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Active Prelude to Calculus

Preface Our Goals

This text is designed for college students who aspire to take calculus and who either need to take a course to prepare them for calculus or want to do some additional self-study. Many of the core topics of the course will be familiar to students who have completed high school. At the same time, we take a perspective on every topic that emphasizes how it is important in calculus. This text is written in the spirit of Active Calculus 6  and is especially ideal for students who will eventually study calculus from that text. The reader will find that the text requires them to engage actively with the material, to view topics from multiple perspectives, and to develop deep conceptual undersanding of ideas.
Many courses at the high school and college level with titles such as “college algebra”, “precalculus”, and “trigonometry” serve other disciplines and courses other than calculus. As such, these prerequisite classes frequently contain wide-ranging material that, while mathematically interesting and important, isn’t necessary for calculus. Perhaps because of these additional topics, certain ideas that are essential in calculus are under-emphasized or ignored. In Active Prelude to Calculus, one of our top goals is to keep the focus narrow on the following most important ideas.
  • Functions as processes. The mathematical concept of function is sophisticated. Understanding how a function is a special mathematical process that converts a collection of inputs to a collection of outputs is crucial for success in calculus, as calculus is the study of how functions change.
  • Average rate of change. The central idea in differential calculus is the instantaneous rate of change of a function, which measures how fast a function’s output changes with respect to changes in the input at a particular location. Because instantaneous rate of change is defined in terms of average rate of change, it’s essential that students are comfortable and familiar with the idea, meaning, and applications of average rate of change.
  • Library of basic functions. The vast majority of functions in calculus come from an algebraic combination of a collection of familiar basic functions that include power, circular, exponential, and logarithmic functions. By developing understanding of a relatively small family of basic functions and using these along with transformations to consider larger collections of functions, we work to make the central objects of calculus more intuitive and accessible.
  • Families of functions that model important phenomena. Mathematics is the language of science, and it’s remarkable how effective mathematics is at representing observable physical phenomena. From quadratic functions that model how an object falls under the influence of gravity, to shifted exponential functions that model how coffee cools, to sinusoidal functions that model how a spring-mass system oscillates, familiar basic functions find many important applications in the world around us. We regularly use these physical situations to help us see the importance of functions and to understand how families of functions that depend on different parameters are needed to represent these situations.
  • The sine and cosine are circular functions. Many students are first introduced to the sine and cosine functions through right triangles. While this perspective is important, it is more important in calculus and other advanced courses to understand how the sine and cosine functions arise from a point traversing a circle. We take this circular function perspective early and first, and do so in order to develop deep understanding of how the familiar sine and cosine waves are generated.
  • Inverses of functions. When a function has an inverse function, the inverse function affords us the opportunity to view an idea from a new perspective. Inverses also play a crucial role in solving algebraic equations and in determining unknown parameters in models. We emphasize the perspective that an inverse function is a process that reverses the process of the original function, as well as important basic functions that arise as inverses of other functions, especially logarithms and inverse trigonometric functions.
  • Exact values versus approximate ones. The ability to represent numbers exactly is a powerful tool in mathematics. We regularly and consistently distinguish between a number’s exact value, such as \(\sqrt{2}\text{,}\) and its approximation, say \(1.414\text{.}\) This idea is also closely tied to functions and function notation: \(e^{-1}\text{,}\) \(\cos(2)\text{,}\) and \(\ln(7)\) are all symbolic representations of exact numbers that can only be approximated by a computer.
  • Finding function formulas in applied settings. In applied settings with unknown variables, it’s especially useful to be able to represent relationships among variables, since such relationships often lead to functions whose behavior we can study. We work throughout Active Prelude to Calculus to ready students for problems in calculus that ask them to develop function formulas by observing relationships.
  • Long-term trends, unbounded behavior, and limits. By working to study functions as objects themselves, we often focus on trends and overall behavior. In addition to introducing the ideas of a function being increasing or decreasing, or concave up or concave down, we also focus on using algebraic approaces to comprehend function behavior where the input and/or output increase without bound. In anticipation of calculus, we use limit notation and work to understand how this shorthand summarizes key features of functions.