Resumen de Discovering Calculus on the Surface

Aaron Wangberg, Ben Jonhson

• 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.