Agency is a relationship based on an agreement authorizing one person, the agent, to act for another, the principal. For example an agent may negotiate and make contracts with third persons on behalf of the principal. Actions of an agent can obligate the principal to third persons. Actions of an agent may also give a principal rights against third persons.
An agency can be created for the purpose of doing almost any act the principal could do. However, there are some acts that must be done in person and cannot be done by an agent. Examples would be: testifying in court for another individual, making a will, and voting.
A general agent is authorized by the principal to transact all the affairs of a particular kind of business. For example, a person appointed as manager of a store is a general agent.
A special agent is authorized by the principal to handle a particular business transaction or perform a specific act. For example, a specific power of attorney appointing an agent (attorney-in-fact) to sell a particular piece of real estate or a certain car would be the appointment of a special agent.
A universal agent is authorized by the principal to do any act that can be delegated to a representative. An example would be giving a person a general power of attorney. This form is such a general power of attorney.
General power formula in integral calculus is a fundamental concept that allows us to evaluate integrals involving powers of a variable. It is an important technique used to find the antiderivative or integral of a polynomial function. The general power formula, also known as the power rule, states that if we have a function of the form f(x) = XSN, where n is any real number except -1, then the integral of f(x) with respect to x can be evaluated as: oxen DX = (1/(n+1))X(n+1) + C Here, ∫ represents the integralDXdx denotes the differential of x, and C is the constant of integration. The general power formula is applicable to a wide range of functions involving powers of x. For example, if we have f(x) = x^2, then the integral of f(x) can be found using the power formula as: ∫x^DXdx = (1/(2+1Xx^(2+1) + C = (1/3)x^3 + C Similarly, for f(x) = x^3, the integral becomes: ∫x^DXdx = (1/(3+1Xx^(3+1) + C = (1/4)x^4 + C In general, the formula allows us to easily compute integrals of various polynomial functions. Different types of general power formulas can be applied depending on the nature of the problem. Some variations include: 1. Definite integral: While the general power formula provides a way to find an indefinite integral, where the result is a function, it can also be adapted to evaluate definite integrals. In definite integrals, the lower and upper limits of integration are specified, resulting in a numerical value instead of a function. 2. Negative powers: The general power formula is not valid for n = -1. However, by applying the natural logarithm, techniques such as substitution or partial fractions can be used to evaluate integrals with negative powers. 3. Irrational powers: The formula can also be extended to integrals involving irrational powers of x, such as square roots or fractions. By manipulating the power to an equivalent rational form, the general power formula can still be employed. 4. Trigonometric powers: Trigonometric functions raised to powers can be integrated using methods like trigonometric identities, Euler's formula, or specialized techniques such as trigonometric substitution. Overall, the general power formula in integral calculus is a powerful tool that enables us to find the antiderivative of functions involving powers of a variable. It simplifies the evaluation process and forms the foundation for more advanced techniques in integration.
