Rate of change
The rate of change of a time-dependent variable describes the extent of the change in over a certain period of time in relation to the duration of this period. In vivid terms, it is a measure of how quickly the size changes. By referring to the duration, the unit of measurement in the denominator contains a unit of time; in the numerator there is a unit of . If the change is also related to the size itself, one speaks of a relative rate of change or growth .
A distinction is also made between the mean rate of change between two measurements and the current (also local ) rate of change as an abstract quantity of a model.
Calculation and use
Medium rate of change
The mean rate of change is the average change in a time-dependent measured variable between two points in time and , that is, in the period . It is calculated as the quotient of the difference between the two values at these points in time and the duration of the period:
In the time-quantity diagram ( function graph, diagram) of , the mean rate of change between and is the slope of the secant through the points and on the diagram.
Current rate of change
The current rate of change is the change in a measured variable related to a “moment” (very short period of time) . It can be mathematical as a result of the boundary process
can be represented as a derivative of their time function .
For time-linear changes, the instantaneous rate of change is constantly equal to the mean rate of change.
Rates of change in a broader sense
If the terms are used figuratively for quantities that depend on a parameter other than time, then:
- the mean rate of change is equivalent to the difference quotient
- the current rate of change is equivalent to the differential quotient
If the parameter is a vector quantity, the term “ gradient ” is also used instead of the term “rate” , such as temperature gradient or air pressure gradient .
Examples
- In the case of a linear movement, the speed is the current rate of change of the time-distance function . The article speed in the section definition of speed makes the difference between mean and instantaneous rate of change clear.
- The climb performance of an aircraft indicates how much altitude can be gained in a certain time.
literature
- Harro Heuser: Textbook of Analysis Part 1 . 5th edition. Teubner-Verlag, 1988, ISBN 3-519-42221-2
- Christian Gerthsen, Hans O. Kneser, Helmut Vogel: Physics: a textbook for use alongside lectures . 16th edition. Springer-Verlag, 1992, ISBN 3-540-51196-2
Remarks
- ↑ Helga Lohöfer: Table of the usual change terms for variables and functions. Script for the exercise Mathematical and Statistical Methods for Pharmacists , University of Marburg. 2006.