The Calculus is the area of mathematics concerned with continuous change.  It is a subject with a long and rich history and prehistory, but its key features appear first in the independent works of Newton and Leibniz from the late 17th-century.  The development of the subject lasted into the 19th-century, and it took nearly another hundred years to solidify the basic mathematical foundations that underly the subject.  In fact, like so many subjects in mathematics with a long history, it is still actively studied and developed today in various guises and in more modern forms.  New ideas are being discovered all the time!

This course, The Principles of Calculus, is a reimagining of the standard freshman calculus curriculum from a "principle-based" perspective.  The perspective of our course is that learning a subject in mathematics is learning to reason using principles in the framework of a sufficiently expressive language.  This course introduces eight principles of reasoning, a mathematical language that supports their application, and a collection of spatial skills to support concept learning and information processing.  

The course will take you through a landscape of ideas that begins with learning how to visualize the decomposition of sets into subsets and a line into line segments, and culminates in the construction of a simple but reasonably accurate model for planetary motion.  At the end of this course, you will deeply understand our place in the solar system.  Developing this understanding is one of the greatest triumphs of human thought and creativity.

Each chapter of our course focuses on the development of one principle of reasoning, the language necessary for the utilization of the principle, and the interaction of the newly introduced principle with the principles that preceed it in the course.  The chapter titles reflect the principle that the chapter introduces:

• Chapter I  Decomposition

• Chapter II  Transformation

• Chapter III  Rigidity

• Chapter IV  Symmetry

• Chapter V  Finite Approximation

• Chapter VI  Local Linear Apprixomation

• Chapter VII  Higher Order Approximation

• Chapter VIII  Integration

The workbook contains both instructional slides and unworked examples as exercises. Work through the exercises as you do your readings.  Annotated slides are provided and can be found below the links to the slides. The annotated slides sketch solutions to the examples with varying degrees of completeness. Students should be sure to independently read through the slides and work through the examples.

These are supplementary problems to give you more practice.

This is a minimal list of exercises that a student should work through in order to develop an understanding of the material at both a basic an intermediate level.

These are computer graded problems to help students build basic skills.  The more you work on these, the stronger your basic skills will become!

These exercises, presented in slide format for in-class use, form the basis for a pedagogical approach to teaching the course that centers on problem solving with language-building reinforcement.

These exercises come from the full list of core exercises, but focus more on spatial thinking.  Students should solve these exercises before class and then review other students' work in class.

Worksheets are intended for instructors, teaching asssistants, and tutors to develop their own skills and for students who seek to go above and beyond the basic course to develop a deeper level of mastery.