ESSAY

"Complete investigation of a function and plotting it".

INTRODUCTION

Studying the properties of a function and plotting its graph is one of the most wonderful applications of the derivative. This way of examining 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 the beginning of analysis in grades 10-11 is the systematic study of functions, the disclosure of the applied value of the general methods of mathematics related to the study of functions.

The development of functional representations in the course of studying algebra and the beginning of analysis at the senior level helps high school students to get a visual representation of the continuity and discontinuities of functions, learn about the continuity of any elementary function in the area of ​​its application, learn how to build their graphs and generalize information about the basic elementary functions and realize them role in the study of the phenomena of reality, in human practice.

Function Ascending and Decreasing

The solution of various problems from the field of mathematics, physics and technology leads to the establishment of a functional relationship between the variables involved in this phenomenon.

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

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

The application of the differential calculus to the study of a function relies 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.

Consider how you can find the intervals of increase or decrease of a function, that is, the intervals of its monotonicity. Based on the definition of a monotonically decreasing and increasing function, we can formulate theorems that allow us to relate the value of the first derivative of a given function with 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 monotonically decreases, then at any point of the interval ( x )<0.

Proof. Let the functiony = f ( x ) increases monotonically by( a , b ) , This means that for any sufficiently small > 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 that the limit is positive, that is, ( x )>0 , which was to be proved. The second part of the theorem, which is related to the monotone decrease of the function, is proved similarly.

Theorem 1.2. If the function y = f ( x ) , continuous on the segment[ a , b ] and differentiable at all its interior points, and, moreover, ( x ) >0 for anyone x ϵ ( a , b ) , then this function is monotonically increasing 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< . By Lagrange's theorem

( c ) = .

But ( c )>0 and > 0, so ( > 0, i.e.

(. The result obtained indicates a monotonous increase in the function, which was to be proved. The second part of the theorem is proved similarly.

Function extremes

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

Definition 2.1. Dot is called the maximum point of the function

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

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

Rice. 2.1

Theorem 2.1 (necessary condition for the existence of an extremum). If differentiable on the interval( a , b ) the function has at the 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 .

Theorem 2.1 implies 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 point x = 0 the derivative is equal to 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 breaks are called critical points of the given function..

Therefore, Theorem 2.1 is not sufficient for determining extremal points.

Theorem 2.2 (sufficient condition for the 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 this point passes from left to right, the sign of the derivative changes from plus to minus, then this is the maximum point, and, conversely, from minus to plus, the minimum point.

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

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

1) find the scope of the function;

2) calculate the derivative;

3) find critical points;

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

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

Intervals of convexity and concavity of a function

Another characteristic of a function graph that can be determined using a 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, 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 ) 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. Take an arbitrary pointϵ ( a , b ) and draw at this point the tangent to the graph of the functiony = f ( x ) (Fig. 3.1).

The theorem will be proved 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 ordinates 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 - ) .

Compose the difference and :

- = f(x) – f( ) - ( )(x- ).

Apply to differencef ( x ) – f ( ) Lagrange's mean 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, by the hypothesis of the theorem, ( )<0.

Multiplying these three factors, we get that , which was to be proved.

Definition 3.2. The point separating the interval of convexity from the interval of concavity is called the inflection point..

From Definition 3.1 it follows that at a given point the tangent intersects the curve, that is, on the one hand, 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 the inflection point.

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

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

4. Function asymptotes

In the previous paragraphs, methods for studying the behavior of a function with the help of a derivative were considered. However, among the questions concerning the complete study of the function, there are those that are not related to the derivative.

So, for example, it is necessary to know how the function behaves when the point of its graph is infinitely removed from the origin. Such a problem can arise in two cases: when the argument of the function goes to infinity, and when the function itself goes to infinity at the break of the second kind at the end point. 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 ) a straight line is called, which has the property that the distance from the graph to this straight line tends to zero with an unlimited removal of the graph point from the origin.

There are two types of asymptotes: vertical and oblique.

The vertical asymptotes are straight linesx = , which have the property that the graph of the function in their neighborhood goes to infinity, that is, the condition is met: .

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

Oblique asymptotes are described by the general equation of a straight line in a plane, i.e.y = kx + b . Hence, in contrast to the vertical asymptotes, here it is necessary to determine the numbersk and b .

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

How to investigate a function and plot its graph?

It seems that I am beginning to understand the soulful face of the leader of the world proletariat, the author of collected works in 55 volumes .... The long journey began with elementary information about functions and graphs, and now work on a laborious topic ends with a natural result - an article about the full function study. The long-awaited task is formulated as follows:

Investigate the function by methods of differential calculus and, based on the results of the study, build its graph

Or in short: examine the function and plot it.

Why explore? In simple cases, it will not be difficult for us to deal with elementary functions, draw a graph obtained using elementary geometric transformations etc. However, the properties and graphic representations of more complex functions are far from obvious, which is why a whole study is needed.

The main steps of the solution are summarized in the reference material Function Study Scheme, this is your section guide. Dummies need a step-by-step explanation of the topic, some readers don't know where to start and how to organize the study, and advanced students may be interested in only a few points. But whoever you are, dear visitor, the proposed summary with pointers to various lessons will orient and direct you in the direction of interest in the shortest possible time. The robots shed a tear =) The manual was made up in the form of a pdf file and took its rightful place on the page Mathematical formulas and tables.

I used to break the study of the function into 5-6 points:

6) Additional points and graph based on the results of the study.

As for the final action, I think everyone understands everything - it will be very disappointing if in a matter of seconds it is crossed out and the task is returned for revision. A CORRECT AND ACCURATE DRAWING is the main result of the solution! It is very likely to "cover up" analytical oversights, while an incorrect and/or sloppy schedule will cause problems even with a perfectly conducted study.

It should be noted that in other sources, the number of research items, the order of their implementation and the design style may differ significantly from the scheme proposed by me, but in most cases it is quite enough. The simplest version of the problem consists of only 2-3 steps and is formulated something like this: “explore the function using the derivative and plot” or “explore the function using the 1st and 2nd derivative, plot”.

Naturally, if another algorithm is analyzed in detail in your training manual or your teacher strictly requires you to adhere to his lectures, then you will have to make some adjustments to the solution. No more difficult than replacing a fork with a chainsaw spoon.

Let's check the function for even / odd:

This is followed by a template unsubscribe:
, so this function is neither even nor odd.

Since the function is continuous on , there are no vertical asymptotes.

There are no oblique asymptotes either.

Note : I remind you that the higher order of growth than , so the final limit is exactly " plus infinity."

Let's find out how the function behaves at infinity:

In other words, if we go to the right, then the graph goes infinitely far up, if we go to the left, infinitely far down. Yes, there are also two limits under a single entry. If you have difficulty deciphering the signs, please visit the lesson about infinitesimal functions.

So the function not limited from above and not limited from below. Considering that we do not have break points, it becomes clear and function range: is also any real number.

USEFUL TECHNIQUE

Each task step brings new information about the graph of the function, so in the course of the solution it is convenient to use a kind of LAYOUT. Let's draw a Cartesian coordinate system on the draft. What is known for sure? Firstly, the graph has no asymptotes, therefore, there is no need to draw straight lines. Second, we know how the function behaves at infinity. According to the analysis, we draw the first approximation:

Note that in effect continuity function on and the fact that , the graph must cross the axis at least once. Or maybe there are several points of intersection?

3) Zeros of the function and intervals of constant sign.

First, find the intersection point of the graph with the y-axis. It's simple. It is necessary to calculate the value of the function when:

Half above sea level.

To find the points of intersection with the axis (zeroes of the function), you need to solve the equation, and here an unpleasant surprise awaits us:

At the end, a free member lurks, which significantly complicates the task.

Such an equation has at least one real root, and most often this root is irrational. In the worst fairy tale, three little pigs are waiting for us. The equation is solvable using the so-called Cardano's formulas, but paper damage is comparable to almost the entire study. In this regard, it is wiser orally or on a draft to try to pick up at least one whole root. Let's check if these numbers are:
- does not fit;
- there is!

It's lucky here. In case of failure, you can also test and, and if these numbers do not fit, then I'm afraid there are very few chances for a profitable solution to the equation. Then it is better to skip the research point completely - maybe something will become clearer at the final step, when additional points will break through. And if the root (roots) are clearly “bad”, then it is better to remain modestly silent about the intervals of constancy of signs and to more accurately complete the drawing.

However, we have a beautiful root, so we divide the polynomial for no remainder:

The algorithm for dividing a polynomial by a polynomial is discussed in detail in the first example of the lesson. Complex Limits.

As a result, the left side of the original equation expands into a product:

And now a little about a healthy lifestyle. Of course I understand that quadratic equations need to be solved every day, but today we will make an exception: the equation has two real roots.

On the number line, we plot the found values and interval method define the signs of the function:


og Thus, on the intervals chart located
below the x-axis, and at intervals - above this axis.

The resulting findings allow us to refine our layout, and the second approximation of the graph looks like this:

Please note that the function must have at least one maximum on the interval, and at least one minimum on the interval. But we don't know how many times, where and when the schedule will "wind around". By the way, a function can have infinitely many extremes.

4) Increasing, decreasing and extrema of the function.

Let's find the critical points:

This equation has two real roots. Let's put them on the number line and determine the signs of the derivative:


Therefore, the function increases by and decreases by .
At the point the function reaches its maximum: .
At the point the function reaches its minimum: .

The established facts drive our template into a rather rigid framework:

Needless to say, differential calculus is a powerful thing. Let's finally deal with the shape of the graph:

5) Convexity, concavity and inflection points.

Find the critical points of the second derivative:

Let's define signs:


The function graph is convex on and concave on . Let's calculate the ordinate of the inflection point: .

Almost everything cleared up.

6) It remains to find additional points that will help to more accurately build a graph and perform a self-test. In this case, they are few, but we will not neglect:

Let's execute the drawing:

The inflection point is marked in green, additional points are marked with crosses. The graph of a cubic function is symmetrical about its inflection point, which is always located exactly in the middle between the maximum and minimum.

In the course of the assignment, I gave three hypothetical intermediate drawings. In practice, it is enough to draw a coordinate system, mark the points found, and after each point of the study, mentally figure out what the graph of the function might look like. It will not be difficult for students with a good level of preparation to carry out such an analysis solely in their minds without involving a draft.

For a standalone solution:

Example 2

Explore the function and build a graph.

Everything is faster and more fun here, an approximate example of finishing at the end of the lesson.

A lot of secrets are revealed by the study of fractional rational functions:

Example 3

Using the methods of differential calculus, investigate the function and, based on the results of the study, construct its graph.

Decision: the first stage of the study does not differ in anything remarkable, with the exception of a hole in the definition area:

1) The function is defined and continuous on the entire number line except for the point , domain: .

, so this function is neither even nor odd.

Obviously, the function is non-periodic.

The graph of the function consists of two continuous branches located in the left and right half-plane - this is perhaps the most important conclusion of the 1st paragraph.

2) Asymptotes, the behavior of a function at infinity.

a) With the help of one-sided limits, we study the behavior of the function near the suspicious point, where the vertical asymptote must clearly be:

Indeed, the functions endure endless gap at the point
and the straight line (axis) is vertical asymptote graphic arts .

b) Check if oblique asymptotes exist:

Yes, the line is oblique asymptote graphics if .

It makes no sense to analyze the limits, since it is already clear that the function in an embrace with its oblique asymptote not limited from above and not limited from below.

The second point of the study brought a lot of important information about the function. Let's do a rough sketch:

Conclusion No. 1 concerns intervals of sign constancy. At "minus infinity" the graph of the function is uniquely located below the x-axis, and at "plus infinity" it is above this axis. In addition, one-sided limits told us that both to the left and to the right of the point, the function is also greater than zero. Please note that in the left half-plane, the graph must cross the x-axis at least once. In the right half-plane, there may be no zeros of the function.

Conclusion No. 2 is that the function increases on and to the left of the point (goes “from bottom to top”). To the right of this point, the function decreases (goes “from top to bottom”). The right branch of the graph must certainly have at least one minimum. On the left, extremes are not guaranteed.

Conclusion No. 3 gives reliable information about the concavity of the graph in the vicinity of the point. We cannot yet say anything about convexity/concavity at infinity, since the line can be pressed against its asymptote both from above and from below. Generally speaking, there is an analytical way to figure this out right now, but the shape of the chart "for nothing" will become clearer at a later stage.

Why so many words? To control subsequent research points and avoid mistakes! Further calculations should not contradict the conclusions drawn.

3) Points of intersection of the graph with the coordinate axes, intervals of constant sign of the function.

The graph of the function does not cross the axis.

Using the interval method, we determine the signs:

, if ;
, if .

The results of the paragraph are fully consistent with Conclusion No. 1. After each step, look at the draft, mentally refer to the study, and finish drawing the graph of the function.

In this example, the numerator is divided term by term by the denominator, which is very beneficial for differentiation:

Actually, this has already been done when finding asymptotes.

- critical point.

Let's define signs:

increases by and decreases to

At the point the function reaches its minimum: .

There were also no discrepancies with Conclusion No. 2, and, most likely, we are on the right track.

This means that the graph of the function is concave over the entire domain of definition.

Excellent - and you don't need to draw anything.

There are no inflection points.

The concavity is consistent with Conclusion No. 3, moreover, it indicates that at infinity (both there and there) the graph of the function is located higher its oblique asymptote.

6) We will conscientiously pin the task with additional points. Here we have to work hard, because we know only two points from the study.

And a picture that, probably, many have long presented:

In the course of the assignment, care must be taken to ensure that there are no contradictions between the stages of the study, but sometimes the situation is urgent or even desperately dead-end. Here the analytics "does not converge" - and that's it. In this case, I recommend an emergency technique: we find as many points belonging to the graph as possible (how much patience is enough), and mark them on the coordinate plane. Graphical analysis of the found values ​​in most cases will tell you where is the truth and where is the lie. In addition, the graph can be pre-built using some program, for example, in the same Excel (it is clear that this requires skills).

Example 4

Using the methods of differential calculus, investigate the function and build its graph.

This is a do-it-yourself example. In it, self-control is enhanced by the evenness of the function - the graph is symmetrical about the axis, and if something in your study contradicts this fact, look for an error.

An even or odd function can only be investigated for , and then the symmetry of the graph can be used. This solution is optimal, but it looks, in my opinion, very unusual. Personally, I consider the entire numerical axis, but I still find additional points only on the right:

Example 5

Conduct a complete study of the function and plot its graph.

Decision: rushed hard:

1) The function is defined and continuous on the entire real line: .

This means that this function is odd, its graph is symmetrical with respect to the origin.

Obviously, the function is non-periodic.

2) Asymptotes, the behavior of a function at infinity.

Since the function is continuous on , there are no vertical asymptotes

For a function containing an exponent, typically separate the study of "plus" and "minus infinity", however, our life is facilitated just by the symmetry of the graph - either there is an asymptote on the left and on the right, or it is not. Therefore, both infinite limits can be arranged under a single entry. In the course of the solution, we use L'Hopital's rule:

The straight line (axis) is the horizontal asymptote of the graph at .

Pay attention to how I cleverly avoided the full algorithm for finding the oblique asymptote: the limit is quite legal and clarifies the behavior of the function at infinity, and the horizontal asymptote was found "as if at the same time."

It follows from the continuity on and the existence of a horizontal asymptote that the function limited from above and limited from below.

3) Points of intersection of the graph with the coordinate axes, intervals of constancy.

Here we also shorten the solution:
The graph passes through the origin.

There are no other points of intersection with the coordinate axes. Moreover, the intervals of constancy are obvious, and the axis can not be drawn: , which means that the sign of the function depends only on the "x":
, if ;
, if .

4) Increasing, decreasing, extrema of the function.


are critical points.

The points are symmetrical about zero, as it should be.

Let's define the signs of the derivative:


The function increases on the interval and decreases on the intervals

At the point the function reaches its maximum: .

Due to the property (oddity of the function) the minimum can be omitted:

Since the function decreases on the interval , then, obviously, the graph is located at "minus infinity" under with its asymptote. On the interval, the function also decreases, but here the opposite is true - after passing through the maximum point, the line approaches the axis from above.

It also follows from the above that the function graph is convex at "minus infinity" and concave at "plus infinity".

After this point of the study, the area of ​​\u200b\u200bvalues ​​of the function was also drawn:

If you have a misunderstanding of any points, I once again urge you to draw coordinate axes in your notebook and, with a pencil in your hands, re-analyze each conclusion of the task.

5) Convexity, concavity, inflections of the graph.

are critical points.

The symmetry of the points is preserved, and, most likely, we are not mistaken.

Let's define signs:


The graph of the function is convex on and concave on .

Convexity/concavity at extreme intervals was confirmed.

At all critical points there are inflections in the graph. Let's find the ordinates of the inflection points, while again reducing the number of calculations, using the oddness of the function: