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Mathematical Modeling and Applied Calculus

Joel Kilty and Alex M. McAllister
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Jason M. Graham
, on

It seems that it is now fairly common for institutions to offer courses in mathematical modeling or applied calculus to undergraduate students or advanced high school students. Mathematical Modeling and Applied Calculus by Kilty and McAllister offers an interesting, perhaps even novel, textbook option for such a course, and also manages to synthesize calculus and mathematical modeling in a very nice way. In fact, I believe that the book has some very valuable features that perhaps are even seriously lacking in other books for meant such a course.

The two standout features of Mathematical Modeling and Applied Calculus are its synthesized approach to mathematical modeling and calculus and its use of real-world data to motivate mathematical modeling in a manner that is interesting and relevant. Of course, the book covers the usual concepts: limits, derivatives, and integrals of (real) functions, the use of functions for mathematical modeling, and the use of calculus in the analysis of such models. The real-world data in brought-in and used via an accompanying package for the R statistical computing environment. Furthermore, the book makes use of other R packages such as mosaic, which aids student learning by making computations and visualizations simpler to perform. Among other unusual topics, Mathematical Modeling and Applied Calculus covers dimensional analysis, the method of least squares, optimization, and certain aspects of multivariable functions.

The book contains a good number of practice exercises, both “by-hand” and computer-based (using R). The problems test the readers’ understanding of the main concepts and give students an opportunity to apply what they have learned.

I find the choice of topics really interesting. There is an excellent balance of theory and application. Furthermore, most of the theoretical concepts covered in the textbook are those that are essential for gaining facility in basic mathematical modeling. Furthermore, mastery of the content covered in Mathematical Modeling and Applied Calculus would make the learning of many common statistical modeling techniques much easier for students, especially those students majoring in degrees from the life and social sciences.

For those students majoring in degrees from the life and social sciences, this book has a lot to offer. After mastering both the calculus and modeling concepts and use of R as presented in the text, such students should be well-prepared to actually use calculus and mathematical modeling in their major coursework and likely even in research. I definitely recommend this book for consideration as a textbook for a course on calculus and/or mathematical modeling for students majoring in life or social science disciplines.

After reading through Mathematical Modeling and Applied Calculus, I was struck by a thought. One can debate what is the appropriate post-algebra/geometry course for high school students to take in their quantitative curriculum. A (traditional) course in calculus is of course a common option, but recently arguments have been given for doing away with high school calculus courses. Proponents of this have said that courses in statistics or computer programming are better suited for providing the necessary quantitative skills for “today’s students”. The main line of reasoning for the argument against high school calculus being that we live in a highly data-centric world.

What struck me while reading Mathematical Modeling and Applied Calculus is that calculus (particularly via its relation to mathematical modeling) has a lot to do with data, perhaps just as much as statistics and programming do. Furthermore, using R (or some similar computing environment) as is done in Mathematical Modeling and Applied Calculus for the purpose of mathematical modeling adds additional benefit. Perhaps a course based on this book would be an additional interesting and valuable option for a post-algebra/geometry mathematics course for high school students, one that does not require eliminating a high school calculus experience but still provides valuable skills for students entering into a data-centric world.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

1. Functions for Modeling Data
1.1. Functions
1.2. Multivariable Functions
1.3. Linear Functions
1.4. Exponential Functions
1.5. Inverse Functions
1.6. Logarithmic Functions
1.7. Trigonometric Functions
2. Mathematical Modeling
2.1. Modeling with Linear Functions
2.2. Modeling with Exponential Functions
2.3. Modeling with Power Functions
2.4. Modeling with Sine Functions
2.5. Modeling with Sigmoidal Functions
2.6. Single Variable Modeling
2.7. Dimensional Analysis
3. The Method of Least Squares
3.1. Vectors and Vector Operations
3.2. Linear Combinations of Vectors
3.3. Existence of Linear Combinations
3.4. Vector Projection
3.5. The Method of Least Squares
4. Derivatives
4.1. Rates of Change
4.2. The Derivative as a Function
4.3. Derivatives of Modeling Functions
4.4. Product and Quotient Rules
4.5. The Chain Rule
4.6. Partial Derivatives
4.7. Limits and the Derivative
5. Optimization
5.1. Global Extreme Values
5.2. Local Extreme Values
5.3. Concavity and Extreme Values
5.4. Newton's Method and Optimization
5.5. Multivariable Optimization
5.6. Constrained Optimization
6. Accumulation and Integration
6.1. Accumulation
6.2. The Definite Integral
6.3. First Fundamental Theorem
6.4. Second Fundamental Theorem
6.5. The Method of Substitution
6.6. Integration by Parts