How to find the coordinates of the vector ab. Formula for determining vector coordinates for spatial problems

Scientific adviser:

1. Introduction 3

2. Historical sketch 4

3. “Place” of ODZ when solving equations and inequalities 5-6

4. Features and dangers of ODZ 7

5. ODZ – there is a solution 8-9

6. Finding ODZ is extra work. Equivalence of transitions 10-14

7. ODZ in the Unified State Exam 15-16

8. Conclusion 17

9. Literature 18

1. Introduction

Problem: equations and inequalities in which it is necessary to find ODZ did not find a place in the algebra course for systematic presentation, which is probably why my peers and I often make mistakes when solving such examples, spending a lot of time solving them, while forgetting about ODZ.

Target: be able to analyze the situation and draw logically correct conclusions in examples where it is necessary to take into account DL.

Tasks:

1. Study theoretical material;

2. Solve many equations, inequalities: a) fractional-rational; b) irrational; c) logarithmic; d) containing inverse trigonometric functions;

3. Apply the studied materials in a situation that differs from the standard one;

4. Create a work on the topic “Area of acceptable values: theory and practice”

Project work: I started working on the project by repeating the functions I knew. The scope of many of them is limited.

ODZ occurs:

1. When deciding fractional rational equations and inequalities

2. When deciding irrational equations and inequalities

3. When deciding logarithmic equations and inequalities

4. When solving equations and inequalities containing inverse trigonometric functions

Having solved many examples from various sources(Unified State Exam manuals, textbooks, reference books), I systematized the solution of examples according to the following principles:

· you can solve the example and take into account the ODZ (the most common method)

· it is possible to solve the example without taking into account the ODZ

· it is only possible to come to the right decision by taking into account the ODZ.

Methods used in the work: 1) analysis; 2) statistical analysis; 3) deduction; 4) classification; 5) forecasting.

Studied the analysis Unified State Exam results over the past years. Many mistakes were made in examples in which it is necessary to take into account DL. This once again emphasizes relevance my topic.

2. Historical sketch

Like other concepts of mathematics, the concept of a function did not develop immediately, but passed long haul development. P. Fermat’s work “Introduction and Study of Flat and Solid Places” (1636, published 1679) says: “Whenever in final equation There are two unknown quantities, there is a place.” Essentially what we are talking about here is functional dependence and her graphic representation(“place” in Fermat means line). The study of lines according to their equations in R. Descartes’ “Geometry” (1637) also indicates a clear understanding of the mutual dependence of the two variables. In I. Barrow (“Lectures on Geometry”, 1670) in geometric shape the mutual inverse nature of the actions of differentiation and integration is established (of course, without using these terms themselves). This already indicates a completely clear mastery of the concept of function. In geometric and mechanical form We also find this concept in I. Newton. However, the term “function” first appears only in 1692 with G. Leibniz and, moreover, not quite in its modern understanding. G. Leibniz calls various segments associated with a curve (for example, the abscissa of its points) a function. In the first printed course, “Analysis of infinitesimals for the knowledge of curved lines” by L'Hopital (1696), the term “function” is not used.

The first definition of a function in a sense close to the modern one is found in I. Bernoulli (1718): “A function is a quantity composed of a variable and a constant.” This not entirely clear definition is based on the idea of specifying a function by an analytical formula. The same idea appears in the definition of L. Euler, given by him in “Introduction to the Analysis of Infinites” (1748): “The function of a variable quantity is an analytical expression composed in some way from this variable quantity and numbers or constant quantities.” However, L. Euler is no longer alien to modern understanding function, which does not connect the concept of a function with any analytical expression of it. In his " Differential calculus” (1755) says: “When some quantities depend on others in such a way that when the latter change they themselves are subject to change, then the first are called functions of the second.”

WITH early XIX centuries, more and more often they define the concept of a function without mentioning its analytical representation. In "Treatise on Differential and integral calculus"(1797-1802) S. Lacroix says: “Every quantity whose value depends on one or many other quantities is called a function of these latter.” IN " Analytical theory heat" by J. Fourier (1822) there is a phrase: "Function f(x) denotes a completely arbitrary function, that is, a sequence of given values, subordinate or not general law and corresponding to all values x contained between 0 and some value x" The definition of N. I. Lobachevsky is close to modern: “... General concept function requires that the function from x name the number that is given for each x and together with x gradually changes. The function value can be given or analytical expression, or a condition that provides a means of testing all numbers and choosing one of them, or, finally, a dependence may exist and remain unknown.” It is also said there a little lower: “The broad view of the theory allows for the existence of dependence only in the sense that numbers one with another in connection are understood as if given together.” Thus, modern definition functions, free from references to analytical task, usually attributed to P. Dirichlet (1837), was repeatedly proposed before him.

The domain of definition (admissible values) of a function y is the set of values of the independent variable x for which this function is defined, i.e., the domain of change of the independent variable (argument).

3. “Place” of the range of acceptable values when solving equations and inequalities

1. When solving fractional rational equations and inequalities the denominator must not be zero.

2. Solving irrational equations and inequalities.

2.1..gif" width="212" height="51"> .

IN in this case there is no need to find the ODZ: from the first equation it follows that the obtained values of x satisfy the following inequality: https://pandia.ru/text/78/083/images/image004_33.gif" width="107" height="27 src=" > is the system:

Since they enter into the equation equally, then instead of inequality, you can include inequality https://pandia.ru/text/78/083/images/image009_18.gif" width="220" height="49">

https://pandia.ru/text/78/083/images/image014_11.gif" width="239" height="51">

3. Solving logarithmic equations and inequalities.

3.1. Scheme for solving a logarithmic equation

But it is enough to check only one condition of the ODZ.

3.2..gif" width="115" height="48 src=">.gif" width="115" height="48 src=">

4. Trigonometric equations kind are equivalent to the system (instead of inequality, you can include inequality in the system https://pandia.ru/text/78/083/images/image024_5.gif" width="377" height="23"> are equivalent to the equation

4. Features and dangers of the range of permissible values

In mathematics lessons we are required to finding ODZ in each example. At the same time mathematical essence In this case, finding the ODZ is not at all mandatory, often not necessary, and sometimes impossible - and all this without any damage to the solution of the example. On the other hand, it often happens that after solving an example, schoolchildren forget to take into account the DL, write it down as the final answer, and take into account only some conditions. This circumstance is well known, but the “war” continues every year and, it seems, will continue for a long time.

Consider, for example, the following inequality:

Here, the ODZ is sought and the inequality is solved. However, when solving this inequality, schoolchildren sometimes believe that it is quite possible to do without searching for ODZ, or more precisely, it is possible to do without the condition

In fact, to obtain the correct answer it is necessary to take into account both the inequality , and .

But, for example, the solution to the equation: https://pandia.ru/text/78/083/images/image032_4.gif" width="79 height=75" height="75">

which is equivalent to working with ODZ. However, in this example, such work is unnecessary - it is enough to check the fulfillment of only two of these inequalities, and any two.

Let me remind you that any equation (inequality) can be reduced to the form . ODZ is simply the domain of definition of the function on the left side. The fact that this area must be monitored follows from the definition of the root as a number from the domain of definition of a given function, thereby from the ODZ. Here is a funny example on this topic..gif" width="20" height="21 src="> has a domain of definition of a set of positive numbers (this, of course, is an agreement to consider a function with, but reasonable), and then -1 is not is the root.

5. Range of acceptable values – there is a solution

And finally, in a lot of examples, finding the ODZ allows you to get the answer without bulky layouts, or even verbally.

1. OD3 is empty set, which means the original example has no solutions.

1) 2) 3)

2. B ODZ one or more numbers are found, and a simple substitution quickly determines the roots.

1) , x=3

2)Here in the ODZ there is only the number 1, and after substitution it is clear that it is not a root.

3) There are two numbers in the ODZ: 2 and 3, and both are suitable.

4) > In the ODZ there are two numbers 0 and 1, and only 1 is suitable.

ODZ can be used effectively in combination with analysis of the expression itself.

5) < ОДЗ: Но в правой части неравенства могут быть только positive numbers, so we leave x=2. Then we substitute 2 into the inequality.

6) From the ODZ it follows that, where we have ..gif" width="143" height="24"> From the ODZ we have: . But then and . Since, there are no solutions.

From the ODZ we have: https://pandia.ru/text/78/083/images/image060_0.gif" width="48" height="24">>, which means . Solving the last inequality, we get x<- 4, что не входит в ОДЗ. Поэтому решения нет.

3) ODZ: . Since then

On the other hand, https://pandia.ru/text/78/083/images/image068_0.gif" width="160" height="24">

ODZ:. Consider the equation on the interval [-1; 0).

It fulfills the following inequalities https://pandia.ru/text/78/083/images/image071_0.gif" width="68" height="24 src=">.gif" width="123" height="24 src="> and there are no solutions. With the function and https://pandia.ru/text/78/083/images/image076_0.gif" width="179" height="25">. ODZ: x>2..gif" width="233" height ="45 src="> Let's find the ODZ:

An integer solution is only possible for x=3 and x=5. By checking we find that the root x=3 does not fit, which means the answer is x=5.

6. Finding the range of acceptable values is extra work. Equivalence of transitions.

You can give examples where the situation is clear even without finding DZ.

Equality is impossible, because when subtracting a larger expression from a smaller one, the result must be a negative number.

2. .

The sum of two non-negative functions cannot be negative.

I will also give examples where finding ODZ is difficult, and sometimes simply impossible.

And finally, searches for ODZ are very often just extra work, which you can do without, thereby proving your understanding of what is happening. There are a huge number of examples that can be given here, so I will choose only the most typical ones. The main solution method in this case is equivalent transformations when moving from one equation (inequality, system) to another.

1.. ODZ is not needed, because, having found those values of x for which x2 = 1, we cannot obtain x = 0.

2. . ODZ is not needed, because we find out when the radical expression is equal to a positive number.

3. . ODZ is not needed for the same reasons as in the previous example.

ODZ is not needed, because the radical expression is equal to the square of some function, and therefore cannot be negative.

6. ..gif" width="271" height="51"> To solve, only one restriction for the radical expression is sufficient. In fact, from the written mixed system it follows that the other radical expression is non-negative.

8. DZ is not needed for the same reasons as in the previous example.

9. ODZ is not needed, since it is enough for two of the three expressions under the logarithm signs to be positive to ensure the positivity of the third.

10. .gif" width="357" height="51"> ODZ is not needed for the same reasons as in the previous example.

It is worth noting, however, that when solving using the method of equivalent transformations, knowledge of the ODZ (and properties of functions) helps.

Here are some examples.

1. . OD3, which implies that the expression on the right side is positive, and it is possible to write an equation equivalent to this one in this form https://pandia.ru/text/78/083/images/image101_0.gif" width="112" height="27 "> ODZ: But then, and when solving this inequality, it is not necessary to consider the case when the right side is less than 0.

3. . From the ODZ it follows that, and therefore the case when https://pandia.ru/text/78/083/images/image106_0.gif" width="303" height="48"> Go to general view looks like that:

https://pandia.ru/text/78/083/images/image108_0.gif" width="303" height="24">

There are two possible cases: 0 >1.

This means that the original inequality is equivalent to the following set of systems of inequalities:

The first system has no solutions, but from the second we obtain: x<-1 – решение неравенства.

Understanding the conditions of equivalence requires knowledge of some subtleties. For example, why are the following equations equivalent:

And finally, perhaps most importantly. The fact is that equivalence guarantees the correctness of the answer if some transformations of the equation itself are made, but is not used for transformations in only one of the parts. Abbreviations and the use of different formulas in one of the parts are not covered by the equivalence theorems. I have already given some examples of this type. Let's look at some more examples.

1. This decision is natural. On the left side by property logarithmic function let's move on to the expression ..gif" width="111" height="48">

Having solved this system, we get the result (-2 and 2), which, however, is not an answer, since the number -2 is not included in the ODZ. So, do we need to establish ODS? Of course not. But since we used a certain property of the logarithmic function in the solution, then we are obliged to provide the conditions under which it is satisfied. Such a condition is the positivity of expressions under the logarithm sign..gif" width="65" height="48">.

2. ..gif" width="143" height="27 src="> numbers are subject to substitution in this way . Who wants to do such tedious calculations?.gif" width="12" height="23 src="> add a condition, and you can immediately see that only the number https://pandia.ru/text/78/083/ meets this condition images/image128_0.gif" width="117" height="27 src=">) was demonstrated by 52% of test takers. One of the reasons for such low rates is the fact that many graduates did not select the roots obtained from the equation after squaring it.

3) Consider, for example, the solution to one of the problems C1: “Find all values of x for which the points of the graph of the function lie above corresponding points graph of the function ". The task boils down to solving fractional inequality containing logarithmic expression. We know the methods for solving such inequalities. The most common of them is the interval method. However, when using it, test takers make various mistakes. Let's look at the most common mistakes using inequality as an example:

X< 10. Они отмечают, что в первом случае решений нет, а во втором – корнями являются числа –1 и . При этом выпускники не учитывают условие x < 10.

8. Conclusion

To summarize, we can say that there is no universal method for solving equations and inequalities. Every time, if you want to understand what you are doing and not act mechanically, a dilemma arises: what solution should you choose, in particular, should you look for ODZ or not? I think that the experience I have gained will help me solve this dilemma. I will stop making mistakes by learning how to use ODZ correctly. Whether I can do this, time, or rather the Unified State Examination, will tell.

9. Literature

And others. “Algebra and the beginnings of analysis 10-11” problem book and textbook, M.: “Prosveshchenie”, 2002. “Handbook for elementary mathematics" M.: “Nauka”, 1966. Newspaper “Mathematics” No. 46, Newspaper “Mathematics” No. Newspaper “Mathematics” No. “History of mathematics in school grades VII-VIII”. M.: “Enlightenment”, 1982. etc. “The most complete edition of options real tasks Unified State Exam: 2009/FIPI" - M.: "Astrel", 2009. etc. "Unified State Exam. Mathematics. Universal materials for preparing students/FIPI" - M.: "Intelligence Center", 2009. etc. "Algebra and the beginnings of analysis 10-11." M.: “Enlightenment”, 2007. , “Workshop on solving problems school mathematics(algebra workshop).” M.: Education, 1976. “25,000 mathematics lessons.” M.: “Enlightenment”, 1993. “Preparing for the Olympiads in mathematics.” M.: “Exam”, 2006. “Encyclopedia for children “MATHEMATICS”” volume 11, M.: Avanta +; 2002. Materials from the sites www. *****, www. *****.

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Fractional equations. ODZ.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

We continue to master the equations. We already know how to work with linear and quadratic equations. The last view left - fractional equations . Or they are also called much more respectably - fractional rational equations . It is the same.

Fractional equations.

As the name implies, these equations necessarily contain fractions. But not just fractions, but fractions that have unknown in denominator. At least in one. For example:

Let me remind you that if the denominators are only numbers, these are linear equations.

How to decide fractional equations? First of all, get rid of fractions! After this, the equation most often turns into linear or quadratic. And then we know what to do... In some cases it can turn into an identity, such as 5=5 or an incorrect expression, such as 7=2. But this rarely happens. I will mention this below.

But how to get rid of fractions!? Very simple. Applying the same identical transformations.

We need to multiply the entire equation by the same expression. So that all denominators are reduced! Everything will immediately become easier. Let me explain with an example. Let us need to solve the equation:

As taught in junior classes? We move everything to one side, bring it to a common denominator, etc. Forget how horrible dream! This is what you need to do when you add or subtract fractions. Or you work with inequalities. And in equations, we immediately multiply both sides by an expression that will give us the opportunity to reduce all denominators (i.e., in essence, by common denominator). And what is this expression?

On the left side, reducing the denominator requires multiplying by x+2. And on the right, multiplication by 2 is required. This means that the equation must be multiplied by 2(x+2). Multiply:

This ordinary multiplication fractions, but I’ll write it down in detail:

Please note that I am not opening the bracket yet (x + 2)! So, in its entirety, I write it:

On the left side it contracts entirely (x+2), and on the right 2. Which is what was required! After reduction we get linear the equation:

And everyone can solve this equation! x = 2.

Let's solve another example, a little more complicated:

If we remember that 3 = 3/1, and 2x = 2x/ 1, we can write:

And again we get rid of what we don’t really like - fractions.

We see that to reduce the denominator with X, we need to multiply the fraction by (x – 2). And a few are not a hindrance to us. Well, let's multiply. All left side And all right side:

Parentheses again (x – 2) I'm not revealing. I work with the bracket as a whole as if it were one number! This must always be done, otherwise nothing will be reduced.

With a feeling of deep satisfaction we reduce (x – 2) and we get an equation without any fractions, with a ruler!

Now let’s open the brackets:

We bring similar ones, move everything to the left side and get:

But before that we will learn to solve other problems. On interest. That's a rake, by the way!

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

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In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.

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Any expression with a variable has its own range of valid values, where it exists. ODZ must always be taken into account when making decisions. If it is absent, you may get an incorrect result.

This article will show how to correctly find ODZ and use examples. The importance of indicating the DZ when making a decision will also be discussed.

Yandex.RTB R-A-339285-1

Valid and invalid variable values

This definition is related to the allowed values of the variable. When we introduce the definition, let's see what result it will lead to.

Starting from grade 7, we begin to work with numbers and numerical expressions. Initial definitions with variables jumps to the meaning of expressions with the selected variables.

When there are expressions with selected variables, some of them may not satisfy. For example, an expression of the form 1: a, if a = 0, then it does not make sense, since it is impossible to divide by zero. That is, the expression must have values that are suitable in any case and will give an answer. In other words, they make sense with the existing variables.

Definition 1

If there is an expression with variables, then it makes sense only if the value can be calculated by substituting them.

Definition 2

If there is an expression with variables, then it does not make sense when, when substituting them, the value cannot be calculated.

That is, this implies a complete definition

Definition 3

Existing admissible variables are those values for which the expression makes sense. And if it doesn’t make sense, then they are considered unacceptable.

To clarify the above: if there is more than one variable, then there may be a pair of suitable values.

Example 1

For example, consider an expression of the form 1 x - y + z, where there are three variables. Otherwise, you can write it as x = 0, y = 1, z = 2, while another entry has the form (0, 1, 2). These values are called valid, which means that the value of the expression can be found. We get that 1 0 - 1 + 2 = 1 1 = 1. From this we see that (1, 1, 2) are unacceptable. The substitution results in division by zero, that is, 1 1 - 2 + 1 = 1 0.

What is ODZ?

Range of acceptable values – important element when calculating algebraic expressions. Therefore, it is worth paying attention to this when making calculations.

Definition 4

ODZ area is the set of values allowed for a given expression.

Let's look at an example expression.

Example 2

If we have an expression of the form 5 z - 3, then the ODZ has the form (− ∞, 3) ∪ (3, + ∞) . This is the range of valid values that satisfies the variable z for a given expression.

If there are expressions of the form z x - y, then it is clear that x ≠ y, z takes any value. This is called ODZ expressions. It must be taken into account so as not to obtain division by zero when substituting.

The range of permissible values and the range of definition have the same meaning. Only the second of them is used for expressions, and the first is used for equations or inequalities. With the help of DL, the expression or inequality makes sense. The domain of definition of the function coincides with the range of permissible values of the variable x for the expression f (x).

How to find ODZ? Examples, solutions

Finding ODZ means finding everything valid values, suitable for given function or inequality. Failure to meet these conditions may result in incorrect results. To find the ODZ, it is often necessary to go through transformations in a given expression.

There are expressions where their calculation is impossible:

if there is division by zero;
taking the root of a negative number;
the presence of a negative integer indicator – only for positive numbers;
calculating the logarithm of a negative number;
domain of definition of tangent π 2 + π · k, k ∈ Z and cotangent π · k, k ∈ Z;
finding the value of the arcsine and arccosine of a number for a value not belonging to [ - 1 ; 1 ] .

All this shows how important it is to have ODZ.

Example 3

Find the ODZ expression x 3 + 2 x y − 4 .

Solution

Any number can be cubed. This expression does not have a fraction, so the values of x and y can be anything. That is, ODZ is any number.

Answer: x and y – any values.

Example 4

Find the ODZ of the expression 1 3 - x + 1 0.

Solution

It can be seen that there is one fraction where the denominator is zero. This means that for any value of x we will get division by zero. This means that we can conclude that this expression is considered undefined, that is, it does not have any additional liability.

Answer: ∅ .

Example 5

Find the ODZ of the given expression x + 2 · y + 3 - 5 · x.

Solution

Availability square root indicates that this expression must be greater than or equal to zero. At negative value it doesn't make sense. This means that it is necessary to write an inequality of the form x + 2 · y + 3 ≥ 0. That is, this is the desired range of acceptable values.

Answer: set of x and y, where x + 2 y + 3 ≥ 0.

Example 6

Determine the ODZ expression of the form 1 x + 1 - 1 + log x + 8 (x 2 + 3) .

Solution

By condition, we have a fraction, so its denominator should not be equal to zero. We get that x + 1 - 1 ≠ 0. The radical expression always makes sense when greater than or equal to zero, that is, x + 1 ≥ 0. Since it has a logarithm, its expression must be strictly positive, that is, x 2 + 3 > 0. The base of the logarithm must also have positive value and different from 1, then we add the conditions x + 8 > 0 and x + 8 ≠ 1. It follows that the desired ODZ will take the form:

x + 1 - 1 ≠ 0, x + 1 ≥ 0, x 2 + 3 > 0, x + 8 > 0, x + 8 ≠ 1

In other words, it is called a system of inequalities with one variable. The solution will lead to the following ODZ notation [ − 1, 0) ∪ (0, + ∞) .

Answer: [ − 1 , 0) ∪ (0 , + ∞)

Why is it important to consider DPD when driving change?

During identity transformations, it is important to find the ODZ. There are cases when the existence of ODZ does not occur. To understand whether a given expression has a solution, you need to compare the VA of the variables of the original expression and the VA of the resulting one.

Identity transformations:

may not affect DL;
may lead to the expansion or addition of DZ;
can narrow the DZ.

Let's look at an example.

Example 7

If we have an expression of the form x 2 + x + 3 · x, then its ODZ is defined over the entire domain of definition. Even when bringing similar terms and simplification of the expression ODZ does not change.

Example 8

If we take the example of the expression x + 3 x − 3 x, then things are different. We have fractional expression. And we know that division by zero is unacceptable. Then the ODZ has the form (− ∞, 0) ∪ (0, + ∞) . It can be seen that zero is not a solution, so we add it with a parenthesis.

Let's consider an example with the presence of a radical expression.

Example 9

If there is x - 1 · x - 3, then you should pay attention to the ODZ, since it must be written as the inequality (x − 1) · (x − 3) ≥ 0. It is possible to solve by the interval method, then we find that the ODZ will take the form (− ∞, 1 ] ∪ [ 3 , + ∞) . After transforming x - 1 · x - 3 and applying the property of roots, we have that the ODZ can be supplemented and everything can be written in the form of a system of inequalities of the form x - 1 ≥ 0, x - 3 ≥ 0. When solving it, we find that [ 3 , + ∞) . This means that the ODZ is completely written as follows: (− ∞, 1 ] ∪ [ 3 , + ∞) .

Transformations that narrow the DZ must be avoided.

Example 10

Let's consider an example of the expression x - 1 · x - 3, when x = - 1. When substituting, we get that - 1 - 1 · - 1 - 3 = 8 = 2 2 . If we transform this expression and bring it to the form x - 1 · x - 3, then when calculating we find that 2 - 1 · 2 - 3 the expression makes no sense, since the radical expression should not be negative.

Should be adhered to identity transformations, which ODZ will not change.

If there are examples that expand on it, then it should be added to the DL.

Example 11

Let's look at the example of a fraction of the form x x 3 + x. If we cancel by x, then we get that 1 x 2 + 1. Then the ODZ expands and becomes (− ∞ 0) ∪ (0 , + ∞) . Moreover, when calculating, we already work with the second simplified fraction.

In the presence of logarithms, the situation is slightly different.

Example 12

If there is an expression of the form ln x + ln (x + 3), it is replaced by ln (x · (x + 3)), based on the property of the logarithm. From this we can see that the ODZ from (0 , + ∞) to (− ∞ , − 3) ∪ (0 , + ∞) . Therefore for ADL definitions ln (x · (x + 3)) it is necessary to carry out calculations on the ODZ, that is, the (0 , + ∞) set.

When solving, it is always necessary to pay attention to the structure and type of the expression given by the condition. At correct location area of definition the result will be positive.

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