By Vinogradov V.
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Definition 32 The derivative of f (x) at x0 is f (x0 + ∆x) − f (x0 ) f (x0 ) = x→x lim . 0 ∆x If the limit exists, f is called differentiable at x. An alternative notation for the derivative often found in textbooks is df df (x) f (x) = y = = . e if z = z(t) then z˙ = , z¨ = 2 . e. the argument of a function may take any value from D) is denoted by C (1) (D). Geometrically speaking, the derivative represents the slope of the tangent line to f at x. Using the derivative, the equation of the tangent line to f (x) at x0 can be written as y = f (x0 ) + f (x0 )(x − x0 ).
2 2 2 x (by parts, f (x) = x, g(x) = e ) c) xex dx = xex − ex dx = xex − ex + C. Definition 37 (the Newton-Leibniz formula). The definite integral of a continuous function f is b f (x)dx = F (x)|ba = F (b) − F (a), a (2) for F (x) such that F (x) = f (x) for all x ∈ [a, b] The indefinite integral is a function. The definite integral is a number! We understand the definite integral in Riemann sense: Given a partition a = x0 < x1 < . . < xn = b and numbers ζi ∈ [xi , xi+1 ], i = 0, 1, . . , n − 1, n−1 b f (x)dx = a f (ζi )(xi+1 − xi ).
G m are defined on an open convex set S ⊂Rn . Let x∗ be a stationary point of the Lagrangian. Then, • if the Lagrangian is concave, x∗ maximizes (4); • if the Lagrangian is convex, x∗ minimizes (4). 2 The Case of Inequality Constraints Classical methods of optimization (the method of Lagrange multipliers) deal with optimization problems with equality constraints in the form of g(x1 , . . , xn ) = c. Nonclassical optimization, also known as mathematical programming, tackles problems with inequality constraints like g(x1 , .
A cookbook of mathematics by Vinogradov V.