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Knowledge map of differential calculus of functions of one variable
A mind map for learning the knowledge content of differential calculus of functions of one variable, including differentiability, definitions of differentials, definitions of derivatives, derivative functions, properties of derivatives and functions, four operations of differential calculus of one variable, calculation of derivatives, and geometric applications of derivatives , demonstrative applications, etc., hurry up and collect the pictures below to learn!
Edited at 2021-08-01 09:36:31
Knowledge map of differential calculus of functions of one variable
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Differential Calculus of Functions of One Variable
Derivatives and Differentials
Differentiable
Definition of differential
①
②
③
Definition of derivative (problem of derivative at one point)
Derivative must be continuous
Derivative changes parity
Define and judge the differentiability of f(x) at x=a
Protect both sides
Can't cross
note: It is still not possible to cross when continuous.
same order
The problem of derivative at one point
abstract function at one point
Piecewise functions (including absolute value functions) at the piecewise point
Special points in the four arithmetic operations
Too complicated
point that does not hold
Define with derivatives
note
A discrete point must be defined by knowing that the point is differentiable.
Even if it is not a piecewise function, sometimes it is necessary to use derivatives to define the derivation, and some parts of the expression are not differentiable at a certain point, but the overall expression may also be differentiable at that point.
derivative function
Geometric meaning
Derivatives and properties of functions
stationary point
The point where the derivative is equal to 0
Concave-convexity
Extreme point
This point is a stationary point and the left derivative of this point > 0 and the right derivative < 0, then this point is a maximum value point, otherwise it is a minimum value point.
Unary differential calculus
Arithmetic
Derivative calculation
Basic derivation formula
Definition of derivation
Symbol writing
Explicit function derivation
Implicit function derivation
Treat y as a function of x, pay attention to compound derivation
Piecewise function derivation (including absolute value)
Critical points are discussed separately
When the second derivative exists, the first derivative is continuous
Inverse function derivation
That is, the product of the derivatives of two inverse functions is 1
A monotonic function must have an inverse function
Multinomial multiplication and division, square root, power (logarithmic derivation method)
Derivation of functions determined by parametric equations
Derivative of composite functions
chain rule
higher order derivatives
Induction
Find first-order, second-order and third-order rules
Find it to be true when n=0
Suppose it is true when n=m
Prove that it is true when n=m 1
formula method
Leibniz formula
①
②
note
When you see the higher-order derivative of the product of two functions, you can generally use Leibniz's formula, and combine it with the commonly used formulas in the formula method;
Sometimes it is difficult to find a higher-order derivative of a function. If it can be converted into the product form of two functions, Leibniz's formula can also be used.
Taylor expansion
Use the uniqueness of Taylor's formula to find specific order derivatives
a (abstract expansion)
b (specific expansion)
c
The Taylor formula expansion of odd (even) functions will never have even (odd) terms
Geometric applications of derivatives
Research object
three generations
①
②
③
④
Piecewise function (including absolute value)
parametric equations
①
②
Implicit function F(x,y)=0
research content
Tangent, normal, intercept (not distance, can be positive or negative)
Find the first derivative
extremum
Extreme value judgment
Determine function domain
Derive and find the stationary point and non-differentiable point of f(x)
judge
first sufficient condition
Using the monotonicity of centroid neighborhoods
second sufficient condition
note
The first-order derivative is monotonically increasing
Extreme value, monotonic
Find the first derivative
draw number line
inflection point, concavity and convexity
Find the second derivative
The point where the second derivative is 0 or does not exist is the inflection point (the junction of concave and convex)
Concave-convexity
Asymptote
vertical asymptote
horizontal asymptote
oblique asymptote
note: Horizontal asymptotes and oblique asymptotes cannot exist at the same time on the same side.
best value
Points and endpoints where the first derivative is 0
curvature, relative rate of change
Simply put, the correlation rate of change is an independent variable x with two dependent variables. When one of the dependent variables is A, the relationship between the other dependent variable B and the independent variable x is found.
Curvature is the degree to which a curve bends at a point
curvature
radius of curvature
R = 1/k
circle of curvature
The curvature of any point on a circle of curvature with radius R is equal
The curvature circle passes through point M and has a common tangent with the original curve at point M, so y' is the same
The curvature circle passes through point M. At point M, it has the same k and the same concave direction as the original curve, so y'' is also the same.
Problems with bounded and unbounded interval primitive functions and derivative functions
finite interval
infinite interval
Can't deduce the relationship between the two
Proof application
Mean value theorem (Zhang)
Determine the interval
Mark all possible points on the number line
Determine the auxiliary function (not the point)
simple case
complex situation
①
a
b
c
Always consider the following situations
②
a
b
c
③
④
Determine the theorem to use
(1)
zero point theorem
(2)
Intermediate value theorem
(3)
Fermat's theorem
(4)
Rolle's theorem
Commonly used
(5)
Lagrange's mean value theorem
Commonly used
(6)
Taylor formula
Commonly used
(7)
Cauchy's mean value theorem
Commonly used
Summary of common key points
①
②
Use limits (continuous, differentiable, guaranteed number, calculate limits)
③
Use zero point and intermediate value theorem
④
Use integrals (mean value theorem, sign preservation, original function definition, calculating integrals)
⑤
Use Fermat's theorem
⑥
Use parity and even properties (in big questions, especially the given function expression, you must observe whether parity can be used)
⑦
Use geometric conditions
⑧
Use determinant conditions
other problems
mean value theorem
Fermat's theorem
stationary point
Rolle median
three conditions
The closed interval is continuous, the open interval is differentiable, f(a) = f(b)
prove
Use the maximum value theorem to divide the two cases of M=m and M>m and discuss and prove that we can get
Lagrange
two conditions
Closed intervals are continuous, open intervals are differentiable
Auxiliary function If f(x) is linear, the auxiliary function is always equal to 0
equivalent form
a and b can be variables
Cauchy median
Taylor median
condition:
Lagrangian remainder
Peano remainder
Points related to the first derivative
interval midpoint
other
Select x priority
Interval endpoint
any point x
Generalization of Taylor's Mean Value Theorem
condition
in conclusion
median score
prove
Supplementary theorem
bounded theorem
note
Maximum value theorem
note: When you see the sum of function values, think of using the intermediate value theorem.
Intermediate value theorem
zero point theorem
Median value theorem problem-solving skills
About θ
It exists in Lagrangian, Taylor and integral median (theta in the integral median belongs to the closed interval)
in principle
f(x) expresses concretely
Solve for θ
f(x) abstract
Unable to solve for θ
To find multiple equal points, use Roll
Indirect use of Roll through other relationships
median proof
Only ξ
reduction method
Two terms, derivative difference, first order
grouping method
Tinkering with tricks
There are a, b, ξ
separable
a, b side
Lagrange
Cauchy's mean value theorem
inseparable
Tinkering with tricks
There are ξ, η
note
Derived from number security
constructor helper function
Find three points and use Lagrangian once each
Mean value theorem selection
Twice Lagrangian
note: The intermediate value theorem only recognizes continuity, regardless of the order of derivatives
inequality problem
Proof of inequality
(1)
monotonicity
①
②
note
(2)
Use the best value
Ideas
function with parameters
There are no parameters in the derivative
Derivatives contain parameters
(3)
Use concave and convex
①
②(Promotion)
(4)
Use the mean value theorem
Use the Lagrangian median formula
Use the Cauchy median formula
Using Taylor's formula with Lagrangian remainder
The feasibility of the above method lies in whether the corresponding "if" can be realized
equation problem
Problems with function zeros or equation roots
Theoretical basis
(1)
Zero point theorem and its extension
generalized zero point theorem
(2)
Use derivative tools to study functional behavior
(3)
Rolle's original words (corollary to Rolle's theorem)
(4)
(5)
monotonicity method
Find the extreme value first
Then find the function values at both ends of the domain (if you can’t get it, find the limit)
Determine the number of zero points
test method
(1)
prove the identity
(2)
The number of zero points of the function (the number of equation roots, the number of curve intersections)
at least a few
at most a few
Exactly a few
note: Common discussions involving references
(3)
Equation (column) problems
(4)
Interval (column) problem
Physics applications
"The rate of change of A to B" is the core
arc differential
arc differential pythagorean