Calculus is an important tool for building mathematical models of the world around us and is thus used in a variety of disciplines, such as physics and engineering. These disciplines rely on calculus courses to provide the mathematical foundation needed for success in their courses. Unfortunately, due to the basal conceptions of what it means to understand calculus, many students leave their calculus course(s) with an understanding misaligned with what is needed in the follow-on discipline courses and are thus ill-prepared. By working with presumed experts (undergraduate mathematics and other discipline faculty members) to develop a small number of prototype tasks that elicit, document, and measure students' understanding of a few calculus concepts they believe are essential to successful academic pursuits within their respective disciplines, this study documents how the faculty participants' underlying conceptions about understanding changed and converged. At the end of the study, the faculty participants exhibited a greater preference for conceptual aspects of understanding rather than procedural aspects in their articulations of what it means to understand calculus and what it means to apply calculus: application is not just a melding of understanding and procedural knowledge, it is taking that understanding and internalizing it and using it in a new way (i.e., in situations with structural similarities). Even though the faculty participants converged to this unified definition of application, their definitions of understanding remained separate. Teachers of calculus push students to develop a deep understanding of calculus concepts such that they can explain it to another student well enough to make that student understand; whereas the users of calculus expect the students to walk into their classrooms with a deep understanding of the concepts such that they can recognize a concept within the given situation or context, select a calculus tool that will efficiently get them an answer, compute the answer, and make an accurate determination or prediction. As this study revealed, the end goal of the calculus courses and the beginning goal of the client disciple courses do not align. Implications for calculus instruction and curriculum are mentioned. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]