The point function value is the maximum/minimum value of all function values ​​in the neighborhood of the point

premise

in conclusion

premise

Conclusion P88

1. stationary point

2. The first-order partial derivative has at least one non-existent point (an undifferentiable point of a function of one variable)

Find the possible extreme points of f(x,y) inside region D

Find the maximum and minimum points of f(x,y) on the boundary of area D (conditional extreme value)

Establish the objective function z=f(x,y)

Follow the above steps to find the maximum value of f(x,y) on the bounded closed area D

It is required that point (x, y) approaches point (x0, y0) in any way within D, and f (x, y) approaches the same certain constant A.

1. Ratio of the numerator to the power of the denominator (preliminary judgment)

2. Taking the absolute value and forcing it - easy to scale (generalization of the limit conclusion of the sequence)

1. Take the special path y=kx

2. Find the original weight limit

If f(x,y) approaches the point (x0,y0) along any two directions in D, the limit does not exist if the function values ​​are not the same.

Multivariable functions without Lupida

Just be able to judge whether a multivariate function is continuous or not

The limit value of the function at this point = function value

The sum, difference and product quotient of a multivariate continuous function (the denominator is not 0) is still a continuous function

The composite function of a multivariate continuous function is still a continuous function

A multivariate elementary function is continuous within its defined region (unary - defined interval)

maximum value theorem

Intermediate value theorem

Multivariate functions do not discuss discontinuity types (complex)

definition

First seek (define) descendants

Find first (specific point) and then find

Notice

The tangent of the angle between the tangent direction and the direction vector in the ∥ coordinate axis direction

The n-order mixed partial derivatives are continuous in region D --->The n-order mixed partial derivatives are equal in region D

Contain abstract functions to find higher-order partial derivatives --> merge mixed partial derivatives

I. definition

II. sufficient conditions for differentiability

The relationship between continuous, deflectable and differentiable

Prove several classic counterexamples

Understand the definition of total differential

The inner and outer layers are differentiable--->the composite function is differentiable

Derivable

Derivative method (chain derivative rule)

unary function

Regardless of whether u and v are intermediate variables or independent variables, the differential form of the multivariate function f(u,v) is the same.

If the implicit function takes the partial derivative of a certain variable ≠0---> then the variable is the dependent variable

1. Invariance in total differential form (impermanence changes)

2. Find partial derivatives of the independent variables taken on both sides (constant or variable)

3. Derivative formula of implicit function