The geometric meaning of the derivative. Presentation on the topic "geometric meaning of the derivative of a function" Presentation on the topic geometric meaning of the derivative

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Sooner or later every correct mathematical idea finds application in this or that business. A.N. Krylov

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The purpose of the lesson

1) find out what the geometric meaning of the derivative is, derive the equations of the tangent to the graph of the function 2) Develop the OUUN of mental activity: analysis, generalization and systematization, logical thinking, conscious perception of educational material 3) form the ability to assess your level of knowledge and the desire to improve it, contribute to the development of the need for self-education. Education of responsibility, collectivism.

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Lesson vocabulary

derivative, linear function, slope, continuity, tangents of angles (acute, obtuse).

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Make a pair of 3 minutes each student works independently, 2 minutes - work in pairs. Discussion of the results and recording in the answer card. (Card number 1 remains with the student for self-control, card number 2 must be handed over to the teacher)

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Answer.

Make a couple

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Definition

The function given by the formula y=kx+b is called linear. The number k=tg is called the slope of the line.

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y x -1 0 1 2 y=kx+b

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y x -1 0 1 2 y=kx+b

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y x 0 y=yₒ+k(х-xₒ)   x-xₒ y-yₒ xₒ x Mₒ(xₒ;yₒ) M(x;y) A(x;yₒ)

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Equation of a straight line with slope k passing through the point (x0;y0) y=y0+k(x-x0) Equation of a straight line with slope k passing through the point (x0;y0) y=y0+k(x-x0) (1) Slope of a straight line passing through the points (x1; y1) and (x0; y0) (2)

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y x -1 0 1 2 Find the slope of the line y=kx+b

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Definition

The tangent to the graph of the function y \u003d f (x) is the limiting position of the secant. picture

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tangent secant

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Practical research work The geometric meaning of the derivative

Purpose: Using the data of practical work, determine what the geometric meaning of the derivative is Equipment: Rulers, protractors, microcalculators, graph paper with a graph

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Exercise

1. Plot the tangent to the graph of the function ... at the point with the abscissa xₒ=2 2. Measure the angle formed by the tangent and the positive direction of the x-axis. 3. Write down =…. 4. Calculate with the help of a microcalculator tg=…. 5. Calculate f´(xₒ), to do this, find f´(x) 6. Write down: f´(x)=…. ; f´(xₒ)=…. 7. Select two points on the tangent graph, write down their coordinates. 8. Calculate the slope of the straight line k using the formula 9. Enter the results of the calculation in the table

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The geometric meaning of the derivative

The value of the derivative of the function y=f(x) at the point x0 is equal to the slope of the tangent to the graph of the function y=f(x) at the point (x0;f(x0))

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The equation of the tangent to the graph of the function

1. Write the equation of a straight line with slope k passing through the point 2. Replace k with, and y=y0+k(x-x0)

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Slides captions:

The geometric meaning of the derivative. Tangent equation. f(x)

Using formulas and differentiation rules, find the derivatives of the following functions:

one . What is the geometric meaning of the derivative? 2. Can a tangent be drawn at any point on the graph? Which function is called differentiable at a point? 3 . The tangent is inclined at an obtuse angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function? four . The tangent is inclined at an acute angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function? 5 . The tangent is inclined at right angles to the positive direction of the x-axis. What can be said about the derivative?

for differentiable functions: 0 ° ≤ α ≤ 180 ° , α ≠ 90 ° α - obtuse tg α 0 f ´(x 1) >0 position of the tangent is not defined tg α n.a. f ´(x 3) n.a. α = 0 tg α =0 f ´(x 2) = 0

y \u003d f / (x 0) (x - x 0) + f (x 0) (x 0; f (x 0)) - coordinates of the touch point f ´ (x 0) \u003d tg α \u003d k - slope angle tangent tangent at a given point or slope (x; y) - coordinates of any point of the tangent Tangent equation

No. 1. Find the slope of the tangent to the curve at the point with the abscissa x 0 = - 2. Task B8 FBTZ USE

No. 2. Specify the value of the coefficient k at which the graphs of linear functions y = 8x+12 and y = k x - 3 are parallel. Answer: 8. Task B8 FBTZ USE

0 Y X 1 -1 1 -1 №3. The function y \u003d f (x) is defined on the interval (-7; 7). The figure below shows a graph of its derivative. Find the number of tangents to the graph of the function y \u003d f (x) that are parallel to the x-axis. Answer: 3. Task B8 FBTZ USE

No. 4. The figure shows a straight line that is tangent to the graph of the function y \u003d p (x) at the point (x 0; p (x 0)). Find the value of the derivative at the point x 0. Answer: -0.5. Task B8 FBTZ USE

0 Y X 1 -1 1 -1 №5. All tangents parallel to the straight line y=2x+5 or coinciding with it were drawn to the graph of the function f(x). Specify the number of touch points. Answer: 4. Task B8 FBTZ USE

Write the equations of tangents to the graph of the function at the points of its intersection with the x-axis. Independent work

Last name, first name Testing Creative task Lesson +,-, :), :(, : |

1 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. Write an equation for the tangent to the graph of the function f (x) \u003d 0.5 -4, if the tangent forms an angle of 45 degrees with the positive direction of the x-axis.

2 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. Write the equation of the tangent to the graph of the function f (x) \u003d, parallel to the straight line y \u003d 9 x - 7.

3 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. The straight line passing through the origin touches the graph of the function y \u003d f (x) at point A (-7; 14). Find.

4 group number 1. What is the geometric meaning of the derivative? No. 2. What properties should the function y \u003d f (x) defined on the interval (a; b) have, so that at the point with the abscissa x 0 Є (a; b) its graph has a tangent? No. 3. What is the tangent equation? No. 4. The straight line y \u003d -4x-11 is tangent to the graph of the function. Find the abscissa of the point of contact.

Preview:

Lesson script
in algebra and the beginnings of analysis in the 10th grade.

Topic: “The geometric meaning of the derivative. Tangent Equation»

Objectives: 1) to continue the formation of a system of mathematical knowledge and skills on the topic "Tangential Equation", necessary for application in practical activities, study of related disciplines, continuing education;

2) develop computer and multimedia skills curricula to organize their own cognitive activity;

3) develop logical thinking, algorithmic culture, critical thinking;

4) to cultivate tolerance, communication.

During the classes.

1. Organizing time.
2. Message topics, setting goals for the lesson.
3. Checking homework.

1. Tasks basic level(scanned work)
2. The students solved the task of practical content of an increased level of complexity by choice. One of the students presents his solution in the form of a multimedia project: “A parabolic bridge is being built connecting points A and B, the distance between which is 200 m. The entrance to the bridge and the exit from the bridge should be straight sections of the path, these sections are directed to the horizon at an angle 150. The indicated lines must be tangent to the parabola. Equate the bridge profile in the given coordinate system"

1. Updating of basic knowledge.

1. Differentiate functions:

• ()
• y=4()
• y=7x+4()
• y=tg x+ ()
• y=x 3 sinx()
• y=()

1. Answer the questions:

• What is the geometric meaning of the derivative?
• Can a tangent be drawn at any point on the graph? Which function is called differentiable at a point?
• The tangent is inclined at an obtuse angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function?
• The tangent is inclined at an acute angle to the positive direction of the x-axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function?
• The tangent is inclined at right angles to the positive direction of the OX axis. What can be said about the sign of the derivative and the nature of the monotonicity of the function?
• What should the graph of a function differentiable at a point look like?

1. What is the tangent equation? Explain that in this equation (x 0; f (x 0 )) , f ’ (x 0 ), (x; y)
2. Find the slope of the tangent to the curve y=2x 2 +x at the point with abscissa x 0 =-2 (-7).
3. Specify the value of the coefficient k at which the graphs of linear functions y = 8x+12 and y = kx – 3 are parallel. (eight)
4. The function y \u003d f (x) is defined on the interval (-7; 7). The figure below shows a graph of its derivative. Find the number of tangents to the graph of the function y \u003d f (x) that are parallel to the x-axis. (3)
5. The figure shows a straight line that is tangent to the graph of the function y \u003d p (x) at the point (x 0; p(x 0 )). Find the value of the derivative at the point x 0 . (-0,5)
6. All tangents parallel to the straight line y=2x+5 or coinciding with it were drawn to the graph of the function f(x). Specify the number of touch points. (four)

1. Independent work with selective checking (one student performs the task at the blackboard). Write the equations of tangents to the graph of a function f(x) \u003d 4 - x 2 at the points of its intersection with the x-axis. (y \u003d - + 4x + 8). Demonstration illustration.
2. Work in creative groups of 5-6 people.

1. Pass computer testing in turn (Additional testing for lesson 5, options 1 and 2 "Lessons of Cyril and Methodius Algebra"). The results are entered into the diagnostic card.
2. Complete tasks in notebooks:

1 group

y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b

No. 4. Write the equation of the tangent to the graph of the function f(x) = 0.5 x 2 -4 if the tangent forms an angle of 45 with the x-axis 0 .

2 group

No. 1. What is the geometric meaning of the derivative?

No. 2. What properties should a function have y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b ) did its graph have a tangent?

No. 3. What is the tangent equation?

№ 4. Write the equation of the tangent to the graph of the function f (x) \u003d x 3 /3 parallel to the line y \u003d 9 x - 7.

3 group

No. 1. What is the geometric meaning of the derivative?

No. 2. What properties should a function have y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b ) did its graph have a tangent?

No. 3. What is the tangent equation?

No. 4. The straight line passing through the origin touches the graph of the function
y \u003d f (x) at point A (-7; 14). Find . (Assignment from KIM to prepare for the exam)

4 group

No. 1. What is the geometric meaning of the derivative?

No. 2. What properties should a function have y = f(x ) defined on the interval ( a; b ) so that at the point with the abscissa x 0 Є (a; b ) did its graph have a tangent?

No. 3. What is the tangent equation?

No. 4. The line y=-4x-11 is tangent to the graph of the function f(x)=x 3+7x2 +7x-6. Find the abscissa of the point of contact. (Assignment from KIM to prepare for the exam)

A report on the work done is carried out at the blackboard by one of the group. It is chosen by the teacher or the group. The mark of the respondent and the self-assessment of each member of the group are entered in the diagnostic card.

1. Summing up the lesson. Reflection.
2. Homework consists of exercises B8 FBTZ FIPI.

Municipal budgetary educational institution

Glukhiv secondary school

Abstract open lesson in algebra

on the topic:

Derivative and its geometric meaning. Derivative in the exam "

teacher of mathematics and computer science

Dikalov Dmitry Gennadievich

2015

Lesson summary on the topic: Derivative and its geometric meaning

Lesson Objectives:

Tutorials:

• Repeat the basic concepts of the section "Derivative"
• To teach students how to quickly solve problems on the topic "Derivative" from the USE options

Developing:

• Development of cognitive interest, logical thinking, development of memory, mindfulness.
• educate interest in the structure of computer networks.

Educational:

• to cultivate a conscientious attitude to work, initiative;
• education of discipline and organization

Lesson type:

• lesson of repetition and consolidation of knowledge

Lesson structure:

• Organizing time;
• updating of basic knowledge
• problem solving
• homework

Equipment : presentation program Microsoft Office PowerPoint, presentation, computer, multimedia projector, interactive whiteboard.

Lesson plan:

1. Organizational moment (1 min)
2. Updating knowledge (5 min)
3. Problem solving (34 min)
4. Summing up the lesson (4 min)
5. Homework (1 min)

During the classes:

I. Organizational moment

The teacher greets, introduces the topic, objectives and course of the lesson.

II. Knowledge update

1. 1. What is the geometric meaning of the derivative?
2. How are the intervals of increasing (decreasing) functions?
3. What is the algorithm for finding extremum points?
4. How do stationary points differ from extremum points?

III. Problem solving.

Solving problems on finding the derivative at a point, finding intervals of increase and decrease, finding points at which the derivative \u003d 0, finding the largest and smallest values ​​of the function.

Students solve these tasks using an interactive whiteboard, each task is depicted on a separate slide.

Students discuss the nuances of solving problems as the slides move.

The following tasks are offered to students for independent solution.

IV. Summing up the lesson.

To summarize the lesson, 1-2 students are called to the board to solve problems from textbook No. 956 (1,2): find the intervals of increase and decrease of the function y \u003d 2x 3 +3x 2 -2

Student decision:

To find the intervals of increase and decrease of a function, let's find its derivative:

y`=6x 2 +6x

To find stationary points, we equate the derivative to 0 and solve this equation, we get the points x=0 and x=-1. Let's find the extremum points among these points. To do this, we determine the sign of the derivative on each of the three intervals. On the interval x0, the derivative is positive, which means that the function increases on these intervals. On the interval

1

The student writes down the answer.

V. Homework

No. 957, No. 956 (finish)

Grading students who actively showed themselves in the lesson.