Advanced Problem Solving with Maple

A First Course

William P. Fox, William C. Bauldry

• Publisher: Chapman and Hall/CRC
• Year: 2019
• eBook ISBN: 9780429891342
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Problem Solving is essential to solve real-world problems. Advanced Problem Solving with Maple: A First Course applies the mathematical modeling process by formulating, building, solving, analyzing, and criticizing mathematical models. It is intended for a course introducing students to mathematical topics they will revisit within their further studies.

The authors present mathematical modeling and problem-solving topics using Maple as the computer algebra system for mathematical explorations, as well as obtaining plots that help readers perform analyses. The book presents cogent applications that demonstrate an effective use of Maple, provide discussions of the results obtained using Maple, and stimulate thought and analysis of additional applications.

Highlights:

• The book's real-world case studies prepare the student for modeling applications
• Bridges the study of topics and applications to various fields of mathematics, science, and engineering
• Features a flexible format and tiered approach offers courses for students at various levels
• The book can be used for students with only algebra or calculus behind them

About the authors:

Dr. William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. Currently, he is an adjunct professor, Department of Mathematics, the College of William and Mary. He received his Ph.D. at Clemson University and has many publications and scholarly activities including twenty books and over one hundred and fifty journal articles.

William C. Bauldry, Prof. Emeritus and Adjunct Research Prof. of Mathematics at Appalachian State University, received his PhD in Approximation Theory from Ohio State. He has published many papers on pedagogy and technology, often using Maple, and has been the PI of several NSF-funded projects incorporating technology and modeling into math courses. He currently serves as Associate Director of COMAP's Math Contest in Modeling (MCM).

*Please note that the Maple package, "PSM", is now on the public area of the Maple Cloud. To access it:

• From the web:

1. Go to the website https://maple.cloud

2. Click on "packages" in the left navigation pane

3. Click on "PSM" in the list of packages.

4. Click the "Download" button to capture the package.

• From Maple:

1. Click on the Maple Cloud icon (far right in the Maple window toolbar).

Or click on the Maple Cloud button on Maple's Start page to go to the website.

2. Click on the "packages" in the navigation pane

3. Click on "PSM" in the list of packages.

The package then downloads into Maple directly.

1

Introduction to Problem Solving and Maple

Objectives

(1) Understand the nature of problem solving.

(2) Understand the use of Maple commands.

(3) Understand the Maple Applications Center and its uses.

1.1 Problem Solving

What do we mean by problem solving? We interpret this as having a real problem whose understanding and solution requires quantitative analysis and one or more solution techniques using mathematics. To put into context, we say we need a well-defined problem. After we have a well-defined problem, we must brainstorm variables and assumptions that might impact the problem. We build or select a known model and choose a solution technique or combinations of techniques to obtain answer. We solve and perform sensitivity analysis. We interpret all results, implement, and if necessary, refine the entire process.

In many ways, this process is very similar to the mathematical modeling processes described in other texts: Albright [A2011], Giordano et al [GFH2013], and Meerscheart [M2007] to name a few. Readers may want to examine these texts for a more detailed approach. As a co-author of the Giordano text, my approach is most similar to the approach we describe in that text.

There are four- and five-step processes for problem solving. We present and describe a simple five-step method.

Step 1. Define and understand the problem.

Step 2. Develop strategies to solve the problem. This includes a problemsolving formulation including a methodology to obtain a solution. If data is available, examine the data, plot it, and look for patterns.

Step 3. Solve the problem formulated in Step 2.

Step 4. Perform a self-reflection of your process. You want to make sure the solution answers the problem from Step 1. You also want to ensure the results pass the “commonsense” test. If not go back to Step 2 and reformulate the strategy.

Step 5. If necessary, extend the problem.

We will not concentrate on the modeling portion but on the selection of the model and the solution technique processes including the use of technology in the solution process.

One key point is that our results must pass the “commonsense” test. For example, we were conducting spring-mass experiments in a classroom on the 3rd floor of our mathematics and science building. The simple purpose was to investigate Hooke’s Law. The springs were small and the weights varied from a fraction to about 50 grams. After the experiments, we asked the students to calculate the stretch of their spring if it were attached to a seat, and they sat on the seat. Every student found an answer but none said the spring would most likely break long before it stretched that far.

Let’s preview a problem we will see in Volume I and II. We have data for time (t) and an index (y) from [0, 100]. Our plot shows a negative linear trend. We compute the correlation which is −0.94, and is interpreted as a strong negative linear relationship. We use linear regression to build a regression equation which has some very good diagnostics, but one questionable diagnostic from the residual plot. The main goal is predicting the future, which is why the problem is being solved in the first place. The answer for y comes out negative which is not a possible answer for y. So we continue our problem solving and correct the residual plot issue by adding a quadratic term to the regression equation. Again, our diagnostics are all excellent this time. We attempt to use the model to predict but our answer does not pass the commonsense test as it is too large. A simple plot shows that for the time value in the future we are on the increasing past of the quadratic polynomial. If we cannot use our regression equation then our work is useless. Now, we continue on the nonlinear regression and use an exponential function to fit our data. Finally, not only are all the diagnostics excellent, but our use of the new regression equation passes the commonsense test.

We also believe that in the 21st-century, technology is a key element in all problem solving. Technolo...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction to Problem Solving and Maple
• 2 Introduction, Basic Concepts, and Techniques in Problem Solving with First-Order, Ordinary Differential Equations
• 3 Introduction, Basic Concepts, and Techniques in Problem Solving with Systems of Ordinary Differential Equations
• 4 Problem Solving with Linear, Integer, and Mixed Integer Programming
• 5 Model Fitting and Linear Regression
• 6 Statistical and Probabilistic Problem Solving with Maple
• 7 Problem Solving with Simulation
• Index