Examples of exponential function in the following topics:
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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 $y = e^{x}$ 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.
- Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
- 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.
- Exponential function is the function $e^x$ the number (approximately 2.718281828) such that the function $e^x$ is its own derivative .
- The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
- The exponential function $e^x$ 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.
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- A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives.
- Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
- Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
- Let's assume that the integration of a function ($f(x)$) cannot be performed analytically.
- 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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- Taylor series represents a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.
- A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
- The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists.
- A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.
- 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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- A transcendental function is a function that is not algebraic.
- Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
- A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
- Because of this, transcendental functions can be an easy-to-spot source of dimensional errors.
- Bottom panel: Graph of sine function versus angle.
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- Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.
- Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.
- Exponential decay occurs in the same way, providing the growth rate is negative.
- If $\tau > 0$ and $b > 1$, then $x$ has exponential growth.
- Apply the exponential growth and decay formulas to real world examples
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- Hyperbolic function is an analog of the ordinary trigonometric function, also called circular function.
- The basic hyperbolic functions are the hyperbolic sine "$\sinh$," and the hyperbolic cosine "$\cosh$," from which are derived the hyperbolic tangent "$\tanh$," and so on, corresponding to the derived functions.
- The hyperbolic functions take real values for a real argument called a hyperbolic angle.
- In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine.
- When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence meromorphic.
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- An inverse function is a function that undoes another function: For a function $f(x)=y$ the inverse function, if it exists, is given as $g(y)= x$.
- Inverse function is a function that undoes another function: If an input $x$ into the function $f$ produces an output $y$, then putting $y$ into the inverse function $g$ produces the output $x$, and vice versa. i.e., $f(x)=y$, and $g(y)=x$.
- A function $f$ that has an inverse is called invertible; the inverse function is then uniquely determined by $f$ and is denoted by $f^{-1}$ (read f inverse, not to be confused with exponentiation).
- Not all functions have an inverse.
- A function $f$ and its inverse $f^{-1}$.
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- This is essentially exponential growth based on a constant rate of compound interest: $P(t)=P_0e^{rt}$ where $P_0=P(0)=\text{initial population}$, $r$ is the growth rate, and $t$ is the time.
- A slightly more realistic and largely used population growth model is the logistic function that may be defined by the formula: $P(t) = \frac{1}{1 + e^{-t}}$.
- In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time.
- The potential field is given by a function $V:R^3 \rightarrow R$ and the trajectory is a solution of a differential equation.
- The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.