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.
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Yes calculus is a type of math that deals with the study of continuously changing quantities.
An understanding of the math behind the problem will help you solve the problem. High-level Computer Science is about modeling natural sciences which is where you will need to understand the basics of calculus.
Complex analysis.
yes it is
Some high schools have a class called "math analysis" which is before calculus, so yes. To mathematicians, however, "analysis" is a synonym for "modern" calculus, and as such, there are classes usu. taught at universities with a similar name, but far beyond basic calculus.