# Calculus Index

Calculus is a subject in mathematics that is all about making curvy things look straight.

## Notation

The syntax for a limit is a the abbreviation "lim" followed an expression. Underneath the letters "lim" is the value the variable approaches within the expression denoted as the variable with an arrow to the value it is approaching.

The first derivative of a function is denoted by a apostrophe after the function name. Alternatively the partial symbol can be used to represent the derivative with respect to a variable.

The notation for an integral in mathematics is a slanted vertical line with a start and end value that describe the range of the integral. This is followed by the function being integrated and the variable with respect to which the integral is being evaluated.

The nabla symbol is used to represent taking the gradient of a function.

## Operators

The derivative is a concept related to calculus and the slope of a function.

The gradient operator returns a vector representing the change in a function at a point. The operator is similar to the derivative operator of calculus, the difference being that it operates on functions of higher dimension.

An integral can be geometrically interpretted as the area under the curve of a function between the two points a and b. The concept is used throughout physics and higher level mathematics.

The limit operator is used to evaluate the behavior of functions and series.

## Concepts

The fundamental theorem of calculus relates integration to differentiation by defining the integral of a continuous function on a closed bounded interval.

The intermediate value theorem states that if f is a continuous function an contains the interval [a,b], then f takes on every value at least once between f(a) and f(b) at some point in the interval.

The mean value theorem states that for a function f that is continuous on the closed interval [a,b] and differentiable on the interval (a,b) there exists some point c where a < c < b whose slope is equal to the slope formed by the points a and b.

Riemann Sum is a method for approximating the area underneath a continuous function. To find the Riemann's sum, divide the area under the curve into n equal width rectangles. Then calculate the area of each rectangle and sum the results together.

A series is the operation of adding an infinite number expressions together. A series is denoted using the summation operator and the index variable k.

## Examples

To find the line tangent to a curve at a point.

To calculate the Riemann sum of a function using the left approximation method. negative x squared plus four x from zero to four using the left approximation method.