It depends on your degree. If it is required by your degree, often an opportunity is provided to take it at the college. Math college classes proceed at a faster pace, so don't procrastinate if you are interested in math oriented degree.
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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
Once you've completed differential and integral calculus, multivariable calculus is often next step, and beyond that there is advanced calculus which generalizes calc to multidimensional spaces and uses vector-valued functions. Often concurrent with high level calculus in college courses is linear algebra and differential equations. There's nothing really 'after' calculus, because any topic in mathematics has a myriad of problems, theories, and potential applications to be explored. Calculus is, however, normally the highest level of math taught in US high schools and is a basic required course for any science/engineering major in college.
Trigonometry isn't required to learn calculus, but it does help you to understand some of the concepts. Geometry, however, is usually required before taking a course in trigonometry.
Yes, most premed students do take 3 semesters of calculus in college.
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.