Examples of exponential in the following topics:
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- Exponential decay occurs in the same way, providing the growth rate is negative.
- 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.
- This graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
- Apply the exponential growth and decay formulas to real world examples
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- Now that we have derived a specific case, let us extend things to the general case of exponential function.
- Here we consider integration of natural exponential function.
- Note that the exponential function y=ex is defined as the inverse of ln(x).
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- The derivative of the exponential function is equal to the value of the function.
- The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative.
- If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
- Graph of the exponential function illustrating that its derivative is equal to the value of the function.
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- Both exponential and logarithmic functions are widely used in scientific and engineering applications.
- The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
- The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
- The exponential function ex can be characterized in a variety of equivalent ways.
- The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function.
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- The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red).
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- The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx.
- γy represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.
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- The exponential function (in blue) and the sum of the first n+1 terms of its Taylor series at 0 (in red) up to n=8.
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- This is essentially exponential growth based on a constant rate of compound interest: P(t)=P0ert where P0=P(0)=initial population, r is the growth rate, and t is the time.
- The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
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- More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative t, which slows to linear growth of slope 41 near t=0, then approaches y=1 with an exponentially decaying gap.
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- The exponential function (in blue), and the sum of the first n+1 terms of its Maclaurin power series (in red).