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Differential calculus mind map
This is a mind map about differential calculus. Differential calculus refers to the study of derivatives and differentials of functions and their application in the study of functions. Differential calculus and integral calculus are closely related and together form calculus, a basic branch of analysis.
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Differential calculus mind map
- bacteria
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
- Plant asexual reproduction
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- Reproductive development of animals
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- Outline
Differential calculus
Function properties
Boundedness
Monotonicity
monotonically increasing or decreasing
Parity
cyclical
Derivatives and Differentials
Derivative
One-sided derivative
Differentiable at x, both left and right derivatives exist and are equal
Geometric meaning
slope
physical meaning
Velocity is the derivative of displacement with respect to time, the derivative of electric quantity with respect to time is current intensity, and the derivative of mass with respect to lines and planes
Basic elementary function derivative formula
Derivative rule
Compound derivation
Derivation of functions determined by parametric equations
higher order derivatives
Differentiable and continuous relations
Differentiable must be continuous, continuous may not be differentiable (corner points are not differentiable)
differential
definition
Necessary and sufficient conditions, derivable at x
Geometric meaning
Robetta's Law
Differential calculus of multivariate functions
Partial derivative
definition
Seeking the law
Taking the partial derivative of x means treating y as a constant
Geometric meaning
Find the partial derivative of x
The slope of the curve intercepted by y=y0 to the x-axis
Higher order partial derivatives
application
The partial derivative of function F is the normal vector,
Total differential
Differentiable in (x, y), partial derivatives must exist
Continuous, differentiable, differentiable relationships
One yuan
Differentiable is equal to differentiable. Differentiable must be continuous. Continuous is not necessarily differentiable or differentiable.
Diverse
Continuity has nothing to do with differentiability. Differentiation must be continuous and differentiable. Partial derivatives must be continuous and differentiable.
Composite multivariate function
Implicit function derivation rule
Derivative Differential Difference
application
Monotonicity
The derivative is greater than 0 and increases monotonically.
mean value theorem
Rolle's theorem
At least one tangent line is parallel to A
Lagrange's mean value theorem
At least one tangent line is parallel to AB
Extreme value and maximum value of function
Judgment of extreme value
necessary conditions
Extreme point
non-derivable point
Obtained from stationary points and undifferentiable points
It must be a stagnation point, and the stagnation point is not necessarily an extreme point.
stationary point
Derivative=0
sufficient conditions
The left and right derivatives at x0 have opposite signs and have extreme values.
If the second derivative is less than 0, it is a maximum value, otherwise it is a minimum value.
best value
Endpoint, derivative 0 point (stationary point), underivable point, comparison size
Extreme value of multivariate small function
There are extreme extreme values, and the partial derivative is 0, which must be a stationary point.
sufficient conditions
Bumps and inflection points
Concave and convex
definition
determination
If the second derivative is greater than 0, it is concave
inflection point
definition
Boundary point of concave and convex, the tangent line must pass through the curve
theorem
necessary conditions
Second derivative = 0
not a sufficient condition
There are inflection points on both sides of the second-order derivative change sign at x0
The third derivative is not equal to 0
The concept of function continuity and discontinuity points
definition
Three conditions
discontinuity
Discontinuities of the first kind
There are both left and right limits
jump break point
Can remove discontinuities
Type II discontinuities
At least one of the left and right limits does not exist
infinite discontinuity
Oscillation break point
basic elementary functions
Against Pluto Three
Basic elementary functions form elementary functions
Elementary functions are continuous within a defined interval
Properties of continuous functions on closed intervals
There must be a maximum and minimum value
Must be a closed interval and continuous
There must be a boundary
zero point theorem
Intermediate value theorem
It is deduced that in a closed interval, the continuous value must be between the maximum value and the minimum value.
Function limit definition and properties
There is x that tends to infinity and x0
A necessary and sufficient condition for the existence of a limit is that the left and right limits exist and are equal
Two important limits
You can also enter the numbers and press the calculator
extreme algorithm
Some formulas of trigonometric functions
sin(2x)=2sinx.cosx
sinx2 cosx2=1
cos(2x)=cosx2-sinx2=1-2sinx2=2cosx2-1
Infinitely Small and Infinitely Large
Properties of infinitesimal operations
The sum of a finite number of infinitesimal algebras is infinitesimal, but an infinite number may not
The product of bounded functions (limited variables, constants, finite infinitesimals) and infinitesimals is infinitesimal,
infinitesimal comparison