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More generally, for each value of , we can calculate the corresponding likelihood. The result of such calculations is displayed in Figure 1. The result of such calculations is displayed in Figure 1. The integral of L {\textstyle {\mathcal {L}}} over [0, 1] is 1/3; likelihoods need not integrate or sum to one over the parameter space.
Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called differentiation .
The mean value theorem gives a relationship between values of the derivative and values of the original function. If f ( x ) is a real-valued function and a and b are numbers with a < b , then the mean value theorem says that under mild hypotheses, the slope between the two points ( a , f ( a )) and ( b , f ( b )) is equal to the slope of the ...
Differential quadrature is the approximation of derivatives by using weighted sums of function values. [ 22 ] [ 23 ] Differential quadrature is of practical interest because its allows one to compute derivatives from noisy data.
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}
The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is applicable (see methods of contour integration).
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]