The instantaneous speed of an object moving in a straight line

If the function y=f(x) is differentiable, then the function is continuous at that point. Continuity is not a sufficient condition for differentiability.

An infinitesimal quantity is a variable whose limit is zero.

The geometric meaning of differentials

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Geometric meaning

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Geometric meaning

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Geometric meaning

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Geometric meaning

Taylor's Mean Value Theorem with Lagrangian Remainder

Taylor's Mean Value Theorem with Peano Remainder

Find the best value

Find the limit

Find higher-order derivatives at specific points

Higher order derivatives of abstract functions

Prove the inequality

graphic method

theorem method

Steps to find the monotonic interval of a function

definition

determination

first sufficient condition

second sufficient condition

Steps to find extreme value

continuous on a closed interval

There is only one stationary point that is differentiable in the interval and is an extreme point.

There is a unique stationary point in the definition interval and the function must have a maximum or minimum value

A multivariate function is a function that contains multiple independent variables.

Element function, domain, value, range, independent variable, dependent variable, graph

The essence of the N-ary function: a mapping from a set in R^n to R -> Generalization: a mapping from a set in R^n to R^m

Example: Plane curve parametric equation/coordinate transformation on the plane

Triangle Inequality: The sum of two sides of a triangle is greater than the third side

Open set: Every point in the set is an interior point. Necessary and sufficient conditions: There are no boundary points in E. Closed set: the set contains all its boundary points

Connected: Any two points in E can be connected by a curve falling in E

Region: connected non-empty open set closed area

Arithmetic

The limit of the larger function ≥ the limit of the smaller function

"Pinch Theorem"

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The cumulative limit and the comprehensive limit are two different concepts, and generally speaking there is no necessary connection between them. When finding the limit, the comprehensive limit cannot be defined by the cumulative limit

u=f(x,y) is defined in area D and is continuous at every point in D. It is said that u=f(x,y) is continuous in area D.

f(x,y), that is, g(x,y) is continuous at one point (x0,y0), then ± * /(g(x0,y0) are all continuous

z=f(x,y) is defined and continuous near point (x0,y0), u=g(z) is continuous at point z0=f(x0,y0), then u=g(f(x,y) ) is continuous at point (x0, y0)

Necessary and sufficient conditions for mapping f:D->R^m at a point P0∈D: each component fi of f is continuous at P0

boundedness theorem

Maximum (minimum) value theorem

Intermediate value theorem

The geometric meaning of partial derivatives

Two second-order mixed partial derivatives being continuous in a region ensure that they are equal to each other

Differentiable: The function f (x, y) is differentiable at every point in the point area D. It is said that the function is differentiable.

If a function is differentiable, it must be continuous

If the function is differentiable, then partial derivatives exist

The existence of partial derivatives does not mean continuity, that is, the existence of partial derivatives does not necessarily mean that they are differentiable.

Sufficient conditions for differentiability: two partial derivatives exist and are continuous

For elementary functions: if partial derivatives exist, they must be differentiable

Lagrangian median formula for binary functions

If the function z=f(x,y) has continuous partial derivatives in the region D and satisfies ∂f/∂x≡0, ∂f/∂y≡0, then f(x,y) is a constant in D

Taylor's formula with Lagrangian remainder

Taylor's formula with Peano-type remainder

Taylor polynomial

chain rule

algorithm

unary function

binary function

formula method

differential method

Derivative method

definition

The relationship between directional derivatives and partial derivatives

calculate

definition

Simple geometry application - the relationship between gradient and isosurface

Gradient and Directional Derivatives

Definition of extreme value

Necessary conditions for extreme values

Sufficient conditions for extreme values

step

Convert to unconditional extreme value

Lagrange multiplier method