Find out the most important function of the number. The most and least important functions of the few changes in the region. Functions of many changes

Appointment 1.11 Let the function of two changers be set z = z (x, y), (x, y) D . Krapka M 0 (x 0 ;y 0 ) - internal point of the area D .

Yakscho in D є such a neighborhood UM 0 specks M 0 , which for all points

then a speck M 0 is called the local maximum point. And the meaning z(M 0 ) - local maximum.

And as for all points

then a speck M 0 is called the point of the local minimum of the function z(x,y) . And the meaning z(M 0 ) - Local minimum.

The local maximum and local minimum are called local extrema of the function z(x,y) . On fig. 1.4 explained geometric zmist local maximum: M 0 - point to the maximum, to what is on the surface z = z(x, y) clear point C 0 to know better for any other reason C (Which has the maximum locality).

Respectfully, there are dots on the surface (for example, At ), if you know more C 0 , ale qi dots (for example, At ) not є "judicial" with a dot C 0 .

Zocrema, point At confirms the understanding of the global maximum:

Similarly, the global minimum is determined:

The knowledge of global maximums and minimums will be discussed in paragraph 1.10.

Theorem 1.3 (required extremum).

Let the function be set z = z (x, y), (x, y) D . Krapka M 0 (x 0 ;y 0 D - Local extremum point.

What do you have z" x і z" y , then

Geometric confirmation is "obviously". What's next C 0 on (Fig. 1.4) to draw a dotically flat area, there "naturally" pass horizontally, i.e. under the hood 0° to axis Oh i to axis OU .

The same goes for a geometric change of private relatives (Fig. 1.3):

what it was necessary to bring.

Appointment 1.12.

What's next M 0 think (1.41), then it is called the stationary point of the function z (x, y) .

Theorem 1.4 (sufficient mind for the extremum).

Let me ask z = z (x, y), (x, y) D , as there may be private events of a different order in the vicinity of the point M 0 (x 0 ,y 0 ) D . And why M 0 - Stationary point Let's calculate:

The proof of the Vicorist theorem by those (Taylor's formula of the function of a number of variables and the theory of quadratic forms), which is not considered by any helper.

butt 1.13.

Go to the extremum:

1. We know the stationary points that break the system (1.41):

so we have found some stationary points. 2.

after Theorem 1.4, points have a minimum. And why

according to Theorem 1.4 at the point

Maximum. And why

§10 The largest and smallest value of the function of two variables in a closed area

Theorem 1.5 Let go near a closed region D function is set z = z(x, y) , that can be without interruption private trips of the first order. Cordon G regions D є shmatkovo smooth (that is folded from shmatkіv "smooth on dotik" curves or straight lines). Todi in the region D function z(x,y) reach your greatest M and least m value.

Without confirmation.

You can propagate the next plan of rebuking M і m . 1. We will be chairs, we can see all parts of the cordon of the region D and we know all the "kutovі" points of the cordon. 2. We know the stationary points in the middle D . 3. Stationary points of the skin from the cordons are known. 4. Calculate at all stationary and apex points, and then choose the most M and least m meaning.

Case 1.14 Know more M and least m function value z = 4x2-2xy+y2-8x near the closed area D , circumscribed: x=0, y=0, 4x+3y=12 .

1. Let's move the area D (Fig. 1.5) on the flat Ohu .

Kutovі points: Pro (0; 0), B (0; 4), A (3; 0) .

Cordon G regions D consists of three parts:

2. We know the stationary points in the middle of the region D :

3. Stationary points on cordons l 1 ,l 2 ,l 3 :

4. Six values ​​are counted:

From omitting six values, choose the most and the least.

Theorem 1.5 Let go near a closed region D function is set z = z(x, y) , that can be without interruption private trips of the first order. Cordon G regions D є shmatkovo smooth (that is folded from shmatkіv "smooth on dotik" curves or straight lines). Todi in the region D function z (x, y) reach your greatest M and least m value.

Without confirmation.

You can propagate the next plan of rebuking M і m .
1. We will be chairs, we can see all parts of the cordon of the region D and we know all the "kutovі" points of the cordon.
2. We know the stationary points in the middle D .
3. Stationary points of the skin from the cordons are known.
4. Calculate at all stationary and apex points, and then choose the most M and least m meaning.

Case 1.14 Know more M and least m function value z = 4x2-2xy+y2-8x near the closed area D , circumscribed: x = 0, y = 0, 4x + 3y = 12 .

1. Let's move the area D (Fig. 1.5) on the flat Ohu .

Kutovі points: Pro (0; 0), B (0; 4), A (3; 0) .

Cordon G regions D consists of three parts:

2. We know the stationary points in the middle of the region D :

3. Stationary points on cordons l 1 , l 2 , l 3 :

4. Six values ​​are counted:

Apply

example 1.

This function is assigned at all changing values x і y , crim the cob of coordinates, de znamennik turns to zero.

Rich Member x2+y2 uninterrupted usudi, and therefore i uninterrupted square root of an uninterrupted function.

Drib will be uninterrupted everywhere, Crimea dot, de banner to zero. That function, which is being looked at, is uninterrupted on the entire coordinate plane Ohu , including the cob of coordinates.

butt 2.

Follow the function for safety z=tg (x, y) . Tangent of values ​​and without interruption for all final meanings argument, crim value, equal to unpaired number of magnitude π /2 , then. including points, de

With cutaneous fixed "k" Equation (1.11) signifies a hyperbole. Therefore, the function є uninterrupted function x and y including points that lie on curves (1.11).

example 3.

Know private outdoor functions u=z-xy , z > 0 .

butt 4.

Show what the function

satisfied with the sameness:

– this equality is valid for all points M(x; y; z) cream points M 0 (a; b; c) .

Let's look at the function z=f(x, y) of two independent variables and install the geometric substitution of private variables z" x = f" x (x, y) і z" y = f" y (x, y) .

Whose mind is equal z=f (x, y) є leveling of the surface (Fig. 1.3). Held flat y = const . At pererizі tsієї superficial surfaces z=f (x, y) vide deyka line l 1 peretina, vzdovzh that change less than size X і z .

Private trip z" x (її geometric shift without a middle vyplyaє z known to us geometric sense of a similar function of one variable) is numerically superior to the tangent of the kuta α sickly, by extension to the axis Oh , shodo L1 to the curve l 1 , scho to go near the surface z=f (x, y) flat y = const at the point M (x, y, f (xy)): z "x \u003d tgα .

At the retina and the surface z=f (x, y) flat X = const widde line peretina l 2 , vzdovzh that change less than magnitude at і z . Todi private fun z" y numerically superior to the tangent of the kuta β nahilu by extension to the axle OU , shodo L2 to the specified line l 2 peretina in dots M (x, y, f (xy)): z "x \u003d tgβ .

Example 5.

What kind of kutvoruє іz vіssyu Oh dotichna to the line:

at the point M(2,4,5) ?

Vikoristovuєmo geometric replacement of a private replacement for a replacement X (at fast at ):

Example 6.

Zgidno (1.31):

Example 7.

Vvayayuchi, scho equal

implicitly define a function

know z" x , z" y .

For this reason (1.37) we need evidence.

Example 8.

Go to the extremum:

1. We know the stationary points that break the system (1.41):

so we have found some stationary points.
2.

after Theorem 1.4, points have a minimum.

And why

4. Six values ​​are counted:

From omitting six values, choose the most and the least.

List of literature:

ü Belko I. V., Kuzmich K. K. Great math for economists I semester: Express course. - M.: New knowledge, 2002. - 140 p.

ü Gusak A. A. Mathematical analysis and differential alignment. - Minsk: TetraSystems, 1998. - 416 p.

ü Gusak A. A. Vishcha mathematics. Heading guide for university students in 2 volumes. - Mn., 1998. - 544 p. (1 vol.), 448 p. (2 tons).

ü Kremer N. Sh., Putko B. A., Trishin I. M., Fridman M. N. Mathematics for Economists: A Handbook for Universities / Ed. prof. N. Sh. Kremer. - M.: UNITI, 2002. - 471 p.

ü Yablonsky A. I., Kuznetsov A. V., Shilkina E. I. that in. Vishcha mathematics. Zagalniy course: Pidruchnik / Zag. ed. S. A. Samal. - Mn.: Vish. school, 2000. - 351 p.

More and less meaning

The function, which is surrounded in a closed area, reaches its largest and smallest value, either at stationary points, or at points that lie on the boundary area.

To find the largest and smallest value of the function, it is necessary:

1. Find stationary points that lie in the middle of this region, and calculate the values ​​of the function for them.

2. Know the most (least) value of the function of the inter-region.

3. Equalize all the negative values ​​of the function: the largest (less) and will be the largest (smallest) values ​​of the function for this gallery.

butt 2. Find the largest (least) value of the function: y .

Solution.

the point is stationary; .

2 . The border of the closed area is the ring, de.

The function of the inter-region becomes the function of one change: , de . We know the most and least important functions.

For x = 0; (0,-3) and (0,3) are critical points.

Calculate the value of the function on the ends of the wreath

3 . Porivnyuyuchi mizh himself otrimuemo,

At points A and B.

At points C and D.

example 3. Find the largest and smallest value of the function in the closed area, given the unevenness:

Solution. The area є trikutnik, we will surround the axes of coordinates і with a straight line x + y = 1.

1. We know stationary points in the middle of the region:

; ; y = - 1/8; x = 1/8.

Stationary point does not belong to this area, so the value of z in it is not calculated.

2 .Doslіdzhuєmo function on the cordon. The shards of the boundary are formed from three dіlyanki, described by three different equals, doslіdzhuєmo function of the skin dіlantsі okremo:

a) div 0A: y=0- equal 0A, then ; from equal it is clear that the function increases by 0A from 0 to 1. Mean .

b) on the distance 0B: x = 0 - the distance 0B, then; -6y + 1 = 0; - Critical point.

in) to the direct x + y = 1: y = 1-x, then we take the function

We calculate the value of the function z at the point B(0,1).

3 .Perіvnyuyuchi numbers otrimuemo, scho

To straight AB.

At point B.

Test for self-control knowledge.

one . Function extremum – ce

a) її pokhіdnі first order

b) її equal

c) її schedule

d) її maximum and minimum

2. The extremum of the function as many as possible can be reached:

a) only at the points that lie in the middle of the designated area, in which case the private values ​​of the first order are greater than zero

b) only at points that lie in the middle of the designated area, in which case the private values ​​of the first order are less than zero

c) only at the points that lie in the middle of the designated area, in which case the private values ​​of the first order are not equal to zero

d) only at points that lie in the middle of the designated area, in which case private similarities of the first order are equal to zero

3. A function that is uninterrupted in a closed area, reaching its highest and lowest values:

a) at stationary points

b) either at stationary points, or at points that lie on the inter-region

c) at points that lie on the inter-region

d) at all points

4. Stationary points for the function of how many variables are called points:

a) for some u

b) some of them have private first-order differences greater than zero

c) for some of them, the first-order private changes are equal to zero

d) for some of them, private behaviors of the first order are less than zero

Let the function y = f (x) be interrupted by the wind. Apparently, such a function reaches its greatest. that hiring. value. This function can be taken at the inner point of the window, or on the boundary of the window, tobto. at = a or = b. Like a point to trace the middle of the critical points of a given function.

We take the rule of value of the largest and smallest value of the function to:

1) determine the critical points of the function on the interval (a, b);

2) calculate the values ​​of the function at the found critical points;

3) calculate the value of the function of the kintsyah vіdrіzka, tobto. at points x=a and x=b;

4) the average of the calculated values ​​of the function is to choose the most and the least.

Respect:

1. If the function y = f (x) has more than one critical point per vdrіzku and won є the point of maximum (minimum), then at this point the function gains the largest (least) value.

2. Since the function y=f(x) has no critical points, it means that the function monotonously increases and decreases for the new one. Also, the function takes its maximum value (M) to one end of the stroke, and the least (m) to the other.

60. Complex numbers. Formula de Moivre.
complex number name viraz mind z = x + iy, de x and y - dіysnі numbers, and i - so called. obvious loneliness. If x=0, then the number 0+iy=iy ranks. let's show it by number; even though y=0, the number x+i0=x is mapped to the current number x, but it means that the impersonal R of all functions. numbers yavl. under the multiplicity of the impersonal Z usikh complex numbers, then. . Number x names the decimal part z, . Two complex numbers і are called equal (z1=z2) even and only once, if equal parts and equal parts are equal: x1=x2, y1=y2. Zocrema, the complex number Z=x+iy equals zero and then if x=y=0. The concepts of "greater" and "less" for complex numbers are not introduced. Two complex numbers z \u003d x + iy і, which are considered only by the sign of the explicit part, are called obtained.

Geometric representation of complex numbers.

Whether a complex number z = x + iy can be represented by a point M(x,y) of the plane Oxy such that x=Re z, y=Im z. Firstly, the skin point M(x;y) of the coordinate plane can be used as the image of the complex number z = x + iy. The area, where complex numbers are displayed, is called the complex area, because he has to lie the real numbers z = x + 0i = x. All ordinates are called explicit vertices, to the fact that on it lie the apparent complex numbers z = 0 + iy. The complex number Z=x+iy can be inserted behind the auxiliary radius vector r=OM=(x,y). The length of the vector r, which represents the complex number z, is called the modulus of this number and is denoted by | z | or r. Rozmir kuta mizh poklade. Directly on the real axis, the vector r, which represents a complex number, is called the argument of the complex number, denoted by Arg z or . The complex number argument Z = 0 is not assigned. The argument of a complex number - the value is richly significant and is measured with accuracy up to the dodanku, de arg z - the main value of the argument, put in the space (), then. - (Sometimes as the head value of the argument, take the value that should contain the gap (0; )).

Writing the number z as z=x+iy is called the algebraic form of a complex number.

Dії over complex numbers

Addendum. The sum of two complex numbers z1=x1+iy1 and z2=x2+iy2 is a complex number that is equal: z1+z2=(x1+x2) + i(y1+y2). Addition of complex numbers can change and change power: z1+z2=z2+z1. (Z1 + Z2) + Z3 = Z1 + (Z2 + Z3). Vіdnіmannya. Vіdnіmannya vyznaєtsya yak dіya, zvorotne dodavannya. The difference between complex numbers z1 and z2 is called such a complex number z that, being added to z2, gives the number z1, that is. z = z1-z2, so z + z2 = z1. Like z1=x1+iy1, z2=x2+iy2, it is easy to take z out of this assignment: z=z1-z2=(x1-x2) + i(y1-y2). plural. The complement of complex numbers z1=x1+iy1 and z2=x2+iy2 is a complex number that is equal to z=z1z2= (x1x2-y1y2) + i(x1y2+y1x2). Zvіdsi, zokrema, i vyplyaє: . Like the number of assignments for the trigonometric form: .

When complex numbers are multiplied, their modules are multiplied, and the arguments are added. De Moivre formula(as well as є n multipliers and stinks the same): .

By the end of 2020, NASA is launching an expedition to Mars. Deliver the spacecraft to Mars with an electronic carrier bearing the names of all registered participants in the expedition.

Registration of participants in the vote. Take away your ticket to Mars for the blessings.

Like this post, having solved your problem, or simply being worthy of you, share your strength with your friends in social networks.

You need to copy and paste one of these code options into the code of your web page, between tags і or just after the tag . Behind the first version of MathJax, a smaller and less tacky side is preferred. Natomist another option automatically selects and upgrades to the latest version of MathJax. If you insert the first code, it will need to be updated periodically. If you insert other code, the sides will get more interested, so you won't have to constantly follow MathJax updates.

Enable MathJax in the simplest way in Blogger or WordPress: in the site's navigation panel, add a widget, assignments for inserting third-party JavaScript code, copy the first or another option to the engagement code presented above, and resize the widget closer to the top of the template (before speech, we don't need new 'language) , the MathJax script scripts are invoked asynchronously). From i all. Now check the syntax of MathML, LaTeX and ASCIIMathML, and you are ready to insert mathematical formulas on the web pages of your site.

Chergovy before the New Rock... the weather is frosty, those snizhinki on shibtsі... Everything prompted me to write again about... fractals, and about those who know about Wolfram Alpha. Іz thogo drive є tsіkava stattya, in yakіy є buttocks of two-dimensional fractal structures. Immediately, the world can see folded butts of trivial fractals.

A fractal can be visually manifested (described), like a geometric figure or a body (looming in the air, which is also impersonal, to this particular type, impersonal dot), the details that make such a shape, like the figure itself. Tobto tse self-similar structure, looking at the details as if enlarged, mimic the very form that is without enlargement. Similarly, in a visually striking geometric figure (not a fractal), with more minor details, as if a simple form can be made, the figure is lower. For example, when you finish the great big part of the ellipse, it looks like a straight tree. This is not the case with fractals: for any kind of improvement, we will repeat the same folding form, as if with skin improvements, repeat again and again.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Mystery in the name of science: formal form. That is, if a part of the fractal will be enlarged to the extent of the whole, it will be seen as a whole, or exactly, or, possibly, with a slight deformation.