maths.free › Multivariable Calculus › Optimisation in several variables

Optimisation in several variables

Critical points, the second-derivative test, and Lagrange multipliers.

Critical points are where the gradient vanishes; the Hessian's determinant sorts them into minima, maxima and saddles. With a constraint, Lagrange multipliers say the gradients of f and of the constraint are parallel at the optimum. Picture it: the surface's bowl bottom, hilltop or saddle. The 3D view shows which. Think it: constrained optimisation is the geometry of tangency between level sets.

Ýüklenen mysal · gradient of x^2 + y^2 - 2x - 4y

Multivariable-Calculus · Partial-Derivatives

Gradient of x^2 - 2x + y^2 - 4y

x^{2} - 2 x + y^{2} - 4 y

Adım adım

  1. f(x, y) = x^{2} - 2 x + y^{2} - 4 y

    The gradient is the vector of partial derivatives. Differentiate with respect to each variable, holding the others constant.

  2. \frac{\partial f}{\partial x} = 2 x - 2

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = 2 y - 4

    Treat every variable except y as a constant.

  4. \nabla f = \left[\begin{matrix}2 x - 2\\2 y - 4\end{matrix}\right]

    Assemble the gradient vector. It points in the direction of steepest ascent.

  5. (1, 2)

    Critical points: where every partial derivative is zero.

Jawaby görkez
\nabla f = \left[\begin{matrix}2 x - 2\\2 y - 4\end{matrix}\right]

Indi sen Bir meseläni saýla, ýa-da ýaz ýa-da özüňi surata çek. Her ädim, bir resim, jogap sorag berlençä gizlenýär.

Özüň işle

Mugt hasap her bir ders üçin notlar goşýar, tamamlanan zadyňyzyň ýazgysyny, çözen problemaňyzy bir ýerde, we bu sahypa barada sorap biljek mugallymy. Matematika özüçe her kim üçin açyk, girseň ýa girmeseň.

_Gün _Giriş

Şu ýerde ulanylan simwollar

Eň täze düşündirme, surat, we her harpyň manysy üçin her bir simwoly basyň.

Nädip: Optimisation in several variables

  1. The gradient is the vector of partial derivatives. Differentiate with respect to each variable, holding the others constant.
  2. Treat every variable except x as a constant.
  3. Treat every variable except y as a constant.
  4. Assemble the gradient vector. It points in the direction of steepest ascent.
  5. Critical points: where every partial derivative is zero.

Adamlar soraýan soraglar

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen: the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

_Ýaşa Multivariable Calculus

Functions of several variablesPartial derivatives and the gradientDouble and triple integralsVector fields, line integrals and the big theorems