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# calculus

Calculus notations

## Calculus

 The maximum of the continuous function in the range to . The minimum of the continuous function in the range to . The supremum of the function over the interval , i.e. the smallest real number that is greater than or equal to every value of with . The infimum of the function over the interval , i.e. the biggest real number that is smaller than or equal to every value of with . This denotes an infinite sequence of real numbers , , , ... that are given through some particular formula depending on the index of each term. For example the sequence is the sequence of terms This denotes the limit of the sequence , whenever it exists. Intuitively this is the value that the term approaches as the index gets closer and closer to infinity. It can be shown for example that This is called an infinite series of the sequence and it is defined through a sequence of partial sums whose terms are given by the formula Whenever it exists, the limit of the sequence is called the value of the infinite series and thus This is called an infinite product of the sequence and it is defined through a sequence of partial products whose terms are given by the formula Whenever it exists, the limit of the sequence is called the value of the infinite product and thus This denotes the limit of the function at point , whenever it exists. Intuitively this is the value that the function approaches as the argument gets closer and closer to . It can be shown for example that This denotes the first derivative of the function at point , whenever it exists. It can be defined through the following formula using limits This denotes the -th derivative of the function at point , whenever it exists. It can be defined recurrently through the following formulae The partial derivative of a function , with respect to the -th variable, at a point . This is defined through the following formula Basically this evaluates the first derivative of the function , defined by Therefore it will also be valid to write As an example consider the function given by . Using the basic differentiation rules it is easy to see that If is a function then the -th order partial derivative of with respect to the -th variable is defined through the following recurrence relation For example if is defined by , then while The Riemann integral of the non-negative real-valued function on the interval . This basically gives the area below the graph of calculated from point to point . The improper integral of the non-negative real-valued function defined through This gives the area below the graph of calculated from point to infinity. The improper integral of the non-negative real-valued function defined through This gives the area below the graph of calculated from minus infinity to point .

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