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In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function.
The first three values of the expression x[5]2. The value of 3[5]2 is 7 625 597 484 987; values for higher x, such as 4[5]2, which is about 2.361 × 10 8.072 × 10 153 are much too large to appear on the graph. In mathematics, pentation (or hyper-5) is the fifth hyperoperation.
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or a Schwichtenberg-Wainer hierarchy) [1] is an ordinal-indexed family of rapidly increasing functions f α: N → N (where N is the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal).
For each integer n > 2, the function n x is defined and increasing for x ≥ 1, and n 1 = 1, so that the n th super-root of x, , exists for x ≥ 1. However, if the linear approximation above is used, then y x = y + 1 {\displaystyle ^{y}x=y+1} if −1 < y ≤ 0 , so y y + 1 s {\displaystyle ^{y}{\sqrt {y+1}}_{s}} cannot exist.
In his 1947 paper, [5] Reuben Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g., (,,) = (,), the ...
This is the case, for example, for the Bring radical, which is the function implicitly defined by f ( x ) 5 + f ( x ) + x = 0 {\displaystyle f(x)^{5}+f(x)+x=0} . In more precise terms, an algebraic function of degree n in one variable x is a function y = f ( x ) , {\displaystyle y=f(x),} that is continuous in its domain and satisfies a ...
An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
With exponential functions, increasing the input by one unit causes the output to increase by a fixed multiple, which is known as the base of the exponential function. If both arguments and values of a function are in the logarithmic scale (i.e., when log(y) is a linear function of log(x)), then the straight line represents a power law: