Examples of power law in the following topics:
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- A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
- A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
- Polynomials are made of power functions.
- Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents).
- Power functions are a special case of power law relationships, which appear throughout mathematics and statistics.
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- Next, we will raise both sides to the power of $e$ in an attempt to remove the logarithm from the right hand side:
- We do this by taking the natural logarithm of both sides and re-arranging terms using the following logarithm laws:
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- A power series (in one variable) is an infinite series of the form:
- can be written as a power series around the center $c=1$ as:
- In such cases, the power series takes the simpler form
- All power series $f(x)$ in powers of $(x-c)$ will converge at $x=c$.
- The number $r$ is called the radius of convergence of the power series.
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- Johannes Kepler describes planetary motion with three laws: 1.
- In order to discuss this law, and the laws that follow, we should examine the components of an ellipse a bit more closely.
- Kepler's third law describes the relationship between the distance of the planets from the Sun, and their orbits period.
- Newton derived his theory of the acceleration of a planet from Kepler's first and second laws.
- Illustration of Kepler's second law.
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- Limits of functions can often be determined using simple laws, such as L'Hôpital's rule and squeeze theorem.
- Limits of functions can often be determined using simple laws.
- Calculate a limit using simple laws, such as L'Hôpital's Rule or the squeeze theorem
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- The power series method is used to seek a power series solution to certain differential equations.
- The power series method is used to seek a power series solution to certain differential equations.
- The power series method calls for the construction of a power series solution:
- Using power series, a linear differential equation of a general form may be solved.
- Identify the steps and describe the application of the power series method
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- For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 10 10 = 103.
- The idea of logarithms is to reverse the operation of exponentiation, that is raising a number to a power.
- Raising to integer powers is easy.
- However, the definition also assumes that we know how to raise numbers to non-integer powers.
- For the definition to work, it must be understood that ' raising two to the 0.3219 power' means 'raising the 10000th root of 2 to the 3219th power'.
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- This scalar product of force and velocity is classified as instantaneous power delivered by the force.
- Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.
- where the term $\mathbf{F}\cdot\mathbf{v}$ is the power over the instant $\delta t$.
- Calculate "work" as the integral of instantaneous power applied along the trajectory of the point of application
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- Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest.
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- Many fundamental laws of physics and chemistry can be formulated as differential equations.