Inverse Problems : Activities for Undergraduates (Classroom Resource Material)

Rating:
Summary: Problems from perspectives I has not thought of
Review: While the basics of inverting functions are part of the pre-calculus curriculum, applying inverse operations is not given the attention that it should. Granted that differentiation and integration are inverse operations and that inverting matrices is part of the linear algebra class, this truly does not do justice to how ubiquitous inverse problems are.
Three general principles generally apply when looking at functions and their inverses.

* One direction is much harder than the other.
* One direction does not yield a unique solution.
* It may not be possible to perform the inverse of a function.

For example, while differentiation yields a unique answer, antidifferentiation has an infinite family of solutions. Multiplying two numbers together is easy, factoring is hard.
Many of the problems in this book arise from the study of physical phenomena. This is only fitting, in that collecting data and constructing an equation to model it is a classic form of inverse problem. The range of problems is considerable and quite impressive. While some are fairly standard issue, there were many that took approaches that I have never considered.
There are several levels to the problems: precalculus, calculus, differential equations and linear algebra. With solutions to most included, this is a treasure chest of unusual problems perfect for advanced presentations, extra credit and the occasional examination puzzler. Some of the problems require the use of a symbolic mathematics package and MATLAB is the one used throughout. However, with so many problems, the absence of MATLAB would not be sorely felt.
As a steady reader of math books, I sometimes grow tired of seeing only slight variations of the same old problems. In this book, I saw many different problems from perspectives that I was unaware of. That was exciting, stimulating and refreshing. I wish there were more just like it. I can almost guarantee that if you teach precalculus, calculus, differential equations or linear algebra, there will be something in here that you can use to improve the quality of the course.