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In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: (). The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
Taking the difference of each side between two values = and = and applying the fundamental theorem of calculus gives the definite integral version: ′ = () () ′ (). The original integral ∫ u v ′ d x {\displaystyle \int uv'\,dx} contains the derivative v' ; to apply the theorem, one must find v , the antiderivative of v' , then evaluate ...
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In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
Similar to a Vietnamese pho rice noodle soup, this red curry soup packs big flavor and a bit of heat. The crisp raw vegetables help cool things down. —Monnie Norasing, Mansfield, Texas
This threshold is determined by minimizing intra-class intensity variance, or equivalently, by maximizing inter-class variance. [2] Otsu's method is a one-dimensional discrete analogue of Fisher's discriminant analysis , is related to Jenks optimization method , and is equivalent to a globally optimal k-means [ 3 ] performed on the intensity ...
In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. [citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.