECTIVES: 53a. Explain the geometric interpretation of average and instantaneous rates of change. 53b. Compute average rates of change (slopes of secant lines) and approximate instantaneous rates of change. 53c. Find the equation of a tangent line to a curve.

ASSIGNMENT: READ: 3. Pages 2-5 (Stop prior to “The Limit of a Sequence”) in the Stewart Text4. Section 2.1 in the Stewart Text


THINK ABOUT:In Physics next year, your textbook will define instantaneous velocity as how fast a

particle is moving at a given instant and will use the equation below:

v=lim ¿Δt→0 Δ xΔ t

=dxdt

This equation displays two features of instantaneous velocity v. First, v is the rate at which a particles position x is changing with time at a given instant; that is, v is the derivative of x with respect to t . Second, v at any instant, is the slope of the particle’s position – time curve at the point representing that instant. Velocity is another vector quantity and thus has an associated direction.2

Are these the same thing as what you just read about in Stewart? How are they similar and different?

2 Halliday, Resnick, and Walker, Fundamentals of Physics (New York: John Wiley & Sons, 2001), p. 15.




Functions

Rates of Change II Lesson 55

OBJECTIVES: 54a. Explain what it means for a function to have a limit. 54b. Approximate the limit of a function using graphic and numeric means. 54c. Use the mathematical definition of continuity to determine if a function is continuous at a point.

ASSIGNMENT: READ: 1. Section 2.2 in the Stewart Text 2. Section 2.5 (pages 114-116, stop before Example 3) in the Stewart Text





Functions

Rates of Change III Lesson 56

OBJECTIVES: 54a. Explain what it means for a function to have a limit. 54b. Approximate the limit of a function using graphic and numeric means. 54c. Use the mathematical definition of continuity to determine if a function is continuous at a point.

ASSIGNMENT: READ: 3. Section 2.2 in the Stewart Text 4. Section 2.5 (pages 114-116, stop before Example 3) in the Stewart Text



Functions

Course Review PSL 8

OBJECTIVES:Problems and questions pertinent to the entire MA103 course can be addressed during

this problem solving lab.

ASSIGNMENT:

READ: 1. TEE Memo found under the “Exam Instructions” for TEE on the MA103 course

website. 2. TEE Room Assignments under the “Exam Instructions” for TEE on the MA103 course

website.

DO: Complete and electronically submit course end feedback



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