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They can be taken at the same time.
Pre-calculus is supposed to be a stringent review of trig and algebra in preparation for calculus. So, pre-calculus, I would say.
In short, no. Elementary calculus includes finding limits, basic differentiation and integration, dealing with sequences and series, and simple vector operations, among other concepts. Pre-calculus mostly focuses on the algebra necessary to perform those operations, with perhaps some introduction to limits or other simple ideas from elementary calculus.
Well it depends on which calculus class you are longing for. If you want AB Calculus then you are not required to take pre-calculus; however you must finish all of state's required math courses. Which is probably Algebra, geometry, algebra 2/trig.If you want BC calculus, you need Pre calc and all of the required math classes. BC calculus is a lot more difficult and it will require a lot more time after school. If you are not willing to spend as much time as required , dont dare to take it