For a complete study of the function and plotting its graph, it is recommended to use the following scheme:

1) find the scope of the function;

2) find the discontinuity points of the function and vertical asymptotes (if they exist);

3) investigate the behavior of the function at infinity, find the horizontal and oblique asymptotes;

4) investigate the function for evenness (oddity) and for periodicity (for trigonometric functions);

5) find extrema and intervals of monotonicity of the function;

6) determine the intervals of convexity and inflection points;

7) find points of intersection with the coordinate axes, if possible, and some additional points that refine the graph.

The study of the function is carried out simultaneously with the construction of its graph.

Example 9 Explore the function and build a graph.

1. Domain of definition: ;

2. The function breaks at points
,
;

We investigate the function for the presence of vertical asymptotes.

;
,
─ vertical asymptote.

;
,
─ vertical asymptote.

3. We investigate the function for the presence of oblique and horizontal asymptotes.

Straight
─ oblique asymptote, if
,
.

,
.

Straight
─ horizontal asymptote.

4. The function is even because
. The parity of the function indicates the symmetry of the graph with respect to the y-axis.

5. Find the intervals of monotonicity and extrema of the function.

Let's find the critical points, i.e. points where the derivative is 0 or does not exist:
;
. We have three points
;

. These points divide the entire real axis into four intervals. Let's define the signs on each of them.

On the intervals (-∞; -1) and (-1; 0) the function increases, on the intervals (0; 1) and (1; +∞) it decreases. When passing through a point
the derivative changes sign from plus to minus, therefore, at this point, the function has a maximum
.

6. Let's find convexity intervals, inflection points.

Let's find the points where is 0, or does not exist.

has no real roots.
,
,

points
and
divide the real axis into three intervals. Let's define the sign at every interval.

Thus, the curve on the intervals
and
convex downwards, on the interval (-1;1) convex upwards; there are no inflection points, since the function at the points
and
unspecified.

7. Find the points of intersection with the axes.

with axle
the graph of the function intersects at the point (0; -1), and with the axis
the graph does not intersect, because the numerator of this function has no real roots.

The graph of the given function is shown in Figure 1.

Figure 1 ─ Graph of the function

Application of the concept of derivative in economics. Function elasticity

To study economic processes and solve other applied problems, the concept of function elasticity is often used.

Definition. Function elasticity
is called the limit of the ratio of the relative increment of the function to the relative increment of the variable at
, . (VII)

The elasticity of a function shows approximately how many percent the function will change
when changing the independent variable by 1%.

The elasticity of a function is used in the analysis of demand and consumption. If the elasticity of demand (in absolute value)
, then demand is considered elastic if
─ neutral if
─ inelastic with respect to price (or income).

Example 10 Calculate the elasticity of a function
and find the value of the elasticity index for = 3.

Solution: according to the formula (VII) the elasticity of the function:

Let x=3 then
This means that if the independent variable increases by 1%, then the value of the dependent variable will increase by 1.42%.

Example 11 Let the demand function regarding the price has the form
, where ─ constant coefficient. Find the value of the elasticity index of the demand function at the price x = 3 den. units

Solution: calculate the elasticity of the demand function using the formula (VII)

Assuming
monetary units, we get
. This means that at the price
monetary unit a price increase of 1% will cause a decrease in demand by 6%, i.e. demand is elastic.

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

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

Investigate a Function and Plot: Examples and Solutions Online

We have prepared for you a lot of ready-made feature studies, both paid in the solution book, and free in the Feature Research Examples section. On the basis of these solved tasks, you will be able to get acquainted in detail with the methodology for performing such tasks, by analogy, perform your own research.

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

The solved examples, in any case, will be a good help for 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 in inventing more and more intricate tasks for poor students. So, dear students, qualified assistance will not hurt you.

Solving problems for the study of a function to order

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 the 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 range of values, examine for continuity and discontinuity, set the parity, check your function for periodicity, find the points of intersection with the coordinate axes. And, of course, further with the help of differential calculus: we will find asymptotes, calculate extrema, inflection points, and build the graph itself.

The reference points in the study of functions and the construction of their graphs are characteristic points - points of discontinuity, extremum, inflection, intersection with the coordinate axes. With the help of differential calculus, it is possible to establish the characteristic features of the change in functions: increase and decrease, maxima and minima, the direction of the convexity and concavity of the graph, the presence of asymptotes.

A sketch of the function graph can (and should) be sketched after finding the asymptotes and extremum points, and it is convenient to fill in the summary table of the study of the function in the course of the study.

Usually, the following scheme of function research is used.

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

2.Examine the function for even or odd (axial or central symmetry of the graph.

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

4.Find and investigate 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 points of intersection of the curve with the coordinate axes, if they exist.

7.Compile a summary table of the study.

8.Build a graph, taking into account the study of the function, carried out according to the above points.

Example. Explore Function

and plot it.

7. Let's make a summary table of the study of the function, where we will enter all the characteristic points and the intervals between them. Given the parity of the function, we get the following table:

				Chart Features
[-1, 0[			Increasing	Convex
				(0; 1) – maximum point
]0, 1[			Decreases	Convex
				Inflection point, forms with the axis Ox obtuse angle

Conduct a complete study and plot a function graph

y(x)=x2+81−x.y(x)=x2+81−x.

1) Function scope. Since the function is a fraction, you need to find the zeros of the denominator.

1−x=0,⇒x=1.1−x=0,⇒x=1.

We exclude the only point x=1x=1 from the function definition area and get:

D(y)=(−∞;1)∪(1;+∞).D(y)=(−∞;1)∪(1;+∞).

2) Let us study the behavior of the function in the vicinity of the discontinuity point. Find one-sided limits:

Since the limits are equal to infinity, the point x=1x=1 is a discontinuity of the second kind, the line x=1x=1 is a vertical asymptote.

3) Let's determine the intersection points of the graph of the function with the coordinate axes.

Let's find the points of intersection with the ordinate axis OyOy, for which we equate x=0x=0:

Thus, the point of intersection with the axis OyOy has coordinates (0;8)(0;8).

Let's find the points of intersection with the abscissa axis OxOx, for which we set y=0y=0:

The equation has no roots, so there are no points of intersection with the OxOx axis.

Note that x2+8>0x2+8>0 for any xx. Therefore, for x∈(−∞;1)x∈(−∞;1), the function y>0y>0 (takes positive values, the graph is above the x-axis), for x∈(1;+∞)x∈(1; +∞) function y<0y<0 (принимает отрицательные значения, график находится ниже оси абсцисс).

4) The function is neither even nor odd because:

5) We investigate the function for periodicity. The function is not periodic, as it is a fractional rational function.

6) We investigate the function for extremums and monotonicity. To do this, we find the first derivative of the function:

Let us equate the first derivative to zero and find the stationary points (at which y′=0y′=0):

We got three critical points: x=−2,x=1,x=4x=−2,x=1,x=4. We divide the entire domain of the function into intervals by given points and determine the signs of the derivative in each interval:

For x∈(−∞;−2),(4;+∞)x∈(−∞;−2),(4;+∞) the derivative y′<0y′<0, поэтому функция убывает на данных промежутках.

For x∈(−2;1),(1;4)x∈(−2;1),(1;4) the derivative y′>0y′>0, the function increases on these intervals.

In this case, x=−2x=−2 is a local minimum point (the function decreases and then increases), x=4x=4 is a local maximum point (the function increases and then decreases).

Let's find the values ​​of the function at these points:

Thus, the minimum point is (−2;4)(−2;4), the maximum point is (4;−8)(4;−8).

7) We examine the function for kinks and convexity. Let's find the second derivative of the function:

Equate the second derivative to zero:

The resulting equation has no roots, so there are no inflection points. Moreover, when x∈(−∞;1)x∈(−∞;1) y′′>0y″>0 is satisfied, that is, the function is concave when x∈(1;+∞)x∈(1;+ ∞) y′′<0y″<0, то есть функция выпуклая.

8) We investigate the behavior of the function at infinity, that is, at .

Since the limits are infinite, there are no horizontal asymptotes.

Let's try to determine oblique asymptotes of the form y=kx+by=kx+b. We calculate the values ​​of k,bk,b according to the known formulas:

We found that the function has one oblique asymptote y=−x−1y=−x−1.

9) Additional points. Let's calculate the value of the function at some other points in order to build a graph more accurately.

y(−5)=5.5;y(2)=−12;y(7)=−9.5.y(−5)=5.5;y(2)=−12;y(7)=−9.5.

10) Based on the data obtained, we will build a graph, supplement it with the asymptotes x=1x=1 (blue), y=−x−1y=−x−1 (green) and mark the characteristic points (the intersection with the ordinate axis is purple, extrema are orange, additional points are black) :

Task 4: Geometric, Economic problems (I have no idea what, here is an approximate selection of problems with a solution and formulas)

Example 3.23. a

Decision. x and y y
y \u003d a - 2 × a / 4 \u003d a / 2. Since x = a/4 is the only critical point, let's check whether the sign of the derivative changes when passing through this point. For xa/4 S "> 0, and for x >a/4 S "< 0, значит, в точке x=a/4 функция S имеет максимум. Значение функции S(a/4) = a/4(a - a/2) = a 2 /8 (кв. ед).Поскольку S непрерывна на и ее значения на концах S(0) и S(a/2) равны нулю, то найденное значение будет наибольшим значением функции. Таким образом, наиболее выгодным соотношением сторон площадки при данных условиях задачи является y = 2x.

Example 3.24.

Decision.
R = 2, H = 16/4 = 4.

Example 3.22. Find the extrema of the function f(x) = 2x 3 - 15x 2 + 36x - 14.

Decision. Since f "(x) \u003d 6x 2 - 30x +36 \u003d 6 (x - 2) (x - 3), then the critical points of the function x 1 \u003d 2 and x 2 \u003d 3. Extreme points can only be at these points. So as when passing through the point x 1 \u003d 2, the derivative changes sign plus to minus, then at this point the function has a maximum.When passing through the point x 2 \u003d 3, the derivative changes sign minus to plus, therefore, at the point x 2 \u003d 3, the function has a minimum. Calculating the values ​​of the function in points
x 1 = 2 and x 2 = 3, we find the extrema of the function: maximum f(2) = 14 and minimum f(3) = 13.

Example 3.23. It is necessary to build a rectangular area near the stone wall so that it is fenced off with wire mesh on three sides, and adjoins the wall on the fourth side. For this there is a linear meters of the grid. At what aspect ratio will the site have the largest area?

Decision. Denote the sides of the site through x and y. The area of ​​the site is S = xy. Let be y is the length of the side adjacent to the wall. Then, by condition, the equality 2x + y = a must hold. Therefore y = a - 2x and S = x(a - 2x), where
0 ≤ x ≤ a/2 (the length and width of the area cannot be negative). S "= a - 4x, a - 4x = 0 for x = a/4, whence
y \u003d a - 2 × a / 4 \u003d a / 2. Since x = a/4 is the only critical point, let's check whether the sign of the derivative changes when passing through this point. For xa/4 S "> 0, and for x >a/4 S "< 0, значит, в точке x=a/4 функция S имеет максимум. Значение функции S(a/4) = a/4(a - a/2) = a 2 /8 (кв. ед).Поскольку S непрерывна на и ее значения на концах S(0) и S(a/2) равны нулю, то найденное значение будет наибольшим значением функции. Таким образом, наиболее выгодным соотношением сторон площадки при данных условиях задачи является y = 2x.

Example 3.24. It is required to make a closed cylindrical tank with a capacity of V=16p ≈ 50 m 3 . What should be the dimensions of the tank (radius R and height H) in order to use the least amount of material for its manufacture?

Decision. The total surface area of ​​the cylinder is S = 2pR(R+H). We know the volume of the cylinder V = pR 2 H Þ H = V/pR 2 =16p/ pR 2 = 16/ R 2 . Hence, S(R) = 2p(R 2 +16/R). We find the derivative of this function:
S "(R) \u003d 2p (2R- 16 / R 2) \u003d 4p (R- 8 / R 2). S " (R) \u003d 0 for R 3 \u003d 8, therefore,
R = 2, H = 16/4 = 4.

Similar information.

For some time now in TheBat (it is not clear for what reason) the built-in certificate database for SSL has ceased to work correctly.

When checking the post, an error pops up:

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. You are welcome
contact your server administrator.

And it is offered a choice of answers - YES / NO. And so every time you shoot mail.

Decision

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

Since I needed to merge all the files into one, I first converted all 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. Formats can be absolutely any (source) and doc, and jpg, and even zip archive!

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

Update May 2015

I found another great site! Even more convenient and functional for creating a completely arbitrary collage! This site is http://www.fotor.com/ru/collage/ . Use on health. And I will use it myself.

Faced in life with the repair of electric stoves. I already did a lot of things, learned a lot, but somehow I 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 the electric stove?

The answer turned out to be simple. No need to measure anything, you can calmly determine by eye what size you need.

The smallest burner is 145 millimeters (14.5 centimeters)

Medium burner is 180 millimeters (18 centimeters).

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

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

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