76. Conditional extremum. Lagrange multiplier method . The function z \u003d f (x, y) has a conditional minimum (maximum) at the interior point M 0 (x 0, y 0), if for any points M (x, y) from some neighborhood O (M 0), satisfying the constraint equation ϕ (x, y) \u003d 0, the condition ∆f (x 0, y 0) \u003d f (x, y) -f (x 0, y 0) ≥0, (∆f (x 0, y 0) ≤ 0). In the general case, this problem is reduced to finding the usual Lagrange extremum L (x, y, λ) \u003d f (x, y) \u003d λϕ (x, y) with an unknown Lagrange multiplier λ. The necessary condition for the extremum of the Lagrange function L (x, y, λ) is a system of three equations with three unknowns x, y, λ: ... A sufficient condition for the extremum of the Lagrange function is in the following statement ∆\u003e 0, then the function z \u003d f (x, y) at the point M 0 (x 0, y 0) has a conditional minimum, ∆<0- то условный максимум.

77. Number series. Basic concepts. Series convergence ... Number series is called an expression of the form, where u 1, u 2,…., u n,… are real or complex numbers, called members of a number, u n - common member row. A series is considered given if the common term of the series u n is known, expressed as a function of its number n: u n \u003d f (n). The sum of the first n terms of the series is called the nth partial sum series and is denoted by S n, i.e. S n \u003d u 1 + u 2 + ... + u n. If there is a finite limit of the sequence of partial sums of the series , then this limit is called the sum of the series and they say that a number converges.

78. A necessary criterion for convergence. Harmonic series. Theorem:Let the number series u 1 + u 2 +… + u n +…, (1) converge, and S is its sum. Then, with an unlimited increase in the number n of terms of the series, its common term u n tends to 0. This criterion is a necessary but not sufficient criterion for the convergence of the series, since you can specify a series for which equality is satisfied

In fact, if it converged, it would equal 0. Thus, the theorem we have proved sometimes allows us, without calculating the sum S n, to draw a conclusion about the divergence of one or another series. For example, the series diverge since . Harmonic series - the sum, made up of an infinite number of members, reciprocal of consecutive numbers of natural numbers: The series is called harmonic because it is composed of "harmonics": the (\\ displaystyle k) th harmonic extracted from the violin string is the fundamental tone produced by a string of length (\\ displaystyle (\\ frac (1) (k))) over length original string.

Definition 1

If for each pair $ (x, y) $ of values \u200b\u200bof two independent variables from a certain domain a certain value of $ z $ is associated, then $ z $ is said to be a function of two variables $ (x, y) $ in this domain.

Notation: $ z \u003d f (x, y) $.

Let a function $ z \u003d f (x, y) $ of two independent variables $ (x, y) $ be given.

Remark 1

Since the variables $ (x, y) $ are independent, one of them can be changed, while the other remains constant.

Let's give the variable $ x $ an increment of $ \\ Delta x $, while keeping the value of the variable $ y $ unchanged.

Then the function $ z \u003d f (x, y) $ will receive an increment, which will be called the partial increment of the function $ z \u003d f (x, y) $ with respect to the variable $ x $. Designation:

Definition 2

The partial derivative with respect to the variable $ x $ of the given function $ z \u003d f (x, y) $ is the limit of the ratio of the partial increment $ \\ Delta _ (x) z $ of the given function to the increment $ \\ Delta x $ at $ \\ Delta x \\ Designation: $ z "_ (x), \\, \\, f" _ (x) (x, y), \\, \\, \\ frac (\\ partial z) (\\ partial x), \\, \\, \\ frac ( \\ partial f) (\\ partial x) $.

Remark 2

\\ [\\ frac (\\ partial z) (\\ partial x) \u003d \\ mathop (\\ lim) \\ limits _ (\\ Delta x \\ to 0) \\ frac (\\ Delta _ (x) z) (\\ Delta x) \u003d \\ mathop (\\ lim) \\ limits _ (\\ Delta x \\ to 0) \\ frac (f (x + \\ Delta x, y) -f (x, y)) (\\ Delta x). \\]

Let's give the variable $ y $ the increment $ \\ Delta y $, while keeping the value of the variable $ x $ unchanged.

Then the function $ z \u003d f (x, y) $ will receive an increment, which will be called the partial increment of the function $ z \u003d f (x, y) $ with respect to the variable $ y $. Designation:

Definition 3

The partial derivative with respect to the variable $ y $ of the given function $ z \u003d f (x, y) $ is the limit of the ratio of the partial increment $ \\ Delta _ (y) z $ of the given function to the increment $ \\ Delta y $ at $ \\ Delta y \\ Designation: $ z "_ (y), \\, \\, f" _ (y) (x, y), \\, \\, \\ frac (\\ partial z) (\\ partial y), \\, \\, \\ frac ( \\ partial f) (\\ partial y) $.

Remark 3

By the definition of a partial derivative, we have:

\\ [\\ frac (\\ partial z) (\\ partial y) \u003d \\ mathop (\\ lim) \\ limits _ (\\ Delta y \\ to 0) \\ frac (\\ Delta _ (y) z) (\\ Delta y) \u003d \\ mathop (\\ lim) \\ limits _ (\\ Delta y \\ to 0) \\ frac (f (x, y + \\ Delta y) -f (x, y)) (\\ Delta y). \\]

Note that the rules for calculating the partial derivative of a given function coincide with the rules for calculating the derivatives of a function of one variable. However, when calculating the partial derivative, you need to remember which variable is used to search for the partial derivative.

Example 1

Decision:

$ \\ frac (\\ partial z) (\\ partial x) \u003d (x + y ^ (2)) "_ (x) \u003d 1 $ (by variable $ x $),

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x + y ^ (2)) "_ (y) \u003d 2y $ (by variable $ y $).

Example 2

Determine the partial derivatives of a given function:

at point (1; 2).

By the definition of partial derivatives, we get:

$ \\ frac (\\ partial z) (\\ partial x) \u003d (x ^ (2) + y ^ (3)) "_ (x) \u003d 2x $ (by variable $ x $),

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x + y ^ (2)) "_ (y) \u003d 2y $ (by variable $ y $).

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x ^ (2) + y ^ (3)) "_ (y) \u003d 3y ^ (2) $ (by variable $ y $).

{!LANG-3307b1bcaa47f028f49e5c64aa791be3!}

{!LANG-7b7efbe5b917468a3ed9174a12ab848e!}

\\ [\\ left. \\ frac (\\ partial z) (\\ partial x) \\ right | _ ((1; 2)) \u003d 2 \\ cdot 1 \u003d 2, \\ left. \\ frac (\\ partial z) (\\ partial y) \\ right | _ ((1; 2)) \u003d 3 \\ cdot 2 ^ (2) \u003d 12. \\]

Definition 4

If for each triple $ (x, y, z) $ of values \u200b\u200bof three independent variables from a certain region a certain value of $ w $ is associated, then $ w $ is said to be a function of three variables $ (x, y, z) $ in this area.

Notation: $ w \u003d f (x, y, z) $.

Definition 5

If for each collection $ (x, y, z, ..., t) $ of values \u200b\u200bof independent variables from a certain region, a certain value of $ w $ is associated, then $ w $ is said to be a function of variables $ (x, y, z, ..., t) $ in this domain.

Notation: $ w \u003d f (x, y, z, ..., t) $.

For a function of three or more variables, in the same way as for a function of two variables, partial derivatives with respect to each of the variables are determined:

$ \\ frac (\\ partial w) (\\ partial z) \u003d \\ mathop (\\ lim) \\ limits _ (\\ Delta z \\ to 0) \\ frac (\\ Delta _ (z) w) (\\ Delta z) \u003d \\ mathop ( \\ lim) \\ limits _ (\\ Delta z \\ to 0) \\ frac (f (x, y, z + \\ Delta z) -f (x, y, z)) (\\ Delta z) $;

$ \\ frac (\\ partial w) (\\ partial t) \u003d \\ mathop (\\ lim) \\ limits _ (\\ Delta t \\ to 0) \\ frac (\\ Delta _ (t) w) (\\ Delta t) \u003d \\ mathop ( \\ lim) \\ limits _ (\\ Delta t \\ to 0) \\ frac (f (x, y, z, ..., t + \\ Delta t) -f (x, y, z, ..., t)) ( \\ Delta t) $.

Example 3

By the definition of partial derivatives, we get:

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x + y ^ (2)) "_ (y) \u003d 2y $ (by variable $ y $).

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x ^ (2) + y ^ (3)) "_ (y) \u003d 3y ^ (2) $ (by variable $ y $).

$ \\ frac (\\ partial w) (\\ partial x) \u003d (x + y ^ (2) + 2z) "_ (x) \u003d 1 $ (by variable $ x $),

$ \\ frac (\\ partial w) (\\ partial y) \u003d (x + y ^ (2) + 2z) "_ (y) \u003d 2y $ (by variable $ y $),

$ \\ frac (\\ partial w) (\\ partial z) \u003d (x + y ^ (2) + 2z) "_ (z) \u003d 2 $ (by variable $ z $).

Example 4

By the definition of partial derivatives, we get:

at point (1; 2; 1).

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x + y ^ (2)) "_ (y) \u003d 2y $ (by variable $ y $).

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x ^ (2) + y ^ (3)) "_ (y) \u003d 3y ^ (2) $ (by variable $ y $).

$ \\ frac (\\ partial w) (\\ partial x) \u003d (x + y ^ (2) +2 \\ ln z) "_ (x) \u003d 1 $ (by variable $ x $),

$ \\ frac (\\ partial w) (\\ partial y) \u003d (x + y ^ (2) +2 \\ ln z) "_ (y) \u003d 2y $ (by variable $ y $),

$ \\ frac (\\ partial w) (\\ partial z) \u003d (x + y ^ (2) +2 \\ ln z) "_ (z) \u003d \\ frac (2) (z) $ (by variable $ z $) ...

The values \u200b\u200bof the partial derivatives at a given point:

\\ [\\ left. \\ frac (\\ partial w) (\\ partial x) \\ right | _ ((1; 2; 1)) \u003d 1, \\ left. \\ frac (\\ partial w) (\\ partial y) \\ right | _ ((1; 2; 1)) \u003d 2 \\ cdot 2 \u003d 4, \\ left. \\ frac (\\ partial w) (\\ partial z) \\ right | _ ((1; 2; 1)) \u003d \\ frac (2) (1) \u003d 2. \\]

Example 5

By the definition of partial derivatives, we get:

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x + y ^ (2)) "_ (y) \u003d 2y $ (by variable $ y $).

$ \\ frac (\\ partial z) (\\ partial y) \u003d (x ^ (2) + y ^ (3)) "_ (y) \u003d 3y ^ (2) $ (by variable $ y $).

$ \\ frac (\\ partial w) (\\ partial x) \u003d (3 \\ ln x + y ^ (2) + 2z + ... + t ^ (2)) "_ (x) \u003d \\ frac (3) (x ) $ (by variable $ x $),

$ \\ frac (\\ partial w) (\\ partial y) \u003d (3 \\ ln x + y ^ (2) + 2z + ... + t ^ (2)) "_ (y) \u003d 2y $ (by variable $ y $),

$ \\ frac (\\ partial w) (\\ partial z) \u003d (3 \\ ln x + y ^ (2) + 2z + ... + t ^ (2)) "_ (z) \u003d 2 $ (with respect to the variable $ z $),

$ \\ frac (\\ partial w) (\\ partial t) \u003d (3 \\ ln x + y ^ (2) + 2z + ... + t ^ (2)) "_ (t) \u003d 2t $ (by variable $ t $).

Let us prove (7) for example.

Let be ( x k, y k) → (x 0 , at 0) ((x k, y k) ≠ (x 0 , at 0)); then

Thus, the limit on the left side of (9) exists and is equal to the right side of (9), and since the sequence ( x k, y k) tends to ( x 0 , at 0) according to any law, then this limit is equal to the limit of the function f (x, y) ∙φ (x, y) at point ( x 0 , at 0).

Theorem. if function f (x, y) has a limit that is not equal to zero at the point ( x 0 , at 0), i.e.

then there exists δ\u003e 0 such that for all x, at

it satisfies the inequality

Therefore, for such (x, y)

those. inequality (11) holds. From inequality (12) for the indicated (x, y) should

A< 0 (сохранение знака).

By definition, the function f(x) = f (x 1 , …, x n) = Ahas a limit at the point

(write more f(x) → A (x → x 0)), if it is defined on some neighborhood of the point x 0, except perhaps for herself, and if there is a limit

whatever the striving for x 0 sequence of points x k from the specified neighborhood ( k \u003d 1, 2, ...) other than x 0 .

Another equivalent definition is this: function f has at point x 0 limit equal to ANDif it is defined in some neighborhood of the point x 0, with the possible exception of itself, and for any ε\u003e 0 there is δ\u003e 0 such that

for all xsatisfying the inequalities

0 < |x – x 0 | < δ.

This definition, in turn, is equivalent to the following: for any ε\u003e 0 there is a neighborhood U (x 0 ) points x 0 such that for all x

Obviously, if the number AND there is a limit f(x) at x 0, then AND there is a function limit f(x 0 + h) from h at zero point:

and vice versa.

Consider some function fgiven at all points of the neighborhood of the point x 0, except perhaps the point x 0; let ω \u003d (ω 1, ..., ω P) Is an arbitrary vector of length one (| ω | \u003d 1) and t \u003e 0 is a scalar. View points x 0 + tω (0 < t) form an outgoing x 0 ray in the direction of the vector ω. For each ω, we can consider the function

from scalar variable t, where δ ω is a number depending on ω. The limit of this function (from one variable t)

if it exists, it is natural to call it the limit f at the point x 0 in the direction of the vector ω.

Will write

You can talk about the limit fwhen x → ∞:

For example, in the case of a finite number AND equality (14) must be understood in the sense that for any ε\u003e 0 one can indicate such N\u003e 0, which for points xfor which | x| > N, function f is defined and the inequality

So the function limit f(x) = f(x 1 , ..., x n) from p variables is defined by analogy in the same way as for a function of two variables.

Thus, we turn to the definition of the limit of a function of several variables.

Number AND called the limit of the function f(M) at M → M 0 if for any number ε\u003e 0 there is always such a number δ\u003e 0 that for any points Mother than M 0 and satisfying the condition | MM 0 | < δ, будет иметь место неравенство |f(M) – AND | < ε.

Limit denote

Limit theorems. If functions f 1 (M) and f 2 (M) at M → M 0 each tend to a finite limit, then:

Example 1. Find the limit of a function:

Decision. We transform the limit as follows:

Let be y = kxthen

Example 2. Find the limit of a function:

Decision. Let's use the first remarkable limit

Example 3. Find the limit of a function:

Decision. Let's use the second wonderful limit

Continuity of a function of several variables

By definition, the function f (x, y) is continuous at the point ( x 0 , at 0) if it is defined in some of its neighborhood, including at the point itself ( x 0 , at 0) and if the limit f (x, y) at this point is equal to its value in it:

Continuity condition i.e. function fis continuous at the point ( x 0 , at 0) if the function is continuous f (x 0 + Δ x, at 0 + Δ y) in variables Δ x, Δ at at Δ x = Δ y \u003d0.

You can enter an increment Δ and functions and = f (x, y) at the point (x, y) corresponding to increments Δ x, Δ at arguments

Δ and = f (x + Δ x, at + Δ y) – f (x, y)

and in this language define the continuity f at (x, y) : function f continuous at the point (x, y) , if

Theorem. The sum, difference, product and quotient of continuous at point ( x 0 , at 0) functions f and φ is a continuous function at this point, if, of course, in the case of the quotient φ ( x 0 , at 0) ≠ 0.

Constant from can be viewed as a function f (x, y) = from from variables x, y... It is continuous in these variables, because

The next most complex functions are f (x, y) = x and f (x, y) = at... They can also be viewed as functions of (x, y) and yet they are continuous. For example, the function f (x, y) = x matches each point (x, y) number equal to x... Continuity of this function at an arbitrary point (x, y) can be proven like this.

Continuity of function

A function of two variables f (x, y) defined at the point (x 0, y 0) and in some neighborhood of it is called continuous at the point (x 0, y 0) if the limit of this function at the point (x 0, y 0 ) is equal to the value of this function f (x 0, y 0), i.e. if

A function that is continuous at every point of a certain area is called continuous in this area. Continuous functions of two variables have properties similar to those of continuous functions of one variable.

If at some point (x 0, y 0) the continuity condition is not satisfied, then the function f (x, y) is said to be discontinuous at the point (x 0, y 0).

Differentiation of a function of two variables

Partial derivatives of the first order

An even more important characteristic of the change in function is the limits:

Ratio limit

is called the first order partial derivative of the function z \u003d f (x, y) with respect to the argument x (abbreviated as the partial derivative) and is denoted by the symbols or or

Similarly, the limit

is called the partial derivative of the function z \u003d f (x, y) with respect to the argument y and is denoted by the symbols or or.

Finding partial derivatives is called partial differentiation.

From the definition of a partial derivative, it follows that when it is found by one particular argument, the other particular argument is considered constant. After performing the differentiation, both private arguments are considered variable again. In other words, partial derivatives and are functions of two variables x and y.

Private differentials

The quantity

called the main linear part of the increment? x f (linear with respect to the increment of the private argument? x). This value is called the partial differential, and is denoted by the symbol d x f.

Similarly

Total differential of a function of two variables

By definition, the total differential of a function of two variables, denoted by the symbol d f, is the principal linear part of the total increment of the function:

The total differential turned out to be equal to the sum of the partial differentials. Now the formula for the total differential can be rewritten as follows:

We emphasize that the formula for the total differential is obtained under the assumption that the partial derivatives of the first order

are continuous in some neighborhood of the point (x, y).

A function that has a total differential at a point is called differentiable at that point.

For a function of two variables to be differentiable at a point, it is not enough that it has all the partial derivatives at this point. It is necessary that all these partial derivatives be continuous in some neighborhood of the point under consideration.

Higher-order derivatives and differentials

Consider a function of two variables z \u003d f (x, y). It was already noted above that the partial derivatives of the first

are themselves functions of two variables, and they can be differentiated with respect to x and with respect to y. We get derivatives of the highest (second) order:

There are already four second-order partial derivatives. Without proof, the statement is made: If mixed partial derivatives of the second order are continuous, then they are equal:

Consider now the first-order differential

It is a function of four arguments: x, y, dx, dy, which can take different values.

The differential of the second order is calculated as the differential of the differential of the first order: under the assumption that the differentials of the partial arguments dx and dy are constants: