MindMap Gallery calculus
Mind map about calculus in advanced mathematics, including functions and limits, derivatives and differentials, differential mean value theorem and applications of derivatives, indefinite integrals, etc.
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calculus
Chapter 1 Functions and Limits
1. Mapping and functions
function definition
function type
basic elementary functions
Inverse function
What are the conditions for the existence of an inverse function?
inverse trigonometric function
Domain
The meaning of independent and dependent variables
image
Looking for guidance?
Refers to the power triangle...
explicit function
Implicit function
parametric equations
Polar equation
rounding function
Max Min function
symbolic function
Composite function
piecewise function
properties of functions
Monotonicity
Parity
Boundedness
Does a definition necessarily have a boundary?
Definition 1
Definition 2
How to derive mutual equivalence?
2 The limit of the sequence
Definition of the limit of a sequence (convergence)
Convergence Criteria for Monotone Bounded Sequences Monotone Bounded Sequences Must Have Limits
increase
reduce
an important limit
promotion
It is a model often constructed when seeking limits.
Some properties of convergent sequences
Principle of pinching
A good way to find limits that I often overlook!
Bounded quantities multiplied by infinitesimals?
Convergence of Sequences and Convergence of Odd and Even Subsequences
ultimate uniqueness
Boundedness of convergent sequences
Four extreme arithmetic operations
Conditions of use!
Three limits of functions
Definition (similar to sequence limit)
Necessary and sufficient conditions for the existence of limits
What does the existence of the limit at X0 have to do with?
How to find the limit
split item method
rationalize
Utilize important extreme structures
Fractional exponential power form: grab the bulk of higher order terms
Lópida's Law
Taylor formula
Principle of pinching
Product of sum and difference Product of sum and difference
Equivalent to infinitesimal
sequence limit
Thought: Turn the infinite into the finite
zoom
Stolz theorem
An important formula in definite integrals
Other transformation techniques
The relationship between function limits and sequence limits
inclusion relationship
4. Comparison between infinitesimal, infinitesimal and infinitesimal
infinitesimal definition
properties of infinitesimals
The sum-difference product of infinitesimal terms is still infinitesimal
Bounded quantities multiplied by infinitesimals?
infinitesimal comparison
Advanced
low level
Same level
equivalence
Find the limit of equivalent infinitesimal
important formula
Conditions of Use
Sometimes we have to consider the difference between going to the right and going to 0!
The replaced part is the factor of the entire fraction
Related theorem
High level Low level ~ Low level
Same order theorem
The coefficient cannot be zero after addition and subtraction operations
The concept of main part
5. Continuity of functions
basic knowledge
Definition: The limit of the function at x0 is equal to the function value at x0
Continuous left and right
What are the necessary and sufficient conditions for continuity at a certain point?
Geometric meaning
Basic elementary functions are continuous within their domain
The addition, subtraction, multiplication, division, and compounding of continuous functions are still continuous.
function break point
The first type of discontinuity point exists in both left and right limits.
Can go
jump
The second type of discontinuity point means that at least one of the left and right limits does not exist.
Special: Infinite discontinuities of the second type
Properties of continuous functions on closed intervals
Continuous on an open interval? Continuous in a closed interval?
Maximum value theorem
zero point theorem
Intermediate Value Theorem for Continuous Functions
Chapter 2 Derivatives and Differentials
1. The concept of derivatives
definition
Find derivatives using derivative definitions
Left and right derivatives
Classification discussion!
What are the necessary and sufficient conditions for the existence of derivatives?
Geometric meaning
tangent
Normal
Differentiable on open intervals and differentiable on closed intervals
Differentiable and continuous relations
symbolic representation
2. Four arithmetic derivation rules for functions
3. Derivative Rules for Composite Functions
Ordinary composite functions
exponential
Logarithmic derivation method: take the logarithm of the left and right at the same time, and convert the exponential form into a product form
Note that the independent variable when deriving y is x!
Using the logarithmic derivation method for multiplication and division forms, you can change multiplication and division into addition and subtraction.
4. Derivation of Implicit Functions
Take the derivative of x on both sides of the equal sign at the same time, and then move y' to one side of the equal sign
5. Derivation of inverse functions
First order derivative
Second order derivative
Six Derivatives of Parametric Equations
Change the derivation object
Derivatives of polar coordinate equations
Equivalent to the parameters in the parametric equation are
Seven differentials
Definition of differential
Differentiability and derivability are mutually sufficient and necessary conditions
Calculation of differentials
How to find differential
Using Differentials to Find Approximations
8. Higher order derivatives
several formulas
Chapter 3 Differential Mean Value Theorem and Application of Derivatives
1 Differential Mean Value Theorem
Fermat's Lemma: The derivative of a continuously differentiable point is zero.
Rolle's theorem:
Lagrange's mean value theorem:
Cauchy's mean value theorem:
inference:
2. Lópida's Law
Conditions of Use
The numerator and denominator limits exist
The field of detachment can be guided
an important limit
3. Monotonicity
Four extreme values
The location of the extreme point
Derivable extreme points (stationary points)
non-derivable point
How to determine extreme points
first sufficient condition
left and right domain monotonicity
second sufficient condition
Positive and negative second derivative
Pay attention to the situation where the second derivative is 0! Discuss alone
How to determine the best value point
Consider extreme points and endpoints
5 Taylor formula
Expansion of some common derivatives at x=0
Seeking the limit? Determine which number to expand to
6. Concave-convexity of function
The geometric meaning of concavity and convexity
Judgment of concavity and convexity
inflection point of function
The definition of inflection point: (Xo,f(Xo))
Continuous at Xo
The concavity and convexity of the left and right areas are opposite
The nature of the turning point
If the second derivative exists at the inflection point, the second derivative is 0
The second derivative may not exist
7. Find the asymptote
vertical asymptote
Xo is the function discontinuity point
horizontal asymptote
oblique asymptote
Chapter 4 Indefinite integrals
a concept
Finding indefinite integrals and derivation are reciprocal processes
All primitive functions of f(x)
C represents any constant
2. Calculation of indefinite integrals
24 commonly used points tables
Itemization (decomposition) method
Trigonometric functions
The first type of substitution method (differentiation method)
When the denominators are in similar forms, they can be appropriately deformed and matched.
The second type of substitution method (two substitutions)
Integration by parts
3. Indefinite integrals of rational functions
Convert an improper fraction into a proper fraction by splitting terms or dividing with remainder
The proper fraction is further divided into terms:
Chapter 5 Definite integrals
The definition of a certain integral
Derivation: extreme thinking
An important formula for finding limits
Note: Three steps to use
The geometric meaning of definite integrals
area of geometric figures
Pay attention to the positive and negative!
2. Properties of definite integrals
Linear
Additivity
Reverseness
The definite integral has nothing to do with the integral variable
unequal relationship
Both sides of the inequality sign can be definite integrals at the same time
Note: Both sides of the equal sign can be indefinitely integrated at the same time, but both sides of the inequality sign cannot be indefinitely integrated at the same time.
The relationship between indefinite integrals and the maximum and minimum values in the interval
Integral mean value theorem
3 Newton-Leibniz formula
variable limit integral
variable upper limit
variable lower limit
Leibniz's theorem
Newton-Leibniz formula
4. Calculation of definite integrals
common practice
First find the indefinite integral, then do the subtraction operation
Definite integral substitution method
Symmetric interval function
cyclical
Integration by parts method for definite integrals
A definite integral formula related to sine and cosine