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Exponential growth. Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential ...
Exponential decay. 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. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value.
Use the assumption that e = a b to obtain. The first term is an integer, and every fraction in the sum is actually an integer because n ≤ b for each term. Therefore, under the assumption that e is rational, x is an integer. We now prove that 0 < x < 1. First, to prove that x is strictly positive, we insert the above series representation of e ...
Half-life (symbol t½) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential (or, rarely ...
Biological exponential growth. Biological exponential growth is the unrestricted growth of a population of organisms, occurring when resources in its habitat are unlimited. Most commonly apparent in species that reproduce quickly and asexually, like bacteria, exponential growth is intuitive from the fact that each organism can divide and ...
The Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology named after Patrick H. Leslie. The Leslie matrix (also called the Leslie model) is one of the most well-known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration, growing in an unlimited environment ...
e. -folding. In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e; [1] it is the base- e analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine and theoretical physics, especially when cosmic inflation is investigated.
The number e is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithm function. It is the limit of as n tends to infinity, an expression that arises in the computation of compound interest. It is the value at 1 of the (natural) exponential function, commonly ...
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