private derivative functions z = f(x, y by variable x the derivative of this function is called at a constant value of the variable y, it is denoted or z "x.

private derivative functions z = f(x, y) by variable y called the derivative with respect to y at a constant value of the variable y; it is denoted or z "y.

The partial derivative of a function of several variables with respect to one variable is defined as the derivative of this function with respect to the corresponding variable, provided that the other variables are considered constant.

full differential function z = f(x, y) at some point M(X, y) is called the expression

,

Where and are calculated at the point M(x, y), and dx = , dy = y.

Example 1

Calculate the total differential of the function.

z \u003d x 3 - 2x 2 y 2 + y 3 at the point M (1; 2)

Solution:

1) Find partial derivatives:

2) Calculate the value of partial derivatives at the point M(1; 2)

() M \u003d 3 1 2 - 4 1 2 2 \u003d -13

() M \u003d - 4 1 2 2 + 3 2 2 \u003d 4

3) dz = - 13dx + 4dy

Questions for self-control:

1. What is called an antiderivative? List the properties of an antiderivative.

2. What is called indefinite integral?

3. List the properties of the indefinite integral.

4. List the basic integration formulas.

5. What integration methods do you know?

6. What is the essence of the Newton-Leibniz formula?

7. Give a definition of a definite integral.

8. What is the essence of calculating a definite integral by the substitution method?

9. What is the essence of the method of calculating a definite integral by parts?

10. What function is called a function of two variables? How is it designated?

11. What function is called a function of three variables?

12. What set is called the domain of a function?

13. With the help of what inequalities can one define a closed region D on a plane?

14. What is called the partial derivative of the function z \u003d f (x, y) with respect to the variable x? How is it designated?

15. What is called the partial derivative of the function z \u003d f (x, y) with respect to the variable y? How is it designated?

16. What expression is called the total differential of a function

Topic 1.2 Ordinary differential equations.

Problems leading to differential equations. Differential equations with separable variables. General and private solutions. Homogeneous differential equations of the first order. Linear homogeneous equations second order with constant coefficients.

Practical lesson No. 7 "Finding general and particular solutions differential equations with separable variables"*

Practical lesson No. 8 "Linear and homogeneous differential equations"

Practical lesson No. 9 "Solution of differential equations of the 2nd order with constant coefficients»*

L4, chapter 15, pp. 243 - 256

Guidelines

Practical work №2

"Function Differential"

Purpose of the lesson: Learn to solve examples and problems on a given topic.

Theory questions (initial level):

1. The use of derivatives for the study of functions to an extremum.

2. Differential of a function, its geometric and physical meaning.

3. Full differential functions of many variables.

4. The state of the body as a function of many variables.

5. Approximate calculations.

6. Finding partial derivatives and total differential.

7. Examples of the use of these concepts in pharmacokinetics, microbiology, etc.

(self-training)

1. answer questions on the topic of the lesson;

2. solve examples.

Examples

Find the differentials of the following functions:

1)	2)	3)
4)	5)	6)
7)	8)	9)
10)	11)	12)
13)	14)	15)
16)	17)	18)
19)	20)

Using derivatives to study functions

The condition for the function y = f(x) to increase on the segment [a, b]

The condition for the function y=f(x) to decrease on the segment [a, b]

The condition for the maximum function y=f(x) at x= a

f"(a)=0 and f""(a)<0

If for x \u003d a the derivatives f "(a) \u003d 0 and f "(a) \u003d 0, then it is necessary to investigate f "(x) in the vicinity of the point x \u003d a. The function y \u003d f (x) for x \u003d a has a maximum , if when passing through the point x \u003d and the derivative f "(x) changes sign from "+" to "-", in the case of a minimum - from "-" to "+" If f "(x) does not change sign when passing through point x = a, then at this point the function has no extremum

Function differential.

The differential of an independent variable is equal to its increment:

Function differential y=f(x)

Differential of the sum (difference) of two functions y=u±v

Differential of the product of two functions y=uv

The quotient differential of two functions y=u/v

dy=(vdu-udv)/v 2

Function increment

Δy \u003d f (x + Δx) - f (x) ≈ dy ≈ f "(x) Δx

where Δx: is the increment of the argument.

Approximate calculation of the function value:

f(x + Δx) ≈ f(x) + f "(x) Δx

Application of the differential in approximate calculations

The differential is used to calculate the absolute and relative errors in indirect measurements u = f(x, y, z.). Absolute error of the measurement result

du≈Δu≈|du/dx|Δx+|du/dy|Δy+|du/dz|Δz+…

Relative error of the measurement result

du/u≈Δu/u≈(|du/dx|Δx+|du/dy|Δy+|du/dz|Δz+…)/u

FUNCTION DIFFERENTIAL.

Function differential as the main part of the function increment and. The concept of the differential of a function is closely related to the concept of a derivative. Let the function f(x) continuous for given values X and has a derivative

D f/Dx = f¢(x) + a(Dx), whence the function increment Df = f¢(x)Dx + a(Dx)Dx, where a(Dx) ® 0 at Dx ® 0. Let us define the order of the infinitesimal f¢(x)Dx Dx.:

Therefore, infinitesimal f¢(x)Dx and Dx have the same order of magnitude, that is f¢(x)Dx = O.

Let us define the order of the infinitesimal a(Dх)Dх with respect to the infinitesimal Dx:

Therefore, the infinitesimal a(Dх)Dх has a higher order of smallness than the infinitesimal Dx, that is a(Dx)Dx = o.

Thus, an infinitesimal increment Df differentiable function can be represented in the form of two terms: an infinitesimal f¢(x)Dx of the same order of smallness with Dx and infinitesimal a(Dх)Dх higher order of smallness compared to infinitesimal Dx. This means that in equality Df=f¢(x)Dx + a(Dx)Dx at Dx® 0 the second term tends to zero "faster" than the first, i.e. a(Dx)Dx = o.

First term f¢(x)Dx, linear with respect to Dx, called function differential f(x) at the point X and denote dy or df(read "de game" or "de ef"). So,

dy = df = f¢(x)Dx.

Analytic meaning of the differential lies in the fact that the differential of a function is the main part of the increment of the function Df, linear with respect to the increment of the argument Dx. The differential of a function differs from the increment of a function by an infinitesimal one of a higher order of smallness than Dx. Really, Df=f¢(x)Dx + a(Dx)Dx or Df = df + a(Dx)Dx . Argument differential dx equal to its increment Dx: dx=Dx.

Example. Calculate the value of the differential of a function f(x) = x 3 + 2x, when X varies from 1 to 1.1.

Solution. Let's find a general expression for the differential of this function:

Substituting values dx=Dx=1.1–1= 0.1 and x=1 into the last formula, we get the desired value of the differential: df½ x=1; = 0,5.

PARTIAL DERIVATIVES AND DIFFERENTIALS.

Partial derivatives of the first order. The first-order partial derivative of the function z = f(x,y ) by argument X at the considered point (x; y) called the limit

if it exists.

Partial derivative of a function z = f(x, y) by argument X denoted by one of the following symbols:

Similarly, the partial derivative with respect to at denoted and defined by the formula:

Since the partial derivative is the usual derivative of a function of one argument, it is not difficult to calculate it. To do this, you need to use all the rules of differentiation considered so far, taking into account in each case which of the arguments is taken as a "constant number" and which serves as a "differentiation variable".

Comment. To find the partial derivative, for example, with respect to the argument x – df/dx, it suffices to find the ordinary derivative of the function f(x,y), assuming the latter is a function of one argument X, a at- permanent; to find df/dy- vice versa.

Example. Find the values ​​of partial derivatives of a function f(x,y) = 2x2 + y2 at the point P(1;2).

Solution. Counting f(x,y) single argument function X and using the rules of differentiation, we find

At the point P(1;2) derivative value

Considering f(x; y) as a function of one argument y, we find

At the point P(1;2) derivative value

TASK FOR STUDENT'S INDEPENDENT WORK:

Find the differentials of the following functions:

Solve the following tasks:

1. By how much will the area of ​​a square with side x = 10 cm decrease if the side is reduced by 0.01 cm?

2. The equation of body motion is given: y=t 3 /2+2t 2 , where s is expressed in meters, t is in seconds. Find the path s covered by the body in t=1.92 s from the start of the motion.

LITERATURE

1. Lobotskaya N.L. Fundamentals of Higher Mathematics - M .: "Higher School", 1978.C198-226.

2. Bailey N. Mathematics in biology and medicine. Per. from English. M.: Mir, 1970.

3. Remizov A.N., Isakova N.Kh., Maksina L.G. Collection of problems in medical and biological physics - M .: "Higher School", 1987. C16-20.

Consider changing a function when incrementing only one of its arguments − x i, and let's call it .

Definition 1.7.private derivative functions by argument x i called .

Designations: .

Thus, the partial derivative of a function of several variables is actually defined as the derivative of the function one variable - x i. Therefore, all the properties of derivatives proved for a function of one variable are valid for it.

Comment. In the practical calculation of partial derivatives, we use the usual rules for differentiating a function of one variable, assuming that the argument with respect to which differentiation is carried out is variable, and the remaining arguments are constant.

1. z= 2x² + 3 xy –12y² + 5 x – 4y +2,

2. z = x y ,

Geometric interpretation of partial derivatives of a function of two variables.

Consider the surface equation z = f(x,y) and draw a plane x = const. Let us choose a point on the line of intersection of the plane with the surface M (x, y). If you set the argument at increment Δ at and consider the point T on the curve with coordinates ( x, y+Δ y, z+Δy z), then the tangent of the angle formed by the secant MT with the positive direction of the O axis at, will be equal to . Passing to the limit at , we obtain that the partial derivative is equal to the tangent of the angle formed by the tangent to the resulting curve at the point M with the positive direction of the O axis y. Accordingly, the partial derivative is equal to the tangent of the angle with the O axis X tangent to the curve resulting from the section of the surface z = f(x,y) plane y= const.

Definition 2.1. The full increment of the function u = f(x, y, z) is called

Definition 2.2. If the increment of the function u \u003d f (x, y, z) at the point (x 0, y 0, z 0) can be represented in the form (2.3), (2.4), then the function is called differentiable at this point, and the expression is called the main linear part of the increment or the total differential of the function under consideration.

Notation: du, df (x 0 , y 0 , z 0).

Just as in the case of a function of one variable, the differentials of independent variables are their arbitrary increments, therefore

Remark 1. Thus, the statement "the function is differentiable" is not equivalent to the statement "the function has partial derivatives" - differentiability also requires the continuity of these derivatives at the point under consideration.

4. Tangent plane and normal to the surface. The geometric meaning of the differential.

Let the function z = f(x, y) is differentiable in a neighborhood of the point M (x 0, y 0). Then its partial derivatives are the slopes of the tangents to the lines of intersection of the surface z = f(x, y) with planes y = y 0 and x = x 0, which will be tangent to the surface itself z = f(x, y). Let's write an equation for the plane passing through these lines. The direction vectors of the tangents have the form (1; 0; ) and (0; 1; ), so the normal to the plane can be represented as their vector product: n = (- ,- , 1). Therefore, the equation of the plane can be written as:

where z0 = .

Definition 4.1. The plane defined by equation (4.1) is called tangent plane to the graph of the function z = f(x, y) at the point with coordinates (x 0, y 0, z 0).

From formula (2.3) for the case of two variables it follows that the increment of the function f in the vicinity of the point M can be represented as:

Therefore, the difference between the applicates of the function graph and the tangent plane is an infinitesimal higher order than ρ, at ρ→ 0.

In this case, the differential of the function f looks like:

which corresponds increment of the applicate of the tangent plane to the graph of the function. This is the geometric meaning of the differential.

Definition 4.2. Non-zero vector perpendicular to the tangent plane at a point M (x 0, y 0) surfaces z = f(x, y), is called normal to the surface at that point.

As a normal to the surface under consideration, it is convenient to take the vector - n = { , ,-1}.

Let the function be defined in some (open) domain D points
dimensional space, and
is a point in this area, i.e.
D.

Partial increment of a function of many variables for any variable is called the increment that the function will receive if we give an increment to this variable, assuming that all other variables have constant values.

For example, partial increment of a function over a variable will be

Partial derivative with respect to the independent variable at the point
from the function is called the limit (if it exists) of the partial increment relation
functions to increment
variable while striving
to zero:

The partial derivative is denoted by one of the symbols:

;
.

Comment. Index below in this notation only indicates which of the variables the derivative is taken from, and is not related to at what point
this derivative is calculated.

The calculation of partial derivatives is nothing new compared to the calculation of the ordinary derivative, it is only necessary to remember that when differentiating a function with respect to any variable, all other variables are taken as constants. Let's show this with examples.

Example 1Find Partial Derivatives of Functions
.

Solution. When calculating the partial derivative of a function
by argument consider the function as a function of only one variable , i.e. believe that has a fixed value. At a fixed function
is the power function of the argument . According to the formula for differentiating a power function, we obtain:

Similarly, when calculating the partial derivative we assume that the value is fixed , and consider the function
as an exponential function of the argument . As a result, we get:

Example 2. Hfind partial derivatives and functions
.

Solution. When calculating the partial derivative with respect to given function we will consider as a function of one variable , and expressions containing , will be constant factors, i.e.
acts as a constant factor with a power function (
). Differentiating this expression with respect to , we get:

.

Now, on the contrary, the function considered as a function of one variable , while expressions containing , act as a coefficient
(
).Differentiating according to the rules of differentiation of trigonometric functions, we get:

Example 3 Calculate Partial Derivatives of a Function
at the point
.

Solution. We first find the partial derivatives of this function at an arbitrary point
its domain of definition. When calculating the partial derivative with respect to believe that
are permanent.

when differentiating by will be permanent
:

and when calculating partial derivatives with respect to and by , similarly, will be constant, respectively,
and
, i.e.:

Now we calculate the values ​​of these derivatives at the point
, substituting specific values ​​of variables into their expressions. As a result, we get:

11. Partial and total differentials of a function

If now to a private increment
apply Lagrange's theorem on finite increments with respect to a variable , then, counting continuous, we obtain the following relations:

where
,
is an infinitesimal quantity.

Partial Differential of a Function by variable is called the main linear part of the partial increment
, equal to the product of the partial derivative with respect to this variable and the increment of this variable, and is denoted

Obviously, the partial differential differs from the partial increment by an infinitesimal higher order.

Full function increment many variables is called its increment, which it will receive when we give an increment to all independent variables, i.e.

where is everyone
, depend on and together with them tend to zero.

Under differentials of independent variables agreed to mean arbitrary increments
and label them
. Thus, the expression of the partial differential will take the form:

For example, a partial differential on is defined like this:

.

full differential
functions of many variables is called the main linear part of the total increment
equal to, i.e. the sum of all its partial differentials:

If the function
has continuous partial derivatives

at the point
, then she differentiable at a given point.

For sufficiently small for a differentiable function
there are approximate equalities

,

which can be used for approximate calculations.

Example 4Find the full differential of a function
three variables
.

Solution. First of all, we find the partial derivatives:

Noting that they are continuous for all values
, we find:

For differentials of functions of several variables, all theorems on the properties of differentials are true, which have been proved for the case of functions of one variable, for example: if and are continuous functions of variables
, which have continuous partial derivatives with respect to all variables, and and are arbitrary constants, then:

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transcript

1 LECTURE N Total differential, partial derivatives and differentials of higher orders Total differential Partial differentials Partial derivatives of higher orders Higher order differentials 4 Derivatives of complex functions 4 Total differential Partial differentials If a function z=f(,) is differentiable, then its total differential dz is equal to dz= a +B () z z Noting that A=, B =, we write the formula () in the following form z z dz= + () We extend the concept of a function differential to independent variables, setting the differentials of independent variables equal to their increments: d= ; d= After that, the formula for the total differential of the function will take the form z z dz= d + d () d + d n variables, then du= d (d =) = The expression d z=f (,)d (4) is called the partial differential of the function z=f(,) with respect to the variable; the expression d z=f (,)d (5) is called the partial differential of the function z=f(,) with respect to the variable It follows from formulas (), (4) and (5) that the total differential of a function is the sum of its partial differentials: dz=d z+d z the increment z= z z + + α (,) + β (,) differs from its linear part dz= z z + only by the sum of the last terms α + β, which at 0 and 0 are infinitesimal higher order than the terms of the linear part Therefore when dz 0, the linear part of the increment of the differentiable function is called the main part of the increment of the function and the approximate formula z dz is used, which will be the more accurate, the smaller the absolute value of the increments of the arguments,97 Example Calculate approximately arctan(),0

2 Solution Consider the function f(,)=arctg() Using the formula f(x 0 + x, y 0 + y) f(x 0, y 0) + dz, we get arctg(+) arctg() + [ arctg() ] + [ arctg()] or + + arctg() arctg() () + () Let =, =, then =-0.0, =0.0 Therefore, (0.0 0.0 arctg) arctg( ) + (0.0) 0.0 = arctan 0.0 = + 0.0 + () + () π = 0.05 0.0 0.75 4 It can be shown that the error resulting from the application of the approximate formula z dz does not exceed the number = M (+), where M is the largest value of the absolute values ​​of the second partial derivatives f (,), f (,), f (,) when the arguments change from to + and from to + Partial derivatives of higher orders If the function u =f(, z) has a partial derivative with respect to one of the variables in some (open) domain D, then the found derivative, being itself a function of, z, can, in turn, have partial derivatives at some point (0, 0, z 0) with respect to the same or any other variable For the original function u=f(, z), these derivatives will be partial derivatives of the second order If the first derivative was taken, for example ep, in, then its derivative with respect to, z is denoted as follows: f (0, 0, z0) f (0, 0, z0) f (0, 0, z0) = ; = ; = or u, u, u z z z Derivatives of the third, fourth, and so on orders are determined similarly. Note that the higher-order partial derivative taken with respect to various variables, for example, ; called mixed partial derivative Example u= 4 z, then, u =4 z ; u = 4z; u z = 4 z; u = z u=64z; uzz = 4; u = z u = z u z = 4 z; u z =8 z; u z =6 4 z; u z =6 4 z the function f(,) is defined in an (open) domain D,) in this domain there are first derivatives f and f, as well as second mixed derivatives f and f, and finally,) these last derivatives f and f, as functions of u, are continuous in some point (0, 0) of the region D Then at this point f (0, 0)=f (0, 0) Proof Consider the expression

3 f (0 +, 0 f (0 +, 0) f (0, 0 + f (0, 0) W=, where, are non-zero, for example, are positive, and, moreover, are so small that D contains the entire rectangle [ 0, 0 +; 0, 0 +] 0 +) (, 0) ()= and therefore continuous With this function f (0 +, 0 f (0 +, 0) f (0, 0 f (0, 0) expression W, which is equal to W= can be rewritten in the form: ϕ (0 +) ϕ (0) W= so: W=ϕ (0 + θ, 0 f (0 + θ, 0) (0 + θ)= (0<θ<) Пользуясь существованием второй производной f (,), снова применим формулу конечных приращений, на этот раз к функции от: f (0 +θ,) в промежутке [ 0, 0 +] Получим W=f (0 +θ, 0 +θ), (0<θ <) Но выражение W содержит и, с одной стороны, и и, с другой, одинаковым образом Поэтому, можно поменять их роли и, введя вспомогательную функцию: Ψ()= f (0 +,) f (0,), путем аналогичных рассуждений получить результат: W=f (0 +θ, 0 +θ) (0<θ, θ <) Из сопоставления () и (), находим f (0 +θ, 0 +θ)=f (0 +θ, 0 +θ) Устремив теперь и к нулю, перейдем в этом равенстве к пределу В силу ограниченности множителей θ, θ, θ, θ, аргументы и справа, и слева стремятся к 0, 0 А тогда, в силу (), получим: f (0, 0)=f (0, 0), что и требовалось доказать Таким образом, непрерывные смешанные производные f и f всегда равны Общая теорема о смешанных производных Пусть функция u=f(, n) от переменных определена в открытой n-мерной области D и имеет в этой области всевозможные частные производные до (n-)-го порядка включительно и смешанные производные n-го порядка, причем все эти производные непрерывны в D При этих условиях значение любой n-ой смешанной производной не зависит от того порядка, в котором производятся последовательные дифференцирования Дифференциалы высших порядков Пусть в области D задана непрерывная функция u=f(, х), имеющая непрерывные частные производные первого порядка Тогда, du= d + d + + d

4 We see that du is also a function of, If we assume the existence of continuous partial derivatives of the second order for u, then du will have continuous partial derivatives of the first order and we can talk about the total differential of this differential du, d(du), which is called second-order differential (or second differential) of u; it is denoted by d u We emphasize that the increments d, d, d are considered constant and remain the same when moving from one differential to the next (moreover, d, d will be zero) So, d u=d(du)=d(d + d + + d) = d() d + d() d + + d() d or d u = (d + d + d + + d) d + + (d + d + = d + d + + d + dd + dd + + dd + + Similarly, the third-order differential d u is defined, and so on. If the function u has continuous partial derivatives of all orders up to and including the nth one, then the existence of the nth differential is guaranteed. But the expressions for them become more and more complex We can simplify the notation Let's take out the “letter u” in the expression of the first differential Then, the notation will be symbolic: du=(d + d + + d) u ; d u=(d + d + + d) u ; d n n u=(d + d + + d) u, which should be understood as follows: first, the “polynomial” in brackets is formally raised to a power according to the rules of algebra, then all the resulting terms are “multiplied” by u (which is added to n in the numerators at) , and only after that all symbols return their value as derivatives and differentials u d) d u on the variable t in some interval: =ϕ(t), =ψ(t), z=λ(t) Let, in addition, as t changes, the points (, z) do not go beyond the region D Substituting the values, and z into function u, we get a complex function: u=f(ϕ(t), ψ(t), λ(t)) Suppose that u has continuous partial derivatives u, u and u z in and z and that t, t and z t exist Then it is possible to prove the existence of a derivative of a complex function and calculate it. We give the variable t some increment t, then, and z will receive increments, respectively, and z, the function u will receive an increment u Let us represent the increment of the function u in the form: (this can be done, since we assumed the existence of continuous partial derivatives u, u and u z) u=u +u +u z z+α +β +χ z, where α, β, χ 0 at, z 0 We divide both part of the equality on t, we obtain u z z = u + u + uz + α + β + χ t t t t t t t 4

5 Let us now let the increment t approach zero: then, z will tend to zero, since the functions, z of t are continuous (we assumed the existence of derivatives t, t, z t), and therefore, α, β, χ also tend to zero In the limit we obtain u t =u t +u t +u z z t () We see that under the assumptions made, the derivative of the complex function really exists , z in several variables t: =ϕ(t, v), =ψ(t, v), z=χ(t, v) Besides the existence and continuity of partial derivatives of the function f(, z), we assume here the existence of derivatives of functions, z with respect to t and v This case does not differ significantly from the one already considered, since when calculating the partial derivative of a function of two variables, we fix one of the variables, and we are left with a function of only one variable, the formula () will be the same z, and () must be rewritten as: = + + (a) t t t z t z = + + (b) v v v z v Example u= ; =ϕ(t)=t ; =ψ(t)=cos t u t = - t + ln t = - t- ln sint 5

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Lectures Chapter Functions of several variables Basic concepts Some functions of several variables are well known Let's give some examples To calculate the area of ​​a triangle, Heron's formula S is known

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First-order differential equations resolved with respect to the derivative Existence and uniqueness theorem for a solution In the general case, a first-order differential equation has the form F ()

Lecture 3 Extremum of a function of several variables Let a function of several variables u = f (x, x) be defined in the domain D, and the point x (x, x) = belongs to this domain The function u = f (x, x) has

Module Topic Function sequences and series Properties of uniform convergence of sequences and series Power series Lecture Definitions of function sequences and series Uniformly

9 Derivative and differential 91 Basic formulas and definitions for solving problems Definition Let the function y f () is defined on some f (Δ) f () Δy neighborhood of the point Relation limit for Δ Δ Δ, if

1 Topic 1. First order differential equations 1.0. Basic definitions and theorems First order differential equation: independent variable; y = y() is the desired function; y = y () its derivative.

Lecture 8 Differentiation of a complex function Consider a complex function t t t f where ϕ t t t t t t t f t t t t t t t t t

MOSCOW STATE TECHNICAL UNIVERSITY OF CIVIL AVIATION V.M. Lyubimov, E.A. Zhukova, V.A. Ukhova, Yu.A. Shurinov

II DIFFERENTIAL EQUATIONS First Order Differential Equations Definition Relationships in which unknown variables and their functions are under the derivative or differential sign are called

6 Problems leading to the concept of a derivative Let a material point move in a straight line in one direction according to the law s f (t), where t is time and s is the path traveled by the point in time t Note a certain moment

Lecture 3. Indefinite integral. Antiderivative and indefinite integral In differential calculus, the problem is solved: for a given function f () find its derivative (or differential). Integral calculus

1 Lecture 7 Derivatives and differentials of higher orders Abstract: The concept of a differentiable function is introduced, a geometric interpretation of the first differential is given and its invariance is proved

Functions of several arguments The concept of a function for each element x from the set X according to some law y \u003d f (x) is associated with a single value of the variable y from the set Y to each pair of numbers

Compiled by VPBelkin 1 Lecture 1 Function of several variables 1 Basic concepts Dependence \u003d f (1, n) of a variable on variables 1, n is called a function of n arguments 1, n In what follows, we will consider

DIFFERENTIAL EQUATIONS General concepts Differential equations have numerous and very diverse applications in mechanics, physics, astronomy, technology, and in other branches of higher mathematics (for example,

I Definition of a function of several variables Domain of definition When studying many phenomena, one has to deal with functions of two or more independent variables. For example, body temperature at a given moment

Lecture 8 Fermat, Rolle, Cauchy, Lagrange and L'Hospital's theorems

SA Lavrenchenko wwwlawrencenkoru Lecture 4 Differentiation of complex functions Implicit differentiation Recall the differentiation rule for functions of one variable, also called the chain rule (see

Section Differential calculus of functions of one and several variables Real argument function Real numbers Positive integers are called natural numbers Add to natural numbers

Workshop: “Differentiability and differential of a function” If the function y f () has a finite derivative at a point, then the increment of the function at this point can be represented as: y (,) f () () (), where () at

Lecture Differential equations of the th order The main types of differential equations of the th order and their solution Differential equations are one of the most common means of mathematical

TOPIC 1 DERIVATIVE FUNCTION DIFFERENTIAL FUNCTION PROGRAM QUESTIONS: 11 Functional connection Function limit 1 Function derivative 1 Mechanical physical and geometric meaning of the derivative 14 Basic

M I N I S T E R S T O E D U R A O V A N I A I A N A U K I R O S S I O Y F E D E R A T I O N FEDERAL STATE AUTONOMOUS EDUCATIONAL INSTITUTION OF HIGHER EDUCATION "National Research

DISCIPLINE "Higher Mathematics" course, semester Correspondence form of study TOPIC Matrix Algebra

V.V. Zhuk, A.M. Kamachkin Differentiability of functions of several variables. Differentiability of a function at a point. Sufficient conditions for differentiability in terms of partial derivatives. Complex differentiation

Chapter 4 The Limit of a Function 4 1 THE CONCEPT OF THE LIMIT OF A FUNCTION This chapter focuses on the concept of the limit of a function. Defined what is the limit of a function at infinity, and then the limit at a point, limits

LECTURE 23 CANONICAL TRANSFORMATIONS. THEOREM OF LIOUVILLE ON CONSERVATION OF PHASE VOLUME. FREE TRANSFORMATION GENERATING FUNCTION We continue to study canonical transformations. Let us first recall the main

Department of Mathematics and Informatics Mathematical analysis Educational and methodological complex for HPE students studying with the use of distance technologies Module 3 Differential calculus of functions of one

55 is at an infinitesimal value of a higher order of smallness compared to ρ n (,), where ρ () + (), then it can be represented in the Peano form n R, ρ Example Write the Taylor formula for n with

Topic Definite integral Definite integral Problems leading to the concept of a definite integral The problem of calculating the area of ​​a curvilinear trapezoid In the Oxy coordinate system, a curvilinear trapezoid is given,

5 Power series 5 Power series: definition, domain of convergence Function series of the form (a + a) + a () + K + a () + K a) (, (5) numbers are called power series Numbers

Numerical series Numerical sequence Opr A numerical sequence is a numerical function defined on the set of natural numbers x - a common member of the sequence x =, x =, x =, x =,

Differential equations lecture 4 Equations in total differentials. Integrating factor Lecturer Anna Igorevna Sherstneva 9. Equations in total differentials The equation d + d = 14 is called the equation

Faculty of Metallurgy Department of Higher Mathematics

Mathematical analysis Section: Function of several variables Topic: Differentiability of FNP (end. Partial derivatives and differentials of complex FNP. Differentiation of implicit functions Lecturer Rozhkova S.V.

(Fermat's theorem - Darboux's theorem - Rolle's theorem - Lagrange's theorem mean value theorem - geometric interpretation of the mean value theorem - Cauchy's theorem - finite increment formula - L'Hopital's rule

Chapter 4 Fundamental theorems of differential calculus Disclosure of uncertainties Fundamental theorems of differential calculus Fermat's theorem (Pierre Fermat (6-665) French mathematician) If the function y f

LECTURE 7 DIFFERENTIAL CALCULATION OF A FUNCTION OF ONE VARIABLE 1 The concept of a derivative of a function

Ministry of Education of the Republic of Belarus Vitebsk State Technological University Topic. "Rows" Department of Theoretical and Applied Mathematics. developed by Assoc. E.B. Dunina. Main

Lecture 3 Taylor and Maclaurin series Application of power series Expansion of functions into power series Taylor and Maclaurin series For applications, it is important to be able to expand a given function into a power series, those functions

58 Definite integral Let the function () be given on the interval. We will consider the function continuous, although this is not necessary. We choose arbitrary numbers on the interval, 3, n-, satisfying the condition:

Higher order differential equations. Konev V.V. Lecture outlines. Contents 1. Basic concepts 1 2. Equations that allow order reduction 2 3. Linear differential equations of higher order

Lecture 20 THEOREM ON THE DERIVATIVE OF A COMPLEX FUNCTION. Let y = f(u) and u= u(x). We get a function y depending on the argument x: y = f(u(x)). The last function is called a function of a function or a complex function.

Differentiation of an implicit function Consider the function (,) = C (C = const) This equation defines an implicit function () Suppose we have solved this equation and found an explicit expression = () Now we can

Moscow Aviation Institute (National Research University) Department of Higher Mathematics Limits Derivatives Functions of several variables Guidelines and control options

LABORATORY WORK 7 GENERALIZED FUNCTIONS I. BASIC CONCEPTS AND THEOREMS Denote by D the set of all infinitely differentiable finite functions of a real variable. it

Chapter 3. Investigation of functions with the help of derivatives 3.1. Extremums and monotonicity Consider a function y = f () defined on some interval I R. It is said that it has a local maximum at the point

Moscow State Technical University named after N.E. Bauman Faculty of Fundamental Sciences Department of Mathematical Modeling А.Н. Kanatnikov,

Guidelines and variants of the RGR on the topic The function of several variables for students of the specialty Design. If the quantity is uniquely determined by setting the values ​​of the quantities and independent of each other,

Moscow State Technical University named after N.E. Bauman Faculty of Fundamental Sciences Department of Mathematical Modeling А.Н. Kanatnikov, A.P. Kryshenko

METHODOLOGICAL INSTRUCTIONS FOR CALCULATION TASKS ON THE COURSE OF HIGHER MATHEMATICS "ORDINARY DIFFERENTIAL EQUATIONS SERIES DOUBLE INTEGRALS" PART III THEME SERIES Contents Series Numerical series Convergence and divergence

Function limit. Number Sequence Limit Definition. An infinite numerical sequence (or simply a numerical sequence) is a function f f (, defined on the set of all

Lecture 19 DERIVATIVE AND ITS APPLICATIONS. DEFINITION OF DERIVATIVE. Let we have some function y=f(x) defined on some interval. For each value of the argument x from this interval, the function y=f(x)

Differential calculus of functions of several variables Functions of several variables A quantity is called a function of variables n if each point M n belonging to some set X is assigned

LECTURE N 7 .Power

Lecture 3 Existence and uniqueness theorem for a solution to a scalar equation Statement of the problem Main result Consider the Cauchy problem d f () d =, () =

Federal Agency for Education Moscow State University of Geodesy and Cartography (MIIGAiK) METHODOLOGICAL INSTRUCTIONS AND TASKS FOR INDEPENDENT WORK on the course HIGHER MATHEMATICS