Calculus is the study of differentials; It has plethoric real-world applications. There are two basic areas of calculus: integral calculus and differential calculus. The latter deals with finding instantaneous and generalized rates of change of a function f(\chi), whereas the former exclusively deals with finding the area between two functions f_{1}(\chi) and f_{2}(\chi). The fundamental theorem of calculus states that \int \frac{d f(\chi)}{d\chi} \, d\chi = f(\chi). Likewise, \frac{d \int f(\chi) \, d\chi}{d\chi} = f(\chi).

Limits Edit

Sometimes, limits are considered to be a part of calculus. The notation is lim_{\chi \rightarrow \alpha} f(\chi) . Limits are used to describe the behaviour of functions as they approach a discontinuity, asymptote, or \pm \infty. The notation for a limit where \chi approaches \alpha from the positive direction is lim_{\chi \rightarrow \alpha^{+}} f(\chi), as the notation for a limit where \chi approaches \alpha from the negative direction is lim_{\chi \rightarrow \alpha^{-}} f(\chi). Limits can be used to describe differentials. Say we have some line h(\chi> that is secant to f(\chi) at points \beta_{1}, \beta_{3}. We want to find the point derivative at a point \beta_{2}, so we take the limit of the slope of h(\chi) as it approaches the function g(\chi) tangent to f(\chi) at \beta_{2}.

Integration Edit

One of the most basic notions of calculus is the differential, d\chi. This represents an extremely small change in \chi, where \Delta \chi \rightarrow 0. The area under a function must be given by \sum\limits_{-\infty}^{\infty} f(\chi) \cdot d\chi, which is usually represented as the indefinite integral \int f(\chi) d\chi. Another type of integral, the definite integral, is used to represent the area under a curve along an interval \left[\chi_{1}, \chi_{2}\right]. Its notation is \int \limits_{\chi_{1}}^{\chi_{2}} f(\chi) d\chi .

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