Lecture: The concept of the derivative of a function, the geometric meaning of the derivative

The concept of the derivative of a function

Consider some function f(x), which will be continuous throughout the entire interval of consideration. On the interval under consideration, we choose the point x 0, as well as the value of the function at this point.

So, let's look at a graph on which we mark our point x 0, as well as the point (x 0 + ∆x). Recall that ∆x is the distance (difference) between two selected points.

It is also worth understanding that each x corresponds to its own value of the function y.

The difference between the values ​​of the function at the point x 0 and (x 0 + ∆x) is called the increment of this function: ∆y \u003d f (x 0 + ∆x) - f (x 0).

Let's pay attention to the additional information that is available on the chart - this is the secant, which is called KL, as well as the triangle that it forms with intervals KN and LN.

The angle at which the secant is located is called its angle of inclination and is denoted by α. It can be easily determined that the degree measure of the angle LKN is also equal to α.

And now let's recall the relations in a right triangle tgα = LN / KN = ∆у / ∆х.

That is, the tangent of the slope of the secant is equal to the ratio of the increment of the function to the increment of the argument.

At one time, the derivative is the limit of the ratio of the increment of the function to the increment of the argument on infinitesimal intervals.

The derivative determines the rate at which the function changes over a certain area.

The geometric meaning of the derivative

If you find the derivative of any function at a certain point, then you can determine the angle at which the tangent to the graph in a given current will be located relative to the OX axis. Pay attention to the graph - the angle of inclination of the tangent is denoted by the letter φ and is determined by the coefficient k in the straight line equation: y \u003d kx + b.

That is, we can conclude that the geometric meaning of the derivative is the tangent of the slope of the tangent at some point of the function.

When solving various problems of geometry, mechanics, physics and other branches of knowledge, it became necessary to use the same analytical process from a given function y=f(x) get a new function called derivative function(or simply derivative) of this function f(x) and are symbolized

The process by which a given function f(x) get a new function f"(x), called differentiation and it consists of the following three steps: 1) we give the argument x increment  x and determine the corresponding increment of the function  y = f(x+ x)-f(x); 2) make up the relation

3) counting x permanent, and  x0, we find
, which is denoted by f"(x), as if emphasizing that the resulting function depends only on the value x, at which we pass to the limit. Definition: Derivative y "=f" (x) given function y=f(x) given x is called the limit of the ratio of the increment of the function to the increment of the argument, provided that the increment of the argument tends to zero, if, of course, this limit exists, i.e. finite. Thus,
, or

Note that if for some value x, for example when x=a, relation
at  x0 does not tend to a finite limit, then in this case we say that the function f(x) at x=a(or at the point x=a) has no derivative or is not differentiable at a point x=a.

2. The geometric meaning of the derivative.

Consider the graph of the function y \u003d f (x), differentiable in the vicinity of the point x 0

f(x)

Let's consider an arbitrary straight line passing through the point of the graph of the function - the point A (x 0, f (x 0)) and intersecting the graph at some point B (x; f (x)). Such a straight line (AB) is called a secant. From ∆ABC: ​​AC = ∆x; BC \u003d ∆y; tgβ=∆y/∆x .

Since AC || Ox, then ALO = BAC = β (as corresponding in parallel). But ALO is the angle of inclination of the secant AB to the positive direction of the Ox axis. Hence, tgβ = k is the slope of the straight line AB.

Now we will decrease ∆x, i.e. ∆x→ 0. In this case, point B will approach point A according to the graph, and the secant AB will rotate. The limiting position of the secant AB at ∆x → 0 will be the straight line (a), called the tangent to the graph of the function y \u003d f (x) at point A.

If we pass to the limit as ∆х → 0 in the equality tgβ =∆y/∆x, then we get
or tg \u003d f "(x 0), since
-angle of inclination of the tangent to the positive direction of the Ox axis
, by definition of a derivative. But tg \u003d k is the slope of the tangent, which means that k \u003d tg \u003d f "(x 0).

So, the geometric meaning of the derivative is as follows:

Derivative of a function at a point x 0 equal to the slope of the tangent to the graph of the function drawn at the point with the abscissa x 0 .

3. Physical meaning of the derivative.

Consider the movement of a point along a straight line. Let the point coordinate at any time x(t) be given. It is known (from the course of physics) that the average speed over a period of time is equal to the ratio of the distance traveled during this period of time to the time, i.e.

Vav = ∆x/∆t. Let us pass to the limit in the last equality as ∆t → 0.

lim Vav (t) = (t 0) - instantaneous speed at time t 0, ∆t → 0.

and lim = ∆x/∆t = x "(t 0) (by the definition of a derivative).

So, (t) = x"(t).

The physical meaning of the derivative is as follows: the derivative of the functiony = f(x) at the pointx 0 is the rate of change of the functionf(x) at the pointx 0

The derivative is used in physics to find the speed from a known function of coordinates from time, acceleration from a known function of speed from time.

 (t) \u003d x "(t) - speed,

a(f) = "(t) - acceleration, or

If the law of motion of a material point along a circle is known, then it is possible to find the angular velocity and angular acceleration during rotational motion:

φ = φ(t) - change in angle with time,

ω \u003d φ "(t) - angular velocity,

ε = φ"(t) - angular acceleration, or ε = φ"(t).

If the distribution law for the mass of an inhomogeneous rod is known, then the linear density of the inhomogeneous rod can be found:

m \u003d m (x) - mass,

x  , l - rod length,

p \u003d m "(x) - linear density.

With the help of the derivative, problems from the theory of elasticity and harmonic vibrations are solved. Yes, according to Hooke's law

F = -kx, x – variable coordinate, k – coefficient of elasticity of the spring. Putting ω 2 \u003d k / m, we obtain the differential equation of the spring pendulum x "(t) + ω 2 x (t) \u003d 0,

where ω = √k/√m is the oscillation frequency (l/c), k is the spring rate (H/m).

An equation of the form y "+ ω 2 y \u003d 0 is called the equation of harmonic oscillations (mechanical, electrical, electromagnetic). The solution to such equations is the function

y = Asin(ωt + φ 0) or y = Acos(ωt + φ 0), where

A - oscillation amplitude, ω - cyclic frequency,

φ 0 - initial phase.

To find out the geometric value of the derivative, consider the graph of the function y = f(x). Take an arbitrary point M with coordinates (x, y) and a point N close to it (x + $\Delta $x, y + $\Delta $y). Let us draw the ordinates $\overline(M_(1) M)$ and $\overline(N_(1) N)$, and draw a line parallel to the OX axis from the point M.

The ratio $\frac(\Delta y)(\Delta x) $ is the tangent of the angle $\alpha $1 formed by the secant MN with the positive direction of the OX axis. As $\Delta $x tends to zero, point N will approach M, and the tangent MT to the curve at point M will become the limiting position of the secant MN. Thus, the derivative f`(x) is equal to the tangent of the angle $\alpha $ formed by the tangent to curve at the point M (x, y) with a positive direction to the OX axis - the slope of the tangent (Fig. 1).

Figure 1. Graph of a function

When calculating the values ​​using formulas (1), it is important not to make a mistake in the signs, because increment can be negative.

The point N lying on the curve can approach M from any side. So, if in Figure 1, the tangent is given the opposite direction, the angle $\alpha $ will change by $\pi $, which will significantly affect the tangent of the angle and, accordingly, the slope.

Conclusion

It follows that the existence of the derivative is connected with the existence of a tangent to the curve y = f(x), and the slope -- tg $\alpha $ = f`(x) is finite. Therefore, the tangent must not be parallel to the OY axis, otherwise $\alpha $ = $\pi $/2, and the tangent of the angle will be infinite.

At some points, a continuous curve may not have a tangent or have a tangent parallel to the OY axis (Fig. 2). Then the function cannot have a derivative in these values. There can be any number of such points on the function curve.

Figure 2. Exceptional points of the curve

Consider Figure 2. Let $\Delta $x tend to zero from negative or positive values:

\[\Delta x\to -0\begin(array)(cc) () & (\Delta x\to +0) \end(array)\]

If in this case relations (1) have a finite aisle, it is denoted as:

In the first case, the derivative on the left, in the second, the derivative on the right.

The existence of a limit speaks of the equivalence and equality of the left and right derivatives:

If the left and right derivatives are not equal, then at this point there are tangents that are not parallel to OY (point M1, Fig. 2). At points M2, M3, relations (1) tend to infinity.

For N points to the left of M2, $\Delta $x $

To the right of $M_2$, $\Delta $x $>$ 0, but the expression is also f(x + $\Delta $x) -- f(x) $

For point $M_3$ on the left $\Delta $x $$ 0 and f(x + $\Delta $x) -- f(x) $>$ 0, i.e. expressions (1) are both positive on the left and right and tend to +$\infty $ both when $\Delta $x approaches -0 and +0.

The case of the absence of a derivative at specific points of the line (x = c) is shown in Figure 3.

Figure 3. Absence of derivatives

Example 1

Figure 4 shows the graph of the function and the tangent to the graph at the point with the abscissa $x_0$. Find the value of the derivative of the function in the abscissa.

Decision. The derivative at a point is equal to the ratio of the increment of the function to the increment of the argument. Let's choose two points with integer coordinates on the tangent. Let, for example, these be points F (-3.2) and C (-2.4).

Synopsis of an open lesson by a teacher at Pedagogical College No. 4 of St. Petersburg

Martusevich Tatyana Olegovna

Date: 12/29/2014.

Topic: The geometric meaning of the derivative.

Lesson type: learning new material.

Teaching methods: visual, partly exploratory.

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 tangent equation and teach how to find it.

Educational tasks:

To achieve an understanding of the geometric meaning of the derivative; derivation of the tangent equation; learn how to solve basic problems;

to provide a repetition of the material on the topic "Definition of a derivative";

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

Development tasks:

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

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

Educational tasks:

to promote the education of interest in mathematics;

education of activity, mobility, ability to communicate.

Lesson type - a combined lesson using ICT.

Equipment – multimedia installation, presentationMicrosoftpowerpoint.

Lesson stage

Time

Teacher activity

Student activities

1. Organizational moment.

Message about the topic and purpose of the lesson.

Topic: The geometric meaning of the derivative.

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 tangent equation and teach how to find it.

Preparing students for work in the classroom.

Preparation for work in class.

Awareness of the topic and purpose of the lesson.

Note-taking.

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

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

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

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

Finding the derivative of these functions by 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 ratio of function increment to 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's questions.

Formulation of 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.

Create an outline.

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 point is the derivative positive?

Negative?

Equal to zero?

Learning to search for an algorithm for answers to the questions posed by the schedule.

Understanding and comprehending and applying new information to solve a problem.

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

Task condition message.

Recording a task condition.

Formulation of 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 of the figure. Discussion and verification of answers in pairs, formulating a question to the teacher in case of difficulty.

7. Working with new material: explanation.

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

A detailed explanation of the derivation of the equation of the tangent to the function graph at a point, using as a visual aid in the form of a multimedia presentation, answers to students' questions.

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

Sketching, drawing.

8. Working with new material: explanation.

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

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

Note-taking.

Task condition message.

Training in the application of acquired knowledge.

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

Recording a task condition.

Making assumptions about possible ways to solve the problem in the implementation of each item of the action plan. Problem solving together with the teacher.

Recording the solution of the problem and the answer.

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

Individual control. Advice and assistance to students as needed.

Verification and explanation of the solution using the presentation.

Application of acquired knowledge.

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

10. Homework.

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

№ 860 (2,4,6,8),

Homework message with comments.

Recording homework.

11. Summing up.

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

We learned what the geometric meaning of the derivative is.

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

Correction and clarification of the results of the lesson.

Enumeration of the results of the lesson.

12. Reflection.

1. Did you have a lesson: a) easy; b) usually; c) difficult.

a) learned (a) completely, I can apply;

b) learned (a), but find it difficult to apply;

c) didn't get it.

3. Multimedia presentation in the lesson:

a) helped the assimilation of the material; b) did not help the assimilation of the material;

c) interfered with the assimilation of the material.

Conducting reflection.

Before reading the information on the current page, we advise you to watch a video about the derivative and its geometric meaning

See also 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 way.

The derivative of the function y \u003d f (x) at the 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 \u003d f (x) at the point x0 and textbooks is denoted by the symbol f "(x0). Therefore, by definition

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

The derivative of the function y \u003d f (x) at x \u003d xo is equal to the slope of the tangent to the graph of this function at the point Mo (ho, f (xo)), i.e.

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

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

The normal to the 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 \u003d f (x) at the point Mo (xo, yo) will be written as follows:

The 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 is expressed using the derivative.

If the ratio dy/dx at 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, the function f(x) defined in some neighborhood of the point x0 has a derivative f'(x) if and only if the one-sided derivatives exist and are equal to each other.

Geometric interpretation of the derivative as the slope 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 called differentiable at that point. A function that has a derivative at every point of a given interval is called differentiable in this interval. If the interval is closed, then there are one-sided derivatives at its ends.

The operation of finding the derivative is called.