Answer: x = 3.

8 method. Application of the derivative in solving irrational equations

The most common method used to solve equations using the derivative method is the estimation method.

EXAMPLE 15:

Solve the equation: (1)

Solution: Since https://pandia.ru/text/78/021/images/image122_1.gif" width="371" height="29">, or (2). Consider the function ..gif" width="400" height="23 src=">.gif" width="215" height="49"> at all and, therefore, increases. Therefore the equation is equivalent to an equation having a root that is the root of the original equation.

Answer:

EXAMPLE 16:

Solve the irrational equation:

The domain of a function is a segment. Let's find the largest and smallest values ​​of this function on the segment. To do this, we find the derivative of the function f(x): https://pandia.ru/text/78/021/images/image136_1.gif" width="37 height=19" height="19">. Let's find the values ​​of the function f(x) at the ends of the segment and at the point: So, But and, therefore, equality is possible only if https://pandia.ru/text/78/021/images/image136_1.gif" width="37" height="19 src=" >. Checking shows that the number 3 is the root of this equation.

Answer: x = 3.

9 method. Functional

In exams, they sometimes ask you to solve equations that can be written in the form , where is a function.

For example, some equations: 1) 2) . Indeed, in the first case , in the second case . Therefore, solve irrational equations using the following statement: if a function is strictly increasing on the set X and for any , then the equations, etc. are equivalent on the set X .

Solve the irrational equation: https://pandia.ru/text/78/021/images/image145_1.gif" width="103" height="25"> strictly increases on the set R, and https://pandia.ru/text/78/021/images/image153_1.gif" width="45" height="24 src=">..gif" width="104" height="24 src=" > which has a single root. Therefore, equation (1) equivalent to it also has a single root

Answer: x = 3.

EXAMPLE 18:

Solve the irrational equation: (1)

By virtue of the definition of a square root, we obtain that if equation (1) has roots, then they belong to the set https://pandia.ru/text/78/021/images/image159_0.gif" width="163" height="47" >.(2)

Consider the function https://pandia.ru/text/78/021/images/image147_1.gif" width="35" height="21"> strictly increases on this set for any ..gif" width="100" height ="41"> which has a single root Therefore, and its equivalent on the set X equation (1) has a single root

Answer: https://pandia.ru/text/78/021/images/image165_0.gif" width="145" height="27 src=">

Solution: This equation is equivalent to a mixed system

1.1 Irrational equations

Irrational equations are often found in entrance exams in mathematics, since with their help it is easy to diagnose knowledge of concepts such as equivalent transformations, domain of definition, and others. Methods for solving irrational equations are usually based on the possibility of replacing (with the help of some transformations) an irrational equation with a rational one, which is either equivalent to the original irrational equation or is its consequence. Most often, both sides of the equation are raised to the same power. Equivalence is not violated when both sides are raised to an odd power. Otherwise, it is necessary to check the solutions found or evaluate the sign of both sides of the equation. But there are other techniques that may be more effective in solving irrational equations. For example, the trigonometric substitution method.

Example 1: Solve the equation

Since, then. Therefore we can put . The equation will take the form

Let's put where, then

.

.

Answer: .

Algebraic solution

Since then . Means, , so you can expand the module

.

Answer: .

Solving an equation algebraically requires good skills in carrying out identity transformations and competent handling of equivalent transitions. But in general, both decision methods are equivalent.

Example 2: Solve the equation

.

Solution using trigonometric substitution

The domain of definition of the equation is given by the inequality, which is equivalent to the condition, then. Therefore, you can put . The equation will take the form

Since, then. Let's open the internal module

Let's put , Then

.

The condition is satisfied by two values ​​and .

.

.

Answer: .

Algebraic solution

.

Let us square the equation of the first system of the population and obtain

Let it be then. The equation will be rewritten as

By checking we establish that is a root, then by dividing the polynomial by a binomial we obtain the decomposition of the right side of the equation into factors

Let's move from variable to variable, we get

.

Condition satisfy two values

.

Substituting these values ​​into the original equation, we find that is the root.

Solving in a similar way the equation of the second system of the original set, we find that it is also a root.

Answer: .

If in the previous example the algebraic solution and the solution using trigonometric substitution were equivalent, then in this case the solution by substitution is more profitable. When solving an equation using algebra, you have to solve a set of two equations, that is, square it twice. After this unequal transformation, we obtain two equations of the fourth degree with irrational coefficients, which can be eliminated by substitution. Another difficulty is checking the solutions found by substituting them into the original equation.

Example 3: Solve the equation

.

Solution using trigonometric substitution

Since, then. Note that a negative value of the unknown cannot be a solution to the problem. Indeed, let us transform the original equation to the form

.

The factor in parentheses on the left side of the equation is positive, the right side of the equation is also positive, so the factor on the left side of the equation cannot be negative. That's why, then, that's why you can put The original equation will be rewritten as

Since , then and . The equation will take the form

Let . Let's move from the equation to an equivalent system

.

The numbers and are the roots of the quadratic equation

.

Algebraic solution Let's square both sides of the equation

Let's introduce the replacement , then the equation will be written in the form

The second root is superfluous, so consider the equation

.

Since, then.

In this case, the algebraic solution is technically simpler, but it is imperative to consider the given solution using trigonometric substitution. This is due, firstly, to the non-standard nature of the substitution itself, which destroys the stereotype that the use of trigonometric substitution is possible only when. It turns out that trigonometric substitution also finds application. Secondly, it is difficult to solve the trigonometric equation , which is reduced by introducing a substitution to a system of equations. In a certain sense, this replacement can also be considered non-standard, and familiarity with it allows you to enrich your arsenal of techniques and methods for solving trigonometric equations.

Example 4: Solve the equation

.

Solution using trigonometric substitution

Since the variable can take any real value, we put . Then

,

Because .

The original equation, taking into account the transformations carried out, will take the form

Since we divide both sides of the equation by , we get

Let , Then . The equation will take the form

.

Given the substitution , we obtain a set of two equations

.

Let's solve each equation of the set separately.

.

Cannot be a sine value, since for any values ​​of the argument.

.

Because and the right side of the original equation is positive, then . From which it follows that .

This equation has no roots, since .

So, the original equation has a single root

.

Algebraic solution

This equation can easily be “transformed” into a rational equation of the eighth degree by squaring both sides of the original equation. Finding the roots of the resulting rational equation is difficult, and you need to have a high degree of ingenuity to cope with the problem. Therefore, it is advisable to know another way of solving, less traditional. For example, the substitution proposed by I. F. Sharygin.

Let's put , Then

Let's transform the right side of the equation :

Taking into account the transformations, the equation will take the form

.

Let's introduce the replacement , then

.

The second root is superfluous, therefore, and .

If the idea for solving the equation is not known in advance , then solving the standard solution by squaring both sides of the equation is problematic, since the result is an equation of the eighth degree, the roots of which are extremely difficult to find. The solution using trigonometric substitution looks cumbersome. It may be difficult to find the roots of an equation if you do not notice that it is reciprocal. The solution to this equation occurs using the apparatus of algebra, so we can say that the proposed solution is a combined one. In it, information from algebra and trigonometry work together for one goal - to obtain a solution. Also, solving this equation requires careful consideration of two cases. The solution by substitution is technically simpler and more beautiful than using trigonometric substitution. It is advisable that students know this method of substitution and use it to solve problems.

We emphasize that the use of trigonometric substitution to solve problems must be conscious and justified. It is advisable to use substitution in cases where the solution in another way is more difficult or completely impossible. Let's give another example, which, unlike the previous one, can be solved easier and faster using the standard method.

Real numbers. Approximation of real numbers by finite decimal fractions.

A real number, or a real number, is a mathematical abstraction that arose from the need to measure geometric and physical quantities of the surrounding world, as well as to carry out operations such as extracting roots, calculating logarithms, and solving algebraic equations. If natural numbers arose in the process of counting, rational numbers - from the need to operate with parts of a whole, then real numbers are intended to measure continuous quantities. Thus, the expansion of the stock of numbers under consideration led to a set of real numbers, which, in addition to the rational numbers, also includes other elements called irrational numbers .

Absolute error and its limit.

Let there be a certain numerical value, and the numerical value that is assigned to it is considered accurate, then under error in the approximate value of a numerical value (mistake) understand the difference between the exact and approximate value of a numerical value: . The error can take both positive and negative values. The quantity is called known approximation to the exact value of a numerical quantity - any number that is used instead of the exact value. The simplest quantitative measure of error is absolute error. Absolute error approximate value is a quantity about which it is known that: Relative error and its limit.

The quality of the approximation significantly depends on the accepted units of measurement and scales of quantities, therefore it is advisable to correlate the error of a quantity and its value, for which the concept of relative error is introduced. Relative error approximate value is a quantity about which it is known that: . Relative error is often expressed as a percentage. The use of relative errors is convenient, in particular, because they do not depend on the scale of quantities and units of measurement.

Irrational equations

Equations that contain a variable under the root sign are called irrational. When solving irrational equations, the resulting solutions require verification, because, for example, an incorrect equality when squaring can give a correct equality. In fact, an incorrect equality when squared gives the correct equality 1 2 = (-1) 2, 1=1. Sometimes it is more convenient to solve irrational equations using equivalent transitions.

Let's square both sides of this equation; After transformations we arrive at a quadratic equation; and let's substitute.

Complex numbers. Operations on complex numbers.

Complex numbers are an extension of the set of real numbers, usually denoted by . Any complex number can be represented as a formal sum x + iy, Where x And y- real numbers, i- imaginary unit Complex numbers form an algebraically closed field - this means that a polynomial of degree n with complex coefficients has exactly n complex roots, that is, the fundamental theorem of algebra is true. This is one of the main reasons for the widespread use of complex numbers in mathematical research. In addition, the use of complex numbers makes it possible to conveniently and compactly formulate many mathematical models used in mathematical physics and in the natural sciences - electrical engineering, hydrodynamics, cartography, quantum mechanics, vibration theory and many others.

Comparison a + bi = c + di means that a = c And b = d(two complex numbers are equal if and only if their real and imaginary parts are equal).

Addition ( a + bi) + (c + di) = (a + c) + (b + d) i .

Subtraction ( a + bi) − (c + di) = (a − c) + (b − d) i .

Multiplication

Numeric function. Methods for specifying a function

In mathematics, a number function is a function whose domains and values ​​are subsets of number sets—typically the set of real numbers or the set of complex numbers.

Verbal: Using natural language, Y is equal to an integer part of X. Analytical: Using an analytical formula f (x) = x !

Graphic Using a graph Fragment of a graph of a function.

Tabular: Using a table of values

Basic properties of a function

1) Function domain and function range . Function Domain x(variable x), for which the function y = f(x) determined.

Function Range y, which the function accepts. In elementary mathematics, functions are studied only on the set of real numbers.2 ) Zero function) Monotonicity of the function . Increasing function Decreasing function . Even function X f(-x) = f(x). Odd function- a function whose domain of definition is symmetrical with respect to the origin and for any X f (-x) = - f (x. The function is called limited unlimited .7) Periodicity of the function. Function f(x) - periodic period of the function

Function graphs. The simplest transformations of graphs using the function

Graph of a function- a set of points whose abscissas are valid argument values x, and the ordinates are the corresponding values ​​of the function y .

Straight line- graph of a linear function y = ax + b. The function y monotonically increases for a > 0 and decreases for a< 0. При b = 0 прямая линия проходит через начало координат т.0 (y = ax - прямая пропорциональность)

Parabola- graph of the quadratic trinomial function y = ax 2 + bx + c. It has a vertical axis of symmetry. If a > 0, has a minimum if a< 0 - максимум. Точки пересечения (если они есть) с осью абсцисс - корни соответствующего квадратного уравнения ax 2 + bx +c =0

Hyperbola- graph of the function. When a > O it is located in the I and III quarters, when a< 0 - во II и IV. Асимптоты - оси координат. Ось симметрии - прямая у = х (а >0) or y - x (a< 0).

Logarithmic function y = log a x(a > 0)

Trigonometric functions. When constructing trigonometric functions we use radian measure of angles. Then the function y= sin x is represented by a graph (Fig. 19). This curve is called sinusoid .

Graph of a function y=cos x shown in Fig. 20; this is also a sine wave resulting from moving the graph y= sin x along the axis X left to /2.

Basic properties of functions. Monotonicity, evenness, oddness, periodicity of functions.

Function Domain and Function Domain . Function Domain is the set of all valid valid values ​​of the argument x(variable x), for which the function y = f(x) determined.

Function Range is the set of all real values y, which the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.2 ) Zero function- the value of the argument at which the value of the function is zero.3 ) Intervals of constant sign of a function- such sets of argument values ​​on which the function values ​​are only positive or only negative.4 ) Monotonicity of the function .

Increasing function(in a certain interval) - a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

Decreasing function(in a certain interval) - a function for which a larger value of the argument from this interval corresponds to a smaller value of the function.5 ) Even (odd) function . Even function- a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). The graph of an even function is symmetrical about the ordinate. Odd function- a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f (-x) = - f (x). The graph of an odd function is symmetrical about the origin.6 ) Limited and unlimited functions. The function is called limited, if there is a positive number M such that |f (x) | ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited .7) Periodicity of the function. Function f(x) - periodic, if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f (x+T) = f (x). This smallest number is called period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

Periodic functions. Rules for finding the main period of a function.

Periodic function- a function that repeats its values ​​after some non-zero period, that is, it does not change its value when a fixed non-zero number (period) is added to the argument. All trigonometric functions are periodic. Are unfaithful statements regarding the sum of periodic functions: Sum of 2 functions with commensurate (even basic) periods T 1 and T 2 is a function with LCM period ( T 1 ,T 2). The sum of 2 continuous functions with incommensurable (even fundamental) periods is a non-periodic function. There are no periodic functions that are not equal to a constant whose periods are incommensurable numbers.

Plotting graphs of power functions.

Power function. This is the function: y = axn, Where a, n- permanent. At n= 1 we get direct proportionality : y =ax; at n = 2 - square parabola; at n = 1 - inverse proportionality or hyperbole. Thus, these functions are special cases of the power function. We know that the zero power of any number other than zero is 1, therefore, when n= 0 the power function turns into a constant value: y =a, i.e. its graph is a straight line parallel to the axis X, excluding the origin (please explain why?). All these cases (with a= 1) are shown in Fig. 13 ( n 0) and Fig. 14 ( n < 0). Отрицательные значения x are not covered here, since then some functions:

Inverse function

Inverse function- a function that reverses the dependence expressed by this function. A function is inverse to a function if the following identities are satisfied: for all for all

Limit of a function at a point. Basic properties of the limit.

The nth root and its properties.

The nth root of a number is a number whose nth power is equal to a.

Definition: An arithmetic root of the nth power of a is a non-negative number whose nth power is equal to a.

Basic properties of roots:

A power with an arbitrary real exponent and its properties.

Let a positive number and an arbitrary real number be given. The number is called the power, the number is the base of the power, and the number is the exponent.

By definition they believe:

If and are positive numbers and are any real numbers, then the following properties hold:

.

.

Power function, its properties and graphs

Power function complex variable f (z) = z n with an integer exponent is determined using the analytical continuation of a similar function of the real argument. For this purpose, the exponential form of writing complex numbers is used. a power function with an integer exponent is analytic in the entire complex plane, as the product of a finite number of instances of the identity map f (z) = z. According to the uniqueness theorem, these two criteria are sufficient for the uniqueness of the resulting analytic continuation. Using this definition, we can immediately conclude that the power function of a complex variable has significant differences from its real counterpart.

This is a function of the form , . The following cases are considered:

A). If, then. Then , ; if the number is even, then the function is even (that is in front of everyone); if the number is odd, then the function is odd (that is in front of everyone).

Exponential function, its properties and graphs

Exponential function- mathematical function.

In the real case, the base of the degree is some non-negative real number, and the argument of the function is the real exponent.

In the theory of complex functions, a more general case is considered when the argument and exponent can be an arbitrary complex number.

In the most general form - u v, introduced by Leibniz in 1695

Particularly noteworthy is the case when the number e acts as the base of the degree. Such a function is called an exponential (real or complex).

Properties ; ; .

Exponential equations.

Let's move directly to exponential equations. In order to solve an exponential equation, you must use the following theorem: If the powers are equal and the bases are equal, positive and different from one, then their exponents are equal. Let's prove this theorem: Let a>1 and a x =a y.

Let us prove that in this case x=y. Let us assume the opposite of what needs to be proven, i.e. let us assume that x>y or that x<у. Тогда получим по свойству показательной функции, что либо a х ay. Both of these results contradict the conditions of the theorem. Therefore, x = y, which is what needed to be proven.

Definition 1. A perpendicular to a given line is a line segment perpendicular to a given line, which has one of its ends at their intersection point. The end of a segment lying on a given line is called the base of the perpendicular.

Definition 2. An inclined line drawn from a given point to a given line is a segment connecting a given point with any point on a line that is not the base of a perpendicular drawn from the same point to a given line. AB is perpendicular to the plane α.

AC - oblique, CB - projection.

C is the base of the inclined, B is the base of the perpendicular.

The angle between a straight line and a plane.

The angle between a straight line and a plane Any angle between a straight line and its projection onto this plane is called.

Dihedral angle.

Dihedral angle- a spatial geometric figure formed by two half-planes emanating from one straight line, as well as a part of space limited by these half-planes. Half-planes are called edges dihedral angle, and their common straight line is edge. Dihedral angles are measured by a linear angle, that is, the angle formed by the intersection of a dihedral angle with a plane perpendicular to its edge. Every polyhedron, regular or irregular, convex or concave, has a dihedral angle on each edge.

Perpendicularity of two planes.

SIGN OF PERPENDICULARITY OF PLANES.

If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.