matesha
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VfubSet of numbers:
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Definition of function:
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Intersection is
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Union is A∪B
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Disjoint is A∩B=ø
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Sequence is bounded below if
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Sequence is strictly decreasing if
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Sequence is divergent if sequence is oscillating or tending to infinity
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Bounded sequence
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Definition of limit of sequence is
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Sequence is monotone if
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Monotone convergence principle If is an increasing sequence and bounded above, then it is convergent sequence
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Theorem of Bolzano-Weiertrasse: every bounded sequence of complex numbers has A convergent subsequence
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Caushy sequence is Convergent, i.e. if given any ε>0, there exist N=N(ε)∈ℝ, depending on ε, such that whenever m>n≥N
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Composite function is
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Inverse function for is
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Limit of function in since “ε-δ” is
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is infinitesimal value
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is infinitely large value
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Two Remarkable limits
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Function f(x) is continuous in point a
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Point of discontinuity of first kind (common definition) There are the finite limits exist:
, ,
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Points of discontinuity of second kind: If at least one limit does not exist or equals to ∞
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Points of discontinuity of first kind – jump point:
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Points of removable discontinuity third condition is not satisfied
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BOLZANO-WEIERSTRASS THEOREM 1 Suppose function y=f(x) continued on interval [a,b], so function y=f(x) is bounded on interval [a,b]
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BOLZANO-WEIERSTRASS THEOREM 2 Suppose function y=f(x) continued on interval [a,b], so function y=f(x) reaches maximum and minimum on interval [a,b]
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BOLZANO-CAUCHY THEOREM Suppose function y=f(x) continued on interval [a,b] and f(a)<0 and f(b)>0 or f(a)>0 and f(b)<0, so there exist such ξ∊(a,b): f(ξ)=0
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Derivative of function f(x) in point x=a is
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, where v is speed, s is distance, t is time Physical interpretation of derivative
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Differentiation rules
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Differentiation rules
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Differentiation rules
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Differentiation rules
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If for the inverse function exists and , then
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Differentiation rules
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Differentiation rules
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Differentiation rules
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Chain rule of differentiation
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Derivative of n order Differential is main part of increment linearly with regard to 𝛥x: dy=f’(x)𝛥x
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Form of differential dy (in contradistinction to derivative) Is invariable
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L’hopital’s rule
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A function f(x) is said to have stationary point at x=a if
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A function f(x) is said to have a local maximum at x=a if
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A function f(x) is said to have a local minimum at x=a if
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If f’(x)>0 for every x<a in I and f’(x)<0 for every x>a in I, then
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If f’(x)<0 for every x<a in I and f’(x)>0 for every x>a in I, then
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Function f(x) is differentiable at every x∈I, and that f’(x)=0 and if f’’(x)<0, then
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Function f(x) is differentiable at every x∈I, and that f’(x)=0
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Function y=f(x) is concave upwards in interval [A,B], if for any x₁, x₂∈[A,b}
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Function y=f(x) is concave down in interval [A,B], if for any x₁, x₂∈[A,b}
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Inflection point is point of continued function dividing intervals of concavity (upwards and down)
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Prerequisite of inflection point
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Sufficient condition of inflection point: Recall (x₁, x₂, …, x𝓃) is critical point of function z and partial derivatives of second order exist and continued.
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Asymptote of y=f(x) is
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Let function y=f(x) is defined in ε of x₀ and lim f(x)=∞ as x→x₀-0 or lim f(x)=∞ as x→x₀+0
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Antiderivative is function F(x) such that F’(x)=f(x)
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Standard integrals
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Standard integrals
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Standard integrals
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Standard integrals
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Standard integrals
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Integration by substitution: If we maks a substitution x=g(u), then dx=g’du
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Integration by substitution
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Integration by parts
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Integration of irrationalities of kind can be rationalize by substitution
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Integration of irrationalities of kind can be rationalize by substitution:
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Integration of trigonometric functions can be rationalized by substitution
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Integration of trigonometric functions can be rationalized by substitution
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Integration of trigonometric functions can be rationalized by substitution
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Integration of trigonometric function can be rationalized by substitution
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Non-integrable (in finite form) functions
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Non-integrable (in finite form) functions:
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Non-integrable (in finite form) functions:
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Non-integrable (in finite form) functions:
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Non-integrable (in finite form) functions:
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Non-integrable (in finite form) functions:
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Non-integrable (in finite form) functions:
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By the definite integral
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Newton-Leibniz Formulae: Suppose that a function F(x) satisfies F’(x)=f(x) for every x∈[A,B]. Then
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Property of definite integralLinearity
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Property of definite integral
Additivity:
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Property of definite integral
Monotonness: If f(x)≤g(x), and a<b, then
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Property of definite integral
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Non-eigenvalue integral in half-interval [a,+∞)
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If this limit exists and finite, then non-eigenvalue integral is convergent
If this limit does not exists or infinite, then non-eigenvalue integral is divergent
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Particularly if non-eigenvalue integral is divergent, but exists, so is Value Principal of Integral
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Multiplication of matrix
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Transpose matrix
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Row echelon form of matrix is
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Determinant of matrix 3x3 - Rule of triangles or Rule of Sarrus
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Determinant of matrix 2x2 :A=
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det(A)=0 singular
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det(A)≠0 non-singular
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This matrix Is augmented
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An nxn matrix A is said to be invertible if if there exists an nxn matrix B= such that AB=BA=I
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Matrix I is Identity matrix if
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Matrix with members is is Identity matrix
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, then multiplicative inverse is
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Minor is
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Cofactor , where
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where K is the adjoint of matrix A
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Trace of matrix is the sum of diagonal elements
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Rank of a matrix is
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Theorem of Kronecker-Capelli (Solution of system of linear equations)
System is inconsistent
System is consistent
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Cramer’s Rule (solution of system of linear equation using matrix analysis). Suppose that the matrix A is invertible. Then the unique solution of the system Ax=b, where A, x and b are variables of system of linear equations where
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Gaussian Elimination (Matrix Analysis)
The method of Gaussian Elimination reduces any set of linear equations to this triangular form by adding or subtracting suitable multiples of pairs of the equations.
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A vector is an object which has magnitude and direction
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For any vector u=(u₁, u₂) in ℝ², Norm of vector is real non-negative number
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Scalar products of vectors
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Economic interpretation of scalar products of vector
- summary value of goods
where - vector of volume of different goods, - vector of prices of different goods
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Orthogonality condition for vectors: cos θ = 0 (by θ=π⁄2) ⇒ u₁v₁+u₂v₂+u₃v₃=0
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Colinearity condition for vectors v₁/u₁=v₂/u₂=v₃/u₃
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Orthogonal projection of vector. Suppose that u, a∈ℝ³. Then
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Scalar triple product of vector is
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Perpendicular distance D of a plane ax+by+cx+d=0 from a point (x₀, y₀, z₀) is given by
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Suppose that are vectors in a vector space V over R. By a linear combination of the vectors , we mean an expression of the type , where ∈ℝ
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Suppose that are vectors in a vector space V over ℝ: are linearly dependent if there exist , not all zero, such that
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Suppose that are vectors in a vector space V over ℝ: are linearly independent if the only solution of is given by
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Suppose that are vectors in vector space V or R. We say that {is a basis for V if the followig two conditions are satisfied:
a){ =V
b)The vectors are linearly independent.
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Dimension dim(V) of vector space V is maximum number of linearly independent vectors
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THEOREM of matrix of transformation . Matrixes A and A* of linear operator over basis () and basis (() can be coupled as: , where C is matrix of transformation from old basis to new basis.
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DEFINITION. “Quadratic form “ of n variables or where every element is either squared variable, or scalar multiplication of 2 different variables.
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Eigenvalue is solution of characteristics equation
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Quadratic form has canonical form, if
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Elements of Analytical Geometry: Relationship between Polar and Cartesian Co-ordinates: Superimpose one diagram upon the other a) X=r cosθ and y=r sin θ
b) and
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Elements of Analytical Geometry: A straight line is a set of points with cartesian co-ordinates (x,y) satisfying an equation of the form ax+by+c=0, where a, b and c are const.
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Elements of Analytical Geometry: Equation of a line passing through 2 given points
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Elements of Analytical Geometry: Distance between 2 points
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First order differential equation y’=f(x,y) is called incomplete if it does not contain (obviously) or the function itself with, or independent variable x: y’=f(x) or variable y: y’=f(y).
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The differential equation of the form y’=f(x)g(y) called a differential equation with multiple variables
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Linear differential equation of first order is an equation that is linear in the unknown function and its derivative, if
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First-Order Homogeneous Equations:
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Differential equation y’’+a₁y’+a₀y=f(x), where coefficients a₁, a₀ are const, linear differential equations of second order with fixed variables .
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Differential equation y’’+a₁y’+a₀y=f(x) where coefficients a₁, a₀ are constants, called linear differential equations of second order with fixed variables. If f(x)=0, then this equation called non homogeneous
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Differential equation y’’+a₁y’+a₀y=f(x) where coefficients a₁, a₀ are const, called called linear differential equations of second order with fixed variables If f(x)≠0, then this equation called homogeneous
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Fundamental system of solutions
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Functions y₁(x), y₂(x) called linear depended in interval (a,b) if there are exist constant numbers λ₁, λ₂ not equal to zero, such a λ₁y₁(x)+λ₂y₂(x)=0 for every x∈(a,b) . If this condition executes in a case when λ₁=0 and λ₂=0, so functions y₁(x), y₂(x) called linear independed in interval (a,b).
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Solution of homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=0: characteristic equation( to change in a homogeneous equation derivatives from initial function to K in appropriate power) a₂=0 has 2 solution: k₁ and k₂, then general solution is
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Solution of homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=0: characteristic equation( to change in a homogeneous equation derivatives from initial function to K in appropriate power) a₂=0 has 1 solution: k₁=k₂=k, then general solution is
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Solution of homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=0
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Solution of non-homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=f(x):solution of system of equations
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Caushy problem for differential equations: Caushy task – is a one of the main task in the theory of differential equations (ordinary and partial); it is consisted in
searching a solution (integral) of the differential equation, satisfying, so-called, initial conditions (initial data).
Let the function f(x) is a well defined value in D for . We should find a solution which satisfying initial conditions.
Caushy Task: If in D, function f (x, y) is continuous and has continuous partial derivatives , so for any pointin a neighborhood of corresponds only an unique solution
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Common member of series : Number is sum of numerical series
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Sum of numerical series is common member of numerical series
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Numerical series is convergent, if exists and finite, then
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Numerical series is divergent, if (lim is not exists or =∞)
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Property of convergence of numerical series
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Property of convergence of numerical series
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Property of convergence of numerical series:
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Prerequisite of convergence of numerical series
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Harmonic series is sum of infinite quantity of members reversed to serial natural numbers
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Series is harmonic if because of each three members starting second satisfies to one rule
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Generalized harmonic series
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THEOREM OF D’ALAMBER (for series) If for series with positive members
is valid, then the numerical series is convergent by l<1 and divergent by l>1
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THEOREM OF CAUSHY (for series) Let for positive defined series the limit exists then if l<1, series is convergent, if l>1, series is divergent, if l=1, convergence question is open.
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THEOREM OF LEIBNIZ (criteria of series convergence). Sign-alternative series is convergent if
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Sign-variable series Its members can be positive and negative
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THEOREM (sufficient condition of convergence for sign-variable series).
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Series is absolute convergent
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Series is conditional convergent
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Consequence of Theorem of Leibniz Following Theorem of Leibniz we can evaluate error of sum calculation
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Determine the domain of the function: =[1;+∞)
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Determine the domain of the function: x≠±2
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Determine the domain of the function: =x>4 x≠7
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Determine the domain of the function: x>1 x≠3
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Determine the domain of the function: (0;2)
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Find the limit of the function: =4
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Find the limit of the function: =10
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Find the limit of the function: =2/5
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Find the limit of the function: =-1/2
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Find the limit of the function: =3
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Find the limit of the function: =0
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Find the limit of the function: =5
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Find the limit of the function: =4
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Find the limit of the function: =8
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Take derivative of the function: =42xcos(7x^2-1)
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Take derivative of the function: =5/2*√(x+3)
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Take derivative of the function: =2xlnx+x
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Take derivative of the function: =ctgx
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Take derivative of the function: =-tgx
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Take second order derivative of the function: =24x+10
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Take second order derivative of the function: y = sinx=-sinx
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Take derivative of the function: y = (sinx)3 =3(sinx)^2*cosx
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Take derivative of the function: y = (lnx)2 =2lnx/x
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Take derivative of the function: y = =1/2√x
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Take derivative of the function: y = =24cos(6x-1)
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Take derivative of the function: y = =3*23x+2*ln2
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Take derivative of the function: y = =2*72x+5 *ln7
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Take derivative of the function: y = =3(5x^2 -4x +1)^2*(10x-4)
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Find the interval where the function is increasing: =(-∞;-3) (1;+∞)
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Find the interval where the function is decreasing: (1;3)
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Find the extreme points of the function : =xmax= 3 xmin=1 o y(1)-max and y(3) -min
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Find the extreme points of the function : xmax=4 xmin=2 y(2)-max and y(4) -min
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Find the extreme points of the function : x=3 y=3
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Find the interval where the function is concave up: (-1;+∞)
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Find the interval where the function is concave down: (-∞;-1)
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Find the points of inflection of the function: =-1
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Find the interval where the function is concave up: =(2;+∞)
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Find the interval where the function is concave down: (-∞;2)
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Find the points of inflection of the function: =2
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Find the equations of the vertical asymptotes of the function:
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x= -3 and x=3
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Find the equations of the vertical asymptotes of the function: =x= -5 and x=5
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Find the equations of the horizontal asymptotes of the function: =Y=2
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Find the equations of the horizontal asymptotes of the function: =Y=3
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Find the partial derivative of the function of two variables:
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Find the partial derivative of the function of two variables:
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Find the partial derivative of the function of two variables:
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Find the partial derivative of the function of two variables:
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Find the partial derivative of the function of two variables:
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Find the partial derivative of the function of two variables:
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Find the partial derivative of the function of two variables: