The smallest value is derivative. Derived function
The derivative function is one of the difficult topics in school Program. Not every graduate will answer the question of what is derived.
This article is simply clearly talking about what a derivative is and for what it needs. We will not strive to strive for the mathematical strictness of the presentation. The most important thing is to understand the meaning.
We remember the definition:
The derivative is the speed of change of function.
In the picture - graphics of three functions. What do you think is growing faster?
The answer is obvious - the third. She has the most high speed Changes, that is, the greatest derivative.
Here is another example.
Kostya, Grisha and Matvey simultaneously got a job. Let's see how their income changed during the year:
On the schedule immediately everything can be seen, is not it? The bone's income for half a year has grown more than twice. And Grisha revenue has also grown, but quite a little bit. And Matthew's income decreased to zero. Starting conditions are the same, and the speed of change of function, that is derivative- Different. As for Matthew - his income is negatively derived.
Intuitively, we are easily assessing the speed of change of function. But how do you do it?
In fact, we look at how cool the graph of the function goes up (or down). In other words, how quickly changes y with a change in x. Obviously, the same feature at different points may have miscellaneous The derivative is that there may be faster or slower.
The derivative function is indicated.
Show how to find using the graph.
A graph is drawn some function. Take a point with an abscissa on it. We draw at this point tangent to the graphics function. We want to evaluate how cool up a graph of a function. Comfortable value for this - tangent tilt angle.
The derivative of the function at the point is equal to the tangent of the tilt angle, carried out to the graph of the function at this point.
Please note - as an angle of tagging tangent, we take an angle between the tangent and positive direction of the axis.
Sometimes students ask what tangent to the function graphics. This is a straight line having the only one in this area. total point With a schedule, and as shown in our figure. Looks like a tangent to the circumference.
We will find. We remember that the tangent of acute angle in rectangular triangle It is equal to the attitude of the opposite catech to the adjacent one. From the triangle:
We found a derivative with the help of a graph, not even knowing the formula function. Such tasks are often found in the exam in mathematics at the number.
There is another important ratio. Recall that the direct is given by the equation
The value in this equation is called angular coefficient direct. It is equal to the tangent of the angle of inclination direct to the axis.
.
We get that
We remember this formula. It expresses the geometric meaning of the derivative.
The derivative of the function at the point is equal to the angular coefficient of tangent, carried out to the graph of the function at this point.
In other words, the derivative is equal to tangent tilt angle.
We have already said that the same function at different points may have a different derivative. Let's see how the derivative is associated with the behavior of the function.
Draw a graph of some function. Let this function be increasing on some sections, on others - decreases, with different speeds. And even if this feature there will be a point of maximum and a minimum.
At the point, the function increases. Tangent to the graph, carried out at the point, forms a sharp angle with a positive axis direction. So, at the point the derivative is positive.
At the point, our function decreases. Tanner at this point forms a stupid angle with a positive axis direction. Since the dull angle tangent is negative, a derivative is negative at the point.
That's what it turns out:
If the function increases, its derivative is positive.
If decreases, its derivative is negative.
And what will be at the points of the maximum and minimum? We see that at points (maximum point) and (minimum point) tangent horizontal. Consequently, the tangent tangent tilt angle at these points is zero, and the derivative is also zero.
Point is a maximum point. At this point, the increasing function is replaced by descending. Consequently, the sign of the derivative changes at a point with a "plus" to "minus".
At the point - the point of the minimum - the derivative is also zero, but its sign changes from "minus" to the "plus".
Conclusion: With the help of a derivative, you can learn about the behavior of the function all that interests us.
If the derivative is positive, then the function increases.
If the derivative is negative, the function decreases.
At the point of the maximum, the derivative is zero and changes the sign from the "plus" to "minus".
At the point of the minimum, the derivative is also zero and changes the sign from "minus" to the "plus".
We write these conclusions in the form of a table:
| increases | maximum point | decrease | point of minimum | increases | |
| + | 0 | - | 0 | + |
We will make two small clarifications. One of them will need you when solving the tasks of the USE. Other - in the first year, with a more serious study of functions and derivatives.
A case is possible when the derivative of the function at some point is zero, but no maximum, no minimum function at this point at this point. This is the so-called :
At the point tangent to the graphics of the horizontal, and the derivative is zero. However, the function of the function increased - and after the point continues to increase. The sign of the derivative does not change - it has been positive and remained.
It also happens that at the point of the maximum or minimum, the derivative does not exist. On the chart, it corresponds to a sharp breaking when the tangent is impossible at this point.
And how to find a derivative if the function is not specified by the schedule, but by the formula? In this case, applied
This section contains tasks of the EGE In mathematics on topics related to the study of functions and their derivatives.
IN demonstration options Ege 2020. years they can meet at number 14 for basic level and at number 7 For profile level.
Look carefully on these three graphics of functions.
Did you notice that these functions in a sense "relatives"?
For example, in those areas where the graph of the green function is located above zero, the red function increases. In those sites where the graph of the green function is below zero, the red function decreases.
Similar comments can be made regarding red and blue graphs.
You can also notice that zeros of the green function (points x.
\u003d -1 I. x.
\u003d 3) coincide with the points of the red chart extremums: when x.
\u003d -1 on red graph we see a local maximum, with h.
\u003d 3 On the red schedule is a local minimum.
It is easy to see that local maxima and minima of the blue graph are achieved at the same points where the red schedule passes through the value. y.
= 0.
You can take a few more conclusions about the peculiarities of the behavior of these graphs, because they are really connected with each other. Look at the formulas of the functions located under each of the graphs, and by calculations, make sure that each previous one is derived for subsequent and, accordingly, each next is one of the pre-educated previous functions.
φ
1 (x.
) = φ"
2 (x.
) φ
2 (x.
) = Φ
1 (x.
)
φ
2 (x.
) = φ"
3 (x.
)
φ
3 (x.
) = Φ
2 (x.
)
Recall that we know about the derivative:
Derived function y. = f.(x.) At point h. expresses the speed of change of function at the point x..
Physical sense derivative It is that the derivative expresses the rate of proceeding of the process described by the dependence y \u003d f (x).
Geometric meaning of the derivative It is that its value in the considered point is equal to the angular coefficient of tangential, conducted to the graph of the differentiable function at this point.
And now let the red graphics in the drawing are not. Suppose that both formulas are unknown to us. 
Can I ask you about something related to the behavior of the function φ
2 (x.
) if it is known that it is a derived function φ
3 (x.
) and primitive function φ
1 (x.
)?
Can. And you can give an accurate answer to many questions, because we know that the derivative is the characteristic of the function of change of change, so we can judge some of the behaviors of one of these functions, looking at the schedule of the other.
Before answering the following questions, scroll up the page up so that the top pattern containing the red schedule is hidden. When the answers are given, return it back to check the result. And only after that, see my decision.
Attention: To enhance the learning effect answers and solutions Loading separately for each task to serially press buttons on a yellow background. (When there are many tasks, the buttons may appear with a delay. If the buttons are not visible at all, check whether it is allowed in your browser Javascript.)1) Using the graph of the derivative φ" 2 (x. ) (In our case, this is a green schedule), define which of the 2 values \u200b\u200bof the function more φ 2 (-3) or φ 2 (−2)?
According to the graph of the derivative, it can be seen that it is strictly positive in the [-3; -2 section], it means that the function in this area is only increasing, so the value of the function in the left end x. \u003d -3 less than its value at the right end x. = −2.
Answer: φ 2 (−3) φ 2 (−2)
2) Using the primary graph Φ 2 (x. ) (In our case, this is a blue schedule), determine which of the 2 values \u200b\u200bof the function more φ 2 (-1) or φ 2 (4)?
According to the graphics, it is clear that the point x. \u003d -1 is in the area of \u200b\u200bincreasing, therefore the value of the corresponding derivative is positive. Point x. \u003d 4 is located on the decrease site and the value of the corresponding derivative negatively. Since the positive value is more negative, we conclude - the value of an unknown function, which is just a derivative, at point 4 less than at point -1.
Answer: φ 2 (−1) > φ 2 (4)
There can be a lot of such questions on missing graphics, which causes a large variety of tasks with a brief response, built according to the same scheme. Try to solve some of them.
Tasks for determining the characteristics derivative on the function graphics.
Picture 1.
Figure 2.
Task 1.
y. = f. (x. ), determined at the interval (-10,5; 19). Determine the number of integers in which the derivative function is positive.
The derivative function is positive in those areas where the function increases. Figure shows that these intervals (-10.5; -7,6), (-1; 8.2) and (15.7, 19). We list all the points within these intervals: "-10", "- 9", "-8", "0", "1", "2", "3", "4", "5", "6", "7", "8", "16", "17", "18". Total 15 points.
Answer: 15
Comments.
1. When the charts in the charts require the name "points", as a rule, we mean only the values \u200b\u200bof the argument x.
which are the abscissions of the corresponding points located on the chart. The ordents of these points are the values \u200b\u200bof the function, they are dependent and can be easily calculated if necessary.
2. When listing points, we did not take into account the edges of the intervals, since the function at these points does not increase and does not decrease, but "unfolds". The derivative at such points is not positive and not negative, it is zero, so they are called stationary points. In addition, we do not consider the boundaries of the definition area here, because the condition is said that this is the interval.
Task 2.
Figure 1 shows a graph graph y. = f. (x. ), determined at the interval (-10,5; 19). Determine the number of integers in which the derived function f " (x. ) Negative.
The derivative function is negative in those areas where the function decreases. Figure shows that these intervals (-7.6; -1) and (8.2; \u200b\u200b15,7). Whole points within these intervals: "-7", "- 6", "-5", "- 4", "-3", "- 2", "9", "10", "11", "12 "," 13 "," 14 "," 15 ". Total 13 points.
Answer: 13
See comments to the previous task.
To solve the following tasks, you need to recall another definition.
Maximum and minimum features are combined with a common name - points of extremum .
At these points, the derived function is either zero or does not exist ( required Extremum condition).
However, the necessary condition is a sign, but not a guarantee of the existence of an extremum function. A sufficient condition for extremum It is a change of sign of the derivative: if the derivative at the point changes the sign from "+" to "-", then this is the point of the maximum function; If the derivative at the point changes the sign from "-" on "+", then this is the point of a minimum function; If at the point the derivative function is zero, or does not exist, but the sign of the derivative during the transition through this point does not change to the opposite, then the specified point is not an extremma point of the function. This may be a bending point, a break point or a break point of a function of a function.
Task 3.
Figure 1 shows a graph graph y. = f. (x. ), determined at the interval (-10,5; 19). Find the number of points in which the function tangent to the function is parallel to the direct y. \u003d 6 or coincides with it.
Recall that the direct equation has the view y. = kX. + b. where k. - Tilt coefficient of this direct to the axis OX.. In our case k. \u003d 0, i.e. straight y. \u003d 6 not tilted, but parallel to the axis OX.. It means the desired tangents should also be parallel to the axis OX. And there must also have a tilt factor 0. This property of tangents possesses at the points of extremums of functions. Therefore, to answer the question you just need to count all the points of extremums on the chart. Here they are 4 - two points of maximum and two points of minimum.
Answer: 4
Task 4.
Functions y. = f. (x. ), determined on the interval (-11; 23). Find the amount of extremum points functions on the segment.
On the specified segment, we see 2 points of extremum. Maximum function is achieved at the point x.
1 \u003d 4, minimum at point x.
2 = 8.
x.
1 + x.
2 = 4 + 8 = 12.
Answer: 12
Task 5.
Figure 1 shows a graph graph y. = f. (x. ), determined at the interval (-10,5; 19). Find the number of points in which the derived function f " (x. ) Equal to 0.
The derivative function is zero at the extremum points, which are seen on the chart 4:
2 points of maximum and 2 points minimum.
Answer: 4
Tasks for determining the characteristics of the function on the graph of its derivative.
Picture 1.
Figure 2.
Task 6.
Figure 2 shows a graph f " (x. ) - derived function f. (x. ), determined on the interval (-11; 23). At what point is the segment [-6; 2] function f. (x. ) takes the greatest value.
On the specified section, the derivative was not positive, therefore the function did not increase. It declined or passed through stationary points. In this way, the greatest value The function reached on the left segment of the segment: x. = −6.
Answer: −6
Comment: According to the graph, the derivative shows that on the segment [-6; 2] it is zero three times: at points x. = −6, x. = −2, x. \u003d 2. But at the point x. \u003d -2 She did not change the sign, then at this point could not be extremum function. Most likely there was a point of inflection of the graph of the original function.
Task 7.
Figure 2 shows a graph f " (x. ) - derived function f. (x. ), determined on the interval (-11; 23). At what point of the segment, the function takes the smallest value.
On the segment, the derivative is strictly positive, therefore the function in this area has just increased. Thus, the smallest function reached on the left border of the segment: x. = 3.
Answer: 3
Task 8.
Figure 2 shows a graph f " (x. ) - derived function f. (x. ), determined on the interval (-11; 23). Find the number of features of the maximum function f. (x. ), belonging to the segment [-5; 10].
According to such a prerequisite Extremum maximum function may be At points where its derivative is zero. On a given segment, this dots: x. = −2, x. = 2, x. = 6, x. \u003d 10. But according to a sufficient condition, he for sureonly in those of them where the sign of the derivative changes with "+" to "-". On the graph of the derivative we see that only the point is from the listed points x. = 6.
Answer: 1
Task 9.
Figure 2 shows a graph f " (x. ) - derived function f. (x. ), determined on the interval (-11; 23). Find the number of extremum points features f. (x. ) belonging to the segment.
Extreme functions can be at those points where its derivative is 0. On a given segment of the graph, we see 5 such points: x. = 2, x. = 6, x. = 10, x. = 14, x. \u003d 18. But at the point x. \u003d 14 The derivative did not change the sign, therefore it must be excluded from consideration. Thus, 4 points remain.
Answer: 4
Task 10.
Figure 1 shows a graph f " (x. ) - derived function f. (x. ), determined at the interval (-10,5; 19). Find the rates of increasing function f. (x. ). In response, specify the length of the greatest of them.
Gaps of increasing function coincide with the gaps of positivity derivative. On the chart, we see them three - (-9; -7), (4; 12), (18; 19). The longest of them is the second. His length l. = 12 − 4 = 8.
Answer: 8
Task 11.
Figure 2 shows a graph f " (x. ) - derived function f. (x. ), determined on the interval (-11; 23). Find the number of points in which a function tangent f. (x. ) Parallel direct y. = −2x. − 11 or coincides with it.
The angular coefficient (it is the tangent of the angle of inclination) of the specified direct K \u003d -2. We are interested in parallel or coinciding tangents, i.e. Straight with the same slope. Based on the geometrical meaning of the derivative - angular coefficient of tangential in the considered point of the graph of the function, we translate points in which the derivative is equal to -2. Figure 2 of such points 9. It is convenient to count on the intersections of the graph and the coordinate grid line passing through the value of -2 on the axis Oy..
Answer: 9
As you can see, one and the same schedule you can ask a wide variety of questions about the behavior of the function and its derivative. Also, one question can be attributed to the graphs of different functions. Be careful when solving this task on the exam, and it will seem very easy to you. Other types of tasks of this task - on the geometric meaning of primitive - will be considered in another section.
Sergey Nikiforov
If the derivative of the function is adjusted on the interval, and the function itself is continuous on its borders, the boundary points are connected both to the increasing gaps and to the decrease gaps, which fully corresponds to the definition of increasing and decreasing functions.
Fritu Yamaev 26.10.2016 18:50
Hello. How (on what basis) it can be argued that at a point where the derivative is zero, the function increases. Give arguments. Otherwise, it's just someone's whim. What kind of theorem? As well as evidence. Thank you.
Support
The value of the derivative at the point is not directly attributed to the increase in the function at the interval. Consider, for example, functions - they all increase on the segment
Owned Pisarev 02.11.2016 22:21
If the function increases on the interval (A; b) and is defined and continuous at points A and B, it increases on the segment. Those. Point x \u003d 2 is included in this gap.
Although, as a rule, increasing and decrease is considered not on the segment, but on the interval.
But at the point x \u003d 2, the function has a local minimum. And how to explain to children that when they are looking for points of increasing (descending), then the points of local extremum do not consider, and in the gaps of increasing (descending).
Considering that the first part of the ege for " middle group kindergarten", then probably such nuances are busting.
Separately, thank you very much for the "solid ege" to all employees - excellent allowance.
Sergey Nikiforov
A simple explanation can be obtained if you repel from the definition of an increasing / decreasing function. Let me remind you that it sounds like this: the function is called increasing / decreasing in the interval if the greater argument of the function corresponds to a greater / less function value. Such a definition does not use the concept of a derivative, so there can be no questions about points where the derivative does not appear.
Irina Ishmakova 20.11.2017 11:46
Good day. Here in the comments, I see the belief that the boundaries need to include. Suppose I agree with this. But please look, your decision to task 7089. There, when specifying the gaps of increasing the boundary, do not turn on. And it affects the answer. Those. Decision of tasks 6429 and 7089 contradict each other. Please clarify this situation.
Alexander Ivanov
In tasks 6429 and 7089 completely different questions.
In one pro increase in increasing, and in a different interval with a positive derivative.
There is no contradiction.
Extremes are among the gaps of increasing and descending, but the points in which the derivative is zero are not included in the intervals on which the derivative is positive.
A Z. 28.01.2019 19:09
Colleagues, there is an increasing concept at point
(see Fihtendulz for example)
and your understanding of increasing at point X \u003d 2 is opposed to classical definition.
Ascending and decrease is the process and I would like to adhere to this principle.
In any interval, which contains the point x \u003d 2, the function is not increasing. Therefore, inclusion this point x \u003d 2 Process is special.
Usually, to avoid confusion about the inclusion of the end of the intervals, they say separately.
Alexander Ivanov
The function y \u003d f (x) is called increasing at some interval, if the greater value of the argument from this gap corresponds to the greater value of the function.
At point x \u003d 2, the function is differentiable, and on the interval (2; 6), the derivative is positive, it means on the interval)
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