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Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.
That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed monotonically increasing (also increasing or non-decreasing) [3] if for all and such that one has (), so preserves the order (see Figure 1).
A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant (λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
In linear programming, reduced cost, or opportunity cost, is the amount by which an objective function coefficient would have to improve (so increase for maximization problem, decrease for minimization problem) before it would be possible for a corresponding variable to assume a positive value in the optimal solution.
For example, when inputs (labor and capital) increase by 100%, the increase in output is less than 100%. The main reason for the decreasing returns to scale is the increased management difficulties associated with the increased scale of production, the lack of coordination in all stages of production, and the resulting decrease in production ...
The primary reason for the increase is specialization and division of labor. [6] At the point the AP L reaches its maximum value AP L equals the MP L. [7] Beyond this point the AP L falls. During the early stages of production MP L is greater than AP L. When the MP L is above the AP L the AP L will increase.
The Heaviside step function is an often-used step function.. A constant function is a trivial example of a step function. Then there is only one interval, =. The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.