Mathematics Foundation

This can be used as a app or an additional reference app by university students attending a course in algebra, trigonometry, geometry, or calculus. As a calculus app, this app is unique in that it contains all the mathematics (and more) that students will need to know in order to be successful in a calculus course. For this reason, the app will be invaluable for students who may need to fill in some “gaps” in their mathematical background, or review certain topics, while attending a university level calculus course. (Calculus professors will normally not have the time to do this!)

There is a strong emphasis in this app on the unfolding and development of the concepts, and there are comments throughout the app to help the reader trace the historical development of mathematics. Of course, students also need to develop their skills. For this reason, there are many exercises included at the ends of the chapters, and students are encouraged to do all of them as an essential part of working through the app. The range of topics in this app, and examples that demonstrate their interplay and interconnectedness is another unique aspect of the app. This is a rewarding aspect of learning mathematics, which university students are typically not exposed to because of the way current-day university courses are organized.

The four chapters that make up the introduction to algebra are Chapters 1 (The Laws of Algebra), 2 (The Cartesian Plane), 3 (Solving Equations and Factorizing Polynomials), and 6 (Techniques of Algebra). The chapters dealing with trigonometry are Chapters 4 (Trigonometry) and 10 (Spherical Trigonometry). They can be read independently (after the first two chapters have been read, if needed). Chapter 5 (Functions), which is an introduction to functions, can be regarded as the beginning of calculus because the operations of calculus are applied to functions. The concepts and methods relating to the calculation of limits, which underlie the operations of calculus, are introduced in Chapter 7 (Limits). In Chapter 8 (Differential Calculus), the definition of the derivative, the rules for computing derivatives, and the formulas for derivatives of all the standard types of function (i.e., polynomial, rational, trigonometric, root, absolute value, exponential, and logarithmic functions) are introduced. Chapter 10 (Euclidean Geometry) presents many of the theorems that are important in Euclidean Geometry, along with some guides to solving problems in geometry.

Because many students have difficulties with algebra when they enroll in a university calculus course, a first semester course in calculus may well consist of Chapters 5–8. It should also be mentioned that Chapter 6 includes an introduction to partial fractions, a topic that is usually not taught to students until they reach their second semester of calculus.

It is usual, for the rules for limits to be taken as the starting point for the evaluation of limits. In Chapter 7, the slightly different approach to evaluating limits is to take as a fact the continuity of all of the standard functions and any algebraic combinations and compositions of the standard functions. (This fact can be proved using methods of real analysis, which is too advanced for this app.) This means that a limit to any point in the domain of any of these functions can be evaluated simply by making a substitution into the function (i.e., by an application of the equation of continuity). The rules for limits are introduced, instead, at the end of the chapter, where they are used to evaluate certain limits involving trigonometric functions.