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Pre-calculus is supposed to be a stringent review of trig and algebra in preparation for calculus. So, pre-calculus, I would say.
Simple answer: Calculus involves derivation and integration, precal doesn't. Pre calculus gives you some of the algebraic, geometric and trigonometric understanding that is required to comprehend the concepts in calculus. Without the knowledge from precal, calculus would not be easily understood, as it is taught in schools today.
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You must have a strong basis in Algebra, Algebra II, Geometry and Trigonometry. Usually high schools offer a pre-Calculus course which is somewhat of a conglomeration of the aforementioned courses. Then you would move into differential calculus, integral calculus, vector (multi-variable) calculus, and finally differential equations, which is considered to be at the top of the hierarchy of the calculus courses. So take Algebra, Algebra II, Geometry and Trigonometry to get your strong foundation before begining the calculus sequence.
I think by "regular calculus" it is meant simple derivatives and integrations. Regular calculus would be first year calculus probably not including multi-variable calculus or calculus of variations or vector calculus. Wikipedia gives a good explanation of calculus. If you want to sound smart, call it "The Calculus". It is the study of the rate of change (how things change, in relation to other things, often time) In most Universities, regular calculus are the standard analysis of Calculus, concentrating more on the application of it rather than the concept. in comparison is either called "advanced calculus" or in my U, "Honours Calculus" which takes analysis to a whole new level. Both first year course, but the advanced one concentrates on the understanding of concepts, theorems rather than applications alone. It comes in the form of "mathematical proof". Regular Calculus does proofs too, but not as often. --------------------------------------------- Regular calculus is most probably calculus taught in high school or university level, which is simple, mostly single-variable calculus. But then, there are also different calculi called non-Newtonian calculi. These are the non-standard, non-regular calculi, in which different operators are defined. For example, "regular calculus" might mean an additive calculus (where the integral is defined by adding up extremely small pieces), while an integral in multiplicative calculus might involve multiplying infinitely many pieces close to 1.