Know private behaviors and other differential functions. Private holidays and the latest differential of the functions of a small number of changes. Stopping similar functions

Let's take a look at the change of the function when given an increase in only one of the її arguments - x i and we call it yoga.

Appointment 1.7.Private trip functions behind argument x i called.

Designation: .

In this way, the private functions one change - x i. Therefore, for her, all the powers of the like, brought to the function of one change, are fair.

Respect. With a practical calculation of private, similar ones, the strong rules of differentiation of the function of one change, respecting the argument, which differentiate, change, and other arguments are constant.

1. z= 2x² + 3 xy –12y² + 5 x – 4y +2,

2. z = x y ,

Geometrical interpretation of other similar functions of two changeable ones.

Let's look at the smooth surface z = f(x, y) i draw the area x = const. Vibero on the line of the crossbar of the surface from the top speck M (x, y). How to specify an argument at increment Δ at and look at the point T on the curve with coordinates ( x, y+Δ y, z+Δy z), then the tangent of the cut, established by the sіchny MT with the positive direct axis at, dorivnyuvateme . Passing to the boundary when it is taken, it is privately similar to the good tangent of the kuta, fixed to the dot of the curve at the point M with positive forward axis y. Vіdpovіdno private pkhіdna dorіvnyuє tangent kuta z vіssyu X dotichny to a curve, cut off as a result of recutting the surface z = f(x, y) flat y= const.

Appointment 2.1. The new extension of the function u = f(x, y, z) is called

Appointment 2.2. If an increase in the function u = f (x, y, z) at the point (x 0 , y 0 , z 0) can be applied to look at (2.3), (2.4), then the function is called differentiated at the tsij point, and the function is called the head linear Part of the improvement is the new differential of the analyzed function.

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

So, just as in the case of a function of one variable, the differentials of independent variables are taken into account by their sufficient increase, to that

Respect 1. Later, the firmness of "the function is differentiated" is not the same as the firmness of "the function is not private" - for differentiation, it is necessary to have the continuity of these similar ones at the point you are looking at.

4. Dotichnaya plane and normal to the surface. Geometric differential lock.

Come on function z = f(x, y)є differentiated on the outskirts of the point M (x 0, y 0). These are private faucets and kutovymi koefitsients, scho suyuyutsya line peretina surface. z = f(x, y) with flats y = y 0і x = x 0, yakі will be dotichnі and to the very surface z = f(x, y). We store the flatness of the plane, which should pass through straight lines. Direct vectors of dotichnyh can be seen (1; 0; ) i (0; 1; ), so the normal to the plane can be seen vector creative: n = (-, -, 1). Again, the flatness of the area can be written as follows:

de z0 = .

Appointment 4.1. The area that is assigned to equal (4.1) is called dotty flat to the schedule of the function z = f(x, y) at point with coordinates (x 0, y 0, z 0).

Three formulas (2.3) for two different variables, which increase the function f on the outskirts of the point M you can see at a glance:

Otzhe, the difference between the applications of the graph of the function and the dotic area is infinitely small. high order chim ρ, at ρ→ 0.

For which differential function f may look:

what do you think increment of the application of the dotty area to the graph of the function. Why do we have a geometrical value of the differential.

Appointment 4.2. Non-zero vector perpendicular to the dot- ic plane at the point. M (x 0, y 0) surface z = f(x, y), called normal to the surface at this point.

In the capacity of the normal to the expanded surface, manually accept the vector - n = { , ,-1}.

For the sake of simplicity, the recording and presentation of the material is mixed with a breakdown of the functions of the two substantives. Everything further away is also fair for the functions, be it a small number of change.

Appointment. Private trip functions z = f(x, y) by independent change X called pokhidna

calculated at regular at.

Similarly, it is designated privately for a replacement at.

For okremih pokhіdnyh fair zvichaynі rules and formulas of differentiation.

Appointment. Dobutok private similar to the argument X(y) called private differential by change X(at) functions of two changing z = f(x, y) (Signature: ):

Yakshcho under the differential of the independent change dx(dy) understand better X(at), then

For the function z = f(x, y) z'yasuєmo geometric zmіst її frequency similar ta .

Let's look at the dot, dot P 0 (X 0 ,y 0 , z 0) on the surface z = f(x,at) and curve L yak weide when overcut by a surface plane y = y 0 . Qiu curve can be seen as a graph of the function of one change z = f(x, y) at the flat y = y 0 . How to spend at the point R 0 (X 0 , y 0 , z 0) reach the curve L, then, zgіdno with a geometric zmіst of similar functions of one zminnoy , de a – Kut Oh.

Abo: similarly fix another change, tobto. we will carry out a resection of the surface z = f(x, y) flat x = x 0 . Same function

z = f(x 0 ,y) can be considered as a function of one variable at:

de b- Kut, dotistic dots M 0 (X 0 , y 0) with positive direct axis Ouch(Fig. 1.2).

Rice. 1.2. An illustration of the geometric meaning of private holidays.

butt 1.6. given function z = x 2 – 3hu - 4at 2 - x + 2y + 1. Know that.

Solution. Looking out at as a constant value, taken away

In respect X fast, we know

Private trip functions z = f(x, y by change x it is called a random function, with a constant value of a change, it is indicated abo z "x.

Private trip functions z = f(x, y) according to change it is called pokhіdna z y with a constant value of change y; won is denoted by z".

The private functions of a number of people change according to one change are considered as a function of the most important change for the mind, which other changes are respected permanently.

New differential function z = f(x, y)

,

De y are counted at the points M(x, y), and dx = , dy = y.

butt 1

Calculate the last differential of the function.

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

Solution:

1) We know private trips:

2) Calculate the values ​​of private relatives at points 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 primary? Recalculate the power of the first.

2. What is called an insignificant integral?

3. Recalculate power sing integral.

4. List the main formulas for integration.

5. What methods of integration do you know?

6. Why do you think the essence of the formulas of Newton - Leibnitz?

7. Determining the value of the sing integral.

8. Why is the essence of the calculation of the singing integral a way of substitution?

9. What is the essence of the method of counting a single integral by particles?

10. How is the function called the function of two variables? How does she appear?

11. What function is called the function of the three changes?

12. How is the multiplier called the area of ​​function?

13. For the help of such irregularities, can you put a closed lot D on the flat?

14. What is the private func- tion z = f(x, y) called after changing x? How does she appear?

15. What is called a private random function z = f(x, y) behind a replacement y? How does she appear?

16. Any viraz is called the same differential of the function

Topic 1.2. Zvichayny differential rіvnyannya.

Zavdannya, scho to produce differential equals. Differential alignment with the changes that are divided. Zagalni and private decisions. Homogeneous differential alignment of the first fret. Linear uniform equal to another order of post-coefficients.

Practical lesson No. 7 "Knowledge of global and private solutions of differential equals with changes that are divided" *

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

Practical lesson No. 9 "Decision of differential equalities of the 2nd order from constant coefficients" *

L4, section 15, side 243 - 256

Methodical statements

Private similar functions in that case, as if they were based not on one point, but on a real set, these functions are assigned on this set. These functions can be uninterrupted and in certain situations they can have private trips in different points of the designated area.

Private functions of these functions are called private functions of a different order than other private functions.

Private holidays of a different order are divided into two groups:

· other private trips from the change;

· Changes of private holidays in the presence of change.

With a slight differentiation, you can assign okremі pokhіdnі of the third order and so on. Analogous records are designated and recorded privately in higher orders.

Theorem. If everything enters before the calculation of private losses, if they are viewed as functions of their independent variables, without interruption, then the result of private differentiation cannot be left behind in the sequence of differentiation.

Often blamed for the need to complete the turning task, as if it were the case with the appointed one, which is the top differential of the function, by the way, de uninterrupted functions with no interruptions of the first order.

The need for a full differential can be formulated as a theorem, which is acceptable without proof.

Theorem. In order for the differential viraz buv in the area of ​​​​the top differential of the function, assigned and differentiated in tsіy galusi, it is necessary that in tsіy galuzi it was also vikonano to the mind whether it was a pair of independent changes.

The task of calculating the total differential of a function of a different order can be written like this. If the virase of a new differential is also differentiating, then another full differential (or a new differential of a different order) can add a virase, subtracting as a result of the stoppage of the differentiation operation to the first full differential, tobto. . Analytic analysis of another full differential can be of the form:

In order to ensure that zmіshanі pokhіdnі do not lie in the order of differentiation, the formula can be grouped and manifested in a seemingly quadratic form:

Matrix of quadratic form is more expensive:

Let there be given a superposition of functions assigned to i

Pevnih st. When tsimu. Then, as you can without interruption private shifts to a different order at points i, then use another new differential folding function offensive type:

As you can see, another new differential cannot have the power of form invariance. In the case of another differential of a folding function, there are addendums, like in the formulas of another differential of a simple function.

Pobudova of private similar functions of higher orders can be continued, violating the following differentiation of functions:

De indexes add up the meaning of before, tobto. pokhіdna order is looked at as private pokhіdna of the first order in the order of the order. Similarly, you can make and understand the full differential of the order of the function, like the first differential of the first order in the order of the differential: .

In times of simple functions of two substitutions, the formula for calculating the total differential of the order of the function may look

Setting the differentiation operator allows you to take a compact form of writing, which is easy to remember, for calculating the total differential to the order of the function, similar to Newton's binomial formula. I can see the two-world attitude.

Let the function be assigned to the singing (voidcritical) region D dot
peaceful space, that
- a point at tsіy galuzі, tobto.
D.

Private zbіlshennyam functions a lot of changes for some kind of change are called those zbіlshennya, as if we take away the function, as if we had a zbіlshennya tієї zmіnnoї, regardless, that all other changes may have permanent meaning.

For example, a private increase in the function of the change will

Private freelance independent change at the point
as a function, the boundary (as it is) is called the blue of private expansion
functions to zbіlshennya
serpentine during exercise
to zero:

Private pokhіdnu signify one of the symbols:

;
.

Respect. Index below, in these signs, it’s not enough to indicate, for which of the changes it’s worth taking, and it’s not connected with them, at some points
tsya pokhіdna be counted.

The calculation of private casualties does not represent anything new in the calculation of the most significant casualties, it is only necessary to remember that with a differentiating function, be it as a change, all other changes are accepted for staying. Shown in examples.

example 1.Know private outdoor functions
.

Solution. With the calculation of the private casual function
behind the argument we consider the function as a function of only one change , then. mind you what may have a fixed value. With fixed function
є static function of the argument . For the formula of differentiation of the static function, it is necessary:

Similarly, when calculating private tax Please note that the value is fixed , and consider the function
how to show the function to the argument . At the result we take:

butt 2. Hgo private trips і functions
.

Solution. When calculating private tax for given function we are considered as a function of one change , but virazi, what to revenge , Be constant multipliers, tobto.
acting as a constant coefficient with a static function (
). Differentiation of cei viraz according to , We take:

.

Now, now, the function considered as a function of one change , in that hour, like virazi, what to revenge , act as a coefficient
(
).Differentiation according to the rules of differentiation of trigonometric functions, it is necessary:

example 3. Enumerate private useful functions
at the point
.

Solution. We know a little about private similar functions at a certain point
її areas of appointment. When calculating private tax for mind you what
є imminent.

when differentiating according to will be constant
:

and in case of calculation of private casualties for i by , similarly, will be constant, obviously,
і
, then:

Now we can calculate the values ​​of these similar ones at the point
, presenting the specific meaning of the changes to that. At the result we take:

11. Private and real differentials of a function

What's next for private sbіlshennya
zastosuvat Lagrange's theorem about the end of the change , then, seriously without interruption, we take the following precautions:

de
,
- It's incredibly small.

Private differential function by change called the main line part of the private estate
, rivna dobutka privately similar for the price of change for the increase in the price of change, and is designated

Obviously, a private differential is considered as a private expansion to an infinitely small greater order.

Let's improve the functions riches of change are called її increase, as if they take it away, if all independent change ladies increase, then.

de all
, lie down tі at once і out of it to zero.

Pid differentials of independent changes mothers got home in Ukraine enough increment
i designate їх
. In this order, I’ll look at the private differential in the future:

For example, private differential on appear like this:

.

New differential
functions
, Rivna, tobto. sum of all її private differentials:

What is the function
may be uninterrupted private holidays

at the point
, then won differentiated in this point.

When dosing small for a function that differentiates
cherish the place of the nearness of equanimity

,

for the help of those, you can work closer to the calculation.

butt 4.Find the latest differential of a function
three of them
.

Solution. Nasampered, we know private trips:

Remembering that the stench is uninterrupted for all meanings
, we know:

For differential functions rich in change, all theorems about the power of differentials, brought to the point of a function of one change, for example: і - without interruption functions of changing
, which can be uninterrupted private holidays for all changes, and і - pretty fast, then:

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