exponential

(noun)

any function that has an exponent as an independent variable

Related Terms

  • linear
  • tangent
  • e
  • polynomial

Examples of exponential in the following topics:

  • Exponential Growth and Decay

    • Exponential decay occurs in the same way, providing the growth rate is negative.
    • If τ>0\tau > 0τ>0 and b>1b > 1b>1, then xxx has exponential growth.
    • If τ<0\tau<0τ<0 and b>1b > 1b>1, or τ>0\tau > 0τ>0 and 0<b<10 < b < 10<b<1, then xxx 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
  • The Natural Exponential Function: Differentiation and Integration

    • 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=exy = e^{x}y=e​x​​ is defined as the inverse of ln(x)\ln(x)ln(x).
  • Derivatives of Exponential Functions

    • 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.
  • Exponential and Logarithmic Functions

    • 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 exe^xe​x​​ 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.
  • Taylor Polynomials

    • The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red).
  • Predator-Prey Systems

    • 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\alpha xαx.
    • γy\gamma yγy 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.
  • Taylor and Maclaurin Series

    • The exponential function (in blue) and the sum of the first n+1n+1n+1 terms of its Taylor series at 000 (in red) up to n=8n=8n=8.
  • Essential Functions for Mathematical Modeling

    • This is essentially exponential growth based on a constant rate of compound interest: P(t)=P0ertP(t)=P_0e^{rt}P(t)=P​0​​e​rt​​ where P0=P(0)=initial populationP_0=P(0)=\text{initial population}P​0​​=P(0)=initial population, rrr is the growth rate, and ttt is the time.
    • The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
  • Logistic Equations and Population Grown

    • More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative ttt, which slows to linear growth of slope 14\frac{1}{4}​4​​1​​ near t=0t = 0t=0, then approaches y=1y = 1y=1 with an exponentially decaying gap.
  • Power Series

    • The exponential function (in blue), and the sum of the first n+1n+1n+1 terms of its Maclaurin power series (in red).
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