Calculus I and II
Authors: Mairead Greene, Ph.D. and Paula Shorter Ph.D.
Written while both authors were faculty members at Rockhurst University, Kansas City, MO
Partially funded by the Educational Advancement Foundation through a 2011 AIBL Small Grants, Category 2 grant, "Calculus I and II – Concepts and Applications", we have written a complete set of student-centered, in-class activities for a year-long Calculus I and Calculus II sequence, accompanied by instructor support materials.
The Calculus I and II courses that we teach at Rockhurst University are introductory, first-year courses (with typical, non-proof-based differential and integral calculus content) designed to build strong conceptual understanding, critical reasoning, and problem solving abilities in a diverse population of STEM and business students. We believe that these students are best served by developing their ability to independently reason from a deep understanding of calculus concepts to answer novel questions in any discipline (including math).
In order to help our students achieve this goal we have written a set of student-centered, in-class activities that actively engage students on a daily basis in discovering concepts, connecting their discoveries with their own prior understanding, conjecturing and experimenting to further their understanding, applying concepts to solve both purely mathematical and real world problems, and always communicating their understanding (in both written and oral forms). These in-class activities are comprised of a series of carefully scaffolded questions that prompt extensive student investigation (individually or in small groups), preceding full-class discussion and consensus-building. Often these questions begin in an applied setting (where students are typically more comfortable experimenting and conjecturing), and as the activity progresses students are asked to generalize their understanding to more abstract situations. In addition, questions require students to move between different representations (numerical, graphical, and analytical), deepening their understanding of the concepts. Although as instructors we often bring the class together to discuss results, review understanding, or motivate the next activity, there is no need for us to ever lecture on content as students discover all content in the context of these in-class activities.
Our Calculus I course covers standard rate of change topics, building to instantaneous rate of change from average rate of change (including differentiation rules) and applying these concepts to solve typical optimization and related rates problems. In addition, we develop an understanding of differential equations, learn to build differential equation models and use numerical methods and some basic antiderivatives to estimate or find solutions.
Our Calculus II course begins with an initial study of sequences and series as an introduction to infinite sums. The definite integral is then developed from Riemann sums/integrals and students are asked to compare and contrast these two different types of infinite sums. Fundamental Theorems of Calculus, power series (including Taylor series) and standard techniques of integration are covered. In addition, probability and statistics, area and volume and total change are utilized as applications throughout.
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