Lesson objectives:
Educational: Repeat theoretical information on the topic “Application of derivatives”, generalize, consolidate and improve knowledge on this topic.
To teach how to apply the acquired theoretical knowledge when solving various types of mathematical problems.
Consider methods for solving USE tasks related to the concept of derivative of basic and advanced levels of complexity.
Educational:
Skills training: planning activities, working at an optimal pace, working in a group, summarizing.
Develop the ability to evaluate one’s abilities and the ability to communicate with friends.
Foster feelings of responsibility and empathy. Contribute to the development of the ability to work in a team; skills.. refers to the opinions of classmates.
Developmental: Be able to formulate the key concepts of the topic being studied. Develop group work skills.
Lesson type: combined:
Generalization, consolidation of skills, application of properties of elementary functions, application of already formed knowledge, skills and abilities, application of derivatives in non-standard situations.
Equipment: computer, projector, screen, handouts.
Lesson plan:
1. Organizational activities
Reflection of mood
2. Updating the student’s knowledge
3. Oral work
4. Independent work in groups
5. Protection of completed work
6. Independent work
7. Homework
8. Lesson summary
9. Reflection of mood
During the classes
1. Reflection of mood.
Guys, good morning. I came to your lesson with this mood (showing an image of the sun)!
What's your mood?
On your table there are cards with images of the sun, the sun behind a cloud and clouds. Show what mood you are in.
2. Analyzing the results of trial exams, as well as the results of the final certification of recent years, we can conclude that the tasks mathematical analysis,from Unified State Examination work no more than 30%-35% of graduates cope. And in our class, based on the results of training and diagnostic work, not all of them perform them correctly. This is the reason for our choice. We will practice the skill of using derivatives when solving USE problems.
In addition to the problems of final certification, questions and doubts arise about the extent to which the knowledge acquired in this area can and will be in demand in the future, and how justified is the investment of time and health in studying this topic.
Why is a derivative needed? Where do we meet and use derivative? Is it possible to do without it in mathematics and not only?
Student message 3 minutes -
3. Oral work.
4. Independent work in groups (3 groups)
Group 1 task
) What is geometric meaning derivative?
2) a) The figure shows a graph of the function y=f(x) and a tangent to this graph drawn at the point with the abscissa x0. Find the value of the derivative of the function f(x) at the point x0.
b) The figure shows a graph of the function y=f(x) and a tangent to this graph drawn at the point with the abscissa x0. Find the value of the derivative of the function f(x) at the point x0.
Group 1 answer:
1) The value of the derivative of the function at the point x=x0 is equal to the conditional coefficient of the tangent drawn to the graph of this function at the point with the abscissa x0. The zero coefficient is equal to the tangent of the angle of inclination of the tangent (or, in other words) the tangent of the angle formed by the tangent and... the direction of the Ox axis)
2) A)f1(x)=4/2=2
3) B)f1(x)=-4/2=-2
Group 2 task
1) What is the physical meaning of the derivative?
2) The material point moves rectilinearly according to the law
x(t)=-t2+8t-21, where x is the distance from the reference point in meters, t is the time in seconds, measured from the beginning of the movement. Find its speed (in meters per second) at time t=3 s.
3) The material point moves rectilinearly according to the law
x(t)= ½*t2-t-4, where x is the distance from the reference point in meters, t is the time in seconds, measured from the beginning of the movement. At what point in time (in seconds) was its speed equal to 6 m/s?
Group 2 answer:
1) The physical (mechanical) meaning of the derivative is as follows.
If S(t) is the law of rectilinear motion of a body, then the derivative expresses the instantaneous speed at time t:
V(t)=-x(t)=-2t=8=-2*3+8=2
3) X(t)=1/2t^2-t-4
Group 3 task
1) The straight line y= 3x-5 is parallel to the tangent to the graph of the function y=x2+2x-7. Find the abscissa of the tangent point.
2) The figure shows a graph of the function y=f(x), defined on the interval (-9;8). Determine the number of integer points on this interval at which the derivative of the function f(x) is positive.
Group 3 answer:
1) Since the straight line y=3x-5 is parallel to the tangent, then the angular coefficient of the tangent is equal to the angular coefficient of the straight line y=3x-5, i.e., k=3.
Y1(x)=3 ,y1=(x^2+2x-7)1=2x=2 2x+2=3
2) Integer points are points with integer abscissa values.
The derivative of a function f(x) is positive if the function is increasing.
Question: What can you say about the derivative of the function, which is described by the saying “The further into the forest, the more firewood”
Answer: The derivative is positive throughout the entire domain of definition, because this function increases monotonically
6. Independent work (6 options)
7. Homework.
Training work Answers:
Lesson summary.
“Music can elevate or pacify the soul, painting can delight the eye, poetry can awaken feelings, philosophy can satisfy the needs of the mind, engineering can improve the material side of people’s lives. But mathematics can achieve all these goals."
That's what he said American mathematician Maurice Kline.
Thanks for the work!
Sergey Nikiforov
If the derivative of a function is of constant sign on an interval, and the function itself is continuous on its boundaries, then the boundary points are added to both increasing and decreasing intervals, which fully corresponds to the definition of increasing and decreasing functions.
Farit Yamaev 26.10.2016 18:50
Hello. How (on what basis) can we say that at the point where the derivative is equal to zero, the function increases. Give reasons. Otherwise, it's just someone's whim. By what theorem? And also proof. Thank you.
Support
The value of the derivative at a point is not directly related to the increase in the function over the interval. Consider, for example, functions - they are all increasing on the interval
Vladlen Pisarev 02.11.2016 22:21
If a function is increasing on the interval (a;b) and is defined and continuous at points a and b, then it is increasing on the interval . Those. point x=2 is included in this interval.
Although, as a rule, increase and decrease are considered not on a segment, but on an interval.
But at the point x=2 itself, the function has a local minimum. And how to explain to children that when they are looking for points of increase (decrease), we do not count the points of local extremum, but enter into intervals of increase (decrease).
Considering that the first part of the Unified State Exam For " middle group kindergarten", then perhaps such nuances are too much.
Separately, many thanks to all the staff for “Solving the Unified State Exam” - an excellent guide.
Sergey Nikiforov
A simple explanation can be obtained if we start from the definition of an increasing/decreasing function. Let me remind you that it sounds like this: a function is called increasing/decreasing on an interval if a larger argument of the function corresponds to a larger/smaller value of the function. This definition does not use the concept of derivative in any way, so questions about the points where the derivative vanishes cannot arise.
Irina Ishmakova 20.11.2017 11:46
Good afternoon. Here in the comments I see beliefs that boundaries need to be included. Let's say I agree with this. But please look at your solution to problem 7089. There, when specifying increasing intervals, boundaries are not included. And this affects the answer. Those. the solutions to tasks 6429 and 7089 contradict each other. Please clarify this situation.
Alexander Ivanov
Tasks 6429 and 7089 have completely different questions.
One is about increasing intervals, and the other is about intervals with a positive derivative.
There is no contradiction.
The extrema are included in the intervals of increasing and decreasing, but the points in which the derivative is equal to zero are not included in the intervals in which the derivative is positive.
A Z 28.01.2019 19:09
Colleagues, there is a concept of increasing at a point
(see Fichtenholtz for example)
and your understanding of the increase at x=2 is contrary to the classical definition.
Increasing and decreasing is a process and I would like to adhere to this principle.
In any interval that contains the point x=2, the function is not increasing. Therefore inclusion given point x=2 is a special process.
Usually, to avoid confusion, inclusion of the ends of intervals is discussed separately.
Alexander Ivanov
A function y=f(x) is said to be increasing over a certain interval if a larger value of the argument from this interval corresponds to a larger value of the function.
At the point x=2 the function is differentiable, and on the interval (2; 6) the derivative is positive, which means on the interval .
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Solution
The graph shows that the derivative f"(x) of the function f(x) changes sign from plus to minus (at such points there will be a maximum) at exactly one point (between -5 and -4) from the interval [-6; -2 ] Therefore, on the interval [-6; -2] there is exactly one maximum point.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Condition
The figure shows a graph of the function y=f(x), defined on the interval (-2; 8). Determine the number of points at which the derivative of the function f(x) is equal to 0.
Solution
The equality of the derivative at a point to zero means that the tangent to the graph of the function drawn at this point is parallel to the Ox axis. Therefore, we find points at which the tangent to the graph of the function is parallel to the Ox axis. On this chart, such points are extremum points (maximum or minimum points). As you can see, there are 5 extremum points.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Condition
The straight line y=-3x+4 is parallel to the tangent to the graph of the function y=-x^2+5x-7. Find the abscissa of the tangent point.
Show solutionSolution
The angular coefficient of the straight line to the graph of the function y=-x^2+5x-7 at an arbitrary point x_0 is equal to y"(x_0). But y"=-2x+5, which means y"(x_0)=-2x_0+5. Angular the coefficient of the line y=-3x+4 specified in the condition is equal to -3. Parallel lines have the same slope coefficients. Therefore, we find a value x_0 such that =-2x_0 +5=-3.
We get: x_0 = 4.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Condition
The figure shows a graph of the function y=f(x) and points -6, -1, 1, 4 are marked on the abscissa. At which of these points is the derivative the smallest? Please indicate this point in your answer.
The derivative of a function is one of the difficult topics in the 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
First, try to find the domain of the function:
Did you manage? Let's compare the answers:
Is everything right? Well done!
Now let's try to find the range of values of the function:
Found? Let's compare:
Got it? Well done!
Let's work with graphs again, only now it's a little more complicated - find both the domain of definition of the function and the range of values of the function.
How to find both the domain and range of a function (advanced)
Here's what happened:
I think you've figured out the graphs. Now let’s try to find the domain of definition of a function in accordance with the formulas (if you don’t know how to do this, read the section about):
Did you manage? Let's check answers:
- , since the radical expression must be greater than or equal to zero.
- , since you cannot divide by zero and the radical expression cannot be negative.
- , since, respectively, for all.
- , since you cannot divide by zero.
However, we still have one more unanswered point...
I will repeat the definition once again and emphasize it:
Did you notice? The word “single” is a very, very important element of our definition. I'll try to explain it to you with my fingers.
Let's say we have a function defined by a straight line. . At, we substitute this value into our “rule” and get that. One value corresponds to one value. We can even make a table of the different values and graph this function to see for ourselves.
"Look! - you say, ““ occurs twice!” So maybe a parabola is not a function? No, it is!
The fact that “ ” appears twice is not a reason to accuse the parabola of ambiguity!
The fact is that, when calculating for, we received one game. And when calculating with, we received one game. So that's right, a parabola is a function. Look at the graph:
Got it? If not, here is a life example that is very far from mathematics!
Let's say we have a group of applicants who met while submitting documents, each of whom in a conversation told where he lives:
Agree, it is quite possible for several guys to live in one city, but it is impossible for one person to live in several cities at the same time. This is like a logical representation of our “parabola” - Several different X's correspond to the same game.
Now let's come up with an example where the dependency is not a function. Let’s say these same guys told us what specialties they applied for:
Here we have a completely different situation: one person can easily submit documents for one or several directions. That is one element sets are put into correspondence several elements multitudes. Respectively, this is not a function.
Let's test your knowledge in practice.
Determine from the pictures what is a function and what is not:
Got it? And here it is answers:
- The function is - B, E.
- The function is not - A, B, D, D.
You ask why? Yes, here's why:
In all pictures except IN) And E) there are several for one!
I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the range of permissible values of an argument and the range of definition of a function. Let's move on to the next section - how to set a function?
Methods for specifying a function
What do you think the words mean? "set function"? That's right, this means explaining to everyone what function we are talking about in this case. And explain it in such a way that everyone understands you correctly and the function graphs drawn by people based on your explanation are the same.
How can I do that? How to set a function? The simplest method, which has already been used more than once in this article, is using the formula. We write a formula, and by substituting a value into it, we calculate the value. And as you remember, a formula is a law, a rule by which it becomes clear to us and to another person how an X turns into a Y.
Usually, this is exactly what they do - in tasks we see ready-made functions specified by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” baffles. Let's understand everything in order, and let's start with the analytical method.
Analytical method of specifying a function
The analytical method is to specify a function using a formula. This is the most universal, comprehensive and unambiguous method. If you have a formula, then you know absolutely everything about a function - you can make a table of values from it, you can build a graph, determine where the function increases and where it decreases, in general, study it in full.
Let's consider the function. What's the difference?
"What does it mean?" - you ask. I'll explain now.
Let me remind you that in the notation the expression in brackets is called an argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (the expression in brackets) is, we will write it instead in the expression.
In our example it will look like this:
Let's consider another task related to the analytical method of specifying a function, which you will have on the exam.
Find the value of the expression at.
I'm sure that at first you were scared when you saw such an expression, but there is absolutely nothing scary about it!
Everything is the same as in the previous example: whatever the argument (the expression in brackets) is, we will write it instead in the expression. For example, for a function.
What needs to be done in our example? Instead you need to write, and instead -:
shorten the resulting expression:
That's all!
Independent work
Now try to find the meaning of the following expressions yourself:
- , If
- , If
Did you manage? Let's compare our answers: We are used to the fact that the function has the form
Even in our examples, we define the function in exactly this way, but analytically it is possible to specify the function in an implicit form, for example.
Try building this function yourself.
Did you manage?
This is how I built it.
What equation did we finally derive?
Right! Linear, which means that the graph will be a straight line. Let's make a table to determine which points belong to our line:
This is exactly what we were talking about... One corresponds to several.
Let's try to draw what happened:
Is what we got a function?
That's right, no! Why? Try to answer this question with the help of a drawing. What did you get?
“Because one value corresponds to several values!”
What conclusion can we draw from this?
That's right, a function cannot always be expressed explicitly, and what is “disguised” as a function is not always a function!
Tabular method of specifying a function
As the name suggests, this method is a simple sign. Yes Yes. Like the one you and I have already made. For example:
Here you immediately noticed a pattern - the Y is three times larger than the X. And now the task to “think very carefully”: do you think that a function given in the form of a table is equivalent to a function?
Let's not talk for a long time, but let's draw!
So. We draw the function specified by the wallpaper in the following ways:
Do you see the difference? It's not all about the marked points! Take a closer look:
Have you seen it now? When we define a function in a tabular way, we display on the graph only those points that we have in the table and the line (as in our case) passes only through them. When we define a function analytically, we can take any points, and our function is not limited to them. This is the peculiarity. Remember!
Graphical method of constructing a function
The graphical method of constructing a function is no less convenient. We draw our function, and another interested person can find what the y is equal to at a certain x and so on. Graphical and analytical methods are among the most common.
However, here you need to remember what we talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Do you remember? Just in case, I’ll copy here the definition of what a function is:
As a rule, people usually name exactly the three ways of specifying a function that we have discussed - analytical (using a formula), tabular and graphical, completely forgetting that a function can be described verbally. Like this? Yes, very simple!
Verbal description of the function
How to describe a function verbally? Let's take our recent example - . This function can be described as “every real value of x corresponds to its triple value.” That's all. Nothing complicated. You, of course, will object - “there are such complex functions that it is simply impossible to specify verbally!” Yes, there are such, but there are functions that are easier to describe verbally than to define with a formula. For example: “each natural value of x corresponds to the difference between the digits of which it consists, while the minuend is taken to be the largest digit contained in the notation of the number.” Now let's look at how our verbal description of the function is implemented in practice:
The largest digit in a given number is, respectively, the minuend, then:
Main types of functions
Now let's move on to the most interesting part - let's look at the main types of functions with which you have worked/are working and will work in the course of school and college mathematics, that is, let's get to know them, so to speak, and give them a brief description. Read more about each function in the corresponding section.
Linear function
A function of the form where, are real numbers.
The graph of this function is a straight line, so constructing a linear function comes down to finding the coordinates of two points.
The position of the straight line on the coordinate plane depends on the angular coefficient.
The scope of a function (aka the scope of valid argument values) is .
Range of values - .
Quadratic function
Function of the form, where
The graph of the function is a parabola; when the branches of the parabola are directed downwards, when the branches are directed upwards.
Many properties of a quadratic function depend on the value of the discriminant. The discriminant is calculated using the formula
The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:
Domain
The range of values depends on the extremum of the given function (vertex point of the parabola) and the coefficient (direction of the branches of the parabola)
Inverse proportionality
The function given by the formula, where
The number is called the coefficient of inverse proportionality. Depending on the value, the branches of the hyperbola are in different squares:
Domain - .
Range of values - .
SUMMARY AND BASIC FORMULAS
1. A function is a rule according to which each element of a set is associated with a single element of the set.
- - this is a formula denoting a function, that is, the dependence of one variable on another;
- - variable value, or argument;
- - dependent quantity - changes when the argument changes, that is, according to any specific formula reflecting the dependence of one quantity on another.
2. Valid argument values, or the domain of a function, is what is associated with the possibilities in which the function makes sense.
3. Function range- this is what values it takes, given acceptable values.
4. There are 4 ways to set a function:
- analytical (using formulas);
- tabular;
- graphic
- verbal description.
5. Main types of functions:
- : , where, are real numbers;
- : , Where;
- : , Where.