Summary of an open lesson by a teacher at GBPOU "Pedagogical College No. 4 of St. Petersburg"

Martusevich Tatyana Olegovna

Date: 12/29/2014.

Topic: Geometric meaning of derivatives.

Lesson type: learning new material.

Teaching methods: visual, partly search.

The purpose of the lesson.

Introduce the concept of a tangent to the graph of a function at a point, find out what the geometric meaning of the derivative is, derive the equation of the tangent and teach how to find it.

Educational objectives:

Achieve an understanding of the geometric meaning of the derivative; deriving the tangent equation; learn to solve basic problems;

provide repetition of material on the topic “Definition of a derivative”;

create conditions for control (self-control) of knowledge and skills.

Developmental tasks:

promote the formation of skills to apply techniques of comparison, generalization, and highlighting the main thing;

continue the development of mathematical horizons, thinking and speech, attention and memory.

Educational tasks:

promote interest in mathematics;

education of activity, mobility, communication skills.

Lesson type – a combined lesson using ICT.

Equipment – multimedia installation, presentationMicrosoftPowerPoint.

Lesson stage

Time

Teacher's activities

Student activity

1. Organizational moment.

State the topic and purpose of the lesson.

Topic: Geometric meaning of derivatives.

The purpose of the lesson.

Introduce the concept of a tangent to the graph of a function at a point, find out what the geometric meaning of the derivative is, derive the equation of the tangent and teach how to find it.

Preparing students for work in class.

Preparation for work in class.

Understanding the topic and purpose of the lesson.

Note-taking.

2. Preparation for learning new material through repetition and updating of basic knowledge.

Organization of repetition and updating of basic knowledge: definition of derivative and formulation of its physical meaning.

Formulating the definition of a derivative and formulating its physical meaning. Repetition, updating and consolidation of basic knowledge.

Organization of repetition and development of the skill of finding the derivative of a power function and elementary functions.

Finding the derivative of these functions using formulas.

Repetition of the properties of a linear function.

Repetition, perception of drawings and teacher’s statements

3. Working with new material: explanation.

Explanation of the meaning of the relationship between function increment and argument increment

Explanation of the geometric meaning of the derivative.

Introduction of new material through verbal explanations using images and visual aids: multimedia presentation with animation.

Perception of explanation, understanding, answers to teacher questions.

Formulating a question to the teacher in case of difficulty.

Perception of new information, its primary understanding and comprehension.

Formulation of questions to the teacher in case of difficulty.

Creating a note.

Formulation of the geometric meaning of the derivative.

Consideration of three cases.

Taking notes, making drawings.

4. Working with new material.

Primary comprehension and application of the studied material, its consolidation.

At what points is the derivative positive?

Negative?

Equal to zero?

Training in finding an algorithm for answers to questions posed according to a schedule.

Understanding, making sense of, and applying new information to solve a problem.

5. Primary comprehension and application of the studied material, its consolidation.

Message of the task conditions.

Recording the conditions of the task.

Formulating a question to the teacher in case of difficulty

6. Application of knowledge: independent work of a teaching nature.

Solve the problem yourself:

Application of acquired knowledge.

Independent work on solving the problem of finding the derivative from a drawing. Discussion and verification of answers in pairs, formulation of a question to the teacher in case of difficulty.

7. Working with new material: explanation.

Deriving the equation of a tangent to the graph of a function at a point.

A detailed explanation of the derivation of the equation of a tangent to the graph of a function at a point, using a multimedia presentation for clarity, and answers to student questions.

Derivation of the tangent equation together with the teacher. Answers to the teacher's questions.

Taking notes, creating a drawing.

8. Working with new material: explanation.

In a dialogue with students, the derivation of an algorithm for finding the equation of a tangent to the graph of a given function at a given point.

In a dialogue with the teacher, derive an algorithm for finding the equation of the tangent to the graph of a given function at a given point.

Note-taking.

Message of the task conditions.

Training in the application of acquired knowledge.

Organizing the search for ways to solve a problem and their implementation. detailed analysis of the solution with explanation.

Recording the conditions of the task.

Making assumptions about possible ways to solve the problem when implementing each item of the action plan. Solving the problem together with the teacher.

Recording the solution to the problem and the answer.

9. Application of knowledge: independent work of a teaching nature.

Individual control. Consulting and assistance to students as needed.

Check and explain the solution using a presentation.

Application of acquired knowledge.

Independent work on solving the problem of finding the derivative from a drawing. Discussion and verification of answers in pairs, formulation of a question to the teacher in case of difficulty

10. Homework.

§48, problems 1 and 3, understand the solution and write it down in a notebook, with drawings.

№ 860 (2,4,6,8),

Homework message with comments.

Recording homework.

11. Summing up.

We repeated the definition of the derivative; physical meaning of derivative; properties of a linear function.

We learned what the geometric meaning of a derivative is.

We learned how to derive the equation of a tangent to the graph of a given function at a given point.

Correction and clarification of lesson results.

Listing the results of the lesson.

12. Reflection.

1. You found the lesson: a) easy; b) usually; c) difficult.

a) have mastered it completely, I can apply it;

b) have learned it, but find it difficult to apply;

c) didn’t understand.

3. Multimedia presentation in class:

a) helped to master the material; b) did not help master the material;

c) interfered with the assimilation of the material.

Conducting reflection.

Job type: 7

Condition

The straight line y=3x+2 is tangent to the graph of the function y=-12x^2+bx-10. Find b, given that the abscissa of the tangent point is less than zero.

Solution

Let x_0 be the abscissa of the point on the graph of the function y=-12x^2+bx-10 through which the tangent to this graph passes.

The value of the derivative at point x_0 is equal to the slope of the tangent, that is, y"(x_0)=-24x_0+b=3. On the other hand, the point of tangency belongs simultaneously to both the graph of the function and the tangent, that is, -12x_0^2+bx_0-10= 3x_0 + 2. We obtain a system of equations \begin(cases) -24x_0+b=3,\\-12x_0^2+bx_0-10=3x_0+2. \end(cases)

Solving this system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the abscissa condition, the tangent points are less than zero, so x_0=-1, then b=3+24x_0=-21.

Answer

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The straight line y=-3x+4 is parallel to the tangent to the graph of the function y=-x^2+5x-7. Find the abscissa of the tangent point.

Solution

The angular coefficient of the straight line to the graph of the function y=-x^2+5x-7 at an arbitrary point x_0 is equal to y"(x_0). But y"=-2x+5, which means y"(x_0)=-2x_0+5. Angular the coefficient of the line y=-3x+4 specified in the condition is equal to -3. Parallel lines have the same slope coefficients. Therefore, we find a value x_0 such that =-2x_0 +5=-3.

We get: x_0 = 4.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

Solution

From the figure we determine that the tangent passes through points A(-6; 2) and B(-1; 1). Let us denote by C(-6; 1) the point of intersection of the lines x=-6 and y=1, and by \alpha the angle ABC (you can see in the figure that it is acute). Then straight line AB forms an angle \pi -\alpha with the positive direction of the Ox axis, which is obtuse.

As is known, tg(\pi -\alpha) will be the value of the derivative of the function f(x) at point x_0. notice, that tg \alpha =\frac(AC)(CB)=\frac(2-1)(-1-(-6))=\frac15. From here, using the reduction formulas, we get: tg(\pi -\alpha) =-tg \alpha =-\frac15=-0.2.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The straight line y=-2x-4 is tangent to the graph of the function y=16x^2+bx+12. Find b, given that the abscissa of the tangent point is greater than zero.

Solution

Let x_0 be the abscissa of the point on the graph of the function y=16x^2+bx+12 through which

is tangent to this graph.

The value of the derivative at point x_0 is equal to the slope of the tangent, that is, y"(x_0)=32x_0+b=-2. On the other hand, the point of tangency belongs simultaneously to both the graph of the function and the tangent, that is, 16x_0^2+bx_0+12=- 2x_0-4 We obtain a system of equations \begin(cases) 32x_0+b=-2,\\16x_0^2+bx_0+12=-2x_0-4. \end(cases)

Solving the system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the abscissa condition, the tangent points are greater than zero, so x_0=1, then b=-2-32x_0=-34.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The figure shows a graph of the function y=f(x), defined on the interval (-2; 8). Determine the number of points at which the tangent to the graph of the function is parallel to the straight line y=6.

Solution

The straight line y=6 is parallel to the Ox axis. Therefore, we find points at which the tangent to the graph of the function is parallel to the Ox axis. On this chart, such points are extremum points (maximum or minimum points). As you can see, there are 4 extremum points.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The line y=4x-6 is parallel to the tangent to the graph of the function y=x^2-4x+9. Find the abscissa of the tangent point.

Solution

The slope of the tangent to the graph of the function y=x^2-4x+9 at an arbitrary point x_0 is equal to y"(x_0). But y"=2x-4, which means y"(x_0)=2x_0-4. The slope of the tangent y =4x-7, specified in the condition, is equal to 4. Parallel lines have the same angular coefficients. Therefore, we find a value of x_0 such that 2x_0-4 = 4. We get: x_0 = 4.

Answer

Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Job type: 7
Topic: Geometric meaning of derivatives. Tangent to the graph of a function

Condition

The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x_0. Find the value of the derivative of the function f(x) at point x_0.

Solution

From the figure we determine that the tangent passes through points A(1; 1) and B(5; 4). Let us denote by C(5; 1) the point of intersection of the lines x=5 and y=1, and by \alpha the angle BAC (you can see in the figure that it is acute). Then straight line AB forms an angle \alpha with the positive direction of the Ox axis.

Lesson objectives:

Students should know:

• what is called the slope of a line;
• the angle between the straight line and the Ox axis;
• what is the geometric meaning of the derivative;
• equation of the tangent to the graph of a function;
• a method for constructing a tangent to a parabola;
• be able to apply theoretical knowledge in practice.

Lesson objectives:

Educational: create conditions for students to master a system of knowledge, skills and abilities with the concepts of mechanical and geometric meaning of a derivative.

Educational: to form a scientific worldview in students.

Developmental: to develop students’ cognitive interest, creativity, will, memory, speech, attention, imagination, perception.

Methods of organizing educational and cognitive activities:

• visual;
• practical;
• by mental activity: inductive;
• according to the assimilation of material: partially search, reproductive;
• by degree of independence: laboratory work;
• stimulating: encouragement;
• control: oral frontal survey.

Lesson Plan

1. Oral exercises (find the derivative)
2. Student’s message on the topic “Reasons for the emergence of mathematical analysis.”
3. Learning new material
4. Phys. Just a minute.
5. Solving tasks.
6. Laboratory work.
7. Summing up the lesson.
8. Commenting on homework.

Equipment: multimedia projector (presentation), cards (laboratory work).

“A person only achieves something where he believes in his own strength”

L. Feuerbach

Organization of the class throughout the lesson, students' readiness for the lesson, order and discipline.

Setting learning goals for students, both for the entire lesson and for its individual stages.

Determine the significance of the material being studied both in this topic and in the entire course.

Verbal counting

1. Find derivatives:

" , ()" , (4sin x)", (cos2x)", (tg x)", "

2. Logic test.

a) Insert the missing expression.

The general direction of the development of science is ultimately determined by the requirements of the practice of human activity. The existence of ancient states with a complex hierarchical management system would have been impossible without the sufficient development of arithmetic and algebra, because collecting taxes, organizing army supplies, building palaces and pyramids, and creating irrigation systems required complex calculations. During the Renaissance, connections between different parts of the medieval world expanded, trade and crafts developed. A rapid rise in the technical level of production begins, and new sources of energy that are not associated with the muscular efforts of humans or animals are being used industrially. In the XI-XII centuries, fulling and weaving machines appeared, and in the middle of the XV - a printing press. Due to the need for the rapid development of social production during this period, the essence of the natural sciences, which had been descriptive since ancient times, changed. The goal of natural science is an in-depth study of natural processes, not objects. Mathematics, which operated with constant quantities, corresponded to the descriptive natural science of antiquity. It was necessary to create a mathematical apparatus that would describe not the result of the process, but the nature of its flow and its inherent patterns. As a result, by the end of the 12th century, Newton in England and Leibniz in Germany completed the first stage of creating mathematical analysis. What is “mathematical analysis”? How can one characterize and predict the characteristics of any process? Use these features? To penetrate deeper into the essence of a particular phenomenon?

Let's follow the path of Newton and Leibniz and see how we can analyze the process, considering it as a function of time.

Let us introduce several concepts that will help us further.

The graph of the linear function y=kx+ b is a straight line, the number k is called the slope of the straight line. k=tg, where is the angle of the straight line, that is, the angle between this straight line and the positive direction of the Ox axis.

Picture 1

Consider the graph of the function y=f(x). Let's draw a secant through any two points, for example, secant AM. (Fig.2)

Angular coefficient of the secant k=tg. In a right triangle AMC<МАС = (объясните почему?). Тогда tg = = , что с точки зрения физики есть величина средней скорости протекания любого процесса на данном промежутке времени, например, скорости изменения расстояния в механике.

Figure 2

Figure 3

The term “speed” itself characterizes the dependence of a change in one quantity on a change in another, and the latter does not necessarily have to be time.

So, the tangent of the angle of inclination of the secant tg = .

We are interested in the dependence of changes in quantities over a shorter period of time. Let us direct the increment of the argument to zero. Then the right side of the formula is the derivative of the function at point A (explain why). If x -> 0, then point M moves along the graph to point A, which means straight line AM is approaching some straight line AB, which is tangent to the graph of the function y = f(x) at point A. (Fig.3)

The angle of inclination of the secant tends to the angle of inclination of the tangent.

The geometric meaning of the derivative is that the value of the derivative at a point is equal to the slope of the tangent to the graph of the function at the point.

Mechanical meaning of derivative.

The tangent of the tangent angle is a value showing the instantaneous rate of change of the function at a given point, that is, a new characteristic of the process being studied. Leibniz called this quantity derivative, and Newton said that the derivative itself is called the instantaneous speed.

No. 91(1) page 91 – show on the board.

The angular coefficient of the tangent to the curve f(x) = x 3 at point x 0 – 1 is the value of the derivative of this function at x = 1. f’(1) = 3x 2 ; f’(1) = 3.

No. 91 (3.5) – dictation.

No. 92(1) – on the board if desired.

No. 92 (3) – independently with oral testing.

No. 92 (5) – at the board.

Answers: 45 0, 135 0, 1.5 e 2.

Goal: to develop the concept of “mechanical meaning of a derivative.”

Applications of derivatives to mechanics.

The law of rectilinear motion of the point x = x(t), t is given.

1. Average speed of movement over a specified period of time;
2. Velocity and acceleration at time t 04
3. Moments of stopping; whether the point after the moment of stopping continues to move in the same direction or begins to move in the opposite direction;
4. The highest speed of movement in a specified period of time.

The work is performed according to 12 options, the tasks are differentiated by level of difficulty (the first option is the lowest level of difficulty).

Before starting work, a conversation on the following questions:

1. What is the physical meaning of the derivative of displacement? (Speed).
2. Is it possible to find the derivative of speed? Is this quantity used in physics? What is it called? (Acceleration).
3. The instantaneous speed is zero. What can be said about the movement of the body at this moment? (This is the moment of stopping).
4. What is the physical meaning of the following statements: the derivative of motion is equal to zero at point t 0; does the derivative change sign when passing through point t 0? (The body stops; the direction of movement changes to the opposite).

A sample of student work.

x(t)= t 3 -2 t 2 +1, t 0 = 2.

Figure 4

In the opposite direction.

Let's draw a schematic diagram of the speed. The highest speed is achieved at the point

t=10, v (10) =3· 10 2 -4· 10 =300-40=260

Figure 5

1) What is the geometric meaning of the derivative?
2) What is the mechanical meaning of a derivative?
3) Draw a conclusion about your work.

Page 90. No. 91(2,4,6), No.92(2,4,6,), p. 92 No. 112.

Used Books

• Textbook Algebra and beginnings of analysis.
  Authors: Yu.M. Kolyagin, M.V. Tkacheva, N.E. Fedorova, M.I. Shabunina.
  Edited by A. B. Zhizhchenko.
• Algebra 11th grade. Lesson plans based on the textbook by Sh. A. Alimov, Yu. M. Kolyagin, Yu. V. Sidorov. Part 1.
• Internet resources: http://orags.narod.ru/manuals/html/gre/12.jpg

Derivative(functions at a point) - basic concept differential calculus, characterizing the rate of change of the function (at a given point). Defined as limit the relationship between the increment of a function and its increment argument when the argument increment tends to zero, if such a limit exists. A function that has a finite derivative (at some point) is called differentiable (at that point).

The process of calculating the derivative is called differentiation. Reverse process - finding antiderivative - integration.

If a function is given by a graph, its derivative at each point is equal to the tangent of the tangent to the graph of the function. And if the function is given by a formula, the table of derivatives and the rules of differentiation will help you, that is, the rules for finding the derivative.

4. Derivative of a complex and inverse function.

Let now be given complex function , i.e. a variable is a function of a variable, and a variable is, in turn, a function of an independent variable.

Theorem . If And  differentiable functions of its arguments, then a complex function is a differentiable function and its derivative is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable:

.

The statement is easily obtained from the obvious equality (valid for and ) by passing to the limit at (which, due to the continuity of the differentiable function, implies ).

Let's move on to consider the derivative inverse function.

Let the differentiable function on a set have a set of values ​​and on the set there exist inverse function .

Theorem . If at the point derivative , then the derivative of the inverse function at the point exists and is equal to the reciprocal of the derivative of this function: , or

This formula is easily obtained from geometric considerations.

T Just like there is the tangent of the angle of inclination of the tangent line to the axis, that is, the tangent of the angle of inclination of the same tangent (same line) at the same point to the axis.

If they are sharp, then , and if they are blunt, then .

In both cases . This equality is equivalent to equality

5. Geometric and physical meaning of derivative.

1) Physical meaning of the derivative.

If the function y = f(x) and its argument x are physical quantities, then the derivative is the rate of change of the variable y relative to the variable x at a point. For example, if S = S(t) is the distance covered by a point in time t, then its derivative is the speed at the moment of time. If q = q(t) is the amount of electricity flowing through the cross section of the conductor at time t, then is the rate of change in the amount of electricity at time, i.e. current strength at a moment in time.

2) Geometric meaning of the derivative.

Let be some curve, be a point on the curve.

Any line that intersects at least two points is called a secant.

A tangent to a curve at a point is the limiting position of a secant if the point tends to while moving along the curve.

From the definition it is obvious that if a tangent to a curve exists at a point, then it is the only one

Consider the curve y = f(x) (i.e. the graph of the function y = f(x)). Let at the point it has a non-vertical tangent. Its equation: (equation of a line passing through a point and having a slope k).

By definition of the angular coefficient, where is the angle of inclination of the straight line to the axis.

Let be the angle of inclination of the secant to the axis, where. Since is a tangent, then when

Hence,

Thus, we found that is the angular coefficient of the tangent to the graph of the function y = f(x) at the point (geometric meaning of the derivative of a function at a point). Therefore, the equation of the tangent to the curve y = f(x) at the point can be written in the form

Before reading the information on the current page, we recommend watching a video about the derivative and its geometric meaning

Also see an example of calculating the derivative at a point

The tangent to the line l at the point M0 is the straight line M0T - the limiting position of the secant M0M when the point M tends to M0 along this line (i.e., the angle tends to zero) in an arbitrary manner.

Derivative of the function y = f(x) at point x0 called the limit of the ratio of the increment of this function to the increment of the argument when the latter tends to zero. The derivative of the function y = f(x) at the point x0 and in textbooks is denoted by the symbol f"(x0). Therefore, by definition

The term "derivative"(also "second derivative") introduced by J. Lagrange(1797), in addition, he gave the designations y’, f’(x), f”(x) (1770,1779). The designation dy/dx first appears in Leibniz (1675).

The derivative of the function y = f(x) at x = xo is equal to the slope of the tangent to the graph of this function at the point Mo(xo, f(xo)), i.e.

where a - tangent angle to the Ox axis of the rectangular Cartesian coordinate system.

Tangent equation to the line y = f(x) at the point Mo(xo, yo) takes the form

The normal to a curve at some point is the perpendicular to the tangent at the same point. If f(x0) is not equal to 0, then line normal equation y = f(x) at the point Mo(ho, yo) will be written as follows:

Physical meaning of the derivative

If x = f(t) is the law of rectilinear motion of a point, then x’ = f’(t) is the speed of this motion at time t. Flow rate physical, chemical and other processes are expressed using the derivative.

If the ratio dy/dx for x->x0 has a limit on the right (or on the left), then it is called the derivative on the right (respectively, the derivative on the left). Such limits are called one-sided derivatives.

Obviously, a function f(x) defined in a certain neighborhood of the point x0 has a derivative f’(x) if and only if one-sided derivatives exist and are equal to each other.

Geometric interpretation of the derivative as the angular coefficient of the tangent to the graph also applies to this case: the tangent in this case is parallel to the Oy axis.

A function that has a derivative at a given point is said to be differentiable at that point. A function that has a derivative at each point of a given interval is called differentiable in this interval. If the interval is closed, then at its ends there are one-sided derivatives.

The operation of finding the derivative is called.