Y is 2 cube root of x. Integral of a power function

Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"

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Definition of a power function - cube root

Guys, we continue to study power functions. Today we will talk about the "Cubic root of x" function.
What is a cube root?
The number y is called a cube root of x (root of the third degree) if the equality $y^3=x$ holds.
Denoted as $\sqrt(x)$, where x is a radical number, 3 is an exponent.
$\sqrt(27)=3$; $3^3=$27.
$\sqrt((-8))=-2$; $(-2)^3=-8$.
As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers.
The third root of a negative number is negative number. When raised to an odd power, the sign is preserved; the third power is odd.

Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. Using the notation for roots we obtain the desired identity.

Properties of cube roots

a) $\sqrt(a*b)=\sqrt(a)*\sqrt(6)$.
b) $\sqrt(\frac(a)(b))=\frac(\sqrt(a))(\sqrt(b))$.

Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's build a graph of our function.
1) Domain set real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, from any large number we can calculate the third root, and we can go up to infinity, finding everything large values argument.
5) For $x≥0$ the smallest value is 0. This property is obvious.
Let's build a graph of the function by points at x≥0.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd.

Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.

7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).

Examples of solving power functions

Examples
1. Solve the equation $\sqrt(x)=x$.
Solution. Let's build two graphs on one coordinate plane$y=\sqrt(x)$ and $y=x$.

As you can see, our graphs intersect at three points.
Answer: (-1;-1), (0;0), (1;1).

2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, parallel transfer two units to the right and three units down.

3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function.
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) Greatest value No. Lowest value equals minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).

Problems to solve independently

1. Solve the equation $\sqrt(x)=2-x$.
2. Construct a graph of the function $y=\sqrt((x+1))+1$.
3.Plot a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.

Basic goals:

1) to form an idea of ​​the feasibility of a generalized study of the dependencies of real quantities using the example of quantities, connected by relationship y=

2) to develop the ability to construct a graph y= and its properties;

3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.

Equipment, demonstration material: Handout.

1. Algorithm:

2. Sample for completing the task in groups:

3. Sample for self-test of independent work:

4. Card for the reflection stage:

1) I understood how to graph the function y=.

2) I can list its properties using a graph.

3) I did not make mistakes in independent work.

4) I made mistakes in my independent work (list these mistakes and indicate their reason).

During the classes

1. Self-determination for educational activities

Purpose of the stage:

1) include students in educational activities;

2) determine the content of the lesson: we continue to work with real numbers.

Organization educational process at stage 1:

– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated functions studied in 7th grade).

– Today we will continue to work with a set of real numbers, a function.

2. Updating knowledge and recording difficulties in activities

Purpose of the stage:

1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs

y = kx + m, y = kx, y =c, y =x 2, y = - x 2,

2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;

3) record all repeated concepts and algorithms in the form of diagrams and symbols;

4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.

Organization of the educational process at stage 2:

1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)

2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).

What is the name of x? (Independent variable - argument)

What is the name of y? (Dependent variable).

3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).

Individual task:

What is the graph of the functions y = kx + m, y =x 2, y =?

3. Identifying the causes of difficulties and setting goals for activities

Purpose of the stage:

1) organize communicative interaction, during which the distinctive property a task that caused difficulty in learning activities;

2) agree on the purpose and topic of the lesson.

Organization of the educational process at stage 3:

-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)

– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)

– Can you formulate the topic of the lesson? (Function y=, its properties and graph).

– Write the topic in your notebook.

4. Construction of a project for getting out of a difficulty

Purpose of the stage:

1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;

2) fix new way actions in a symbolic, verbal form and using a standard.

Organization of the educational process at stage 4:

Work at this stage can be organized in groups, asking the groups to construct a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.

5. Primary consolidation in external speech

The purpose of the stage: to record the studied educational content in external speech.

Organization of the educational process at stage 5:

Construct a graph of y= - and describe its properties.

Properties y= - .

1.Domain of definition of a function.

2. Range of values ​​of the function.

3. y = 0, y> 0, y<0.

y =0 if x = 0.

y<0, если х(0;+)

4.Increasing, decreasing functions.

The function decreases as x.

Let's build a graph of y=.

Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.

Answer: at our name. = 1, y max. =3

6. Independent work with self-test according to the standard

The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.

Organization of the educational process at stage 6:

Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.

Let's build a graph of y=.

Using a graph, find the smallest and largest values ​​of the function on the segment.

7. Inclusion in the knowledge system and repetition

The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the next lessons.

Organization of the educational process at stage 7:

Solve the equation graphically: = x – 6.

One student is at the blackboard, the rest are in notebooks.

8. Reflection of activity

Purpose of the stage:

1) record new content learned in the lesson;

2) evaluate your own activities in the lesson;

3) thank classmates who helped get the result of the lesson;

4) record unresolved difficulties as directions for future educational activities;

5) discuss and write down your homework.

Organization of the educational process at stage 8:

- Guys, what was our goal today? (Study the function y=, its properties and graph).

– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)

– Analyze your activities in class. (Cards with reflection)

Homework

paragraph 13 (before example 2) № 13.3, 13.4

Solve the equation graphically:

Construct a graph of the function and describe its properties.

Which is equal to $a.$ In other words, this is the solution to the equation $x^3\; =\; a$(usually real solutions are meant).

Real root

Demonstrative form

The root of complex numbers can be defined as follows:

$x^(1/3)\; =\; \backslash exp\; (\backslash tfrac13\; \backslash ln(x))$

If you imagine $x$ How

$x\; =\; r\backslash exp(i\backslash theta)$

then the formula for a cubic number is:

$\backslash sqrt(x)\; =\; \backslash sqrt(r)\backslash exp\; (\backslash tfrac13\; i\backslash theta).$

This geometrically means that in polar coordinates we take the cube root of the radius and divide the polar angle by three to determine the cube root. So if $x$ complex, then $\backslash sqrt(-8)$ will mean not $-2$, Will be $1\; +\; i\backslash sqrt(3).$

At a constant density of matter, the dimensions of two similar bodies are related to each other as the cube roots of their masses. So, if one watermelon weighs twice as much as another, then its diameter (as well as its circumference) will be only a little more than a quarter (26%) larger than the first; and to the eye it will seem that the difference in weight is not so significant. Therefore, in the absence of scales (sale by eye), it is usually more profitable to buy a larger fruit.

Calculation methods

Column

Before starting, you need to divide the number into triplets (the integer part - from right to left, the fractional part - from left to right). When you reach the decimal point, you must add a decimal point at the end of the result.

The algorithm is as follows:

1. Find a number whose cube is smaller than the first group of digits, but when it increases by 1 it becomes larger. Write down the number you find to the right of the given number. Write the number 3 below it.
2. Write the cube of the number found under the first group of numbers and subtract. Write the result after subtraction under the subtrahend. Next, take down the next group of numbers.
3. Next, we replace the found intermediate answer with the letter $a$. Calculate using the formula $$ such a number $x$ that its result is less than the lower number, but when increased by 1 it becomes larger. Write down what you find $x$ to the right of the answer. If the required accuracy is achieved, stop calculations.
4. Write down the result of the calculation under the bottom number using the formula $300\backslash times\; a^2\backslash times\; x+30\backslash times\; a\backslash times\; x^2+x^3$ and do the subtraction. Go to step 3.

see also

Write a review about the article "Cubic root"

Literature

• Korn G., Korn T. 1.3-3. Representation of sum, product and quotient. Powers and roots // Handbook of mathematics. - 4th edition. - M.: Nauka, 1978. - P. 32-33.

An excerpt characterizing the cube root

By nine o'clock in the morning, when the troops had already moved through Moscow, no one else came to ask the count's orders. Everyone who could go did so of their own accord; those who remained decided with themselves what they had to do.
The count ordered the horses to be brought in to go to Sokolniki, and, frowning, yellow and silent, with folded hands, he sat in his office.
In calm, not stormy times, it seems to every administrator that it is only through his efforts that the entire population under his control moves, and in this consciousness of his necessity, every administrator feels the main reward for his labors and efforts. It is clear that as long as the historical sea is calm, the ruler-administrator, with his fragile boat resting his pole against the ship of the people and himself moving, must seem to him that through his efforts the ship he is resting against is moving. But as soon as a storm arises, the sea becomes agitated and the ship itself moves, then delusion is impossible. The ship moves with its enormous, independent speed, the pole does not reach the moving ship, and the ruler suddenly goes from the position of a ruler, a source of strength, into an insignificant, useless and weak person.
Rastopchin felt this, and it irritated him. The police chief, who was stopped by the crowd, together with the adjutant, who came to report that the horses were ready, entered the count. Both were pale, and the police chief, reporting the execution of his assignment, said that in the count’s courtyard there was a huge crowd of people who wanted to see him.
Rastopchin, without answering a word, stood up and quickly walked into his luxurious, bright living room, walked up to the balcony door, grabbed the handle, left it and moved to the window, from which the whole crowd could be seen more clearly. A tall fellow stood in the front rows and with a stern face, waving his hand, said something. The bloody blacksmith stood next to him with a gloomy look. The hum of voices could be heard through the closed windows.
- Is the crew ready? - said Rastopchin, moving away from the window.
“Ready, your Excellency,” said the adjutant.
Rastopchin again approached the balcony door.
- What do they want? – he asked the police chief.
- Your Excellency, they say that they were going to go against the French on your orders, they shouted something about treason. But a violent crowd, your Excellency. I left by force. Your Excellency, I dare to suggest...
“If you please, go, I know what to do without you,” Rostopchin shouted angrily. He stood at the balcony door, looking out at the crowd. “This is what they did to Russia! This is what they did to me!” - thought Rostopchin, feeling an uncontrollable anger rising in his soul against someone who could be attributed to the cause of everything that happened. As often happens with hot-tempered people, anger was already possessing him, but he was looking for another subject for it. “La voila la populace, la lie du peuple,” he thought, looking at the crowd, “la plebe qu"ils ont soulevee par leur sottise. Il leur faut une victime, [“Here he is, people, these scum of the population, the plebeians, whom they raised with their stupidity! They need a victim."] - it occurred to him, looking at the tall fellow waving his hand. And for the same reason it came to his mind that he himself needed this victim, this object for his anger.
- Is the crew ready? – he asked another time.
- Ready, Your Excellency. What do you order about Vereshchagin? “He’s waiting at the porch,” answered the adjutant.
- A! - Rostopchin cried out, as if struck by some unexpected memory.
And, quickly opening the door, he stepped out onto the balcony with decisive steps. The conversation suddenly stopped, hats and caps were taken off, and all eyes rose to the count who had come out.
- Hello guys! - the count said quickly and loudly. - Thank you for coming. I’ll come out to you now, but first of all we need to deal with the villain. We need to punish the villain who killed Moscow. Wait for me! “And the count just as quickly returned to his chambers, slamming the door firmly.
A murmur of pleasure ran through the crowd. “That means he will control all the villains! And you say French... he’ll give you the whole distance!” - people said, as if reproaching each other for their lack of faith.

The basic properties of the power function are given, including formulas and properties of the roots. The derivative, integral, power series expansion, and complex number representation of a power function are presented.

Definition

Definition
Power function with exponent p is the function f (x) = xp, the value of which at point x is equal to the value of the exponential function with base x at point p.
In addition, f (0) = 0 p = 0 for p > 0 .

For natural values ​​of the exponent, the power function is the product of n numbers equal to x:
.
It is defined for all valid .

For positive rational values ​​of the exponent, the power function is the product of n roots of degree m of the number x:
.
For odd m, it is defined for all real x. For even m, the power function is defined for non-negative ones.

For negative , the power function is determined by the formula:
.
Therefore, it is not defined at the point.

For irrational values ​​of the exponent p, the power function is determined by the formula:
,
where a is an arbitrary positive number not equal to one: .
When , it is defined for .
When , the power function is defined for .

Continuity. A power function is continuous in its domain of definition.

Properties and formulas of power functions for x ≥ 0

Here we will consider the properties of the power function for non-negative values ​​of the argument x. As stated above, for certain values ​​of the exponent p, the power function is also defined for negative values ​​of x. In this case, its properties can be obtained from the properties of , using even or odd. These cases are discussed and illustrated in detail on the page "".

A power function, y = x p, with exponent p has the following properties:
(1.1) defined and continuous on the set
at ,
at ;
(1.2) has many meanings
at ,
at ;
(1.3) strictly increases with ,
strictly decreases as ;
(1.4) at ;
at ;
(1.5) ;
(1.5*) ;
(1.6) ;
(1.7) ;
(1.7*) ;
(1.8) ;
(1.9) .

Proof of properties is given on the page “Power function (proof of continuity and properties)”

Roots - definition, formulas, properties

Definition
Root of a number x of degree n is the number that when raised to the power n gives x:
.
Here n = 2, 3, 4, ... - a natural number greater than one.

You can also say that the root of a number x of degree n is the root (i.e. solution) of the equation
.
Note that the function is the inverse of the function.

Square root of x is a root of degree 2: .

Cube root of x is a root of degree 3: .

Even degree

For even powers n = 2 m, the root is defined for x ≥ 0 . A formula that is often used is valid for both positive and negative x:
.
For square root:
.

The order in which the operations are performed is important here - that is, first the square is performed, resulting in a non-negative number, and then the root is taken from it (the square root can be taken from a non-negative number). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.

Odd degree

For odd powers, the root is defined for all x:
;
.

Properties and formulas of roots

The root of x is a power function:
.
When x ≥ 0 the following formulas apply:
;
;
, ;
.

These formulas can also be applied for negative values ​​of variables. You just need to make sure that the radical expression of even powers is not negative.

Private values

The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .

Example. Root of roots

Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.

y = x p for different values ​​of the exponent p.

Here are graphs of the function for non-negative values ​​of the argument x. Graphs of a power function defined for negative values ​​of x are given on the page “Power function, its properties and graphs"

Inverse function

The inverse of a power function with exponent p is a power function with exponent 1/p.

If, then.

Derivative of a power function

Derivative of nth order:
;

Deriving formulas > > >

Integral of a power function

P ≠ - 1 ;
.

Power series expansion

At - 1 < x < 1 the following decomposition takes place:

Expressions using complex numbers

Consider the function of the complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = |z|):
z = r e i φ .
We represent the complex number t in the form of real and imaginary parts:
t = p + i q .
We have:

Next, we take into account that the argument φ is not uniquely defined:
,

Let's consider the case when q = 0 , that is, the exponent is a real number, t = p. Then
.

If p is an integer, then kp is an integer. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous.

If p is irrational, then the products kp for any k do not produce an integer. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function.

If p is rational, then it can be represented as:
, Where m, n- integers that do not contain common divisors. Then
.
First n values, with k = k 0 = 0, 1, 2, ... n-1, give n different values ​​of kp:
.
However, subsequent values ​​give values ​​that differ from the previous ones by an integer. For example, when k = k 0+n we have:
.
Trigonometric functions whose arguments differ by multiples of 2π, have equal values. Therefore, with a further increase in k, we obtain the same values ​​of z p as for k = k 0 = 0, 1, 2, ... n-1.

Thus, an exponential function with a rational exponent is multivalued and has n values ​​(branches). Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function. After n such revolutions we return to the first branch from which the countdown began.

In particular, a root of degree n has n values. As an example, consider the nth root of a real positive number z = x. In this case φ 0 = 0 , z = r = |z| = x, .
.
So, for a square root, n = 2 ,
.
For even k, (- 1 ) k = 1. For odd k, (- 1 ) k = - 1.
That is, the square root has two meanings: + and -.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Guys, we continue to study power functions. The topic of today's lesson will be the function - the cube root of x. What is a cube root? The number y is called a cube root of x (root of the third degree) if the equality is satisfied. Designated by:, where x is the radical number, 3 is the exponent.

As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers. The third root of a negative number is equal to a negative number. When raised to an odd power, the sign is preserved; the third power is odd. Let's check the equality: Let. Let us raise both expressions to the third power. Then or In the notation of roots we obtain the desired identity.

Guys, let's now build a graph of our function. 1) Domain of definition is the set of real numbers. 2) The function is odd, since Next we will consider our function at x 0, then we will display the graph relative to the origin. 3) The function increases as x 0. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase. 4) The function is not limited from above. In fact, from an arbitrarily large number we can calculate the third root, and we can move upward indefinitely, finding ever larger values ​​of the argument. 5) When x 0 the smallest value is 0. This property is obvious.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd. Properties of the function: 1) D(y)=(-;+) 2) Odd function. 3) Increases by (-;+) 4) Unlimited. 5) There is no minimum or maximum value. 6) The function is continuous on the entire number line. 7) E(y)= (-;+). 8) Convex downward by (-;0), convex upward by (0;+).

Example. Draw a graph of the function and read it. Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For x-1 we build a graph of the cubic root, and for x-1 we build a graph of a linear function. 1) D(y)=(-;+) 2) The function is neither even nor odd. 3) Decreases by (-;-1), increases by (-1;+) 4) Unlimited from above, limited from below. 5) There is no greatest value. The smallest value is minus one. 6) The function is continuous on the entire number line. 7) E(y)= (-1;+)