## FANDOM

3 Pages

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 $\int \limits_{\chi_{1}}^{\chi_{2}} f(\chi) d\chi$.