In July 2020, NASA will launch an expedition to Mars. The spacecraft will deliver to Mars an electronic carrier with the names of all registered members of the expedition.

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One of these code variants should be copied and pasted into the code of your web page, preferably between tags and or right after the tag ... According to the first option, MathJax loads faster and slows down the page less. But the second option automatically tracks and loads the latest versions of MathJax. If you insert the first code, then it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in your site's dashboard, add a widget for inserting third-party JavaScript code, copy the first or second version of the download code above into it, and place the widget closer to the beginning of the template (by the way, this is not necessary at all because the MathJax script is loaded asynchronously). That's all. Now, learn the MathML, LaTeX, and ASCIIMathML markup syntax, and you're ready to embed math formulas into your website's web pages.

Another New Year's Eve ... frosty weather and snowflakes on the window pane ... All this prompted me to write again about ... fractals, and what Wolfram Alpha knows about it. There is an interesting article about this, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of 3D fractals.

A fractal can be visualized (described) as a geometric figure or a body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, it is a self-similar structure, considering the details of which with magnification, we will see the same shape as without magnification. Whereas in the case of a regular geometric shape (not a fractal), when we zoom in, we will see details that have a simpler shape than the original shape itself. For example, at a high enough magnification, part of the ellipse looks like a line segment. This does not happen with fractals: at any increase in them, we will again see the same complex shape, which with each increase will repeat over and over again.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art for Science: “Fractals are geometric shapes that are as complex in their details as in their general form. part of the fractal will be enlarged to the size of the whole, it will look like a whole, or exactly, or perhaps with a slight deformation. "

And to solve it, you need minimal knowledge of the topic. The next school year is coming to an end, everyone wants to go on vacation, and in order to bring this moment closer, I immediately get down to business:

Let's start with the area. The area referred to in the condition is limited closed set of points of the plane. For example, a set of points bounded by a triangle, including the WHOLE triangle (if from boundaries "Gouge" at least one point, then the area will cease to be closed)... In practice, there are also areas of rectangular, round and slightly more complex shapes. It should be noted that strict definitions are given in the theory of mathematical analysis limitations, isolation, boundaries, etc., but I think everyone is aware of these concepts on an intuitive level, and more is not needed now.

A flat area is denoted by a letter as a standard, and, as a rule, it is set analytically - by several equations (not necessarily linear); less often inequalities. Typical turnover: "closed area, bounded by lines."

An integral part of the task under consideration is the construction of an area in the drawing. How to do it? It is necessary to draw all the listed lines (in this case, 3 straight) and analyze what happened. The desired area is usually slightly hatched, and its border is highlighted with a bold line:


The same area can be set and linear inequalities:, which for some reason are more often written as an enumerated list, and not system.
Since the boundary belongs to the region, all inequalities, of course, lax.

And now the essence of the problem. Imagine an axis extending from the origin directly towards you. Consider a function that continuous in each point of the area. The graph of this function represents some surface, and a small happiness lies in the fact that to solve today's problem, we do not need to know what this surface looks like. It can be located higher, lower, intersect the plane - all this is not important. And the following is important: according to weierstrass theorems, continuous in limited closedarea, the function reaches the maximum (the "highest") and the smallest (the "lowest") values \u200b\u200bthat you want to find. Such values \u200b\u200bare achieved or in stationary points, belonging to the regionD , orat points that lie on the border of this area. From what follows a simple and transparent solution algorithm:

Example 1

In a limited enclosed area

Decision: First of all, you need to depict the area in the drawing. Unfortunately, it is technically difficult for me to make an interactive model of the problem, and therefore I will immediately give the final illustration, which shows all the "suspicious" points found during the research. Usually they are affixed one after the other as they are found:

Based on the preamble, it is convenient to split the decision into two points:

I) Find stationary points. This is a standard action that we have repeatedly performed in the lesson. extrema of several variables:

Found stationary point belongs areas: (mark it on the drawing), which means that we should calculate the value of the function at this point:

- as in the article The largest and smallest values \u200b\u200bof the function on the segment, I will highlight the important results in bold. In a notebook it is convenient to outline them with a pencil.

Pay attention to our second happiness - there is no point in checking sufficient condition for extremum... Why? Even if at a point the function reaches, for example, local minimum, then it STILL DOES NOT MEAN that the resulting value will be minimal throughout the region (see the beginning of the lesson about unconditional extrema) .

What if the stationary point does NOT belong to the area? Almost nothing! It should be noted that and go to the next item.

II) Explore the boundary of the region.

Since the border consists of the sides of a triangle, it is convenient to divide the study into 3 sub-items. But it's better not to do it anyhow. From my point of view, at first it is more advantageous to consider the segments parallel to the coordinate axes, and first of all - lying on the axes themselves. To catch the whole sequence and logic of actions, try to study the ending "in one go":

1) Let's deal with the bottom side of the triangle. To do this, we substitute directly into the function:

Alternatively, you can arrange it like this:

Geometrically, this means that the coordinate plane (which is also given by the equation) "Carves" out surfaces A "spatial" parabola, the vertex of which immediately comes under suspicion. Let's find out where is she:

- the obtained value "hit" the area, and it may well be that at the point (mark in the drawing) the function reaches the highest or the lowest value in the entire area. One way or another, we carry out calculations:

Other "candidates" are, of course, the ends of the segment. Let's calculate the values \u200b\u200bof the function at points (mark in the drawing):

Here, by the way, you can perform a verbal mini-check using the "stripped-down" version:

2) To study the right side of the triangle, we substitute it into the function and "put things in order there":

Here we will immediately perform a rough check, "ringing out" the already processed end of the segment:
, well.

The geometric situation is related to the previous point:

- the resulting value is also "included in the scope of our interests", which means that we need to calculate what the function is equal to at the point that appears:

Let's examine the second end of the segment:

Using the function , let's check it out:

3) Probably everyone knows how to explore the remaining side. We substitute in the function and perform simplifications:

Segment ends have already been researched, but on the draft we still check if we found the function correctly :
- coincided with the result of the 1st subparagraph;
- coincided with the result of the 2nd subparagraph.

It remains to find out if there is something interesting inside the segment:

- there is! Substituting a straight line into the equation, we get the ordinate of this "interestingness":

We mark a point in the drawing and find the corresponding value of the function:

Let's check the calculations according to the "budget" version :
, order.

And the final step: CAREFULLY we look through all the "fat" numbers, I recommend that beginners even make a single list:

from which we choose the largest and smallest values. Answer we write in the style of the problem of finding the largest and smallest values \u200b\u200bof the function on the segment:

Just in case, I will once again comment on the geometric meaning of the result:
- here is the highest point of the surface in the area;
Is the lowest surface point in the area.

In the analyzed problem, we identified 7 "suspicious" points, but their number varies from problem to problem. For a triangular area, the minimum "research set" consists of three points. This happens when a function, for example, sets plane - it is quite clear that there are no stationary points, and the function can reach the largest / smallest values \u200b\u200bonly at the vertices of the triangle. But there are a lot of such examples once, twice - usually you have to deal with some surface of the 2nd order.

If you solve such tasks a little, then the head can go round from the triangles, and therefore I have prepared unusual examples for you to make it square :))

Example 2

Find the largest and smallest function values in a closed area bounded by lines

Example 3

Find the largest and smallest values \u200b\u200bof a function in a bounded closed area.

Pay special attention to the rational order and technique for examining the region boundary, as well as to the chain of intermediate checks, which will almost completely avoid computational errors. Generally speaking, you can solve it as you like, but in some problems, for example, in the same Example 2, there is every chance to significantly complicate your life. An approximate example of finishing assignments at the end of the lesson.

Let's systematize the solution algorithm, otherwise, with my diligence as a spider, it somehow got lost in the long thread of comments from the 1st example:

- At the first step, we build an area, it is desirable to shade it, and highlight the border with a bold line. During the solution, points will appear that need to be placed on the drawing.

- Find stationary points and calculate the values \u200b\u200bof the function only in those of themthat belong to the area. We select the obtained values \u200b\u200bin the text (for example, we outline them with a pencil). If the stationary point does NOT belong to the region, then we mark this fact with an icon or verbally. If there are no stationary points at all, then we draw a written conclusion that they are absent. In any case, this item cannot be skipped!

- Let's explore the border of the area. At first, it is beneficial to deal with straight lines that are parallel to the coordinate axes (if any)... We also highlight the values \u200b\u200bof the function calculated at the "suspicious" points. A lot has been said above about the solution technique and something else will be said below - read, re-read, delve into!

- From the selected numbers, select the largest and smallest values \u200b\u200band give the answer. Sometimes it happens that the function reaches such values \u200b\u200bat several points at once - in this case, all these points should be reflected in the answer. Let, for example, and it turned out to be the smallest value. Then we write down that

The final examples are devoted to other useful ideas that will come in handy in practice:

Example 4

Find the largest and smallest values \u200b\u200bof a function in a closed area .

I have kept the author's formulation, in which the region is given as a double inequality. This condition can be written by an equivalent system or in a more traditional form for this problem:

I remind you that since nonlinear inequalities we encountered on, and if you do not understand the geometric meaning of the notation, then please do not postpone and clarify the situation right now ;-)

Decision, as always, it starts with building an area, which is a kind of "sole":

Hmm, sometimes you have to gnaw not only the granite of science….

I) Find stationary points:

System-idiot's dream :)

A stationary point belongs to the region, namely, lies on its boundary.

And so, it is, nothing ... the lesson went cheerfully - that's what it means to drink the right tea \u003d)

II) Explore the boundary of the region. Without further ado, let's start with the abscissa:

1) If, then

Let's find where the vertex of the parabola is:
- appreciate such moments - "hit" right at the point from which everything is already clear. But don't forget about checking:

Let's calculate the values \u200b\u200bof the function at the ends of the segment:

2) We will deal with the lower part of the “sole” “in one sitting” - without any complexes we substitute it into the function, moreover, we will only be interested in the segment:

Control:

This already brings some revival to the monotonous driving on the knurled track. Let's find the critical points:

We solve quadratic equation, remember this one more? ... However, remember, of course, otherwise you would not have read these lines \u003d) If in the two previous examples it was convenient to calculate in decimal fractions (which, by the way, is rare), then here we are waiting for the usual ordinary fractions. We find the “x” roots and use the equation to determine the corresponding “game” coordinates of the “candidate” points:


Let's calculate the values \u200b\u200bof the function at the points found:

Check the function yourself.

Now we carefully study the won trophies and write down answer:

These are "candidates", so "candidates"!

For an independent solution:

Example 5

Find the smallest and largest values \u200b\u200bof a function in a closed area

An entry with curly braces reads like this: "many points, such that".

Sometimes in such examples they use lagrange multiplier method, but the real need to apply it is unlikely to arise. So, for example, if a function is given with the same domain "de", then after substitution into it - with a derivative of no difficulties; moreover, everything is drawn up "in one line" (with signs) without the need to consider the upper and lower semicircles separately. But, of course, there are also more complex cases where, without the Lagrange function (where, for example, the same equation of the circle) it is difficult to manage - how difficult it is to do without a good rest!

It's good for everyone to pass the session and see you soon next season!

Solutions and Answers:

Example 2: Decision: depict the area in the drawing:

With this service you can find the largest and smallest function value one variable f (x) with the design of the solution in Word. If the function f (x, y) is given, therefore, it is necessary to find the extremum of the function of two variables. You can also find the intervals of increasing and decreasing of the function.

Function entry rules:

A necessary condition for the extremum of a function of one variable

Sufficient condition for the extremum of a function of one variable

F "0 (x *) \u003d 0
f "" 0 (x *)\u003e 0

The point x * is the point of the local (global) minimum of the function.

If at point x * the condition is satisfied:

F "0 (x *) \u003d 0
f "" 0 (x *)< 0

Then point x * is the local (global) maximum.

Example # 1. Find the largest and smallest values \u200b\u200bof the function: on the segment.
Decision.

One critical point x 1 \u003d 2 (f '(x) \u003d 0). This point belongs to the line segment. (The point x \u003d 0 is not critical, since 0∉).
We calculate the values \u200b\u200bof the function at the ends of the segment and at the critical point.
f (1) \u003d 9, f (2) \u003d 5/2, f (3) \u003d 3 8/81
Answer: f min \u003d 5/2 at x \u003d 2; f max \u003d 9 at x \u003d 1

Example # 2. Using the derivatives of higher orders, find the extremum of the function y \u003d x-2sin (x).
Decision.
Find the derivative of the function: y ’\u003d 1-2cos (x). Find the critical points: 1-cos (x) \u003d 2, cos (x) \u003d ½, x \u003d ± π / 3 + 2πk, k∈Z. We find y ’’ \u003d 2sin (x), calculate, so x \u003d π / 3 + 2πk, k∈Z are the minimum points of the function; , so x \u003d - π / 3 + 2πk, k∈Z are the maximum points of the function.

Example No. 3. Explore the extremum function in the vicinity of the point x \u003d 0.
Decision. Here it is necessary to find the extrema of the function. If the extremum is x \u003d 0, then find out its type (minimum or maximum). If there is no x \u003d 0 among the found points, then calculate the value of the function f (x \u003d 0).
It should be noted that when the derivative on each side of a given point does not change its sign, the possible situations are not exhausted even for differentiable functions: it may happen that for an arbitrarily small neighborhood on one side of the point x 0 or on both sides the derivative changes sign. At these points, one has to apply other methods to study functions for extremum.

Example No. 4. Divide the number 49 into two terms, the product of which will be the largest.
Decision. Let us denote x as the first term. Then (49-x) is the second term.
The product will be the maximum: x (49-x) → max

Let's see how to explore a function using a graph. It turns out that looking at the chart, you can find out everything that interests us, namely:

• function domain
• function range
• function zeros
• increasing and decreasing intervals
• maximum and minimum points
• the largest and smallest value of the function on the segment.

Let's clarify the terminology:

Abscissa is the horizontal coordinate of the point.
Ordinate is the vertical coordinate.
Abscissa axis - a horizontal axis, most often called an axis.
Y-axis - vertical axis, or axis.

Argument is the independent variable on which the values \u200b\u200bof the function depend. Most often indicated.
In other words, we ourselves choose, substitute functions into the formula and get.

Domain functions - the set of those (and only those) values \u200b\u200bof the argument for which the function exists.
It is indicated by: or.

In our figure, the function domain is a segment. It is on this segment that the function graph is drawn. Only here this function exists.

Function range is the set of values \u200b\u200bthat a variable takes. In our picture, this is a segment - from the lowest to the highest value.

Function zeros - points where the value of the function is equal to zero, that is. In our figure, these are points and.

Function values \u200b\u200bare positive where . In our figure, these are gaps and.
Function values \u200b\u200bare negative where . We have this interval (or interval) from to.

The most important concepts are increasing and decreasing function on some set. As a set, you can take a segment, an interval, a union of intervals, or the entire number line.

Function increases

In other words, the more, the more, that is, the chart goes to the right and up.

Function decreases on a set if for any and belonging to the set, the inequality follows from the inequality.

For a decreasing function, a larger value corresponds to a smaller value. The graph goes to the right and down.

In our figure, the function increases in the interval and decreases in the intervals and.

Let's define what is maximum and minimum points of the function.

Maximum point is an internal point of the domain of definition, such that the value of the function in it is greater than at all points sufficiently close to it.
In other words, the maximum point is such a point, the value of the function at which morethan in the neighboring ones. This is a local "mound" on the chart.

In our picture - the maximum point.

Minimum point - an internal point of the domain of definition, such that the value of the function in it is less than at all points sufficiently close to it.
That is, the minimum point is such that the value of the function in it is less than in the neighboring ones. This is a local “hole” on the chart.

In our picture - the minimum point.

The point is the boundary. It is not an interior point of the domain and therefore does not fit the definition of a maximum point. After all, she has no neighbors on the left. In the same way, it cannot be a minimum point on our chart.

The maximum and minimum points are collectively called extremum points of the function... In our case, this is and.

And what to do if you need to find, for example, minimum function on the segment? In this case, the answer is. Because minimum function is its value at the minimum point.

Likewise, the maximum of our function is. It is reached at a point.

We can say that the extrema of the function are equal to and.

Sometimes in tasks you need to find largest and smallest function values on a given segment. They do not necessarily coincide with extremes.

In our case smallest function value on the segment is equal to and coincides with the minimum of the function. But its greatest value on this segment is equal to. It is reached at the left end of the line.

In any case, the largest and smallest values \u200b\u200bof a continuous function on a segment are achieved either at the extremum points or at the ends of the segment.

The largest value of a function is called at most, the smallest value is the least of all its values.

A function can have only one largest and only one smallest value, or it may not have them at all. Finding the largest and smallest values \u200b\u200bof continuous functions is based on the following properties of these functions:

1) If in some interval (finite or infinite) the function y \u003d f (x) is continuous and has only one extremum and if it is a maximum (minimum), then it will be the largest (smallest) value of the function in this interval.

2) If the function f (x) is continuous on some segment, then it necessarily has the largest and smallest values \u200b\u200bon this segment. These values \u200b\u200bare reached either at the extremum points lying within the segment, or at the boundaries of this segment.

To find the largest and smallest values \u200b\u200bon a segment, it is recommended to use the following scheme:

1. Find the derivative.

2. Find the critical points of the function at which \u003d 0 or does not exist.

3. Find the values \u200b\u200bof the function at critical points and at the ends of the segment and choose from them the largest f naib and the smallest f naim.

When solving applied problems, in particular, optimization problems, the tasks of finding the largest and smallest values \u200b\u200b(global maximum and global minimum) of a function on the interval X are important. To solve such problems, one should, based on the condition, choose an independent variable and express the investigated value through this variable. Then find the desired largest or smallest value of the resulting function. In this case, the interval of variation of the independent variable, which can be finite or infinite, is also determined from the problem statement.

Example. The tank, which has the shape of a rectangular parallelepiped with a square bottom open at the top, needs to be fished out with tin inside. What should be the dimensions of the tank with its capacity of 108 liters. water so that the cost of tinning it is the lowest?

Decision. The cost of coating the tank with tin will be the least if its surface is minimal for a given capacity. Let us denote by a dm - the side of the base, b dm - the height of the tank. Then the area S of its surface is equal to

AND

The resulting relationship establishes the relationship between the surface area of \u200b\u200bthe tank S (function) and the side of the base a (argument). Let us examine the function S for an extremum. Find the first derivative, equate it to zero and solve the resulting equation:

Hence a \u003d 6. (a)\u003e 0 for a\u003e 6, (a)< 0 при а < 6. Следовательно, при а = 6 функция S имеет минимум. Если а = 6, то b = 3. Таким образом, затраты на лужение резервуара емкостью 108 литров будут наименьшими, если он имеет размеры 6дм х 6дм х 3дм.

Example... Find the largest and smallest function values in between.

Decision: The specified function is continuous on the entire number axis. Function derivative

Derivative at and at. Let's calculate the values \u200b\u200bof the function at these points:

.

The function values \u200b\u200bat the ends of the given interval are equal. Therefore, the largest value of the function is at, the smallest value of the function is at.

Self-test questions

1. Formulate L'Hôpital's rule for disclosing the uncertainties of the form. List the different types of uncertainties that L'Hôpital's rule can be used to address.

2. Formulate the signs of increasing and decreasing functions.

3. Give the definition of the maximum and minimum of the function.

4. Formulate the necessary condition for the existence of an extremum.

5. What values \u200b\u200bof the argument (which points) are called critical? How do you find these points?

6. What are the sufficient criteria for the existence of an extremum of a function? Outline the scheme for studying the function for an extremum using the first derivative.

7. Describe the scheme of studying the function for the extremum using the second derivative.

8. Give the definition of convexity, concavity of a curve.

9. What is called the inflection point of the function graph? Specify a way to find these points.

10. Formulate the necessary and sufficient criteria for the convexity and concavity of a curve on a given segment.

11. Give the definition of the curve asymptote. How to find the vertical, horizontal and oblique asymptotes of the graph of a function?

12. Outline the general scheme of the study of the function and the construction of its graph.

13. Formulate a rule for finding the largest and smallest values \u200b\u200bof a function on a given segment.