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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.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · 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

ជំហាន​ដោយ​ជំហាន

  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.

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

ឥឡូវ​អ្នក ជ្រើស​បញ្ហា ឬ​វាយ​បញ្ចូល ឬ​គូរ​របស់​អ្នក​ផ្ទាល់ ។ ជំហាន​នីមួយៗ រូបភាព ចម្លើយ​ត្រូវ​បាន​លាក់​រហូត​ដល់​អ្នក​សួរ ។

រក្សា​ការ​ធ្វើការ​របស់​អ្នក​ផ្ទាល់

គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

ចុះឈ្មោះ ចូល

និមិត្ត​សញ្ញា​ដែល​បាន​ប្រើ​នៅ​ទីនេះ

ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: 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.

សំណួរ​ដែល​មនុស្ស​សួរ

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.

បន្ថែម​ទៀត​ក្នុង Multivariable Calculus

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