Tangent lines are often first introduced to students in geometry during the study of circles. The topic may be repeatedly reintroduced to students in different contexts throughout their schooling, and often each reintroduction is accompanied by a new, nonequivalent definition of tangent lines. In calculus, tangent lines are again reintroduced to…

The purpose of this paper is to describe (a) multivariable calculus students' meanings for the domain and range of single and multivariable functions and (b) how they generalize their meanings for domain and range from single-variable to multivariable functions. We first describe how students think about domain and range of multivariable functions…