T 8 Derivative logarithmic and indicative function. Differentiation of the indicative and logarithmic function. Pred-like indicative function in the tasks of the ET. The output of the formula of the logarithmic derivative

Differentiation of indicative and logarithmic functions

1. Number e. Function y \u003d e x, its properties, schedule, differentiation

Consider indicative function y \u003d a x, where a\u003e 1. For various bases, and we obtain various graphs (Fig. 232-234), but it can be noted that all of them pass through the point (0; 1), they all have horizontal asymptotes y \u003d 0 All of them are addressed by convexity down and finally, they all have tangents in all their points. We carry out for example a tangent schedule Functions y \u003d 2x at point x \u003d 0 (Fig. 232). If you make accurate constructions and measurements, you can make sure that this tangent forms an angle of 35 ° (approximately).

Now we will spend tangent to the graph of the function y \u003d 3 x, too, at point x \u003d 0 (Fig. 233). Here the angle between tangent and axis will be greater than - 48 °. And for the indicative function y \u003d 10 x in a similar
situations we obtain an angle of 66.5 ° (Fig. 234).

So, if the base and the indicative function y \u003d ah gradually increases from 2 to 10, then the angle between the tangent of the function graph at the point x \u003d 0 and the axis of the abscissa gradually increases from 35 ° to 66.5 °. It is logical to assume that there is a base A for which the corresponding angle is 45 °. This base must be concluded between numbers 2 and 3, since for the function of the U-2x the angle of interest to us is 35 °, which is less than 45 °, and for the function y \u003d 3 x it is 48 °, which is a little more than 45 °. The reason for us is customary to designate the letter E. It is established that the number e is irrational, i.e. is an infinite decimal non-periodic fraction:

e \u003d 2,7182818284590 ...;

in practice, it is usually believed that E \u003d 2.7.

Comment(not very serious). It is clear that L.N. Tolstoy no relation to the number e does not have, nevertheless in the recording of the number e, pay attention, the number 1828 is repeated in a row - the year of birth of L.N. Tolstoy.

The graph of the function y \u003d e x is depicted in fig. 235. This is an exhibitor that differs from other exponentials (graphs of indicative functions with other bases) by the fact that the angle between tangent to the graph at the point x \u003d 0 and the abscissa axis is 45 °.

Properties of function y \u003d e x:

1)
2) is neither even nor odd;
3) increases;
4) is not limited from above, limited to below;
5) does not have the greatest nor least values;
6) continuous;
7)
8) convex down;
9) Differential.

Return to § 45, take a look at the existing list of properties of the indicative function y \u003d a x at a\u003e 1. You will find the same properties 1-8 (which is quite natural), and the ninth property associated with
differentiality of the function, we were not mentioned then. Let's discuss it now.

We derive the formula for finding a U-EX derivative. At the same time, we will not use the usual algorithm, which was developed in § 32 and which was successfully used by success. In this algorithm on final stage It is necessary to calculate the limit, and the knowledge of the theory of limits with you is still very limited. Therefore, we will rely on geometric prerequisites, considering, in particular, the fact of the existence of a tangent to the graph of the indicative function is not doubtful (therefore, we have so confidently recorded the ninth property in the above-mentioned list of properties, the differentiability of the function y \u003d e x).

1. Note that for the function y \u003d f (x), where f (x) \u003d ex, the value of the derivative at point x \u003d 0 is already known to us: f / \u003d TG45 ° \u003d 1.

2. We introduce the function y \u003d G (x), where G (x) -f (x - a), i.e. G (x) -eh "a. In fig. 236 shows a graph of the function y \u003d G (x): it is obtained from the function of the function y - Fx) shift along the x axis on | a | units of scale. Tanner to graphics function y \u003d G (x) in point x-a Parallel to the tangent to the graph of the function y \u003d f (x) at the point x -0 (see Fig. 236), it means that it forms an angle of 45 ° with the axis. Using the geometrical meaning of the derivative, we can write down that G (a) \u003d TG45 °; \u003d 1.

3. Let's return to the function y \u003d f (x). We have:

4. We have established that for any value, a ratio. Instead of the letter, and it is possible, naturally, use the letter x; Then we get

The corresponding integration formula is obtained from this formula:

A.G. Mordkovich Algebra Grade 10

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Theme of the lesson: "Differentiation is indicative and logarithmic function. Pred-like indicative function "in the tasks of the UNT

purpose : Develop the cumulative knowledge of theoretical knowledge on the topic "Differentiation of an indicative and logarithmic function. Pred-like indicative function "To solve the tasks of the ET.

Tasks

Educational: systematize theoretical knowledge of students, consolidate the skills of solving problems on this topic.

Developing: develop memory, observation, logical thinking, Mathematical speech of students, attention, self-assessment skills and self-control.

Educational: promote:

formation of students responsible attitudes towards teaching;

development of sustainable interest in mathematics;

creation of positive internal motivation to the study of mathematics.

Teaching methods: Sensual, visual, practical.

Forms of work:individual, frontal, in pairs.

During the classes

Epigraph: "The mind lies not only in knowledge, but also in the ability to apply knowledge in practice" Aristotle (Slide 2)

I. Organizing time.

II. Salmon crossword. (slide 3-21)

The French Mathematician of the XVII century Pierre Farm identified this line so "straight, most closely adjacent to the curve in a small neighborhood of the point."

Tangent

The function that is specified by the formula y \u003d log a. x.

Logarithmic

The function that is given by the formula y \u003d but x.

Indicative

In mathematics, this concept is used when the velocity of the material point and the angular coefficient tangent to the graphics of the function at a specified point.

Derivative

What is the name of the function f (x) for the function f (x) if the condition f "(x) \u003d f (x) is satisfied for any point from the interval I.

PRINTING

What is the relationship between X and y, at which each element x is put in accordance with the only element of the

Derived from movement

Speed

The function that is specified by the formula y \u003d e x.

Exhibitor

If the function f (x) can be represented as f (x) \u003d g (t (x)), then this function is called ...

III. Mathematical dictation. (Slide 22)

1. Write the formula of the derivative indicative function. ( but x) "\u003d but X · LN. a.

2. Write the formula derivative exhibit. (E x) "\u003d E x

3. Write a natural logarithm derivative formula. (ln x) "\u003d

4. Write the formula of the derivative of the logarithmic function. Log a. x) "\u003d

5. Record general form Valid for function f (x) \u003d but x. F (x) \u003d

6. Record a general view of the first to function f (x) \u003d, x ≠ 0. F (x) \u003d ln | x | + c

Check work (answers to slide 23).

IV. Solving Tasks Ent (simulator)

A) №1,2,3,6,10.36 on the board and in the notebook (slide 24)

B) work in pairs №19,28 (simulator) (slide 25-26)

V. 1. Find errors: (slide 27)

1) f (x) \u003d 5 e - 3x, f "(x) \u003d - 3 E - 3x

2) F (x) \u003d 17 2x, f "(x) \u003d 17 2x ln17

3) f (x) \u003d log 5 (7x + 1), f "(x) \u003d

4) f (x) \u003d ln (9 - 4x), f "(x) \u003d
.

Vi. Presentation of students.

Epigraph: "Knowledge is such a precious thing that it is not forced to get it from any source" Thomas Akvinsky (Slide 28)

VII. House. Week №19,20 page 116.

VIII. Test (backup task) (slide 29-32)

IX. The outcome of the lesson.

"If you want to participate in great life, then fill your head with mathematics while there is a possibility. She will give you a huge help in the whole of your life "M. Kalinin (Slide 33)

Algebra and beginning of mathematical analysis

Differentiation of indicative and logarithmic function

Compiler:

mathematics teacher MOU SOSH №203 HET

novosibirsk city

Vividova T. V.

Number e. Function y \u003d E. x. , its properties, schedule, differentiation

1. We construct for various bases a graphics: 1. y \u003d 2 x 3. y \u003d 10 x 2. y \u003d 3 x (2 options) (1 option) "width \u003d" 640 "

Consider an indicative function y \u003d A. x. where and 1.

Build for various bases but Graphics:

1. y \u003d 2. x.

3. y \u003d 10. x.

2. y \u003d 3. x.

(Option 2)

(1 option)

1) all charts pass through the point (0; 1);

2) All graphics have horizontal asymptotes y \u003d 0

for h.  ∞;

3) all of them are convexing down;

4) They all have tangents in all their points.

We will conduct tangent to the graph y \u003d 2. x. At point h. \u003d 0 and measuring the angle that forms a tangent with the axis h.

With the help of accurate constructions of tangents to schedules, it can be noted that if the base but Indicative function y \u003d A. x. Gradually increases the base from 2 to 10, then the angle between the tangent to the graph of the function at the point h. \u003d 0 and the abscissa axis gradually increases from 35 'to 66.5'.

Therefore, there is a foundation but for which the corresponding angle is 45 '. And this value but concluded between 2 and 3, because for but \u003d 2 angle is 35 ', when but \u003d 3 It is equal to 48 '.

In the course of mathematical analysis it is proved that this foundation exists, it is customary to mark the letter e.

Determined that e. - irrational number, i.e. is an infinite non-periodic decimal fraction:

e \u003d 2, 7182818284590 ... ;

In practice, it is usually believed that e. ≈ 2,7.

Graph and function properties y \u003d E. x. :

1) D (F) = (- ∞; + ∞);

3) increases;

4) is not limited from above, limited to below

5) does not have the greatest nor the smallest

values;

6) continuous;

7) E (F) = (0; + ∞);

8) convex down;

9) Differential.

Function y \u003d E. x. Call exhibitor .

In the course of mathematical analysis it is proved that the function y \u003d E. x. has a derivative anywhere h. :

(E. x. ) \u003d E. x.

(E. 5x ) "\u003d 5e 5x

(E. x-3. ) "\u003d E x-3.

(E. -4x + 1. ) "\u003d -4E -4x-1.

Example 1. . Conductance to the graph of the function at point x \u003d 1.

2) f () \u003d f (1) \u003d E

4) y \u003d e + e (x-1); Y \u003d EX.

Answer:

Example 2. .

x. = 3.

Example 3. .

Explore the extremum function

x \u003d 0 and x \u003d -2

h. \u003d -2 - maximum point

h. \u003d 0 - point minimum

If the basis of the logarithm is the number e. , then they say that natural logarithm . For natural logarithov A special designation has been introduced lN. (L - logarithm, n - natural).

Graph and properties of the function y \u003d ln x

Function properties y \u003d lN X:

1) D (F) = (0; + ∞);

2) is neither even nor odd;

3) increases on (0; + ∞);

4) not limited;

5) does not have the greatest nor least values;

6) continuous;

7) E (f) \u003d (- ∞; + ∞);

8) convex over;

9) Differential.

0 FULL DIFFERENTIATION Formula "Width \u003d" 640 "

In the course of mathematical analysis it is proved that for any value x0. Formula Differentiation

Example 4:

Calculate the value of the derivative function at the point x. = -1.

For example:

Internet resources:

• http://egemaximum.ru/pokazatelnaya-funktsiya/
• http://or-gr2005.narod.ru/grafik/sod/gr-3.html
• http://ru.wikipedia.org/wiki/
• http://900igr.net/prezentatsii.
• http://ppt4web.ru/algebra/prizvodnaja-pokazatelnojj-funkcii.html

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