calculus

• the study of change
• calculus is Latin for small pebble (calx), as stones were used for counting and calculations
• branches
  • differential calculus
  • integral calculus
  • stochastic calculus
  • vector calculus
  • lambda calculus

fundamental theorem of calculus (FTC) §

The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related.

The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related.

1. First Part (FTC1): The process of differentiation and integration are inverses of each other. If $F$ is defined as the integral of $f$, then the derivative of $F$ is $f$.
2. Second Part (FTC2): The definite integral of a function $f$ over an interval $[a,b]$ can be computed using any antiderivative $F$ of $f$. Specifically, it is the difference $F(b)-F(a)$.

First FTC §

Let $f$ be a continuous function on $[a,b]$. Define a function $F$ by

$$F(x)={\int }_a^{x}f(t)dt.$$

Then, $F$ is continuous on $[a,b]$, differentiable on $(a,b)$, and

$$F^{\prime }(x)=f(x)$$

for all $x$ in $(a,b)$.

Second FTC §

Let $f$ be a continuous function on $[a,b]$, and let $F$ be any antiderivative of $f$ on $[a,b]$. Then,

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$