Exponential error

Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function of time is proportional to the function, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent.Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time. Exponential growth is exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function of time is proportional to the function, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent.Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time. The formula for exponential growth of a variable x at the growth rate r, as time t goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is where x0 is the value of x at time 0. This formula is transparent when the exponents are converted to multiplication. For instance, with a starting value of 50 and a growth rate of r = 5% = 0.05 per interval, the passage of one interval would give 50 × 1.051 = 50 × 1.05; two intervals would give 50 × 1.052 = 50 × 1.05 × 1.05; and three intervals would give 50 × 1.053 = 50 × 1.05 × 1.05 × 1.05. In this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent. (The order of multiplication does not change the result based on the associative property of multiplication.) Since the time variable, which is the input to this function, occurs as the exponent, this is an exponential function. This contrasts with growth based on a power function, where the time variable is the base value raised to a fixed exponent, such as cubic growth (or in general terms denoted as polynomial growth). A quantity x depends exponentially on time t if where the constant a is the initial value of x, the constant b is a positive growth factor, and τ is the time constant—the time required for x to increase by one factor of b: If τ > 0 and b > 1, then x has exponential growth. If τ < 0 and b > 1, or τ > 0 and 0 < b < 1, then x has exponential decay. Example: If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question implies a = 1, b = 2 and τ = 10 min.