Students taking a Calculus course for the first time at Concordia University are mature students returning to school after an extended period of time away from formal education, or students lacking the prerequisites to enter into a science, technology, engineering, or mathematics (STEM) related field. Thus, an introductory Calculus course is the gateway for many STEM programs, inhibiting students’ academic progression if not passed. Calculus tends to be construed as a very difficult subject. This impression may be due to the fact that this course is taught in a condensed form, with limited class time, new knowledge (concept, type of problem, technique or method) introduced every week, and little practice time. Calculus requires higher order thinking in mathematics, compared to what students have previously encountered, as well as many algebraic techniques.  As will be shown in this thesis, algebra plays an important role in solving problems that usually make up the final examination in this course.  Through detailed theoretical analysis of problems in one typical final examination, and solutions produced by 63 students, we have identified the prerequisite algebraic knowledge for the course and the specific difficulties, misconceptions and false rules experienced and developed by students lacking this knowledge. We have also shown how the results of our analyses can be used in the construction of a “placement test” for the course – an instrument that could serve the goal of lessening the failure rate in the course, and attrition in STEM programs, by avoiding having underprepared students.