How to solve differentiation of exponential and logarithmic functions. Differentiation of exponential and logarithmic functions - Knowledge Hypermarket. VII. Independent work

Lesson topic: “Differentiation of exponential and logarithmic function. Antiderivative of the exponential function" in UNT assignments

Target : develop students’ skills in applying theoretical knowledge on the topic “Differentiation of exponential and logarithmic functions. Antiderivative of the exponential function" for solving UNT problems.

Tasks

Educational: systematize students’ theoretical knowledge, consolidate problem-solving skills on this topic.

Educational: develop memory, observation, logical thinking, students’ mathematical speech, attention, self-esteem and self-control skills.

Educational: contribute:

developing a responsible attitude towards learning among students;

development of sustainable interest in mathematics;

creating positive internal motivation to study mathematics.

Teaching methods: verbal, 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. Solving the crossword puzzle. (slide 3-21)

The 17th century French mathematician Pierre Fermat defined this line as “The straight line most closely adjacent to the curve in a small neighborhood of the point.”

Tangent

A function that is given by the formula y = log a x.

Logarithmic

A function that is given by the formula y = A X.

Indicative

In mathematics, this concept is used to find the speed of movement. material point and the angular coefficient of the tangent to the graph of the function at a given point.

Derivative

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

Antiderivative

What is the name of the relationship between X and Y, in which each element of X is associated with a single element of Y.

Derivative of displacement

Speed

A function that is given by the formula y = e x.

Exhibitor

If a function f(x) can be represented as f(x)=g(t(x)), then this function is called...

III. Mathematical dictation (slide 22)

1. Write down the formula for the derivative of the exponential function. ( A x)" = A x ln a

2. Write down the formula for the derivative of the exponential. (e x)" = e x

3. Write down the derivative formula natural logarithm. (ln x)"=

4. Write down the formula for the derivative of a logarithmic function. (log a x)"=

5. Record general form antiderivatives for the function f(x) = A X. F(x)=

6. Write down the general form of antiderivatives for the function f(x) =, x≠0. F(x)=ln|x|+C

Check your work (answers on slide 23).

IV. Solving UNT problems (simulator)

A) No. 1,2,3,6,10,36 on the board and in the notebook (slide 24)

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

V. 1. Find errors: (slide 27)

1) f(x)=5 e – 3х, f "(x)= – 3 e – 3х

2) f(x)=17 2x, f "(x)= 17 2x ln17

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

4) f(x)= ln(9 – 4х), f "(x)=
.

VI. Student presentation.

Epigraph: “Knowledge is such a precious thing that there is no shame in obtaining it from any source” Thomas Aquinas (slide 28)

VII. Homework No. 19,20 p.116

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

IX. Lesson summary.

"If you want to participate in great life, then fill your head with mathematics while you have the opportunity. She will then provide you with great help throughout your life” M. Kalinin (slide 33)

Algebra lesson in 11th grade on the topic: "Differentiation and integration of exponential and logarithmic functions"

Lesson objectives:

Systematize the material studied on the topic “Exponential and logarithmic functions.”

To develop the ability to solve problems involving differentiation and integration of exponential and logarithmic functions.

Take advantage of opportunities information technologies to develop motivation to study complex topics in mathematical analysis.

State the requirements for completing the test work on this topic in the next lesson.

During the classes

I. Organizational moment (1 – 2 minutes).

The teacher communicates the objectives of the lesson.

The class is divided into 4 groups.

II. Blitz survey using formulas (homework).

Conversation in the form of dialogue with students.

Let's say you deposited 10,000 rubles in a bank at an interest rate of 12% per annum. In how many years will your investment double?

To do this, we need to solve the equation: , that is How?

We need to go to base 10, that is (using a calculator)

Thus, the doubling of the contribution will occur in six years (a little over).

Here we needed a formula for moving to a new base. What formulas related to differentiation and integration of logarithmic and exponential functions do you know? (all formulas are taken from the pages of the textbook, p. 81, p. 86).

Questions to each other in a chain.

Questions for the teacher.

The teacher asks to derive 1–2 formulas.

On separate small leaves mathematical dictation by knowledge of formulas. A mutual check is underway. The seniors in the groups display the average arithmetic score and enter it into the table.

Activity table

Kind of activity
1. Knowledge of formulas.
2. Individual knowledge. Work in pairs.
3. Oral work.
4. Control tests (computer assessment).
5. Independent work(compulsory level tasks).
6. Tasks of increased complexity.

III. Oral work:

Determine the number of solutions to the equations.

A) ;

B) ;

After students answer using the overhead projector, graphs are displayed on the screen.

A) 2 solutions

B) 1 solution

Additional question: Find highest value functions

A decreasing function has the greatest value when the indicator has the smallest value.

(2 ways)

IV. Individual work.

During oral work from each group with individual tasks 2 people work.

1st group: One examines the function, the second on interactive whiteboard is the graph of this function.

Additional question:. Answer: (Number e? See page 86 of the textbook).

Group 2: Find a curve passing through point n (0; 2) if the slope of the tangent at any point on the curve is equal to the product of the coordinates of the point of tangency. One is differential equation and finds a general solution, the second finds a particular solution using the initial conditions.

Answer:

Additional question: Why equal to the angle between the tangent drawn at point X=0 to the graph of the function y = e x and x-axis. (45 o)

The graph of this function is called the “exponent” (Find information about this in the textbook and check your rationale with the explanations in the textbook, page 86).

Group 3:

Compare

One compares using a microcalculator, and the other without.

Additional question: Determine at what x0 the equality ?

Answer: x = 2 0.5.

Group 4: Prove that

Proof in different ways.

Additional question: Find an approximate value e 1.01. Compare your value with the answer in example 2 (page 86 of the textbook).

V. Working with the textbook.

The children are invited to consider examples from example 1 - example 9 (pages 81 - 84 of the textbook). Based on these examples, run control tests.

VI. Control tests.

The task is on the screen. There is a discussion going on. The correct answer is selected and justification is given. The computer gives a score. The eldest in the group notes in the table the activity of his comrades during the test.

1) Given a function f(x)= 2-e 3x . Determine at what value of C the graph of its antiderivative F(x)+C passes through the point M (1/3;-e/3)

Answer: a) e-1 ; b) 5/8; c) -2/3; d) 2.

2) Given a function f(x)= e 3x-2 +ln(2x+3). Find f"(2/3)

Answer: a) -1; b) 45/13; c) 1/3; d) 2.

3) Does the function satisfy y = e ax equation y" = ay.

Answer: a) yes; b) no; c) everything depends on both; d) it is impossible to say definitely.

VII. Independent work.

Mandatory level tasks: Find extremum points of functions.

		III group

The eldest in the group puts points for this task in the table.

At this time, one person from each group works at the board with tasks of increased complexity.

		III group

The teacher shows the complete written documentation of the tasks along the way (it is projected on the screen, this is very important for completing the subsequent test work).

VIII. Homework.

IX. Lesson summary:

Giving grades taking into account the points received. Norms of grades for the upcoming test work in the next lesson.

Differentiating exponential and logarithmic functions

1. Number e. Function y = e x, its properties, graph, differentiation

Let's consider an exponential function y=a x, where a > 1. For various reasons and we get various graphics(Fig. 232-234), but you can notice that they all pass through the point (0; 1), they all have horizontal asymptote y = 0 at , all of them are convexly facing downwards and, finally, they all have tangents at all their points. Let us draw, for example, a tangent to graphics function y=2x at point x = 0 (Fig. 232). If you make accurate constructions and measurements, you can make sure that this tangent forms an angle of 35° (approximately) with the x-axis.

Now let’s draw a tangent to the graph of the function y = 3 x, also at the point x = 0 (Fig. 233). Here the angle between the tangent and the x-axis will be greater - 48°. And for the exponential function y = 10 x in a similar
situation we get an angle of 66.5° (Fig. 234).

So, if the base a of the exponential function y=ax gradually increases from 2 to 10, then the angle between the tangent to the graph of the function at the point x=0 and the x-axis 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 enclosed between the numbers 2 and 3, since for the function y-2x the angle of interest to us is 35°, which is less than 45°, and for the function y=3 x it is equal to 48°, which is already a little more than 45 °. The base we are interested in is usually denoted by the letter e. It has been established that the number e is irrational, i.e. represents an infinite decimal non-periodic fraction:

e = 2.7182818284590...;

in practice it is usually assumed that e=2.7.

Comment(not very serious). It is clear that L.N. Tolstoy has nothing to do with the number e, however, in writing the number e, please note that the number 1828 is repeated twice in a row - the year of birth of L.N. Tolstoy.

The graph of the function y=e x is shown in Fig. 235. This is an exponential that differs from other exponentials (graphs of exponential functions with other bases) in that the angle between the tangent to the graph at point x=0 and the x-axis is 45°.

Properties of the function y = e x:

1)
2) is neither even nor odd;
3) increases;
4) not limited from above, limited from below;
5) has neither the largest nor the smallest values;
6) continuous;
7)
8) convex down;
9) differentiable.

Return to § 45, look at the list of properties of the exponential function y = a x for a > 1. You will find the same properties 1-8 (which is quite natural), and the ninth property associated with
we did not mention the differentiability of the function then. Let's discuss it now.

Let us derive a formula for finding the derivative y-ex. In this case, we will not use the usual algorithm, which we developed in § 32 and which has been successfully used more than once. In this algorithm final stage we need to calculate the limit, and our knowledge of the theory of limits is still very, very limited. Therefore, we will rely on geometric premises, considering, in particular, the very fact of the existence of a tangent to the graph of the exponential function beyond doubt (that is why we so confidently wrote down the ninth property in the above list of properties - the differentiability of the function y = e x).

1. Note that for the function y = f(x), where f(x) =ex, we already know the value of the derivative at the point x =0: f / = tan45°=1.

2. Let us introduce the function y=g(x), where g(x) -f(x-a), i.e. g(x)-ex" a. Fig. 236 shows the graph of the function y = g(x): it is obtained from the graph of the function y - fx) by shifting along the x axis by |a| scale units. Tangent to the graph of the function y = g (x) in point x-a is parallel to the tangent to the graph of the function y = f(x) at point x -0 (see Fig. 236), which means it forms an angle of 45° with the x axis. Using geometric meaning derivative, we can write that g(a) =tg45°;=1.

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

4. We have established that for any value of a the relation is valid. Instead of the letter a, you can, of course, use the letter x; then we get

From this formula we obtain the corresponding integration formula:

A.G. Mordkovich Algebra 10th grade

Calendar-thematic planning in mathematics, video in mathematics online, Mathematics at school download

Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year guidelines discussion programs Integrated Lessons

Lesson outline

Subject: Algebra

Date: 04/2/13.

Grade: 11th grade

Teacher: Tyshibaeva N.Sh.

Subject: Differentiation of logarithmic and exponential functions. Antiderivative of the exponential function.

Target:

1) formulate formulas for derivatives of logarithmic and exponential functions; teach how to find the antiderivative of an exponential function

2) develop memory, observation, logical thinking, mathematical speech of students, the ability to analyze and compare, develop cognitive interest in the subject;

3) bring up communicative culture students, skills collective activity, cooperation, mutual assistance.

Lesson type: explanation of new material and consolidation of acquired knowledge, skills and abilities.

Equipment : cards, interactive whiteboard.

Technology: differentiated approach

During the classes :

1.Org. moment .(2min) .

2. Solving a crossword puzzle (8 min)

1. The 17th century French mathematician Pierre Fermat defined this line as “The straight line most closely adjacent to the curve in a small neighborhood of the point.”

Tangent

2.Function, which is given by the formula y = a x.

Indicative

3.Function, which is given by the formula y = log ax.

Logarithmic

4. Derivative of displacement

Speed

5.What is the name of the function F(x) for the function f(x), if the condition F"(x) =f(x) is satisfied for any point from the interval I.

Antiderivative

6.What is the name of the relationship between X and Y, in which each element of X is associated with a single element of Y.

Function

7. If the function f(x) can be represented in the form f(x)=g(t(x)), then this function is called...

Complex

Vertical word surname of a French mathematician and mechanic

Lagrange

3.Explanation of new material: (10 min)

The exponential function at any point in the domain of definition has a derivative and this derivative is found by the formula:

(.ln a in the formula we replace the number and on e, we get

(e x)" = e x_ formula derivative of the exponential
A logarithmic function has a derivative at any point in its domain of definition, and this derivative is found by the formula:

(log a x)" = replace the number in the formula and on e, we get

Exponential function y =(A at any point in the domain of definition it has an antiderivative and this antiderivative is found by the formula F(x) =+ C

4. Consolidation of new material (20 min)

Mathematical dictation.

1.Write the formula for the derivative of the exponential function (a X )"

(a x)" = a x ln a

2. Write down the formula for the derivative of the exponential. (e X )"

(e x )" = e x

3. Write down the formula for the derivative of the natural logarithm

4. Write down the formula for the derivative of the logarithmic function (log a x)"=?

(log a x)" =

5. Write down the general form of antiderivatives for the function f(x) = a X .

F(x) = + C

6. Write down the general form of antiderivatives for the function:, x≠0. F(x)=ln|x|+С

Work at the board

№255,№256,№258,№259(2,4)

6.D/z No. 257, No. 261 (2 min)

7. Lesson summary: (3 min)

- What is the formula for a logarithmic function?

What formula defines the exponential function?

What formula is used to find the derivative of a logarithmic function?

What formula is used to find the derivative of an exponential function