Discovering Calculus on the Surface

Abstract

Students can often solve single-variable calculus problems by connecting expressions like to the graphical features of a function, but the difficulty of sketching and visualizing multivariable functions often leaves students to use only algebraic methods to solve higher-dimensional problems. To help students strengthen their geometric understanding, the authors designed real, tangible surfaces cut with a Computer Numerical Control machining center upon which students could discover key properties of multivariable calculus. This paper describes the surfaces and four activities covering level curves, gradient vectors, directional derivatives, and Lagrange multiplier problems. Student and instructor reflections on using the surfaces in class are also included.

Keywords:

• Multivariable calculus
• visualization
• discovery
• contour plot
• level curves
• gradient
• directional derivatives
• Lagrange multipliers
• manipulative

ACKNOWLEDGEMENT

The authors would like to thank the Engineering Department at Winona State University for the use of equipment.