Course of lectures. Open Library

It's easy to remember.

Well, let's not go far, let's take a look at the turning function. What is the return function for the display function? Logarithm:

Our vipadka has a number as its basis:

Such a logarithm (that is, the logarithm from the basis) is called “natural”, and for the new victorist it is especially important: we write the text.

Why do you care? Obviously, .

The way to look at the natural logarithm is even simpler:

Apply:

1. Find a suitable function.
2. Why do the functions look good?

Suggestions: The exponential and natural logarithm functions are uniquely simple to look at. Showing those logarithmic functions with any other basis will be the mother of the future, as we will analyze with you later, after that, we will go through the rules of differentiation.

Rules of differentiation

What rules? I’m revisiting a new term, I’m revisiting?!

Differentiation- The whole process is bad.

Only and everything. How else can you describe the process in one word? Not the production of ... The differential of mathematics is the name given to the very greatest functions at. Vіdbuvaєtsya tsey termіn vіd latinskogo ryznitsya. Axis.

With all of these rules victorious, we have two functions, for example, i. We also need the formula for their increment:

Usyogo є 5 rules.

A constant to blame for the sign of good.

Yakshcho - like a constant number (constant), etc.

Obviously, this rule applies to retail: .

Let's get it. Let it be easier.

apply.

Find out related features:

1. at the point;
2. at the point;
3. at the point;
4. at the point

Solution:

1. (pokhіdna is the same at all points, shards of the whole function, remember?);

Pokhіdna robot

Everything is similar here: we introduce a new function and we know the improvement:

Pokhidna:

Apply:

1. Know similar functions;
2. Find out similar functions at a point.

Solution:

Pokhіdna display functions

Now your knowledge is enough to learn how to know whether it's a display function, and not just exponential (don't forget what it is?).

Father, de - tse yakes number.

We already know a bad function, so let's try to bring our function to a new foundation:

I will speed it up with a simple rule: . Todi:

Oh well, wow. Now try to find out what's wrong, and don't forget that this function is foldable.

Wiishlo?

Axis, correct yourself:

The wiyshla formula is even more similar to a dead exponent: as it was, so it was lost, it appeared as a multiplier, which is just a number, but not a change.

Apply:
Find cool features:

Suggestions:

It's just a number, it's impossible to figure it out without a calculator, so you can't write it down in a more simple way. That is why such a person looks and feels too much.

Respectfully, since there are two functions here, the rule of differentiation is valid:

This butt has two functions:

Pokhіdna logarithmic function

Here it is similar: you already know how to look at the natural logarithm:

For someone who wants to know enough about the logarithm with another basis, for example:

It is necessary to bring this logarithm to the base. And how to change the basis of the logarithm? I spodіvayus, you remember this formula:

Tіlki now zamіst pisatimemo:

The bannerman had a simple constant (a constant number without a change). It's easy to go out:

Pokhіdnі show and logarithmic functions may not be used in EDI, but we will not know them.

Pokhіdna collapsible function.

What is a "foldable function"? Hі, tse is not a logarithm and not an arc tangent. These functions can be collapsible for understanding (if you want the logarithm to be collapsible, read the topic “Logarithms” and go through everything), but from the point of view of mathematics, the word “foldable” does not mean “very important”.

Show yourself a small conveyor: two people sit and shy away like they are with such objects. For example, the first one burns a chocolate bar into a collar, and the other one ties it with a string. To come out such a warehouse object: a chocolate bar, burnt and tied with a string. In order to have a chocolate bar, you need to grow a healthy diet in a healthy order.

Let's create a similar mathematical pipeline: first of all, we know the cosine of a number, and then we take away the number, we square it. Otzhe, we are given a number (chocolate), I know the yogo cosine (obgorka), and then you make those that I have had a square (tie a line). What happened? function. Tse and є butt of a folding function: if the significance of the її znachennja mi is problyaєєєm dіyu without intermediary z zmіnnoї, that bui che other dіyu about those that have the result of the first.

In other words, collapsible function - the entire function, the argument of which is the other function: .

For example, .

We can work as a whole tі w dії і in the reverse order: on the back of your hand you make a square, and then I look for the cosine of the taken number: . It is not easy to guess that the result may be different forever. The peculiarity of folding functions is important: change in order and the function is changed.

Another butt: (the same). .

Diyu, yaku robimo stop, namememo "outward" function, and diya, what is attributed to the first - obviously "internal" function(call them informally, I live them out only in order to explain the material in my simple way).

Try to figure out for yourself, which function is external, and which is internal:

Suggestions: Podil internal and external functions even similar to the replacement of external ones: for example, functions

1. First vikonuvatimeme yaku diyu? I’m going to do a sine, and then we’ll star into a cube. Otzhe, internal function, but outside.
   And the output function is its composition: .
2. Internal: ; calling: .
   Revision: .
3. Internal: ; calling: .
   Revision: .
4. Internal: ; calling: .
   Revision: .
5. Internal: ; calling: .
   Revision: .

vikonuemo zamіnu zmіnnyh and otrimuєmo funktіuєmo.

Well, well, now we’ll take our chocolate bar - I’ll shut up. The order of the day is always reversed: one by one, it seems to be the same external function, then we multiply the result by the next internal function. Hundreds of times I’ll put it like this:

Second example:

Otzhe, formulate, nareshti, official rule:

Algorithm for the familiar folding function:

Everything is simple, right?

Let's check the butts:

Solution:

1) Internal: ;

Calling: ;

2) Internal: ;

(Just don’t think now to be quick on! Z-pіd cosine don’t blame anything, remember?)

3) Internal: ;

Calling: ;

You can clearly see that there is a three-fold folding function here: adzhe is already a folding function in itself, and from it the root is drawn, so that it wins the third day (chocolate in an obgorttsі and with a string is put in a briefcase). But there are many reasons why: all the same, we will “unpack” this function in the same order that it sounds: from the end.

That's why I first differentiate the root, then the cosine, and then we lose the viraz at the temples. And then we multiply everything.

At times, manually number the dії. Tobto uyavimo, scho us vіdomy. In what order will we work diy, to calculate the value of that virase? Let's take a look at the butt:

Whatever the day is supposed to be, the more “beautiful” the function will be. Sequence dіy - like and before:

Here, the contribution was raised 4-r_vneva. Let's signify order diy.

1. Sub-root viraz. .

2. Korin. .

3. Sinus. .

4. Square. .

5. We choose everything before buying:

VIROBNICH. BRIEFLY ABOUT STUFF

Other functions- extension of the increase in the function to the increase in the argument with an infinitely small increase in the argument:

Basic trips:

Differentiation rules:

The constant to blame for the bad sign:

Pokhіdna sum:

Good job:

The trip is private:

Folding functions:

Algorithm for the familiarity of a similar folding function:

1. We see the "internal" function, we know it's gone.
2. It is obvious that the "beautiful" function, it is known that I will be gone.
3. We multiply the results of the first and the other points.

Don’t let life tell us the exact meanings of any quantities. Sometimes you need to know about the change in the value of the bus, for example, the average speed of the bus, the change in the size of the movement before the interval, etc. To match the value of the function at the current point with the values ​​of the function at other points, it is necessary to manually win such an understanding, like “increment of the function” and “increment of the argument”.

The concept of "increase in function" and "increase in argument"

It is possible that x is a good enough point to lie near the point x0. The increment of the argument at the point x0 is called the difference x-x0. The increase is indicated as follows: ∆x.

• ∆x=x-x0.

In other words, the value is also called the increase in the independent change at the point x0. Three formulas are viable: x = x0 + ∆x. In such situations, it seems that the average value of the independent change x0 took away the increment of ∆x.

If we change the argument, then the value of the function will also change.

• f(x) – f(x0) = f(x0 + ∆х) – f(x0).

Larger functions f at the point x0, the difference f(x0 + ∆х) - f(x0) is called the difference in growth ∆х. The increment of the function is indicated by the advancing rank ∆f. In this rank, we take it for the appointment:

• ∆f = f(x0 + ∆x) - f(x0).

In other words, ∆f is also called an increase in fallow land and for the purpose of understanding vicorist ∆y, as a function of the bula, for example, y \u003d f (x).

Geometric Sensation

Look at the coming little ones.

Like a bachite, the increase shows the change of the ordinate and abscissa of the point. And the extension of the function's increase to the increase of the argument is determined to be sloppy, to pass through the spat and end positions of the point.

Let's look at the larger function and argument

example 1. Find the increase in the argument ∆x and the increase in the function ∆f at the point x0, so f(x) = x 2 , x0=2 a) x=1.9 b) x =2.1

Speeding up with formulas, pointing higher:

a) ∆x = x-x0 = 1.9 - 2 = -0.1;

• ∆f=f(1.9) - f(2) = 1.9 2 - 2 2 = -0.39;

b) ∆x=x-x0=2.1-2=0.1;

• ∆f=f(2.1) - f(2) = 2.1 2 - 2 2 = 0.41.

butt 2. Calculate the increase ∆f for the function f(x) = 1/x at the point x0, as an increase in the argument ∆x.

Well, I know, speeding up with formulas, taking it out more.

• ∆f = f(x0 + ∆x) - f(x0) =1/(x0-∆x) - 1/x0 = (x0 - (x0+∆x))/(x0*(x0+∆x)) = - ∆x/((x0*(x0+∆x)).

from medical and biological physics

Lecture №1

VIROBNICH I DIFFERENTIAL FUNCTION.

PRIVATE VIROBNICHI.

1. Ponyatya pokhіdnoї, її mekhanіchny and geometric zmіst.

but ) Increment to the argument of that function.

Let the function y = f (x) be given, where x is the value of the argument from the area of ​​the assigned function. If you choose two values ​​of the argument x o і x іz the first interval of the area of ​​function, then the difference between the two values ​​of the argument is called the larger argument: x - x o =∆x.

The value of the argument x can be assigned in terms of x 0 and the same increase: x \u003d x pro + ∆x.

The difference between two function values ​​is called the larger function: ∆y = ∆f = f(x pro + ∆x) - f(x o).

The increase in the argument and function can be shown graphically (Fig. 1). An increase in the argument and an increase in function can be both positive and negative. As shown in Fig. 1, the geometric increase in the argument ∆х is depicted as an increase in the abscissa, and the increase in the function ∆y - in the increase in the ordinate. The calculation of the increase in the following functions is carried out in an offensive order:

we give the argument an increase ∆x and take the value - x + Δx;

2) the known value of the function of the value of the argument (х+∆х) – f(х+∆х);

3) significant increase in the function ∆f=f(х + ∆х) - f(х).

Butt: Change the function y=х 2, thus changing the argument from x pro =1 to x=3. For a point x about the value of the function f(x o) = x²; for a point (x o + ∆x) the value of the function f (x o + ∆x) \u003d (x o + ∆x) 2 \u003d x² o +2x o ∆x + ∆x 2, stars ∆f \u003d f (x o + ∆x)–f(x o) \u003d (x o + ∆x) 2 -x² o \u003d x² o + 2x o ∆x + ∆x 2 -x² o \u003d 2x about ∆x + ∆x 2; ∆f = 2х about ∆х+∆х 2; ∆х = 3-1 = 2; ∆f =2 1 2+4 = 8.

b)Zavdannya, scho to produce to understand the ugly. Vznachennya pokhіdnoi, її physіchny zmіst.

The understanding of the argument and the function is necessary for the introduction of the understanding of the poor, as it is historically due to the need to designate the security of quiet and other processes.

Let's look at how the speed of a straight-line move can be seen. Let the body collapse straight out of the law: ∆S=  ∆t. For equal circulation: = ∆S/∆t.

For a variable speed, the value ∆Ѕ/∆t is assigned the value  porіvn. , then  porіvn. =∆S/∆t. However, the average swedishness does not give the possibility of imagining the peculiarity of the body’s movement and the date of the announcement of the true swedishness at the time t. With a change in the hour, that is. at ∆t→0, the average smoothness is right up to its middle - mittevskoy sharpness:

 inst. =
 porіvn. =
∆S/∆t.

This is how the chemical reaction manifests itself and mitteva:

 inst. =
 porіvn. =
∆х/∆t,

de x - the amount of speech that was made during the chemical reaction in an hour t. Similar tasks for the designation of the flexibility of various processes were brought to the introduction in mathematics of the understanding of random functions.

Let the function f(х) be given without interruption, assigned on the interval ]a,b[іє increment ∆f=f(x+∆x)–f(x).
є function ∆x that turns the average speed of change of function.

Mezha vіdnosyn , if ∆х→0, think about what is between, is called a random function :

y" x =

.

Pokhіdna is signified:
- (Igreek stroke on ix); " (x) - (ef stroke on ix) ; y" - (engraving stroke); dy / dх – (de igreek to de iks); - (Igrek with a dot).

Walking away from the fate of the pokhіdnoi, we can say that mitteva shvidkіst prіkіlіynіy ruhu є є khіdny vіdnoj shlyakhu by o'clock:

 inst. \u003d S "t \u003d f " (t).

In this way, you can create a nevtishny vysnovka, which is similar to the function behind the argument x є mitteva change the function f(x):

y" x = f " (x) =  inst.

Whom the Poles have a physical sense of the like. The process of knowing the difference is called differentiation, to which the phrase “to differentiate a function” is equivalent to the phrase “to know the difference of a function”.

in)Geometric sense is similar.

P
the derivative function y \u003d f (x) can be a simple geometric sense, binding to concepts dotichї to a curved line at the deyakіy point M. In this way, dotichno, tobto. a straight line is analytically rotated y looking y \u003d kx \u003d tg x, de? – kut badly dotic (straight) to the x axis. Noticeably bezperervnu curve as a function y = f(x), take on the curve point M_ close to it point M 1 and draw through them s_chnu. Її cut coefficient up to sec =tg β = .To approach the point M 1 to M, then the increase in the argument ∆х will move to zero, and the match at β = α will take the position of dot. From Fig. 2 we see: tgα =
tgβ =
=y" x .

to = tgα =
\u003d y" x \u003d f " (X). Also, the top coefficient, which is worth the graph of the function in this point, the older value is similar in the point of turning. For whom the polygaє geometric sense is similar.

G)Zagalne rule znakhodzhennya pokhіdnoi.

From the appointed time, the process of differentiating a function can be an offensive rank:

f(x+∆x) = f(x)+∆f;

know more functions: ∆f= f(х + ∆х) - f(х);

add up the increase in the function to the increase in the argument:

;

Butt: f(x)=x 2; f " (x) =?.

However, as you can see from this simple butt, zastosuvannya zastosuvannoї zastosuvannoї sledovnostі pіd pokhіdnyh pokhіdnyh - process trudomіstkiy and folding. Therefore, for various functions, general differentiation formulas are introduced, as presented in the table "Basic formulas for the differentiation of functions".

Come on X- Argument (independent change); y=y(x)- Function.

We take a fixed value of the argument x=x 0 that calculable value of the function y 0 = y(x 0 ) . Now we’ll put it in a fair order growth (change) the argument is meaningfully yoga  X ( X maybe be some kind of sign).

Argument from zbіlshennyam - dot X 0 +  X. It is permissible, it also has the value of the function y=y(x 0 +  X)(Div. babies).

In this way, with a sufficient change in the value of the argument, the change of the function was taken away, as it is called for more function values:

and not sufficient, but to lie in the form of a function and magnitude
.

An increment to the argument of that function can be kіntsevimi, then. vyslovlyuvatisya in fast numbers, in different countries they are sometimes called end-of-life.

In the economics of the Kintsev, growths are seen more often. For example, the tables contain data about the dovzhina of the treasury of the deaco state. Obviously, the increase in the length of the borders is counted as a path of forward significance from the offensive.

Let's take a look at the dozhina of the zaliznichnoi merezhi as a function, the argument of which will be an hour (rocks).

	Dovzhina of railway stations on 31.12, ths.km.	Prist	Average growth

By itself, the zbіlshennya funktsії (at times dovzhini railway) railroad lines) badly characterizes the change of functions. Our butt is from what 2,5>0,9 it is impossible to grow wisnovok, that the merezha grew more quickly in 2000-2003 rock, lower 2004 r., to that pririst 2,5 up to the trinity period, and 0,9 - Less than one fate. To that it is quite natural that the zbіlshennya funcії produce up to one change in the argument. The increment of the argument here is period: 1996-1993=3; 2000-1996=4; 2003-2000=3; 2004-2003=1 .

We take away those that are called in the economic literature average growth.

You can omit the operation of reducing the argument to one, so that you can take the value of the function for the value of the argument, which is set to one, which is not possible.

In mathematical analysis, zocrema, in differential calculation, one can look at infinitely small (BM) increases in the argument of that function.

Differential functions of one change (the same differential) Similar functions

Increment to argument and function at point X 0 it is possible as a difference of infinitely small values ​​(div. topic 4, BM difference), tobto. BM one order.

Todi їх vіdshennya will be the mother of the Kіntsev boundary, as it is recognized as a similar function in t X 0 .

Between the increase in the function to BM the increase in the argument at the point x=x 0 called pokhіdny functions at this point.

The symbolic designation of the next stroke (and, in fact, the Roman numeral I) was inspired by Newton. You can beat the lower index, which shows, which change is calculated, for example, . There is also a wide variety of other definitions, proponed by the founder of the calculation of the worst, the German mathematician Leibnitz:
. For the occasion, you need to know the signs and the report better at the distribution Function differential and argument differential.

Tse number is estimated speed change functions to pass through a point
.

We install geometric sense similar functions at points. With this method, we will prompt the schedule of the function y=y(x) that is significant on the new point that signifies the change y(x) at the intermediary

Hundreds of graphics of the function in dots M 0
we will respect the border camp of the current M 0 M wash away
(speck M kovzaє behind the graph of the function to the point M 0 ).

Look at
. Obviously,
.

Like a point M straighten the graph of the function straight to the point M 0 , then the value
be pragnet to the singing boundary, as it is significant
. When tsimu.

border cut  zbіgaєtsya s kutom nahily dotichny, carried out before the schedule of the function. M 0 like that
numerically superior cut coefficient of dose at the designated points.

-

geometrical sense of a similar function at a point.

In this way, one can write down the equality of the dot and the normal ( normal – it is straight, perpendicular to to the graph of the function in the real point X 0 :

Shodo - .

Normal -
.

Cіkavі vpadki, if tsі prіmі raztashovanі horizontally or vertically (div. topic 3, okremі vіpadki positioned straight on the plane). Todi,

yakscho
;

yakscho
.

The appointed pokhіdnoї is called differentiation functions.

 What is the function at the point X 0 May I leave Kintsev, she is called differentiated at this point. A function that differentiates at all points of a given interval is called a differentiated function on that interval.

 Theorem . What is the function y=y(x) differentiated in t.ch. X 0 , then the won at this point is uninterrupted.

in such a manner, uninterruptedness- Necessary (albeit not sufficient) mental differential function.

1. zbіlshennya argument and zbіlshennya function.

Let the function be given. Let's take two meanings for the argument: pochatkove that change, as it is accepted to signify
, de - the value of how to change the argument for the transition from the first value to another, it is called zbіlshennyam argument.

The values ​​of the argument that match the first values ​​of the function: that changed
, value , as the value of the function changes when the argument is changed by the value , is called more functions.

2. Understanding between functions at a point.

Number called boundary function
when, what pragne to yakscho for whatever number
find such a number
, what for all
that satisfies nervousness
,
.

Another designation: The number is called the boundary of the function when, which is pragne to, as for whether there is a number, there is such a point around the point, which for whether there is a circle around. be appointed
.

3. infinitely large and infinitely small functions of a point. The function of a point is infinitely small - a function, between them, that it is not possible for the point to be equal to zero. Infinitely great function in the point - the function of the boundary, if there is a difference, to the point of greater inconsistency.

4. main theorems about between them and their implications (without proof).

consequence: the constant multiplier can be blamed for the boundary sign:

Like a sequence converge and intersequence vіdmіnna vіd zero, then

next: the post multiplier can be blamed for the boundary sign.

11. How to understand between functions
і
and between functions vіdmіnna vіd zero,

then it is also necessary to establish the boundary between the two functions, equal the boundary between the functions and:

.

12. yakscho
, then
, is fair and vicious.

13. theorem about the intermediary sequence. Like a sequence
similar, i
і
then

5. between functions on inconsistency.

The number a is called the boundary of the function on inconsistency, (with x pragne to inconsistency) as for whether there is a sequence, which is pragne to inconsistency
show the sequence of meaning, what to move to but.

6. redels of the numerical sequence.

Number but called the boundary of the numerical sequence, as for any positive number there is a natural number N, so what for all n> N nerіvnіst
.

Symbolically, it stands like this:
fair.

The fact that the number butє boundary sequence, signified by the coming rank:

.

7. number "e". natural logarithms.

Number "e" are between numerical sequences, n- th member
, then.

.

Natural logarithm - logarithm with base tobto. natural logarithms are indicated
without appointment.

Number
allows you to move from the tenth logarithm to the natural and back.

, Yogo is called the modulus of transition from natural logarithms to tens.

8. miracles between
,

.

First miracle boundary:

in such a manner

behind the theorem about the intermediary sequence

other miraculous boundary:

.

To prove the basis of the boundary
vikoristovuyut lema: for be-such a fiery number
і
unevenness is fair
(2) (when
or
nervousness turns to jealousy.)

Sequence (1) can be written as follows:

.

Now let's look at the following sequence from the joint member
perekonaєmosya, that she is changing and fringed from below:
yakscho
, then the sequence changes. Yakscho
the sequence is bordered at the bottom. Let's show:

by virtue of equanimity (2)

tobto.
or
. That is, the sequence changes, and so on. then the sequence is bordered from below. As if the sequence is changing and bordered from below, there may be between. Todi

may be between that sequence (1), ie up to.

і
.

L. Euler naming the boundary .

9. one-sided borders, expanding functions.

the number A livu between, as for whether or not the sequence is victorious like this: .

number A right between, as for whether or not the sequence is vikonuetsya like this: .

What's next but lie in the area of ​​assigned function, or її between, break the mental continuity of the function, point but called a point of expansion or a development of a function. yakscho at the right point

12. the sum of the terms of the unfinished recessive geometric progression. Geometrical progression is a sequence, in which case between the coming ones, that forward members are left permanent, and this change is called the sign of progress. The sum of the first n members of the geometric progression are expressed by the formula
tsyu formula manually vykoristovuvatime recessive geometric progression - progression in that the absolute value of the standard is less than zero. - First member; - a sign of progress; - The number of the taken member of the sequence. The sum of the unrestricted recessive progression is a number that is not limited by the sum of the first members of the recessive progression with an unrestricted increase in the number.
then. The sum of the terms of an inexorably slow geometric progression is more expensive .