Power function
When power functions are called elementary mathematical functions of the form
If one considers only natural or integer exponents, to write for the exponent usually:
Special cases
- Constant function: (for r = 0)
- (homogeneous ) linear function / proportionality: ( r = 1)
- Square function and multiples thereof: ( for r = 2)
- The quite rational functions with integer exponents from which the rational functions are composed of the power functions with natural exponent.
- For with it root functions arise.
Definition and set of values
The maximum possible amount of definition depends on the exponent. If one does not allow roots of negative numbers, then it can be specified with the following table:
The value sets you have to consider additionally the sign of a; if it is, it also comes also depend on whether r is odd or even number:
Graphs
The graphs of exponential functions with natural hot parables - th order. Negative hyperbolas with integer -order The parameter of the graph expresses an extension with respect to the y-axis by a factor of | and also to the reflection axis, is if | a.
Has a power function, the definition set, then its graph consists of two branches, otherwise there is only one branch.
Symmetry
Only the graphs of exponential functions with symmetric; more precisely, they are just for even and odd for odd. In the first case its graph is axisymmetrical to the y- axis, in the second it is point symmetric about the origin.
Behavior for x → ± ∞ and x → 0
All power functions with positive exponents have a zero at, rise ( but still slower than the exponential function ) and go against for. For the behavior results for from the symmetry.
All power functions with negative exponents go against for. You fall and go against for.
Continuity, derivation and integration
Each power function is continuous on its definition set.
The corresponding derivation function (see Power Rule )
This formula is valid for all and all, if it is defined only at the site. It is also at the point when it is. The function is continuous, but not differentiate in the body.
For example, is valid in all (or even in the whole, if you allow odd roots of negative numbers - see below).
For an arbitrary non-negative rational number is the formula
For all intervals that are subsets of the set of definitions are valid.
For example, the following applies:
Power functions with roots of negative numbers
In this section only power functions are considered with rational exponents, where the denominator of the truncated exponent is odd, and it is explained how one can extend the definition of which amount to negative numbers. The following is then discussed, which are the above-mentioned properties of the features are changed by it.
Odd roots of negative numbers
( → See also potency)
In the previous sections, the usual in many textbooks convention was used that roots are only defined for non-negative radicand. However, it may also allow for odd roots of negative numbers. For n odd and arbitrary to define, analogous to the well-known definition for positive radicand:
For example, the solution of the equation would be given by (whereas you would have to write without roots of negative numbers according to the usual definition ) according to this definition.
Definition and set of values
For exponential functions with the properties mentioned above, you can now extend the domain of definition to negative: Be with, it is odd, and let and prime, then:
( Note: If, then once again, that a power function with an integer exponent. )
For the definition of this function is then equal amount, for it is the same.
For the set of values one must again note the sign of a. In addition, there is now also a question of whether one of the numbers or even ( ie the product is even) or whether these two numbers are odd ( ie the product is odd):
Symmetry and behavior for x → ± ∞ and x → 0
For symmetry the same applies as with integer exponents: the function is just for even and odd for odd. Your behavior and is then defined by their symmetry properties and their behavior on the right half- axis.
Applications
Power functions have numerous applications in economy, nature and technology:
- Proportionalities (r = 1 ) appear in many contexts: Cost and quantity of goods (excluding quantity discount)
- Conversion between currencies
- Circumference and radius
- Mass and volume ( at constant density )
- Elapsed time and distance traveled (at constant speed )
- Distance traveled and fuel consumed (at constant consumption)
- Force and acceleration (at constant mass )
- Elongation of a body and applied force ( within certain limits, see Hooke's law )
- Number of workers and labor
- Time required for a distance and (constant ) speed
- Required force and length of a lever ( lever rule )
- Mass and force required for a given acceleration
- Area of a square and its side length
- Area of a circle and its radius
- Tension and strain energy of a body
- Kinetic energy and speed
- Distance traveled and time with uniform acceleration
- Electrical power and resistance
- Drag force and velocity in turbulent flow
- Radius and volume of a sphere
- Side length and volume of a cube
- Radiation power of a black body and its absolute temperature ( Stefan- Boltzmann law )
- Scattering cross-section for light scattering and light frequency (which, inter alia, responsible for the blue color of the sky Rayleigh scattering )
- Volume flow through a thin tube and pipe radius (Hagen -Poiseuille )
- Relationship between pressure, volume, and absolute temperature in adiabatic state changes (also see adiabatic )
- Relationship between semi-major axis and orbital period of planet and moons (3rd Kepler 's law)
- Scaling laws, such as phase transitions, but also in biology
- In geometry applies to the relationship between surface area and volume of a cube :; A similar formula is obtained at a ball.
- In a universe that is filled with a homogeneous substance that meets a state equation of the form, the result for the time dependence of the scale factor of the Friedmann equations.