Abstract open lesson teacher of 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 properties 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 solutions 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),
Message homework with comments.
Recording homework.
11. Summing up.
We repeated the definition of the derivative; physical meaning 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.
We begin this article with an overview of the necessary definitions and concepts.
After this, let's move on to writing the equation of the tangent line and give detailed solutions the most typical examples and tasks.
In conclusion, we will focus on finding the equation of the tangent to second-order curves, that is, to the circle, ellipse, hyperbola and parabola.
Page navigation.
Definitions and concepts.
Definition.
Angle of straight line y=kx+b is the angle measured from the positive direction of the x-axis to the straight line y=kx+b in the positive direction (that is, counterclockwise).
In the figure, the positive direction of the x-axis is shown by a horizontal green arrow, the positive direction of the angle is shown by a green arc, the straight line is shown by a blue line, and the inclination angle of the straight line is shown by a red arc.
Definition.
Slope of a straight line y=kx+b is called the numerical coefficient k.
The slope of a straight line is equal to the tangent of the angle of inclination of the straight line, that is, .
Definition.
Direct AB drawn through two points on the graph of the function y=f(x) is called secant. In other words, secant is a straight line passing through two points on the graph of a function.
In the figure, the secant line AB is shown as a blue line, the graph of the function y=f(x) is shown as a black curve, and the angle of inclination of the secant line is shown as a red arc.
If we take into account that the angular coefficient of the straight line is equal to the tangent of the angle of inclination (this was discussed above), and the tangent of the angle in right triangle ABC is the ratio of the opposite side to the adjacent side (this is the definition of the tangent of an angle), then a series of equalities will be true for our secant 
, where are the abscissas of points A and B, 
- the corresponding function values.
That is, secant angle is determined by equality 
or 
, A secant equation written in the form 
or 
(if necessary, refer to the section).
A secant line divides the graph of a function into three parts: to the left of point A, from A to B and to the right of point B, although it may have more than two points in common with the graph of the function.
The figure below shows three actually different secants (points A and B are different), but they coincide and are given by one equation.
We have never come across any talk about a secant line for a straight line. But still, if we start from the definition, then the straight line and its secant line coincide.
In some cases, a secant may have an infinite number of intersection points with the graph of a function. For example, the secant defined by the equation y=0 has an infinite number of points in common with the sine wave.
Definition.
Tangent to the graph of the function y=f(x) at the point called a straight line passing through a point, with a segment of which the graph of a function practically merges for values of x arbitrarily close to .
Let us explain this definition with an example. Let us show that the straight line y = x+1 is tangent to the graph of the function at the point (1; 2). To do this, we will show graphs of these functions as we approach the point of tangency (1; 2). The graph of the function is shown in black, the tangent line is shown as a blue line, and the point of tangency is shown as a red dot.
Each subsequent drawing is an enlarged area of the previous one (these areas are highlighted with red squares).
It is clearly seen that near the point of tangency, the graph of the function practically merges with the tangent line y=x+1.
Now let's move on to the more meaningful definition of a tangent.
To do this, we will show what will happen to the secant AB if point B is infinitely closer to point A.
The figure below illustrates this process.
The secant AB (shown as a blue dotted line) will tend to take the position of the tangent to the straight line (shown as a blue solid line), the inclination angle of the secant (shown as a red dashed arc) will tend to the inclination angle of the tangent (shown as a red solid arc).
Definition.
Thus, tangent to the graph of the function y=f(x) at point A is the limiting position of the secant AB at .
Now we can move on to describing the geometric meaning of the derivative of a function at a point.
Geometric meaning of the derivative of a function at a point.
Let us consider the secant AB of the graph of the function y=f(x) such that points A and B have coordinates and respectively 
, where is the increment of the argument. Let us denote by the increment of the function. Let's mark everything on the drawing:
From right triangle ABC we have . Since, by definition, a tangent is the limiting position of a secant, then 
.
Let us recall the definition of the derivative of a function at a point: the derivative of a function y=f(x) at a point is the limit of the ratio of the increment of the function to the increment of the argument at , denoted 
.
Hence, 
, where is the slope of the tangent.
Thus, the existence of a derivative of the function y=f(x) at a point is equivalent to the existence of a tangent to the graph of the function y=f(x) at the point of tangency, and the slope of the tangent is equal to the value of the derivative at the point, that is .
We conclude: geometric meaning of the derivative of a function at a point consists in the existence of a tangent to the graph of the function at this point.
Equation of a tangent line.
To write the equation of any straight line on a plane, it is enough to know its angular coefficient and the point through which it passes. The tangent line passes through the point of tangency and its angular coefficient for the differentiable function is equal to the value of the derivative at the point. That is, from the point we can take all the data to write the equation of the tangent line.
Equation of the tangent to the graph of the function y = f(x) at a point looks like .
We assume that there is a finite value of the derivative, otherwise the tangent is straight or vertical (if 
And 
), or does not exist (if 
).
Depending on the angular coefficient, the tangent can be parallel to the abscissa axis (), parallel to the ordinate axis (in this case, the tangent equation will have the form), increase () or decrease ().
It's time to give a few examples for clarification.
Example.
Write an equation for the tangent to the graph of the function 
at point (-1;-3) and determine the angle of inclination.
Solution.
Function defined for everyone real numbers(if necessary, refer to the article). Since (-1;-3) is a point of tangency, then 
.
We find the derivative (for this, the material in the article differentiating a function, finding the derivative may be useful) and calculate its value at the point: 
Since the value of the derivative at the point of tangency is the slope of the tangent, and it is equal to the tangent of the angle of inclination, then 
.
Therefore, the angle of inclination of the tangent is equal to 
, and the equation of the tangent line has the form 
Graphic illustration.
The graph of the original function is shown in black, the tangent line is shown as a blue line, and the point of tangency is shown as a red dot. The picture on the right is a magnified view of the area indicated by the red dotted square in the picture on the left.
Example.
Find out whether there is a tangent to the graph of a function 
at point (1; 1), if yes, then compose its equation and determine its angle of inclination.
Solution.
The domain of a function is the entire set of real numbers.
Finding the derivative: 
When the derivative is not defined, but 
And 
therefore, at point (1;1) there is a vertical tangent, its equation is x = 1, and the angle of inclination is equal to .
Graphic illustration.
Example.
Find all points on the graph of the function at which:
a) the tangent does not exist; b) the tangent is parallel to the x-axis; c) the tangent is parallel to the line.
Solution.
As always, we start with the domain of definition of the function. In our example, the function is defined on the entire set of real numbers. Let's expand the modulus sign; to do this, consider two intervals and : 
Let's differentiate the function: 
At x=-2 derivative does not exist, since the one-sided limits at this point are not equal: 
Thus, having calculated the value of the function at x=-2, we can give the answer to point a): the tangent to the graph of the function does not exist at the point (-2;-2).
b) A tangent is parallel to the x-axis if its slope equal to zero(the tangent of the angle of inclination is zero). Because 
, then we need to find all values of x at which the derivative of the function vanishes. These values will be the abscissa of the tangent points at which the tangent is parallel to the Ox axis.
When we solve the equation 
, and when is the equation 
:
It remains to calculate the corresponding values of the function: 
That's why, 
- the required points of the function graph.
Graphic illustration.
The graph of the original function is depicted with a black line; red dots mark the found points at which the tangents are parallel to the abscissa axis.
c) If two straight lines on a plane are parallel, then their angular coefficients are equal (this is written in the article). Based on this statement, we need to find all points on the graph of the function at which the slope of the tangent is equal to eight-fifths. That is, we need to solve the equation. Thus, when we solve the equation 
, and when is the equation 
.
The discriminant of the first equation is negative, therefore it has no real roots: 
The second equation has two real roots: 
We find the corresponding function values: 
At points 
tangents to the graph of a function are parallel to the line.
Graphic illustration.
The graph of the function is shown with a black line, the red line shows the graph of the straight line, the blue lines show the tangents to the graph of the function at points .
For trigonometric functions due to their periodicity, there can be an infinite number of tangent lines that have the same slope (the same slope).
Example.
Write equations of all tangents to the graph of the function 
which are perpendicular to the line.
Solution.
To create an equation for a tangent to the graph of a function, we only need to know its slope and the coordinates of the point of tangency.
We find the angular coefficient of the tangents from: the product of the angular coefficients of perpendicular straight lines is equal to minus one, that is. Since, by condition, the angular coefficient of a perpendicular straight line is equal to , then 
.
Let's start finding the coordinates of the tangent points. First, let's find the abscissas, then calculate the corresponding values of the function - these will be the ordinates of the tangent points.
When describing the geometric meaning of the derivative of a function at a point, we noted that. From this equality we find the abscissa of the tangent points. 
We have arrived at a trigonometric equation. Please pay attention to it, since later we will use it when calculating the ordinates of the tangent points. We solve it (if you have any difficulties, please refer to the section solving trigonometric equations):
The abscissas of the tangent points have been found, let’s calculate the corresponding ordinates (here we use the equality that we asked you to pay attention to just above): 
Thus, all points of contact. Therefore, the required tangent equations have the form: 
Graphic illustration.
The figure of the black curve shows the graph of the original function on the segment [-10;10], the blue lines depict the tangent lines. It is clearly visible that they are perpendicular to the red line. Touch points are marked with red dots.
Tangent to a circle, ellipse, hyperbola, parabola.
Up to this point, we have been busy finding equations for tangents to graphs of single-valued functions of the form y = f(x) at various points. Canonical equations second order curves are not single-valued functions. But we can represent a circle, ellipse, hyperbola and parabola by a combination of two single-valued functions and after that we can compose tangent equations according to a well-known scheme.
Tangent to a circle.
Circle with center at a point 
and radius R is given by .
Let's write this equality as a union of two functions: 
Here the first function corresponds to the upper semicircle, the second - to the lower one.
Thus, in order to construct the equation of the tangent to the circle at a point belonging to the upper (or lower) semicircle, we find the equation of the tangent to the graph of the function (or) at the specified point.
It is easy to show that at points of a circle with coordinates 
And 
the tangents are parallel to the x-axis and are given by the equations and respectively (in the figure below they are shown as blue dots and blue straight lines), and at the points 
And 
- are parallel to the ordinate axis and have equations and, respectively (in the figure below they are marked with red dots and red lines).
Tangent to an ellipse.
Ellipse centered at a point with semi-axes a and b is given by the equation 
.
An ellipse, just like a circle, can be defined by combining two functions - the upper and lower half-ellipse: 
The tangents at the vertices of the ellipse are parallel to either the abscissa axis (shown as blue straight lines in the figure below) or the ordinate axis (shown as red straight lines in the figure below).
That is, the upper half-ellipse is given by the function 
, and the lower one - 
.
Now we can use the standard algorithm to construct an equation for a tangent to the graph of a function at a point.
First tangent at point: 
Second tangent at a point 
:
Graphic illustration.
Tangent to hyperbole.
Hyperbola centered at a point and peaks 
And 
is given by the equality 
(picture below left), and with vertices 
And 
- equality 
(picture below right).
As a combination of two functions, a hyperbola can be represented as
or 
.
At the vertices of the hyperbola, the tangents are parallel to the Oy axis for the first case and parallel to the Ox axis for the second.
Thus, to find the equation of the tangent to the hyperbola, we find out which function the point of tangency belongs to, and proceed in the usual way.
A logical question arises: how to determine which function a point belongs to. To answer it, we substitute the coordinates into each equation and see which of the equalities turns into an identity. Let's look at this with an example.
Example.
Write an equation for the tangent to the hyperbola 
at point .
Solution.
Let's write the hyperbola in the form of two functions: 
Let's find out which function the tangent point belongs to.
For the first function, therefore, the point does not belong to the graph of this function.
For the second function, therefore, the point belongs to the graph of this function.
Find the angular coefficient of the tangent: 
Thus, the tangent equation has the form .
Graphic illustration.
Tangent to a parabola.
To create an equation for a tangent to a parabola of the form 
at a point we use the standard scheme, and write the equation of the tangent as . The tangent to the graph of such a parabola at the vertex is parallel to the Ox axis.
Parabola 
First we define it by combining two functions. To do this, let's solve this equation for y: 
Now we find out which function the tangent point belongs to and proceed according to the standard scheme.
The tangent to the graph of such a parabola at the vertex is parallel to the Oy axis.
For the second function: 
Getting the touch point 
.
Thus, the equation of the desired tangent has 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 = xо is equal to the slope of the tangent to the graph of this function at the point Mo(xo, f(xо)), 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) - law rectilinear movement points, then x’ = f’(t) is the speed of this movement 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.
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 passage 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
The derivative of a function is one of the difficult topics in school curriculum. Not every graduate will answer the question of what a derivative is.
This article explains in a simple and clear way what a derivative is and why it is needed.. We will not now strive for mathematical rigor in the presentation. The most important thing is to understand the meaning.
Let's remember the definition:
The derivative is the rate of change of a function.
The figure shows graphs of three functions. Which one do you think is growing faster?
The answer is obvious - the third. It has the highest rate of change, that is, the largest derivative.
Here's another example.
Kostya, Grisha and Matvey got jobs at the same time. Let's see how their income changed during the year:
The graph shows everything at once, isn’t it? Kostya’s income more than doubled in six months. And Grisha’s income also increased, but just a little. And Matvey’s income decreased to zero. The starting conditions are the same, but the rate of change of the function, that is derivative, - different. As for Matvey, his income derivative is generally negative.
Intuitively, we easily estimate the rate of change of a function. But how do we do this?
What we're really looking at is how steeply the graph of a function goes up (or down). In other words, how quickly does y change as x changes? Obviously, the same function at different points can have different derivative values - that is, it can change faster or slower.
The derivative of a function is denoted .
We'll show you how to find it using a graph.
A graph of some function has been drawn. Let's take a point with an abscissa on it. Let us draw a tangent to the graph of the function at this point. We want to estimate how steeply the graph of a function goes up. A convenient value for this is tangent of the tangent angle.
The derivative of a function at a point is equal to the tangent of the tangent angle drawn to the graph of the function at this point.
Please note that as the angle of inclination of the tangent we take the angle between the tangent and the positive direction of the axis.
Sometimes students ask what a tangent to the graph of a function is. This is a straight line that has a single common point with the graph in this section, and as shown in our figure. It looks like a tangent to a circle.
Let's find it. We remember that the tangent of an acute angle in a right triangle is equal to the ratio of the opposite side to the adjacent side. From the triangle:
We found the derivative using a graph without even knowing the formula of the function. Such problems are often found in the Unified State Examination in mathematics under the number.
There is another important relationship. Recall that the straight line is given by the equation
The quantity in this equation is called slope of a straight line. It is equal to the tangent of the angle of inclination of the straight line to the axis.
.
We get that
Let's remember this formula. It expresses the geometric meaning of the derivative.
The derivative of a function at a point is equal to the slope of the tangent drawn to the graph of the function at that point.
In other words, the derivative is equal to the tangent of the tangent angle.
We have already said that the same function can have different derivatives at different points. Let's see how the derivative is related to the behavior of the function.
Let's draw a graph of some function. Let this function increase in some areas and decrease in others, and at different rates. And let this function have maximum and minimum points.
At a point the function increases. A tangent to the graph drawn at point forms an acute angle with the positive direction of the axis. This means that the derivative at the point is positive.
At the point our function decreases. The tangent at this point forms an obtuse angle with the positive direction of the axis. Since the tangent of an obtuse angle is negative, the derivative at the point is negative.
Here's what happens:
If a function is increasing, its derivative is positive.
If it decreases, its derivative is negative.
What will happen at the maximum and minimum points? We see that at the points (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the tangent at these points is zero, and the derivative is also zero.
Point - maximum point. At this point, the increase in the function is replaced by a decrease. Consequently, the sign of the derivative changes at the point from “plus” to “minus”.
At the point - the minimum point - the derivative is also zero, but its sign changes from “minus” to “plus”.
Conclusion: using the derivative we can find out everything that interests us about the behavior of a function.
If the derivative is positive, then the function increases.
If the derivative is negative, then the function decreases.
At the maximum point, the derivative is zero and changes sign from “plus” to “minus”.
At the minimum point, the derivative is also zero and changes sign from “minus” to “plus”.
Let's write these conclusions in the form of a table:
| increases | maximum point | decreases | minimum point | increases | |
| + | 0 | - | 0 | + |
Let's make two small clarifications. You will need one of them when solving USE problems. Another - in the first year, with a more serious study of functions and derivatives.
It is possible that the derivative of a function at some point is equal to zero, but the function has neither a maximum nor a minimum at this point. This is the so-called :
At a point, the tangent to the graph is horizontal and the derivative is zero. However, before the point the function increased - and after the point it continues to increase. The sign of the derivative does not change - it remains positive as it was.
It also happens that at the point of maximum or minimum the derivative does not exist. On the graph, this corresponds to a sharp break, when it is impossible to draw a tangent at a given point.
How to find the derivative if the function is given not by a graph, but by a formula? In this case it applies