LECTURE NOTES ON MATANALYSIS

Functions of several variables. Geometric representation of a function of two variables. Level lines and surfaces. Limit and continuity of functions of several variables, their properties. Partial derivatives, their properties and geometric meaning.

Definition 1.1. Variable z (with change area Z) called function of two independent variables x,y in abundance M, if each pair ( x,y) from many M z from Z.

Definition 1.2. Many M, in which the variables are specified x,y, called domain of the function, and themselves x,y- her arguments.

Designations: z = f(x, y), z = z(x, y).

Examples.

Comment. Since a couple of numbers ( x,y) can be considered the coordinates of a certain point on the plane; we will subsequently use the term “point” for a pair of arguments to a function of two variables, as well as for an ordered set of numbers
, which are arguments to a function of several variables.

Definition 1.3. . Variable z (with change area Z) called function of several independent variables
in abundance M, if each set of numbers
from many M according to some rule or law, one specific value is assigned z from Z. The concepts of arguments and domain are introduced in the same way as for a function of two variables.

Designations: z = f
,z = z
.

Geometric representation of a function of two variables.

Consider the function

z = f(x, y) , (1.1)

defined in some area M on the O plane xy. Then the set of points in three-dimensional space with coordinates ( x, y, z) , where , is the graph of a function of two variables. Since equation (1.1) defines a certain surface in three-dimensional space, it will be the geometric image of the function under consideration.

z = f(x,y)

M y

Comment. For a function of three or more variables we will use the term “surface in n-dimensional space,” although it is impossible to depict such a surface.

Level lines and surfaces.

For a function of two variables given by equation (1.1), we can consider a set of points ( x,y) O plane xy, for which z takes on the same constant value, that is z= const. These points form a line on the plane called level line.

Example.

Find the level lines for the surface z = 4 – x² - y². Their equations look like x² + y² = 4 – c (c=const) – equations of concentric circles with a center at the origin and with radii
. For example, when With=0 we get a circle x² + y² = 4 .

For a function of three variables u = u (x, y, z) equation u (x, y, z) = c defines a surface in three-dimensional space, which is called level surface.

Example.

For function u = 3x + 5y – 7z–12 level surfaces will be a family of parallel planes given by the equations

3x + 5y – 7z –12 + With = 0.

Limit and continuity of a function of several variables.

Let's introduce the concept δ-neighborhoods points M 0 (X 0 , y 0 ) on the O plane xy as a circle of radius δ with center at a given point. Similarly, we can define a δ-neighborhood in three-dimensional space as a ball of radius δ with center at the point M 0 (X 0 , y 0 , z 0 ) . For n-dimensional space we will call the δ-neighborhood of a point M 0 set of points M with coordinates
, satisfying the condition

Where
- point coordinates M 0 . Sometimes this set is called a “ball” in n-dimensional space.

Definition 1.4. The number A is called limit functions of several variables f
at the point M 0 if

such that | f(M) – A| < ε для любой точки M from δ-neighborhood M 0 .

Designations:
.

It must be taken into account that in this case the point M may be approaching M 0, relatively speaking, along any trajectory inside the δ-neighborhood of the point M 0 . Therefore, one should distinguish the limit of a function of several variables in the general sense from the so-called repeated limits obtained by successive passages to the limit for each argument separately.

Examples.

Comment. It can be proven that from the existence of a limit at a given point in the usual sense and the existence at this point of limits on individual arguments, the existence and equality of repeated limits follows. The reverse statement is not true.

Definition 1.5. Function f
called continuous at the point M 0
, If
(1.2)

If we introduce the notation

That condition (1.2) can be rewritten in the form

(1.3)

Definition 1.6. Inner point M 0 function domain z = f (M) called break point function if conditions (1.2), (1.3) are not satisfied at this point.

Comment. Many discontinuity points can form on a plane or in space lines or fracture surface.

To

several functions

download chart

Graphing a function online

instantly.

Online service instantly draws a graph

Absolutely supported All mathematical functions

Trigonometric functions
				Cosecant	Cotangent	arcsine	arc cosine	Arctangent	Arcsecant	Arccosecant	Arccotangent
Hyperbolic functions
Other
Natural logarithm		Logarithm		Square root				Round down		Round up
Minimum						Maximum
min(expression1,expression2,…)						max(expression1,expression2,…)

Graph the function

Construction of a 3D surface

Enter the equation

Let us construct a surface defined by the equation f(x, y, z) = 0, where a< x < b, c < y < d, m < z < n.

Other examples:

• y = x^2
• z = x^2 + y^2
• 0.3 * z^2 + x^2 + y^2 = 1
• z = sin((x^2 + y^2)^(1/2))
• x^4+y^4+z^4-5.0*(x^2+y^2+z^2)+11.8=0

Canonical view of curve and surface

You can determine the type of curve and 2nd order surface online with a detailed solution:

Rules for entering expressions and functions

Expressions can consist of functions (notations are given in alphabetical order):

absolute(x) Absolute value x
(module x or |x|) arccos(x) Function - arc cosine of xarccosh(x) Arc cosine hyperbolic from xarcsin(x) Arcsine from xarcsinh(x) Arcsine hyperbolic from xarctan(x) Function - arctangent of xarctgh(x) Arctangent hyperbolic from xee a number that is approximately equal to 2.7 exp(x) Function - exponent of x(which is e^x) log(x) or ln(x) Natural logarithm of x
(To get log7(x), you need to enter log(x)/log(7) (or, for example, for log10(x)=log(x)/log(10)) pi The number is "Pi", which is approximately equal to 3.14 sin(x) Function - Sine of xcos(x) Function - Cosine of xsinh(x) Function - Hyperbolic sine of xcosh(x) Function — Hyperbolic cosine of xsqrt(x) Function - square root from xsqr(x) or x^2 Function - Square xtan(x) Function - Tangent from xtgh(x) Function — Tangent hyperbolic from xcbrt(x) Function - cube root of xfloor(x) Function - rounding x downward (example floor(4.5)==4.0) sign(x) Function - Sign xerf(x) Error function (Laplace or probability integral)

The following operations can be used in expressions:

Real numbers enter as 7.5 , Not 7,5 2*x- multiplication 3/x- division x^3- exponentiation x+7- addition x - 6- subtraction

How to graph a function online on this site?

To plot a function online, you just need to enter your function in a special field and click somewhere outside it. After this, the graph of the entered function will be drawn automatically. Let’s say you want to build a classic graph of the “x squared” function. Accordingly, you need to enter “x^2” in the field.

If you need to plot several functions at the same time, then click on the blue “Add more” button. After this, another field will open in which you will need to enter the second function. Its schedule will also be built automatically.

You can adjust the color of the graph lines by clicking on the square located to the right of the function input field. The remaining settings are located directly above the graph area. With their help, you can set the background color, the presence and color of the grid, the presence and color of the axes, the presence of marks, as well as the presence and color of the numbering of graph segments. If necessary, you can scale the function graph using the mouse wheel or special icons in the lower right corner of the drawing area.

After plotting the graph and making the necessary changes to the settings, you can download chart using the big green "Download" button at the very bottom. You will be prompted to save the function graph as a PNG image.

Why do you need to graph a function?

On this page you can build interactive chart online functions.

Graph a function online

Plotting a function graph allows you to see the geometric image of a particular mathematical function. To make it more convenient for you to build such a graph, we have created a special online application. It's completely free, doesn't require registration, and can be used directly in your browser without any hassle. additional settings and manipulation. Constructing graphs for a variety of functions is most often required by middle and high school students studying algebra and geometry, as well as first and second year students taking higher mathematics courses. As a rule, this process It takes a lot of time and requires a lot of office supplies to draw the graph axes on paper, put down coordinate points, connect them with a straight line, etc. Using this online service you will be able to calculate and graph the function instantly.

How does a graphing calculator work for graphing functions?

Online service It works very simply. The function (i.e. the equation itself, the graph of which needs to be plotted) is entered into the field at the very top. Immediately after entering the application instantly draws a graph in the area below this field. Everything happens without refreshing the page. Next, you can enter various color settings, as well as hide/show some elements of the function graph. After this, ready schedule can be downloaded by clicking on the appropriate button at the very bottom of the application. The drawing will be downloaded to your computer in .png format, which you can print or transfer to a paper notebook.

What features does the graph builder support?

Absolutely supported all mathematical functions, which can be useful when plotting graphs. It is important to emphasize here that, in contrast to the classical language of mathematics adopted in schools and universities, the degree sign within the application is designated international sign"^". This is due to the lack of the ability to write a degree in the usual format on a computer keyboard. Below is a table with full list supported functions.

The application supports the following functions:

Trigonometric functions
				Cosecant	Cotangent	arcsine	arc cosine	Arctangent	Arcsecant	Arccosecant	Arccotangent
Hyperbolic functions
Other
Natural logarithm		Logarithm		Square root				Round down		Round up
Minimum						Maximum
min(expression1,expression2,…)						max(expression1,expression2,…)

Examples. Construct function level lines corresponding to the values

Construct function level lines corresponding to the values .

Assuming , we obtain the equations of the corresponding level lines:

By constructing these lines in the Cartesian coordinate system xOy, we obtain straight lines parallel to the bisector of the second and fourth coordinate angles (Fig. 1)

Let's write the equations of the level lines:

, , , And .

By constructing them in the xOy plane, we obtain concentric circles with the center at the origin of coordinates (Fig. 2)

The level lines of this function , , , and are parabolas symmetrical with respect to Oy with a common vertex at the origin (Fig. 3).

2. Directional derivative

An important characteristic of a scalar field is the rate of change of the field in a given direction.

To characterize the rate of change of the field in the direction of the vector, the concept of the derivative of the field in direction is introduced.

Consider the function at point and point.

Let's draw through the points and the vector. The angles of inclination of this vector to the direction of the coordinate axes x, y, z let's denote a, b, g, respectively. The cosines of these angles are called direction cosines vector

Instructions

When constructing level lines, proceed from the fact that they are projections onto a plane with a zero applicate of the lines of intersection of the graph given function with some horizontal plane. The applicate of this section plane is the constant to which the equation of the function must be equated in order to obtain the coordinates of the points of the line. It can change with the step specified in the task conditions if a set of lines is required to be constructed. And if you need to build only one level line, the conditions can give the coordinates of the point lying on it. Graphs from this page can be saved or edited interactively.

Reduce the function specified in the problem conditions to the form f(x,y) = const. For example, if given z = x² + y² - 4*y, it can be written in an alternative form to better represent the shape of the graph of the function, and equated to the constant c: c+4 = x²+(y-2)². The volume graph of such a function is an infinite , and all its sections by a horizontal plane raised to different , (i.e., the desired level lines) will be concentric circles with a radius determined by the formula √(c+4).

Substitute the value for the level line specified in the conditions for the constant c. If it is not given, choose it yourself based on the range of values ​​of the function. For example, for the example above minimum value the constant can be the number -4. The constant can be equated to 5, and in this case the graph of the function will be a circle with radius √(5+4) = 3 and center at a point with abscissa equal to 0 and ordinate equal to 2.

If you need to build several level lines, repeat the previous step as many times as necessary.

On the Internet you can find services that will help with constructing level lines. For example, below is a link to the WolframAlpha service. In the input field on its page, enter the function formula and click on the button with the equal sign. The function z = x² + y² - 4*y used in the example must be entered in the following form: x^2+y^2-4*y. In a few seconds, two- and three-dimensional color graphs with level lines will appear on the page, as well as the figure described by the formula, alternative forms of writing it, and other functions that can be used when constructing level lines.

Sources:

• WolframAlpha Service

Not everyone wants to be a family despot, but even the most timid and self-sufficient people need their opinion to be at least listened to. How to line up correctly lines influence? You can only influence someone who needs something, so let’s look at how to use your partner’s needs to get what you want from him, using Maslow’s pyramid.

Instructions

The sphere of human needs is based on needs, primarily thirst, hunger and sexual desire. Partners are trained like Pavlov's dog using all methods, but this method is the least subtle. Thus, some wives in their youth deprive their husbands of close relationships for the slightest offense, and husbands do the same in relation to those who did not please. However, it is much more effective to use this method positively, that is, in response to concessions, give your loved one intoxicating, enchanting intimacy.

Higher in the hierarchy is the need for security. Every person wants to live comfortably, with a stable lifestyle, without fearing anything. When an offended wife suddenly refuses to cook for her husband, she unknowingly breaks his household habits, causing pain. This is not always a reasonable policy; in negative situations it is better to behave neutrally, and the slightest positive changes reward your husband with your favorite dish or one with which you have romantic associations.

We will consider the next two levels together, because they are close in meaning - these are the needs for respect and love. Insults hurt, but famous question"Are you me?" with subsequent attempts to manipulate pretty much spoil the blood of both men and women. But at this level, many people are very dependent and vulnerable. Encouraging correct behavior is achieved through sincere praise, especially with strangers, gentle touches and loving glances.

The pyramid is crowned by the need for self-realization. Wrong behavior here is ridiculing the tastes, spiritual needs and aspirations of a loved one. After each decision you need, do not skimp on signs of attention to your partner’s creativity. This can manifest itself in small things, for example, you laugh at his good jokes and retell them to other people with reference to the author. It is also good to create conditions for your loved one to be creative in the area where he is truly talented.

Of course, you can achieve your goals by depriving your partner of what they need. But you can truly strengthen and enrich relationships only by trying to satisfy needs loved one by the highest class. Selfless and unselfish love will help you guess in a particular situation.

Video on the topic

Please note

Using the linearity property of the problem, we connect these points with the so-called transition straight line. The line of influence, composed of two constructed branches of the graph S3−4 (x) and the transition straight line, form the line of influence of the force S3−4, meaning the dependence of this force on the location of the unit load (Fig. 97). We build a line of influence of force in rack 3-8 when a single load moves below.

Sources:

• Kinematic method for constructing influence lines in a beam in 2019

The world that surrounds us all has three dimensions, but the sheet of paper or canvas on which we are trying to depict the surrounding reality is, alas, only two-dimensional. In order for the objects we depict to seem as voluminous and realistic as possible, we need to follow certain rules and arrange them correctly. perspective.

You will need

• sheet of paper, pencil, ruler

Instructions

Next, we determine where the object will be located relative to the horizon line. If it is at eye level (that is, on the horizon), then we are looking at the object directly. If an object is above the horizon line, we look at it from below, respectively, in this case it becomes visible bottom part. If an object is placed below the horizon line, then it will be visible upper part. We build an object, check with a ruler that all parallel lines converge at one point.

Video on the topic

Please note

Also, when constructing a perspective, you need to remember not only that all parallel lines converge at one point, but also that as you move away, all depicted objects become smaller. Very distant objects even turn into dots.

IN lately Clear coated roofing materials are increasingly being used in garage construction. The advantage of a transparent roof is that it allows large number daylight, and the lighting level allows you to work without additional artificial lighting.

You will need

• - roulette;
• - felt-tip pen;
• - drill;
• - screws;
• - screwdriver;
• - transparent plastic;
• - sealing rings;
• - sealant;
• - profiled foam.

Instructions

Measure the roofs using a tape measure. Mark the roofing so that its sheets overlap. The overlap width is one and a half centimeters. Mark the cut line with a colored marker. Please note that the end must be adjacent to the edge at an angle of 90 degrees.

Drill holes for screws in the plastic sheets. The diameter of the hole should be 4 mm larger than the diameter of the hardware. Secure with screws. The fasteners should be located on every second ridge of the relief sheet. Plastic is a fairly fragile material, so when attaching it, limit mechanical impact. It is recommended to use a screwdriver.

When installing the roof covering, it is necessary to install O-rings and plastic caps between the walls. As an additional seal, you can use a profiled one, which is attached with screws in through holes.

Video on the topic

Please note

The garage roof will look correct and beautiful only if the rafter frames have the same shape and are correctly aligned. Therefore, when carrying out preparatory and roofing work, templates should be used. The first prefabricated frame is used as such a template.

Useful advice

To prevent the transparent coating from moving during the cutting process, it must be clamped with a tool, using wooden planks as spacers. It is best to cut plastic roofing with a fine-toothed saw. The tool must be slightly tilted and used without pressure. Otherwise, the hacksaw blade will jam.

Sources:

• Device different types garage roofs in 2019

With the onset of summer, I want to change my wardrobe, add new colors and styles to it. You don’t have to go to the store for this - you can sew some clothing models yourself. A sundress is rightfully considered one of the easiest items of clothing to make. Just choose good light fabric, make a pattern and sew all the parts together.

You will need

• - paper;
• - pencil;
• - measuring tape;
• - ruler;
• - scissors.

Instructions

Take a measuring tape and measure the following distances: DSP - length of the back to the waist, DSB - length of the back to the hips, PG - distance from the shoulder to the top of the chest, OT - waist circumference, OB - hips volume, OG - chest volume, VT - distance between the upper points of the chest, DI - length of the product (from shoulder to hem).

Take a large sheet of paper (preferably special paper for patterns with millimeter markings) and draw a rectangle, the length of which is equal to DI and the width is equal to a quarter of OG. If your hips are larger than your chest, the width of the rectangle should be equal to a quarter of the OB. This will be half the front. Please mark one of the following vertical sides like the middle.

Find your waist, chest and hips. To do this, from the upper border of the rectangle, measure distances equal to PG, DST, and DSB and draw at this level horizontal lines.

Find the top point of your chest. To do this, measure half of the VT along the chest line from the middle of the front. Draw a vertical line from this point across the entire rectangle.

At the intersection of this line with the waist line, make a dart; to do this, set aside 2–4 cm to the right and left of the intersection point. Connect these two points with the upper point of the chest and the hip line. You should end up with a long vertical diamond shape. Make a second dart along the side seam (you will get half a rhombus).

Decorate the top of the sundress as you wish in the shape of the letter “L”. You can make a round, triangular or straight cut. Make the armhole low or high, depending on your figure. At the top of the "L" (at the intersection of the armhole and the neckline), fasten the straps.

Construct the back pattern in the same way. The difference between the back and the front is that the upper part will simply be cut horizontally, at the height of the intersection of the armhole line with the side line.

Cut out the details of the sundress pattern and start sewing.

Stages are platforms near the coastline, as if floating above the water.

They are usually wooden and represent an extension of the garden path. You can put a wooden gazebo or bench on the stage, sitting on which you can enjoy fishing or just admiring the pond. And if you can swim in a pond, then more convenient place there is no place for diving.

Designing and installing scaffolding is an interesting and creative task:

1. First, piles are installed, they can be made from a metal pipe (100x100 mm),

2. Then a wooden or metal frame is attached to them, to which the flooring boards are already attached. Gaps are left between them to ventilate the wood.

3. On the shore, every three meters, foundation pillars are built on which the deck rests. They should rise 20-30 cm above the water, given that during rainy periods the water level rises. According to experts, the stage is made of no more than 25% of the water surface.

So far we have considered the simplest functional model, in which function depends on the only thing argument. But when studying various phenomena of the surrounding world, we often encounter simultaneous changes in more than two quantities, and many processes can be effectively formalized function of several variables, Where - arguments or independent variables. Let's start developing the topic with the most common one in practice. functions of two variables .

Function of two variables called law, according to which each pair of values independent variables(arguments) from domain of definition corresponds to the value of the dependent variable (function).

This function denoted as follows:

Either or the other standard letter:

Since the ordered pair of values ​​"x" and "y" determines point on the plane, then the function is also written through , where is a point on the plane with coordinates . This notation is widely used in some practical tasks.

Geometric meaning of a function of two variables very simple. If a function of one variable corresponds to a certain line on a plane (for example, the familiar school parabola), then the graph of a function of two variables is located in three-dimensional space. In practice, most often we have to deal with surface, but sometimes the graph of a function can be, for example, a spatial line(s) or even a single point.

We are well familiar with the elementary example of a surface from the course analytical geometry- This plane. Assuming that , the equation can easily be rewritten as functional form:

The most important attribute of a function of 2 variables is the already stated domain of definition.

Domain of a function of two variables called a set everyone pairs for which the value exists.

Graphically, the domain of definition is the entire plane or part of it. Thus, the domain of definition of the function is the entire coordinate plane - for the reason that for any point exists value .

But such an idle arrangement does not always happen, of course:

Like two variables?

Considering various concepts functions of several variables, it is useful to draw analogies with the corresponding concepts of functions of one variable. In particular, when figuring out domain of definition we paid special attention to those functions that contain fractions, roots even degree, logarithms, etc. Everything is exactly the same here!

The task of finding the domain of definition of a function of two variables with almost 100% probability will be encountered in your thematic work, so I will analyze a decent number of examples:

Example 1

Find the domain of a function

Solution: since the denominator cannot go to zero, then:

Answer: the entire coordinate plane except points belonging to the line

Yes, yes, it is better to write the answer in this style. The domain of definition of a function of two variables is rarely denoted by any symbol; much more often they use verbal description and/or drawing.

If by condition required make a drawing, then it would be necessary to depict the coordinate plane and dotted line make a straight line. The dotted line indicates that the line not included into the domain of definition.

As we will see a little later, in more difficult examples you cannot do without a drawing at all.

Example 2

Find the domain of a function

Solution: the radical expression must be non-negative:

Answer: half-plane

Graphic representation here it’s also primitive: we draw a Cartesian coordinate system, solid draw a straight line and shade the top half-plane. The solid line indicates the fact that it included into the domain of definition.

Attention! If you don’t understand ANYTHING from the second example, please study/repeat the lesson in detail Linear inequalities– without him it will be very difficult!

Thumbnail for independent decision:

Example 3

Find the domain of a function

Two line solution and answer at the end of the lesson.

Let's continue to warm up:

Example 4

And depict it on the drawing

Solution: it is easy to understand that this is the formulation of the problem requires execution of the drawing (even if the domain of definition is very simple). But first, analytics: the radical of the expression must be non-negative: and, given that the denominator cannot go to zero, the inequality becomes strict:

How to determine the area that the inequality defines? I recommend the same algorithm of actions as in the solution linear inequalities.

First we draw line, which is set corresponding equality. The equation determines circle centered at the origin of a radius that divides the coordinate plane into two parts - the “inside” and “exterior” of the circle. Since we have inequality strict, then the circle itself will certainly not be included in the domain of definition and therefore it must be drawn dotted line.

Now let's take it arbitrary plane point, not belonging to circle, and substitute its coordinates into the inequality. The easiest way, of course, is to choose the origin:

Received false inequality, thus, point does not satisfy inequality Moreover, this inequality is not satisfied by any point lying inside the circle, and, therefore, the desired domain of definition is its outer part. The definition area is traditionally hatched:

Anyone can take any point belonging to the shaded area and make sure that its coordinates satisfy the inequality. By the way, the opposite inequality gives circle centered at the origin, radius .

Answer: outer part of the circle

Let's return to the geometric meaning of the problem: now we have found the domain of definition and shaded it, what does this mean? This means that at each point of the shaded area there is a value “zet” and graphically the function is the following surface:

The schematic drawing clearly shows that this surface is located in places over plane (near and far octants from us), in some places – under plane (left and right octants relative to us). The surface also passes through the axes. But the behavior of the function as such is not very interesting to us now - what is important is that all this happens exclusively in the field of definition. If we take any point belonging to the circle, then there will be no surface there (since there is no “zet”), as evidenced by the round space in the middle of the picture.

Please understand this example well, because in it I in more detail explained the very essence of the problem.

The following task is for you to solve on your own:

Example 5

A short solution and drawing at the end of the lesson. In general, in the topic under consideration among 2nd order lines the most popular is the circle, but, as an option, they can “push” into the problem ellipse, hyperbole or parabola.

Let's move up:

Example 6

Find the domain of a function

Solution: the radical expression must be non-negative: and the denominator cannot be equal to zero: . Thus, the domain of definition is specified by the system.

We deal with the first condition using the standard scheme discussed in the lesson. Linear inequalities: draw a straight line and determine the half-plane that corresponds to the inequality. Because inequality non-strict, then the straight line itself will also be a solution.

With the second condition of the system, everything is also simple: the equation specifies the ordinate axis, and since , then it should be excluded from the domain of definition.

Let's draw the drawing, not forgetting that the solid line indicates its entry into the definition area, and the dotted line indicates its exclusion from this area:

It should be noted that here we are already forced make a drawing. And this situation is typical - in many tasks, a verbal description of the area is difficult, and even if you describe it, you will most likely be poorly understood and forced to depict the area.

Answer: scope of definition:

By the way, such an answer without a drawing really looks damp.

Let's repeat it again geometric meaning obtained result: in the shaded area there is a graph of the function, which represents surface of three-dimensional space. This surface can be located above/below the plane, can intersect the plane - in in this case We have all this in parallel. The very fact of the existence of the surface is important, and it is important to correctly find the region in which it exists.

Example 7

Find the domain of a function

This is an example for you to solve on your own. An approximate example of a final task at the end of the lesson.

It’s not uncommon for seemingly simple functions to produce a far from hasty solution:

Example 8

Find the domain of a function

Solution: using square difference formula, let us factorize the radical expression: .

The product of two factors is non-negative , When both multipliers are non-negative: OR When both non-positive: . This is a typical feature. Thus, we need to solve two systems of linear inequalities And COMBINE received areas. In a similar situation, instead of the standard algorithm, the method of scientific, or more precisely, practical poking works much faster =)

We draw straight lines that divide the coordinate plane into 4 “corners”. We take some point belonging to the upper “corner”, for example, a point and substitute its coordinates into the equations of the 1st system: . The correct inequalities are obtained, which means that the solution to the system is all top "corner". Shading.

Now we take the point belonging to the right “corner”. The 2nd system remains, into which we substitute the coordinates of this point: . The second inequality is not true, therefore and all the right "corner" is not a solution to the system.

A similar story is with the left “corner”, which is also not included in the scope of the definition.

And finally, we substitute the coordinates of the experimental point of the lower “corner” into the 2nd system: . Both inequalities are true, which means that the solution to the system is and all the lower “corner”, which should also be shaded.

In reality, of course, there is no need to describe it in such detail - all the commented actions are easily performed orally!

Answer: the domain of definition is association system solutions .

As you might guess, without a drawing such an answer is unlikely to work, and this circumstance forces you to pick up a ruler and pencil, even though the condition did not require it.

And this is your nut:

Example 9

Find the domain of a function

A good student always misses logarithms:

Example 10

Find the domain of a function

Solution: the argument of the logarithm is strictly positive, so the domain of definition is given by the system.

The inequality indicates the right half-plane and excludes the axis.

With the second condition the situation is more intricate, but also transparent. Let's remember sinusoid. The argument is “Igrek”, but this should not confuse me – Igrek, so Igrek, Zyu, so Zyu. Where is sine greater than zero? Sine is greater than zero, for example, on the interval. Since the function is periodic, there are infinitely many such intervals and in collapsed form the solution to the inequality will be written as follows:
, where is an arbitrary integer.

An infinite number of intervals, of course, cannot be depicted, so we will limit ourselves to the interval and its neighbors:

Let's complete the drawing, not forgetting that according to the first condition, our field of activity is limited strictly to the right half-plane:

hmm...it turned out to be some kind of ghost drawing...a good representation of higher mathematics...

Answer:

The next logarithm is yours:

Example 11

Find the domain of a function

During the solution, you will have to build parabola, which will divide the plane into 2 parts - the “inside” located between the branches, and outer part. The method of finding the required part has appeared repeatedly in the article Linear inequalities and previous examples in this lesson.

Solution, drawing and answer at the end of the lesson.

The final nuts of the paragraph are devoted to “arches”:

Example 12

Find the domain of a function

Solution: The arcsine argument must be within the following limits:

Then there are two technical capabilities: more prepared readers by analogy with latest examples lesson Domain of a function of one variable they can “roll” the double inequality and leave the “Y” in the middle. For dummies, I recommend converting the “locomotive” into an equivalent system of inequalities:

The system is solved as usual - we construct straight lines and find the necessary half-planes. As a result:

Please note that here the boundaries are included in the definition area and straight lines are drawn as solid lines. This must always be carefully monitored to avoid a serious mistake.

Answer: the domain of definition represents the solution of the system

Example 13

Find the domain of a function

The sample solution uses an advanced technique - converting double inequalities.

In practice, we also sometimes encounter problems involving finding the domain of definition functions of three variables The domain of definition of a function of three variables can be All three-dimensional space, or part of it. In the first case the function is defined for any points in space, in the second - only for those points that belong to some spatial object, most often - body. It can be a rectangular parallelepiped, ellipsoid, "inside" parabolic cylinder etc. The task of finding the domain of definition of a function of three variables usually consists of finding this body and making a three-dimensional drawing. However, such examples are quite rare. (I only found a couple of pieces), and therefore I will limit myself to just this overview paragraph.

Level lines

To better understand this term, we will compare the axis with height: the higher the “Z” value, the greater the height, the lower the “Z” value, the lower the height. The height can also be negative.

A function in its domain of definition is a spatial graph; for definiteness and greater clarity, we will assume that this is a trivial surface. What are level lines? Figuratively speaking, level lines are horizontal “slices” of the surface at various heights. These “slices” or, more correctly, sections carried out by planes, after which they are projected onto the plane .

Definition: a function level line is a line on the plane at each point of which the function maintains a constant value: .

Thus, level lines help to figure out what a particular surface looks like - and they help without constructing a three-dimensional drawing! Let's consider specific task:

Example 14

Find and plot several level lines of a function graph

Solution: We examine the shape of a given surface using level lines. For convenience, let’s expand the entry “back to front”:

Obviously, in this case, “zet” (height) obviously cannot take negative values (since the sum of squares is non-negative). Thus, the surface is located in the upper half-space (above the plane).

Since the condition does not say at what specific heights the level lines need to be “cut off,” we are free to choose several “Z” values ​​at our discretion.

We examine the surface at zero height, to do this we put the value in the equality :

The solution to this equation is the point. That is, when the level line represents a point.

We rise to a unit height and “cut” our surface plane (substitute into the surface equation):

Thus, for height, the level line is a circle centered at a point of unit radius.

I remind you that all “slices” are projected onto the plane, and that’s why I write down two, not three, coordinates for points!

Now we take, for example, a plane and “cut” the surface under study with it (substituteinto the surface equation):

Thus, for heightthe level line is a circle centered at the radius point.

And, let's build another level line, say for :

–circle centered at a point of radius 3.

The level lines, as I have already emphasized, are located on the plane, but each line is signed - what height it corresponds to:

It is not difficult to understand that other level lines of the surface under consideration are also circles, and the higher we go up (we increase the “Z” value), the larger the radius becomes. Thus, the surface itself It is an endless bowl with an ovoid bottom, the top of which is located on a plane. This “bowl”, together with the axis, “comes straight out at you” from the monitor screen, that is, you are looking at its bottom =) And this is not without reason! Only I pour it on the road so lethally =) =)

Answer: the level lines of a given surface are concentric circles of the form

Note : when a degenerate circle of zero radius (point) is obtained

The very concept of a level line comes from cartography. To paraphrase the established mathematical expression, we can say that level line is a geographical location of points of the same height. Consider a certain mountain with level lines of 1000, 3000 and 5000 meters:

The figure clearly shows that the upper left slope of the mountain is much steeper than the lower right slope. Thus, level lines allow you to reflect the terrain on a “flat” map. By the way, here negative altitude values ​​also acquire a very specific meaning - after all, some areas of the Earth’s surface are located below the zero level of the world’s oceans.