![\displaystyle \max_{x \in [a,b]} f(x)](/cache/eqns/547b627a788b5bfec2f394c675bedd56_d7d.gif) |
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The maximum of the continuous function in the range to .
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![\displaystyle \min_{x \in [a,b]} f(x)](/cache/eqns/5d9c99489d8a1f5b93c26b957c7f0ad4_b7d.gif) |
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The minimum of the continuous function in the range to .
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![\displaystyle\sup_{x \in [a,b]} f(x)](/cache/eqns/b6d64a962cc36330d7f991d240424786_f6d.gif) |
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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 .
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![\displaystyle \inf_{x \in [a,b]} f(x)](/cache/eqns/903b707eea1cd51f5abdd7be1c64af07_47d.gif) |
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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 .
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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
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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
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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
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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
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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
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This denotes the first derivative of the function at point , whenever it exists. It can be defined through the following formula using limits
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This denotes the -th derivative of the function at point , whenever it exists. It can be defined recurrently through the following formulae
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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
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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
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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 . |
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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. |
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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 . |