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MindMap Gallery 1.1.1 Advanced Mathematics Knowledge Framework—Function Limits, Continuity, and Derivatives

1.1.1 Advanced Mathematics Knowledge Framework—Function Limits, Continuity, and Derivatives

Advanced mathematics knowledge framework - a mind map of function limits, continuity, and derivatives. Frequently asked questions on functions include composite functions and properties of functions. The general method of common test questions on limits: use basic limits to find limits, use equivalent infinitesimal substitutions, and use Rational arithmetic rules, using L'Obitar's rule, using Taylor's formula, using the pinch criterion, using the monotonic bounded criterion, using the definition of definite integrals, and using the mean value theorem.

Edited at 2023-10-29 02:56:15

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1.1.1 Advanced Mathematics Knowledge Framework—Function Limits, Continuity, and Derivatives

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Outline

Functions, Limits and Continuity

function

Two basic elements

1.Definition domain 2. Algorithm

basic elementary functions

rounding function Inverse function Power function exponential function Logarithmic function Trigonometric functions inverse trigonometric function

properties of functions

1. Monotonicity 2. Parity 3. Cyclicity 4. Boundedness

Frequently asked questions, methods and techniques

Question Type 1: Composite Function

Question type 2: Behavior of functions

limit

sequence limit

function limit

nature

1. Boundedness 2. Number retention 3. Limit values and infinitesimal values

Limit exists

Three situations need to be divided into left and right limits

Existence criteria: 1.Principle of pinching 2. Limit existence boundedness criterion

infinitesimal amount

nature

Comparison of infinitesimal quantities

infinite amount

nature

Commonly used infinite number of comparisons

The relationship between infinite quantities and unbounded variables

The relationship between infinite quantities and infinitesimal quantities

Frequently asked questions, methods and techniques

Question Type 1: Concept, Nature and Existence Criteria

Question Type 2: Finding the Limit

sequence limit

The limit of the sequence of infinitives

Method for solving limit infinitives of reference functions

Limit of sequence for sum of n terms

1. Principle of pinching 2.Definition of definite integral

Which one to choose?

3.Series summation

The limit of the sequence of consecutive multiplications of n terms

1. Principle of pinching 2. Take the logarithm and convert it into n times sum

The sequence defined by the recurrence relationship X1=a1, Xn+1=F(Xn) (n=1, 2...)

method one: 1. Prove that {Xn} converges (monotone bounded) 2. Take the limits on both sides of the equation Xn 1=F (Xn) 3. Obtain limF(A)=A

Methods to determine monotonicity

Commonly used inequalities

Method Two: 1. Let limXn=A 2. Take the limits on both sides of the equation Xn 1=F (Xn) to get A 3. Prove that limXn=A

Applicable when F(x) decreases monotonically and Xn does not have monotonicity

function limit

"0/0" type limit

1. Lópida’s Law 2. Equivalent infinitesimal substitution 3. Taylor formula

original simplification

1. Find the factor limit whose limit is not 0 first. 2. Rationalization 3.Variable substitution

"oo/oo" type limit

1. Lópida’s Law 2. The numerator and denominator are the same divided by the numerator and denominator The highest order infinity among the terms

"oo-oo" type limit

1. General differentiation into "0/0" (applicable to fractional differences) 2. Rationalization of radical expressions (applicable to radical differences) 3. Add infinite factors, and then substitute them with equal amounts, Variable substitution or Taylor formula (applies to too high degree)

"0*oo" type limit

into "0/0" or "oo/oo"

"1 of oo" type limit

1. Make up the basic limit 2. Index method 3. Relevant conclusions

"oo to the power of 0" and "0 to the power of oo" type limit

Index method

General method

Use basic limits to find limits

Use equivalent infinitesimal substitution

Use rational arithmetic

Utilize the Law of López

Using Taylor's formula

Use the pinch rule

Utilize the monotonic bounded criterion

Use definite integral definition

Use the mean value theorem

Question Type 3: Determine the parameters in the limit expression

"The key is to seek the limit"

Question Type 4: Comparison of infinitesimal orders

Fixed level question types

1. Luobida’s derivation has a definite order “k+1 order” 2. Equivalent infinitesimal substitution 3. Taylor formula

Sorting question type

Definition method - "two-two comparison"

Determine the order

1. Divide x to determine which order of infinitesimal it is of x 2. Derivation and order reduction 3. Take the lower order of the sum 4. Use equivalent infinitesimal substitution (the same applies to abnormal integral substitution) 5. If f(x) is continuous in a certain area where x=0, and when x→0 f(x) is the m-order infinitesimal of x, φ(x) is the nth order infinitesimal of x, then when x→0: The upper limit of the integral is φ(x) and the definite integral of the rule f(x) is the n(m+1) order infinitesimal of x.

continuous

Three conditions that need to be met continuously

1. Defined in xo 2. When x→xo, limf(x) exists 2. When x→xo, limf(x)=f(xo)

Necessary and Sufficient Condition

discontinuity

Classification

Type 1 discontinuity

Can remove discontinuities

jump break point

limf(x) exists

Type II discontinuities

infinite discontinuity

Oscillation discontinuity point.....

how to find

1. Find undefined points 2. Identify the type

Operation

1. Two continuous functions are still continuous after "addition, subtraction, multiplication and division" 2. The inner function is continuous in the domain of definition The outer function is continuous over the range of the inner function "After compounding" is still continuous 3. Basic elementary functions are continuous within the domain of definition 4. Elementary functions are continuous within the defined interval

Conclusions of continuous and discontinuous correlation operations

nature

Maximum value theorem

bounded value theorem

Intermediate value theorem

zero point theorem

Frequently asked questions, methods and techniques

Question Type 1: Discuss continuity and discontinuity types

Question Type 2: Proofs of the Intermediate Value Theorem, Maximum Value Theorem and Zero Point Theorem

Derivative

What is the maximum level that can be reached by using L'Obitat's Law?

three forms

The left and right derivatives exist and are equal

Necessary and sufficient conditions for derivability

Geometric meaning

Represents the slope of the tangent line to the curve at that point

Derivative formula

Derivative rule

rational arithmetic

Composite function derivation method

Implicit function derivation method

derivative of inverse function

1. Power function 2. Power exponential function 3. Multiplication and division of multiple factors

Logarithmic derivation

Advanced derivation

1.Formula method 2. Induction method 3.Taylor series 4. Taylor formula

differential

Differentiable

Relationship with derivatives (necessary and sufficient conditions)

Geometric meaning

Represents the increment on the tangent to the curve

Continuity, derivatives, differentials The relationship between the three

Frequently asked questions, methods and techniques

Question Type 1: Concepts of Derivatives and Differentials

(1) Use derivative definition to find limits

1. Three forms of definition (addition, subtraction, or multiplication and division)

2. Find special functional expressions that satisfy the conditions

Suitable for multiple choice questions and fill-in-the-blank questions

(2) Use derivative definition to find derivatives

Definition

1. Calculate f'(x)

2. Substitute xo

Assume the derivation method of composite function

Contains non-zero factors g(x)

(3) Use derivatives to define the differentiability of judgment functions

Type 1: Find the necessary and sufficient conditions for differentiation at a certain point

Type 2: Containing absolute value - find the necessary and sufficient conditions for being differentiable at a certain point/differentiable but not differentiable

The relationship between function differentiability and function absolute value differentiability

Question Type 2: Geometric Meaning of Derivatives

Question Type 3: Calculation of Derivatives and Differentials

(1) Derivatives of composite functions

Type 1: Calculate the derivative value of a composite function at a certain point

Type 2: Discuss whether the derivative at a certain point of the composite function exists

1. If f'(x) and g'(x) both exist, then there is (x=dy/dx at xo)=f'(x)g'(x)

If either f'(x) or g'(x) does not exist It cannot be inferred that the composite function does not exist At this time, use method 2

2. Find the expression of f(g(x))

(2) Derivatives of implicit functions

(3) Derivative of inverse function

(4) Logarithmic derivation method

(5) Advanced derivation