**Title:** Introduction to Partial Derivatives in a Business Calculus Course

**Discipline(s) or Field(s): **Mathematics

**Authors:** Erick Hofacker, Ioana Ghenciu, Don Leake, Alexandru Tupan, University of Wisconsin-River Falls

**Submission Date:** March 2, 2009

**Executive Summary: **This introductory lesson to partial derivatives to a class of business and social science majors focuses on conceptual understanding in several different ways. It opens with a couple of questions on car loans aimed at assessing the experience and intuition of the class concerning changes in multivariable functions. Then with the help of a computer applet borrowed from MIT the lesson introduces the concept of partial derivatives through its geometrical meaning. TI-89 calculators provide a way for students to easily compute partial derivatives algebraically for a simple polynomial function. Through these two technological tools students explore the relationship between the 3-D graph of a two-variable function and its partials. The 75 minute lesson ends with a couple of partial derivative applications from the fields of business and economics.

The lesson is based on a laboratory/guided discovery approach. Technology is used as a tool for exploration. The learning activities were ordered to achieve understanding first geometrically, then algebraically, and finally through application. Lower-level computational skills were placed in support of higher-level conceptual understanding. Some later questions were directed toward giving students the opportunity to discover connections with previously-learned material. The application portion of the lesson is designed to help students see connections between the mathematics curriculum and other disciplines.

This lesson study reinforced the notion that discovery learning, supported by technology that helps students visualize and compute, is very helpful in the introduction of a conceptually difficult topic such as partial derivatives. The lesson also highlighted the importance of constant and immediate assessment in the classroom. The gulf between an instructor’s perception of student understanding and what is actually the case can be tremendously broad, especially toward the end of a long semester. A third revelation is that usually simpler is better. It is preferable to focus on understanding a few concepts well in the classroom. Finally, the importance of personal contact, student-to-student or student-to-teacher, cannot be overemphasized. While working in a computer lab, the information is right there in the face of the student on the computer screen. In a lengthy classroom or lecture hall, it is far too easy for the weaker student to disengage. In addition every learning environment needs to provide a way for instructors to get within every student’s “sphere of learning.”. Students that are not easily accessed in the classroom, whether in the back of a long classroom or against the wall in a computer lab are in danger of being lost.

**Mathematics: Introduction to Partial Derivatives in a Business Calculus Course (Final Report)**

*Below are links to the lesson materials used to teach it.*

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