Calculus is mainly about limits, which in turn are used to calculate the slope of a line (known as the "derivative"; lots of applications for that), and to calculate the area under a curve (the "integral" - also lots of applications for that). For more details, read the Wikipedia article on "Calculus", or read an introductory book on calculus. As prerequisites, you should be well-acquainted with high-school algebra.
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The mathematical field known as calculus studies rates of change. Calculus is interesting because it brings together most of the mathematical concepts that you learn before taking calculus, such as algebra, trigonometry, and functions, and gives them very realistic applications. One of the most applicable and understandable rates of change for those who have not taken calculus is speed. Speed is the rate of change in position over time, and is studied in depth in every calculus class.
No. Calculus if a field of mathematics.
Leibniz and Newton are usually considered the most famous in the field of calculus.
It depends on your field of expertise and specialization in your field. If you are under the sciences, then it likely that you must know it since knowledge about calculus is required in higher mathematics used in the field. If, say, your field is Psychology, most likely it is uneeded.
Anti-derivatives are a part of the integrals in the calculus field. According to the site Chegg, it is best described as the "inverse operation of differentiation."