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Nonequivalent definitions and student conceptions of tangent lines in calculus

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 students with the standard definition which is dependent on the derivative. This study will examine how students' experiences with tangent lines inside and outside of the calculus classroom can affect their understanding of the concept.