Understand the funny

Come on function f(x) is assigned to a deyakom advance X. Nadamo meaning to the argument in point x 0 X is more augmented Δ x okay point x 0 + Δ x also lay X. Todi vidpovidne improved f (x) function stock Δ at = f(x 0 + Δ x) - f(x 0).

Value 1. Functions f (x) at point x 0 to be called the difference between the increase in function at the point to the increase in the argument at Δ x 0 (yakscho tsya meza isnuє).

Symbols are used for the meaningful funcion at " (x 0) abo f"(x 0):

Yaksho in deyakiy point x 0 boundary (4.1) is not endless:

then it seems that at the point x 0 function f(x) maє I will never end.

Yaksho function f(x) I will go to the skin point of the plurality X, it is stolen f "(x) also functions as an argument X, valid for X.

Geometric sense of the lean

For the definition of a geometric sense, we need to know the value, but to use the graph of the function in the given point.

Value 2. Hate to the graph of functions y = f(x) at point M be called a borderline situation MN, if the point N pragne point M on curves f(x).

Come on point M on the curve f(x) based on the value of the argument x 0, and the speck N - value to argument x 0 + Δ x(Fig. 4.1). Significant dotted slides, x 0 it is necessary, if you want to get rid of the borderline Ox... 3 trikutnik MNA viplyaє, scho

Functions that are lost f(x) at point x 0 isnu, then, without a doubt (4.1), we will

Zvidsy viplya kidnapped f"(x 0) to the kuta function (tangent kuta nahil to the positive right on the axis Ox) = f(x) at point M(x 0, f(x 0)). For a whole kut nakhil, a similar form starts from the formulas (4.2):

Physical sense of the kid

It is acceptable that the function l = f(t) I will describe the law of the fall of a material point along a straight line l by the hour t. Todi growth Δ l = f (t +Δ t) - f (t) - tse paths, passages per interval hour Δ t, and the performance Δ l/Δ t- average speed per hour Δ t... Todi Meza viznacha mittєvu shvidkіst point at the moment one o'clock t I will go off the road in an hour.

The singing sense has lost functions at = f (x) it is also possible to interpret how different functions are: more value f"(x), tim bolshe kut nahilu dotty to crooked, tim cool graph f(x) and faster growth function.

Rights that liva inherited

For the analogy with the one-sided figureheads between the functions, the figure of the right and the other old functions at the point is introduced.

Value 3. Right (livo) funky functions at = f (x) at point x 0 be called the right (livy) relationship (4.1) at Δ x 0, as well as the border.

For the signified one-sided old vikors, the following symbols are used:

Yaksho function f(x) maє in points x 0 I will go, I won’t have left and the right to go to the same points, as I’ve lost.

Guided butt function, as it can be one-sided lost in points, not equal to one. Tse f(x) = |x|. Spraved, at the point x = 0 maєmo f '+(0) = 1, f "-(0) = -1 (Fig.4.2) ta f '+(0) ≠ f ’-(0), tobto. the function is not obscene when X = 0.

The operation of the familiar function is called її differentiation; function, I will go to the point, be called differentiated.

The link between the differentiation and the uninterrupted function at the point of the theorem is established.

THEOREM 1 ... As the function is differentiated at the point x 0, then it is won and without interruption at the point.

Zvorotne solidified wrong: function f(x), without interruption at the point, I can get away from the point. With such a butt є function at = |x|; won without interruption at the point x= 0, ale mute obscene u tsy point

In such a rank, because of the differentiation of the function, it is strong, it is not possible for interruptions, but only a few of the first automatically boil over to each other.

Rivnyannya shodo graph of functions

Yak bulo is indicated in section 3.9, straight ahead, to pass through the point M(x 0, at 0) with kutovym function k maє view

Let the function be set at = f(x). Todi oskilki її is lost in deyakіy point M(x 0, at 0) є We have a cool function and a graph of the function at the point M, then it’s vidsy viplya f(x) u tsy tochtsi maє viglyad

Date: 20.11.2014

Is it also lost?

Table of old.

Pochidna - one of the heads to understand the essence of mathematics. In a whole urotsі mi knowable iz tsim to understand. We can know it by itself, without strict mathematical formulas and proofs.

Tse knowledge to allow:

Intelligence is the essence of awkward buildings from obscene;

Successfully updated

Get ready for serious lessons from the funniest.

Sphatku is a welcome surprise.)

Suvore vznachennya pohіdnoї polyagaє in theory between and a piece to finish is foldable. I will torment youє. It’s almost practical to store the obscene, as a rule, you don’t need such great and great knowledge!

There is enough nobility for a successful performance of a large building near the school and VNZ all the terms- schob zhrozumati zavdannya, that all the rules- Shcheb yogo virishiti. And everything. Tse is gladє.

Let's get started with the knowledge?)

Terms and conditions.

Elementary mathematics is full of all kinds of mathematical operations. Additional data, multiplication, reduction in steps, logarithm, etc. Even before one of the operations is completed, mathematics is elementary. Qia nova operation be called differentiation. The designation of the change of the operatic will be seen in the course of the lessons.

Here it is important to see intelligence, but to differentiate - not just a mathematical operation over a function. Take a function and, according to the singing rules, will be re-created її. The result is a new function. The qia axis is a new function and is called: kidnapped.

Differentiation- Injection over the function.

Pohidna- The result of the tsієї dії.

So, yak, for example, soum- Folding result. Abo privately- The result is rozpodilu.

Knowing the terms, it is possible, like the minimum, the understanding.) The formula is as follows: make a lost function; take it away; prodifferents_yuvati function; count the lost etc. All one also Zrozumіlo, buvayut and folding dedication, deduction of obedient (differentiation) will be deprived of one of the crocs of the revision of the plant.

Poznachatsya lost by a stroke in the mountains of the right-hander over the function. The axis is like this: y " abo f "(x) abo S "(t) and so far.

Read games barcode, ef barcode from ix, eu barcode from those, Well, you see ...)

The dash can also be used to denote a specific function, for example: (2x + 3) ", (x 3 )" , (sinx) " etc. Often it is lost to be recognized for additional differentials, but it is also not visible in any way.

Albeit, it’s a problem for me. If all this is overwhelmed, it’s all over the place.) re-implementation of the function according to the singing rules. Tsikh ruled, wonderfully, not a lot.

To know the lost function, the need for the nobility is deprived of three speeches. Three whales, on which all the differentiation is worth. The axis of the stench ci three whales:

1. Table of old (differentiation formulas).

3. Fast folding function.

Slightly in order. At the same time, all the urotsi can be seen in the table of older ones.

Table of old.

The light has a functionless function. In the middle of a multitude of functions, which are available for practical storage. The function is to sit by all the laws of nature. Three functions, like a zeglinok, can be formulated for all of them. The whole class of functions is called elementary functions. The very functions and learning at school are linear, quadratic, hyperbole too.

Differentiation of functions "from scratch", tobto. vyhodyachi viznachennya pohidnoy and theory between - a piece to finish a labor worker. And mathematicians are also people, well, well!) The stench virahuvali lost elementary functions before us. A table of old ones came in, even ready.)

Axis won, qya plate for the most popular functions. Zliva is an elementary function, on the right - її is lost.

	Functions y	Similar function y y "
1	C (constant value)	C "= 0
2	x	x "= 1
3	x n (n is a number)	(x n) "= nx n-1
	x 2 (n = 2)	(x 2) "= 2x
4	sin x	(sin x) "= cosx
	cos x	(cos x) "= - sin x
	tg x
	ctg x
5	arcsin x
	arccos x
	arctg x
	arcctg x
4	a x
	e x
5	log a x
	ln x ( a = e)

I recommend that you pay homage to the third group of functions of the table of older ones. It is a good statical function - one of the newest formulas, as long as it doesn’t exist! Ones of zrozumіliy?) So, the table of the old ones was bogged down by the nobility. Before the speech, it’s not so important, as you can get up. Try more applications, the table itself and remember!)

Know the tabular meaning of the obscene, as you know, the zavdannya is not the best. To that, even more often at the junior employees, additional chips are developed. For the formulation of the factory, or for the specific functions, such as in the table - nachebto and dumb.

The butt grip is visible:

1. Know the lost function y = x 3

There are no such functions for tables. Ale is inherited from the statical function of the zagalny viglyad (third group). At times n = 3. Axis and a three-way change n then the result is accurately recorded:

(x 3) "= 3 x 3-1 = 3x 2

Axis and all right.

View: y "= 3x 2

2. Know the value of the common function y = sinx at the point x = 0.

Tse zavdannya means that you need to know how to go from the sinus, and then present the meaning x = 0 I will lose myself. The very same order! And then, buva, immediately put zero at the output function ... We are asked to know not the meaning of the output function, but the meaning її obscene. Pohіdna, I guess - it’s already a new function.

According to the table, the sine is known and I will come up with:

y "= (sin x)" = cosx

Pidstavlyaєmo zero at the end:

y "(0) = cos 0 = 1

Tse will be seen.

3. Provide differentiation functions:

Does it inspire?) Such functions in the table of the older ones are almost dumb.

I guess that the differentiation of the function is - it’s easy to know the lost function. I will forget about the elementary trigonometry, I will lose our function to finish it off. Table of no additional helpє ...

Ale yaksho poachiti, how our function is cosine of the sub-cut, Then all at once will be good!

Well well! Keep in mind that the revision of this function before differentiation generally allowed! I, trawl, lie down well life. For the formula for the cosine of the sub-cut:

Tobto. our cunning function є not scho іnshe, yak y = cosx... And tse is a tabular function. Immediately recognizable:

View: y "= - sin x.

Butt for slipped students and students:

4. Know the lost function:

Obviously, such functions are in the tables of old German. If I can guess some elementary mathematics, go through the steps ... Then in general it is possible to simplify the function. The axis is like this:

And іks steps one ten - the same table function! Third group, n = 1/10. Just behind the formula, that is written:

From and everything. Tse will be seen.

I am encouraged by the differentiation from the first whale - the table of the older ones - everything is clear. Lazy roz_bratisya with two whales, scho zalishilsya. The offensive level has mastered the rules of differentiation.

To know the viraz for the obscene exponential function (y = (e ^ x)), obeying the obscene exponential functions.

Decision.

Pochatkov_ crocs є standard: a list of writeable functions \ (\ Delta y \), which will show the increment of the argument \ (\ Delta x \): \ [(\ Delta y = y \ left ((x + \ Delta x) \ right) - y \ left (x \ right)) = ((e ^ (x + \ Delta x)) - (e ^ x)) = ((e ^ x) (e ^ (\ Delta x)) - (e ^ x)) = ((e ^ x) \ left (((e ^ (\ Delta x)) - 1) \ right).) \] It is possible to count as the difference between the increments: \ [(y "\ left (x \ right)) = \ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ Delta y)) ((\ Delta x))) = (\ lim \ limits _ (\ Delta x \ to 0) \ frac ( ((((e ^ x) \ left (((e ^ (\ Delta x)) - 1) \ right))) ((\ Delta x)).) \] Functions \ (y = (e ^ x) \) the number does not lie in Δ x And this can be blamed for the mark. Todi is missing from such a view: \ [(y "\ left (x \ right) = (\ left (((e ^ x)) \ right) ^ \ prime)) = ((e ^ x) \ lim \ limits_ ( \ Delta x \ to 0) \ frac (((e ^ (\ Delta x)) - 1)) ((\ Delta x)).) \] Meaningfully, I will delineate the boundary through \ (L \) and it is computable її okremo. , uh \ ((e ^ 0) = 1 \) and then you can write \ [(L = \ lim \ limits _ (\ Delta x \ to 0) \ frac (((e ^ (\ Delta x))) - 1 )) ((\ Delta x))) = (\ lim \ limits _ (\ Delta x \ to 0) \ frac (((e ^ (\ Delta x)) - (e ^ 0))) ((\ Delta x )) = e "\ left (0 \ right),) \] that is given between the є values ​​of the same display function at zero. Otzhe, \ We have neglected the relationship, in which it is necessary to rotate through the function itself \ (y = (e ^ x) \) and I will go to the point \ (x = 0 \). Probably, so \ For tsogo guess, uh, the number \ (e \) appears at the endless line \ and the number \ (e \) at the step \ (\ Delta x \) will, apparently, dorіvnyuє \ [(e ^ (\ ) Delta x)) = \ lim \ limits_ (n \ to \ infty) (\ left ((1 + \ frac ((\ Delta x)) (n)) \ right) ^ n). \] The formula binom Newton і can be laid out viraz under the sign of the border in binomial row: \ [(\ left ((1 + \ frac ((\ Delta x)) (n)) \ right) ^ n) = \ sum \ limits_ (k = 0) ^ n (C_n ^ k ((\ left ( (\ frac ((\ Delta x)) (n)) \ right)) ^ k)). \] Here \ ((C_n ^ k) \) denotes the number of \ (n \) elements by \ (k \ )). In European and American handlers, the number will be known as yak \ Turning to our line \ (L \), which can now be written in this view: \ [(L = \ lim \ limits _ (\ Delta x \ to 0) \ frac ((( (e ^ (\ Delta x)) - 1)) ((\ Delta x))) = (\ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ lim \ limits_ (n \ to \ infty) \) left [(\ sum \ limits_ (k = 0) ^ n (C_n ^ k ((\ left ((\ frac ((\ Delta x)) (n)) \ right)) ^ k))) \ right ] - 1)) ((\ Delta x)).) \] We have two additions to the binomial row: for \ (k = 0 \) ma \ (k = 1 \). As a result, we can make \ [(L = \ lim \ limits_ (\ Delta x \ to 0) \ frac ((\ lim \ limits_ (n \ to \ infty) \ left [(\ sum \ limits_ (k = 0) ^ n (C_n ^ k ((\ left ((\ frac ((\ Delta x)) (n)) \ right)) ^ k))) \ right] - 1)) ((\ Delta x))) = (\ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ lim \ limits_ (n \ to \ infty) \ left [(C_n ^ 0 ((\ left ((\ frac ((\ Delta x)))) )) \ right)) ^ 0) + C_n ^ 1 ((\ left ((\ frac ((\ Delta x)) (n)) \ right)) ^ 1) + \ sum \ limits_ (k = 2) ^ n (C_n ^ k ((\ left ((\ frac ((\ Delta x)) (n)) \ right)) ^ k))) \ right] - 1)) ((\ Delta x))) = ( \ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ lim \ limits_ (n \ to \ infty) \ left [(1 + n \ cdot \ frac ((\ Delta x)) (n) + \ sum \ limits_ (k = 2) ^ n (C_n ^ k ((\ left ((\ frac ((\ Delta x)) (n)) \ right)) ^ k))) \ right] - 1)) ( (\ Delta x))) = (\ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ Delta x + \ lim \ limits_ (n \ to \ infty) \ sum \ limits_ (k = 2) ^ n (C_n ^ k ((\ left ((\ frac ((\ Delta x)) (n)) \ right)) ^ k)))) ((\ Delta x))) = (\ lim \ limits _ (\ Delta x \ to 0) \ left [(1 + \ frac (1) ((\ Delta x)) \ lim \ limits_ (n \ to \ infty) \ sum \ limits_ (k = 2) ^ n (C_n ^ k

((\ left ((\ frac ((\ Del) ta x)) (n)) \ right)) ^ k))) \ right]) = (1 + \ lim \ limits_ (n \ to \ infty) \ left [(\ lim \ limits _ (\ Delta x \ to 0) \ left ((\ sum \ limits_ (k = 2) ^ n (C_n ^ k \ frac ((((\ left ((\ Delta x) \ right )) ^ (k - 1))))) (((n ^ k))))) \ right)) \ right]) 0 \). Tom, (L = 1). Tse means that the exponential function is lost \ (y = (e ^ x) \) the most important function: \

Do not go to the outskirts of the point, the function is designated.

Significant loss of function at the point

Table of old

Geometric sense of funky function at point. Clearly sichnu AB function graph y = f (x) taku, scho specksі A V , de - zbіlshennya argument. Significantly through improved functions. Everything on the chair matters:

3 rectangular tricycle ABC maєmo. Oskilki for viznachennyam is exact - tse border camp is very, then .

It is necessary to designate an elementary function at the point: an elementary function function graph the point is called the difference between the increase in function to the increase in the argument when, it is .

Otzhe, , de - Kutoviy kofіtsієnt dotichnoї.

With such a rank, the deprivation of the funeral function function graph at the exact same function function graph at the point torcannya, moreover kutoviy kofіtsієnt dotically dorіvnyu valued obedient at dotsі, tobto.

Layout: geometric sense of elementary function at the point Polyaga at іnuvannі dotnoї to the graph of functions at tsіy dotsі.

20 Differentiation of function at points. There is a need for that sufficient intelligence of differentiation.

The increment of the differentiated at the same point of function can be done as a linear function of increasing the argument from the accuracy to the values ​​of the general order of smallness. This means that it is possible to replace the line function to reach the smallest outskirts of the point of function (the speed of change of function is invariable). The line part of the improved function is called the differential (at the given point).

Necessary, albeit lacking, mental differentiation є uninterrupted function. At different functions from the same speech change, the differentiation is equal to that of the simple one. At the same time, the functions of many speech winners have the necessary (but not sufficient) intellectual differentiation and the knowledge of private, old ones for all the winners. For the differentiation of the functions of the decils of the change, the point is sufficient, for the privacy of the old ones they were numbered in the outskirts of the given point and the boules without interruption in the given point.

21 Differentiation of function at points. The theorem about the continuity of the function, how to differentiate.

Theorem.

Whereas the function of a number of points is differentiated, the point of a function is uninterrupted.

Proof.

Let the function y = f (x) y = f (x) be differentiated at the point x0x0, so that the road function is increased Δy = A⋅Δx + α (Δx) ⋅xΔy = A⋅Δx + α (Δx) ⋅x.

When the argument of the function ΔxΔx is increased to zero, the increase in the function ΔyΔy is also reduced to zero, and all means that the function is uninterrupted.

To do this, we have neglected that the function y = f (x) y = f (x) is differentiated at the point x0x0, and at the point at the point and without interruption. It’s necessary to bring it.

In such a rank, the lack of respect for the function in terms of the point necessary, albeit a lack of intelligence for the differentiation of the function.

butt.

Functions y = | x | y = | x | at point x0x0 є without interruption function, but at point at point, the function is not differentiated.

Fair, improved functions of the road:

Δy = f (x0 + Δx) −f (x0) = | Δx | Δy = f (x0 + Δx) −f (x0) = | Δx |.

In the presence of a multitude of otrimmo:

ΔyΔx = | Δx | Δx = (1, Δx> 0, −1, Δx<0ΔyΔx=|Δx|Δx={1,Δx>0, −1, Δx<0.

Between limΔx → 0ΔyΔxlimΔx → 0ΔyΔx is not ісує, but it means that the function y = | x | y = | x |, is uninterrupted at the point x0x0, not differentiated at all points.

22 Differential function. Geometric sensor to differential.

Differential function at singing points x to be called a head, a line part of an improved function.

Differential functions y = f(x) road to the creation of the obscene for the increase of the independent winter x(To the argument.)

Tse write down like this:

Geometric sensor to differential. Differential functions y = f(x) to the increase in ordinate S, drawn up to the graph of the central function at point M ( x; y), when changing x(argument) by the amount of (div. tiny).

23 Rule of differentiation of sum and dobutku.

To prove another rule of differentiation of the speed of change in relation to the obscene and power between the uninterrupted functions.

We can bring it by the rank, that the sum is stolen (profit) n functiy dorіvnyu sumi (rіznitsі) n old

The rule of differentiation is completed with two functions.

We can write down the difference between the increase in the function to the increase in the argument. Vrahovuvatimemo, scho і (the function grows to zero when the argument is incremented, but not to zero).

It’s necessary to bring it up.

24 The invariance of the form 1 of the differential.

Invariance of the first differential

Yaksho x- Square is not, then dx = x - x 0 (fixed by default). Wu tsomu vipadku maєmo

df(x 0) = f "(x 0)dx. (3)

Yaksho x = φ (t) is a differentiated function, then dx = φ" (t 0)dt... Otzhe,

To be the first differential is the power of invariance when changing the argument.

25 Rohl's theorem.

Rolle's theorem (the null theorem) stverjuє, scho

Proof

As soon as the function has become, then it is more firmly obvious, some of the functions are lost to zero at any point in the interval.

Whenever there are some values ​​of the function at the boundary points of the segment equal, then exactly to the theorem of Veyurstrasse, it’s built up its best or the least value in the same point to the middle, to the same value. ...

Geometric sense

The theorem is similar, as if the ordinates of both ends of a smooth crooked line, then on the curve there is a point, which is close to the curve parallel to the abscis axis.

26 Lagrange's theorem and inheritance.

Kintsev's spice formula abo Lagrange mean value theorem if the function is uninterrupted for the duration and differentiated in the interval, then there is such a point that

.

Geometrically The price can be reformulated as follows: there is a point at the top, which is similarly parallel to the chord, but to pass through the points of the graph, which will show the end of the line.

Mechanic tlumachennya: Do not go to the point at the time of the cob position. Todi є shlyakh, passes from moment to moment, performance is average speed for a whole period of time. This means that if the speed of the matter is assigned at any moment to the hour, then the singing moment is ahead of its mean value at the time of the day.

Proof

For the function of one change:

Introduced function. For her, think of Role's theorems: at the end, the meaning is zero. Having shrunk with the guessed theorem, we can deny that the point at which the function is lost is zero:

it is necessary to bring it.

Heritage and publicity

Lagrange's theorem about gradual growth is one of the most important, the university's theorem for all systems of differential calculation. I have a lot of supplements in numerical mathematics, and many theorems of mathematical analysis as well as inherited ones.

Passion 1. The function, which differentiates into different types, from the old ones, which is expensive to zero, is a constant.

Proof. For be-like and simple point, such a thing.

This means that, with all, the parity is correct.

Naslіdok 2 (Taylor's formula from the surplus term in the Lagrange form). If the function is differentiated once on the outskirts of the point, then for the small ones (to be quiet, for which ones lie at the designated outskirts) the Taylor formula is valid:

de - deyake number z interval.

Passion 3. Also, the function of changing two differentiates at the outskirts of the point About and all of the changes lost without interruption at points O, so in all points the fairness is fair:

Proof for. For the physical meaning and for the operator

According to Lagrange's theorem, there are numbers , taki scho

at through the continuity of other legacy functions.

Similarly, it should be .

Ale oskіlki, (how to get overturned without the middle), and between them get bogged down.

Naslіdok 4 (Formula of Newton-Leibniz). If the function is differentiated on the basis of and inherited from Riman at the same time, then the following formula is valid: .

Proof. Nekhai - more rozbittya vіdrizka. Zastosovuchi Lagrange's theorem;

Pidsumovuchi tsі іvnostі, otrimaєmo:

Lіvoruch costs the integral sum of Riman for the integral of the given designated rosbitty. Go to between the diameter of the rosbit, we can accept the Newton-Leibnitz formula.

Naslіdok 5 (The theorem about the estimation of kintsevs sprouts). Do not let the visualization of differentiation without interruption in the opaque compact areas open to space. Todi.

27 Kashi's theorem.

Cauchy's theorem on the mean.

Do not give any data on two functions, such as: 1. value and without interruption for change; 2. lost and kintsev on the interval; 3. lost and not reset to zero immediately at interval 4.; todі isnu, for whichever is valid: ... (I’ll clean up my mind 4, it’s necessary, for example, I’ll be able to mind 3: g "(x) is not guilty of going to zero to zero in the interval.)

Geometrically, it is possible to reformulate it as follows: if we set the law of collapse on the area (to start the abscis and ordinate through a parameter), then on any bend of such a crooked one, given by the parameters, there will be a similar vector, the collinear vector.