Conduct complete research on the feature online. Full examination of the function and plotting of the graph. Respecting your privacy at the company level

For some time now, TheBat's built-in certificate database for SSL has stopped working correctly (it is not clear for what reason).

When checking the post, an error appears:

Unknown CA certificate
The server did not present a root certificate in the session and the corresponding root certificate was not found in the address book.
This connection cannot be secret. Please
contact your server administrator.

And you are offered a choice of answers - YES / NO. And so every time you remove mail.

Solution

In this case, you need to replace the S/MIME and TLS implementation standard with Microsoft CryptoAPI in the TheBat settings!

Since I needed to combine all the files into one, I first converted everything doc files into a single pdf file(using the Acrobat program), and then transferred it to fb2 through an online converter. You can also convert files individually. The formats can be absolutely any (source) - doc, jpg, and even a zip archive!

The name of the site corresponds to the essence :) Online Photoshop.

Update May 2015

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In my life I came across the problem of repairing an electric stove. I’ve already done a lot of things, learned a lot, but somehow had little to do with tiles. It was necessary to replace the contacts on the regulators and burners. The question arose - how to determine the diameter of the burner on an electric stove?

The answer turned out to be simple. You don’t need to measure anything, you can easily determine by eye what size you need.

Smallest burner- this is 145 millimeters (14.5 centimeters)

Middle burner- this is 180 millimeters (18 centimeters).

And finally, the most large burner- this is 225 millimeters (22.5 centimeters).

It is enough to determine the size by eye and understand what diameter you need the burner. When I didn’t know this, I was worried about these dimensions, I didn’t know how to measure, which edge to navigate, etc. Now I'm wise :) I hope I helped you too!

In my life I faced such a problem. I think I'm not the only one.

ABSTRACT

“Full study of a function and construction of its graph.”

INTRODUCTION

Studying the properties of a function and plotting its graph is one of the most wonderful applications of derivatives. This method of studying function has been repeatedly subjected to careful analysis. The main reason is that in applications of mathematics one had to deal with more and more complex functions that appear when studying new phenomena. Exceptions to the rules developed by mathematics appeared, cases appeared when the rules created were not suitable at all, functions appeared that did not have a derivative at any point.

The purpose of studying the course of algebra and beginnings of analysis in grades 10-11 is the systematic study of functions, disclosure of the applied meaning common methods mathematics related to the study of functions.

The development of functional concepts in the course of studying algebra and the beginnings of analysis at the senior level of education helps high school students to obtain visual ideas about the continuity and discontinuities of functions, learn about the continuity of any elementary function in the field of its application, learn to construct their graphs and generalize information about the main elementary functions and understand them role in the study of phenomena of reality, in human practice.

Increasing and decreasing function

Solving various problems from the fields of mathematics, physics and technology leads to the establishment of a functional relationship between those involved in this phenomenon variable quantities.

If such a functional dependence can be expressed analytically, that is, in the form of one or more formulas, then it becomes possible to study it by means of mathematical analysis.

This refers to the possibility of clarifying the behavior of a function when one or another variable changes (where the function increases, where it decreases, where it reaches a maximum, etc.).

The application of differential calculus to the study of a function is based on a very simple connection that exists between the behavior of a function and the properties of its derivative, primarily its first and second derivatives.

Let's consider how we can find intervals of increasing or decreasing function, that is, intervals of its monotonicity. Based on the definition of a monotonically decreasing and increasing function, it is possible to formulate theorems that allow us to relate the value of the first derivative of a given function to the nature of its monotonicity.

Theorem 1.1. If the function y = f ( x ) , differentiable on the interval( a , b ) , increases monotonically on this interval, then at any point
( x ) >0; if it decreases monotonically, then at any point in the interval ( x )<0.

Proof. Let the functiony = f ( x ) monotonically increases by( a , b ) , This means that for anyone small enough > 0 the following inequality holds:

f ( x - ) < f ( x ) < f ( x + ) (Fig. 1.1).

Rice. 1.1

Consider the limit

.

If > 0, then > 0 if< 0, то

< 0.

In both cases, the expression under the limit sign is positive, which means the limit is positive, that is ( x )>0 , which was what needed to be proven. The second part of the theorem, related to the monotonic decrease of the function, is proved in a similar way.

Theorem 1.2. If the function y = f ( x ) , continuous on the segment[ a , b ] and is differentiable in all its internal points, and besides, ( x ) >0 for anyone x ϵ ( a , b ) , then this function increases monotonically by( a , b ) ; If

( x ) <0 for anyone xϵ ( a , b ), then this function decreases monotonically by( a , b ) .

Proof. Let's take ϵ ( a , b ) And ϵ ( a , b ) , and< . According to Lagrange's theorem

( c ) = .

But ( c )>0 and > 0, which means ( > 0, that is

(. The result obtained indicates a monotonic increase in the function, which was what needed to be proven. The second part of the theorem is proved in a similar way.

Extrema of the function

When studying the behavior of a function, a special role is played by the points that separate from each other the intervals of monotonic increase from the intervals of its monotonic decrease.

Definition 2.1. Dot called the maximum point of the function

y = f ( x ) , if for any, however small , ( < 0 , а точка is called a minimum point if ( > 0.

The minimum and maximum points have common name extremum points. The piecewise monotonic function of such points has a finite number on a finite interval (Fig. 2.1).

Rice. 2.1

Theorem 2.1 ( necessary condition existence of an extremum). If differentiable on the interval( a , b ) function has at point from this interval is the maximum, then its derivative at this point is equal to zero. The same can be said about the minimum point .

The proof of this theorem follows from Rolle’s theorem, in which it was shown that at the points of minimum or maximum = 0, and the tangent drawn to the graph of the function at these points is parallel to the axisOX .

It follows from Theorem 2.1 that if the functiony = f ( x ) has a derivative at all points, then it can reach an extremum at those points where = 0.

However this condition is not sufficient, since there are functions for which the specified condition is satisfied, but there is no extremum. For example, the functiony= at a point x = 0 the derivative is zero, but there is no extremum at this point. In addition, the extremum may be at those points where the derivative does not exist. For example, the functiony = | x | there is a minimum at the pointx = 0 , although the derivative does not exist at this point.

Definition 2.2. The points at which the derivative of a function vanishes or has a discontinuity are called critical points of this function.

Consequently, Theorem 2.1 is not sufficient for determining extreme points.

Theorem 2.2 ( sufficient condition existence of an extremum). Let the function y = f ( x ) continuous on the interval( a , b ) , which contains its critical point , and is differentiable at all points of this interval, with the possible exception of the point itself . Then, if, when moving this point from left to right, the sign of the derivative changes from plus to minus, then this is a maximum point, and, conversely, from minus to plus - a minimum point.

Proof. If the derivative of a function changes its sign when passing a point from left to right from plus to minus, then the function moves from increasing to decreasing, that is, it reaches at the point its maximum and vice versa.

From the above, a scheme for studying a function at an extremum follows:

1) find the domain of definition of the function;

2) calculate the derivative;

3) find critical points;

4) by changing the sign of the first derivative, their character is determined.

The task of studying a function for an extremum should not be confused with the task of determining the minimum and maximum values ​​of a function on a segment. In the second case, it is necessary to find not only the extreme points on the segment, but also compare them with the value of the function at its ends.

Intervals of convex and concave functions

Another characteristic of the graph of a function that can be determined using the derivative is its convexity or concavity.

Definition 3.1. Function y = f ( x ) is called convex on the interval( a , b ) , if its graph is located below any tangent drawn to it on a given interval, and vice versa, it is called concave if its graph is above any tangent drawn to it on a given interval.

Let us prove a theorem that allows us to determine the intervals of convexity and concavity of a function.

Theorem 3.1. If at all points of the interval( a , b ) second derivative of the function ( x ) is continuous and negative, then the functiony = f ( x ) is convex and vice versa, if the second derivative is continuous and positive, then the function is concave.

We carry out the proof for the interval of convexity of the function. Let's take an arbitrary pointϵ ( a , b ) and draw a tangent to the graph of the function at this pointy = f ( x ) (Fig. 3.1).

The theorem will be proven if it is shown that all points of the curve on the interval( a , b ) lie under this tangent. In other words, it is necessary to prove that for the same valuesx curve ordinatesy = f ( x ) less than the ordinate of the tangent drawn to it at the point .

Rice. 3.1

For definiteness, we denote the equation of the curve: = f ( x ) , and the equation of the tangent to it at the point :

- f ( ) = ( )( x - )

or

= f ( ) + ( )( x - ) .

Let's make up the difference And :

- = f(x) – f( ) - ( )(x- ).

Apply to differencef ( x ) – f ( ) Lagrange's mean value theorem:

- = ( )( x - ) - ( )( x - ) = ( x - )[ ( ) - ( )] ,

Where ϵ ( , x ).

Let us now apply Lagrange's theorem to the expression in square brackets:

- = ( )( - )( x - ) , Where ϵ ( , ).

As can be seen from the figure,x > , Then x - > 0 And - > 0 . Moreover, according to the theorem, ( )<0.

Multiplying these three factors, we get that , which was what needed to be proven.

Definition 3.2. The point separating the convex interval from the concave interval is called the inflection point.

From Definition 3.1 it follows that at a given point the tangent intersects the curve, that is, on one side the curve is located below the tangent, and on the other, above.

Theorem 3.2. If at the point second derivative of the function

y = f ( x ) is equal to zero or does not exist, and when passing through a point the sign of the second derivative changes to the opposite, then this point is an inflection point.

The proof of this theorem follows from the fact that the signs ( x ) on opposite sides of the point are different. This means that on one side of the point the function is convex, and on the other it is concave. In this case, according to Definition 3.2, the point is the inflection point.

The study of a function for convexity and concavity is carried out according to the same scheme as the study for an extremum.

4. Asymptotes of the function

In the previous paragraphs, methods for studying the behavior of a function using the derivative were discussed. However, among the questions related to the complete study of a function, there are also those that are not related to the derivative.

So, for example, it is necessary to know how a function behaves when a point on its graph moves infinitely away from the origin. This problem can arise in two cases: when the argument of a function goes to infinity and when, during a discontinuity of the second kind at the end point, the function itself goes to infinity. In both of these cases, a situation may arise when the function tends to some straight line, called its asymptote.

Definition . Asymptote of the graph of a functiony = f ( x ) is a straight line that has the property that the distance from the graph to this straight line tends to zero as the graph point moves indefinitely from the origin.

There are two types of asymptotes: vertical and oblique.

Vertical asymptotes include straight linesx = , which have the property that the graph of the function in their vicinity goes to infinity, that is, the condition is satisfied: .

Obviously, the requirement of the specified definition is satisfied here: the distance from the graph of the curve to the straight linex = tends to zero, and the curve itself goes to infinity. So, at points of discontinuity of the second kind, functions have vertical asymptotes, for example,y= at a point x = 0 . Consequently, determining the vertical asymptotes of a function coincides with finding discontinuity points of the second kind.

Oblique asymptotes are described by the general equation of a straight line on a plane, that isy = kx + b . This means that, unlike vertical asymptotes, here it is necessary to determine the numbersk And b .

So let the curve = f ( x ) has an oblique asymptote, that is, atx the points of the curve come as close as desired to the straight line = kx + b (Fig. 4.1). Let M ( x , y ) - a point located on a curve. Its distance from the asymptote will be characterized by the length of the perpendicular| MN | .

Today we invite you to explore and build a graph of a function with us. After carefully studying this article, you will not have to sweat for long to complete this type of task. It is not easy to study and construct a graph of a function; it is a voluminous work that requires maximum attention and accuracy of calculations. To make the material easier to understand, we will study the same function step by step and explain all our actions and calculations. Welcome to the amazing and fascinating world of mathematics! Go!

Domain

In order to explore and graph a function, you need to know several definitions. Function is one of the main (basic) concepts in mathematics. It reflects the dependence between several variables (two, three or more) during changes. The function also shows the dependence of sets.

Imagine that we have two variables that have a certain range of change. So, y is a function of x, provided that each value of the second variable corresponds to one value of the second. In this case, the variable y is dependent, and it is called a function. It is customary to say that the variables x and y are in For greater clarity of this dependence, a graph of the function is built. What is a graph of a function? This is a set of points on the coordinate plane, where each x value corresponds to one y value. Graphs can be different - straight line, hyperbola, parabola, sine wave, and so on.

It is impossible to graph a function without research. Today we will learn how to conduct research and build a graph of a function. It is very important to take notes during the study. This will make the task much easier to cope with. The most convenient research plan:

1. Domain.
2. Continuity.
3. Even or odd.
4. Periodicity.
5. Asymptotes.
6. Zeros.
7. Sign constancy.
8. Increasing and decreasing.
9. Extremes.
10. Convexity and concavity.

Let's start with the first point. Let's find the domain of definition, that is, on what intervals our function exists: y=1/3(x^3-14x^2+49x-36). In our case, the function exists for any values ​​of x, that is, the domain of definition is equal to R. This can be written as follows xÎR.

Continuity

Now we will examine the discontinuity function. In mathematics, the term “continuity” appeared as a result of the study of the laws of motion. What is infinite? Space, time, some dependencies (an example is the dependence of the variables S and t in movement problems), the temperature of a heated object (water, frying pan, thermometer, etc.), a continuous line (that is, one that can be drawn without lifting it from the sheet pencil).

A graph is considered continuous if it does not break at some point. One of the most obvious examples of such a graph is a sinusoid, which you can see in the picture in this section. A function is continuous at some point x0 if a number of conditions are met:

• a function is defined at a given point;
• the right and left limits at a point are equal;
• the limit is equal to the value of the function at point x0.

If at least one condition is not met, the function is said to fail. And the points at which the function breaks are usually called break points. An example of a function that will “break” when displayed graphically is: y=(x+4)/(x-3). Moreover, y does not exist at the point x = 3 (since it is impossible to divide by zero).

In the function that we are studying (y=1/3(x^3-14x^2+49x-36)) everything turned out to be simple, since the graph will be continuous.

Even, odd

Now examine the function for parity. First, a little theory. An even function is one that satisfies the condition f(-x)=f(x) for any value of the variable x (from the range of values). Examples include:

• module x (the graph looks like a daw, the bisector of the first and second quarters of the graph);
• x squared (parabola);
• cosine x (cosine).

Note that all of these graphs are symmetrical when viewed with respect to the y-axis (that is, the y-axis).

What then is called an odd function? These are those functions that satisfy the condition: f(-x)=-f(x) for any value of the variable x. Examples:

• hyperbola;
• cubic parabola;
• sinusoid;
• tangent and so on.

Please note that these functions are symmetrical about the point (0:0), that is, the origin. Based on what was said in this section of the article, an even and odd function must have the property: x belongs to the definition set and -x too.

Let's examine the function for parity. We can see that she doesn't fit any of the descriptions. Therefore, our function is neither even nor odd.

Asymptotes

Let's start with a definition. An asymptote is a curve that is as close as possible to the graph, that is, the distance from a certain point tends to zero. In total, there are three types of asymptotes:

• vertical, that is, parallel to the y-axis;
• horizontal, that is, parallel to the x axis;
• inclined.

As for the first type, these lines should be looked for at some points:

• gap;
• ends of the domain of definition.

In our case, the function is continuous, and the domain of definition is equal to R. Therefore, there are no vertical asymptotes.

The graph of a function has a horizontal asymptote, which meets the following requirement: if x tends to infinity or minus infinity, and the limit is equal to a certain number (for example, a). In this case, y=a is the horizontal asymptote. There are no horizontal asymptotes in the function we are studying.

An oblique asymptote exists only if two conditions are met:

• lim(f(x))/x=k;
• lim f(x)-kx=b.

Then it can be found using the formula: y=kx+b. Again, in our case there are no oblique asymptotes.

Function zeros

The next step is to examine the graph of the function for zeros. It is also very important to note that the task associated with finding the zeros of a function occurs not only when studying and constructing a graph of a function, but also as an independent task and as a way to solve inequalities. You may be required to find the zeros of a function on a graph or use mathematical notation.

Finding these values ​​will help you graph the function more accurately. In simple terms, the zero of a function is the value of the variable x at which y = 0. If you are looking for the zeros of a function on a graph, then you should pay attention to the points at which the graph intersects with the x-axis.

To find the zeros of the function, you need to solve the following equation: y=1/3(x^3-14x^2+49x-36)=0. After carrying out the necessary calculations, we get the following answer:

Sign constancy

The next stage of research and construction of a function (graph) is finding intervals of constant sign. This means that we must determine at which intervals the function takes a positive value and at which intervals it takes a negative value. The zero functions found in the last section will help us do this. So, we need to build a straight line (separate from the graph) and distribute the zeros of the function along it in the correct order from smallest to largest. Now you need to determine which of the resulting intervals has a “+” sign and which has a “-”.

In our case, the function takes a positive value on intervals:

• from 1 to 4;
• from 9 to infinity.

Negative meaning:

• from minus infinity to 1;
• from 4 to 9.

This is quite easy to determine. Substitute any number from the interval into the function and see what sign the answer turns out to have (minus or plus).

Increasing and decreasing function

In order to explore and construct a function, we need to know where the graph will increase (go up along the Oy axis) and where it will fall (crawl down along the y-axis).

A function increases only if a larger value of the variable x corresponds to a larger value of y. That is, x2 is greater than x1, and f(x2) is greater than f(x1). And we observe a completely opposite phenomenon with a decreasing function (the more x, the less y). To determine the intervals of increase and decrease, you need to find the following:

• domain of definition (we already have);
• derivative (in our case: 1/3(3x^2-28x+49);
• solve the equation 1/3(3x^2-28x+49)=0.

After calculations we get the result:

We get: the function increases on the intervals from minus infinity to 7/3 and from 7 to infinity, and decreases on the interval from 7/3 to 7.

Extremes

The function under study y=1/3(x^3-14x^2+49x-36) is continuous and exists for any value of the variable x. The extremum point shows the maximum and minimum of a given function. In our case there are none, which greatly simplifies the construction task. Otherwise, they can also be found using the derivative function. Once found, do not forget to mark them on the chart.

Convexity and concavity

We continue to further explore the function y(x). Now we need to check it for convexity and concavity. The definitions of these concepts are quite difficult to comprehend; it is better to analyze everything using examples. For the test: a function is convex if it is a non-decreasing function. Agree, this is incomprehensible!

We need to find the derivative of a second order function. We get: y=1/3(6x-28). Now let's equate the right side to zero and solve the equation. Answer: x=14/3. We found the inflection point, that is, the place where the graph changes from convexity to concavity or vice versa. On the interval from minus infinity to 14/3 the function is convex, and from 14/3 to plus infinity it is concave. It is also very important to note that the inflection point on the graph should be smooth and soft, there should be no sharp corners.

Defining additional points

Our task is to investigate and construct a graph of the function. We have completed the study; constructing a graph of the function is now not difficult. For more accurate and detailed reproduction of a curve or straight line on the coordinate plane, you can find several auxiliary points. They are quite easy to calculate. For example, we take x=3, solve the resulting equation and find y=4. Or x=5, and y=-5 and so on. You can take as many additional points as you need for construction. At least 3-5 of them are found.

Plotting a graph

We needed to investigate the function (x^3-14x^2+49x-36)*1/3=y. All necessary marks during the calculations were made on the coordinate plane. All that remains to be done is to build a graph, that is, connect all the dots. Connecting the dots should be smooth and accurate, this is a matter of skill - a little practice and your schedule will be perfect.

The reference points when studying functions and constructing their graphs are characteristic points - points of discontinuity, extremum, inflection, intersection with coordinate axes. Using differential calculus, it is possible to establish the characteristic features of changes in functions: increase and decrease, maximums and minimums, the direction of convexity and concavity of the graph, the presence of asymptotes.

A sketch of the graph of the function can (and should) be drawn after finding the asymptotes and extremum points, and it is convenient to fill out the summary table of the study of the function as the study progresses.

The following function study scheme is usually used.

1.Find the domain of definition, intervals of continuity and breakpoints of the function.

2.Examine the function for evenness or oddness (axial or central symmetry of the graph.

3.Find asymptotes (vertical, horizontal or oblique).

4.Find and study the intervals of increase and decrease of the function, its extremum points.

5.Find the intervals of convexity and concavity of the curve, its inflection points.

6.Find the intersection points of the curve with the coordinate axes, if they exist.

7.Compile a summary table of the study.

8.A graph is constructed, taking into account the study of the function carried out according to the points described above.

Example. Explore function

and build its graph.

7. Let’s compile a summary table for studying the function, where we will enter all the characteristic points and the intervals between them. Taking into account the parity of the function, we obtain the following table:

The study of a function is carried out according to a clear scheme and requires the student to have a solid knowledge of basic mathematical concepts such as the domain of definition and values, continuity of the function, asymptote, extremum points, parity, periodicity, etc. The student must be able to differentiate functions freely and solve equations, which can sometimes be very complex.

That is, this task tests a significant layer of knowledge, any gap in which will become an obstacle to obtaining the correct solution. Particularly often, difficulties arise with constructing graphs of functions. This mistake is immediately noticeable to the teacher and can greatly damage your grade, even if everything else was done correctly. Here you can find online function research problems: study examples, download solutions, order assignments.

Explore a function and plot a graph: examples and solutions online

We have prepared for you a lot of ready-made function studies, both paid in the solution book and free in the section Examples of function studies. Based on these solved tasks, you will be able to familiarize yourself in detail with the methodology for performing similar tasks, and carry out your research by analogy.

We offer ready-made examples of complete research and plotting of functions of the most common types: polynomials, fractional-rational, irrational, exponential, logarithmic, trigonometric functions. Each solved problem is accompanied by a ready-made graph with highlighted key points, asymptotes, maxima and minima; the solution is carried out using an algorithm for studying the function.

In any case, the solved examples will be of great help to you as they cover the most popular types of functions. We offer you hundreds of already solved problems, but, as you know, there are an infinite number of mathematical functions in the world, and teachers are great experts at inventing more and more tricky tasks for poor students. So, dear students, qualified help will not hurt you.

Solving custom function research problems

In this case, our partners will offer you another service - full function research online to order. The task will be completed for you in compliance with all the requirements for an algorithm for solving such problems, which will greatly please your teacher.

We will do a complete study of the function for you: we will find the domain of definition and the domain of values, examine for continuity and discontinuity, establish parity, check your function for periodicity, and find the points of intersection with the coordinate axes. And, of course, further using differential calculus: we will find asymptotes, calculate extrema, inflection points, and construct the graph itself.