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In mathematics, a rate is the quotient of two quantities, often represented as a fraction. [1] If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change ...
[5] [6] The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). [7] [8]: 237 [9] The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change. [9]
[16] For example, in a closed system where immigration and emigration does not take place, the rate of change in the number of individuals in a population can be described as: = = = =, where N is the total number of individuals in the specific experimental population being studied, B is the number of births and D is the number of deaths per ...
A percentage change is a way to express a change in a variable. It represents the relative change between the old value and the new one. [6]For example, if a house is worth $100,000 today and the year after its value goes up to $110,000, the percentage change of its value can be expressed as = = %.
Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
The growth rate of output is the time derivative of the flow of output divided by output itself. The growth rate of the labor force is the time derivative of the labor force divided by the labor force itself. And sometimes there appears a time derivative of a variable which, unlike the examples above, is not measured in units of currency:
The most common way to approach related rates problems is the following: [2] Identify the known variables, including rates of change and the rate of change that is to be found. (Drawing a picture or representation of the problem can help to keep everything in order)
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