Conduct a complete study of the function of two variables. Function research online

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. By using differential calculus can be installed characteristics function changes: 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:

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In this article, we will consider a scheme for studying a function, and also give examples of studying extrema, monotonicity, and asymptotes of a given function.

Scheme

1. The domain of existence (ODZ) of a function.
2. Function intersection (if any) with coordinate axes, function signs, parity, periodicity.
3. Breakpoints (their kind). Continuity. Asymptotes are vertical.
4. Monotonicity and extremum points.
5. Inflection points. Convex.
6. Investigation of a function at infinity, for asymptotes: horizontal and oblique.
7. Building a graph.

Study for monotonicity

Theorem. If the function g continuous on , differentiated by (a; b) and g'(x) ≥ 0 (g'(x)≤0), xє(а; b), then g increasing (decreasing) .

Example:

y = 1: 3x 3 - 6: 2x 2 + 5x.

ODZ: хєR

y' = x 2 + 6x + 5.

Find intervals of constant signs y'. Because the y' is an elementary function, then it can change signs only at the points where it becomes zero or does not exist. Her ODZ: хєR.

Let's find the points where the derivative equals 0 (zero):

y' = 0;

x = -1; -5.

So, y growing on (-∞; -5] and on [-one; +∞), y descending on .

Research for extremes

T. x0 is called the maximum point (max) on the set BUT functions g when the maximum value is taken at this point by the function g(x 0) ≥ g(x), xєA.

T. x0 is called the minimum point (min) of the function g on the set BUT when the smallest value is taken by the function at this point g(x 0) ≤ g(x), xєА.

On the set BUT the maximum (max) and minimum (min) points are called extremum points g. Such extrema are also called absolute extrema on the set .

If a x0- extremum point of the function g in some district, then x0 is called the point of local or local extremum (max or min) of the function g.

Theorem (necessary condition). If a x0- extremum point of the (local) function g, then the derivative does not exist or is equal to 0 (zero) at this point.

Definition. Points with a non-existent or equal to 0 (zero) derivative are called critical. It is these points that are suspicious for an extremum.

Theorem (sufficient condition No. 1). If the function g is continuous in some district. x0 and the sign changes through this point when the derivative passes, then this point is the extremum point g.

Theorem (sufficient condition No. 2). Let the function be twice differentiable in some neighborhood of the point and g' = 0 and g'' > 0 (g''< 0) , then this point is the point of maximum (max) or minimum (min) of the function.

Convexity test

The function is called downward convex (or concave) on the interval (a,b) when the graph of the function is located no higher than the secant on the interval for any x with (a,b) that passes through these points .

The function will be convex strictly down on (a,b), if - the graph lies below the secant on the interval.

The function is called upward convex (convex) on the interval (a,b), if for any t points With (a,b) the graph of the function on the interval lies not lower than the secant passing through the abscissas at these points .

The function will be strictly convex upwards on (a, b), if - the graph on the interval lies above the secant.

If the function is in some neighborhood of the point continuous and through t. x 0 during the transition, the function changes its convexity, then this point is called the inflection point of the function.

Study for asymptotes

Definition. The straight line is called the asymptote g(x), if at an infinite distance from the origin, the point of the graph of the function approaches it: d(M,l).

Asymptotes can be vertical, horizontal or oblique.

Vertical line with equation x = x 0 will be the asymptote of the vertical graph of the function g , if point x 0 has an infinite gap, then there is at least one left or right boundary at this point - infinity.

Investigation of a function on a segment for the value of the smallest and largest

If the function is continuous on , then by the Weierstrass theorem there is the largest value and the smallest value on this segment, that is, there are t glasses that belong such that g(x 1) ≤ g(x)< g(x 2), x 2 є . From theorems about monotonicity and extrema, we obtain the following scheme for studying a function on a segment for the smallest and largest values.

Plan

1. Find derivative g'(x).
2. Look up the value of a function g at these points and at the ends of the segment.
3. Compare the found values ​​and choose the smallest and largest.

Comment. If you need to study a function on a finite interval (a,b), or on an infinite (-∞; b); (-∞; +∞) on the max and min values, then in the plan, instead of the values ​​of the function at the ends of the interval, they look for the corresponding one-sided boundaries: instead of f(a) looking for f(a+) = limf(x), instead of f(b) looking for f(-b). So you can find the ODZ function on the interval, because absolute extrema do not necessarily exist in this case.

Application of the derivative to the solution of applied problems for the extremum of some quantities

1. Express this value in terms of other quantities from the condition of the problem so that it is a function of only one variable (if possible).
2. The interval of change of this variable is determined.
3. Conduct a study of the function on the interval for max and min values.

A task. It is necessary to build a rectangular platform, using a grid meters, near the wall so that on one side it is adjacent to the wall, and on the other three it is fenced with a grid. At what aspect ratio will the area of ​​such a site be the largest?

S=xy is a function of 2 variables.

S = x(a - 2x)- function of the 1st variable ; x є .

S = ax - 2x2; S" = a - 4x = 0, xєR, x = a: 4.

S(a: 4) = a 2: 8- the highest value;

S(0)=0.

Find the other side of the rectangle: at = a: 2.

Aspect Ratio: y:x=2.

Answer. The largest area will be a 2 /8 if the side that is parallel to the wall is 2 times the other side.

Function research. Examples

Example 1

Available y=x 3: (1-x) 2 . Do research.

1. ODZ: хє(-∞; 1) U (1; ∞).
2. A general function (neither even nor odd) is not symmetrical with respect to the point 0 (zero).
3. Function signs. The function is elementary, so it can change sign only at points where it is equal to 0 (zero), or does not exist.
4. The function is elementary, therefore continuous on the ODZ: (-∞; 1) U (1; ∞).

Gap: x = 1;

limx 3: (1- x) 2 = ∞- Discontinuity of the 2nd kind (infinite), so there is a vertical asymptote at point 1;

x = 1- the equation of the vertical asymptote.

5. y' = x 2 (3 - x) : (1 - x) 3 ;

ODZ (y’): x ≠ 1;

x = 1 is a critical point.

y' = 0;

0; 3 are critical points.

6. y'' = 6x: (1 - x) 4 ;

Critical t.: 1, 0;

x= 0 - inflection point, y(0) = 0.

7. limx 3: (1 - 2x + x 2) = ∞- there is no horizontal asymptote, but it can be oblique.

k = 1- number;

b = 2- number.

Therefore, there is an oblique asymptote y=x+2 to + ∞ and to - ∞.

Example 2

Given y = (x 2 + 1) : (x - 1). Produce and investigation. Build a graph.

1. The area of ​​existence is the entire number line, except for the so-called. x=1.

2. y crosses OY (if possible) incl. (0;g(0)). We find y(0) = -1 - point of intersection OY .

Points of intersection of the graph with OX find by solving the equation y=0. The equation has no real roots, so this function does not intersect OX.

3. The function is non-periodic. Consider the expression

g(-x) ≠ g(x), and g(-x) ≠ -g(x). This means that it general view function (neither even nor odd).

4. T. x=1 the discontinuity is of the second kind. At all other points, the function is continuous.

5. Study of the function for an extremum:

(x 2 - 2x - 1) : (x - 1)2=y"

and solve the equation y" = 0.

So, 1 - √2, 1 + √2, 1 - critical points or points of possible extremum. These points divide the number line into four intervals .

On each interval, the derivative has a certain sign, which can be set by the method of intervals or by calculating the values ​​of the derivative at individual points. At intervals (-∞; 1 - √2 ) U (1 + √2 ; ∞) , a positive derivative, which means that the function is growing; if xє(1 - √2 ; 1) U(1; 1 + √2 ) , then the function is decreasing, because the derivative is negative on these intervals. Through t. x 1 during the transition (movement follows from left to right), the derivative changes sign from "+" to "-", therefore, at this point there is a local maximum, we find

y max = 2 - 2 √2 .

When passing through x2 changes the derivative sign from "-" to "+", therefore, there is a local minimum at this point, and

y mix = 2 + 2√2.

T. x=1 not so extremum.

6.4: (x - 1) 3 = y"".

On the (-∞; 1 ) 0 > y"" , consequently, the curve is convex on this interval; if xє (1 ; ∞) - the curve is concave. In t point 1 no function is defined, so this point is not an inflection point.

7. It follows from the results of paragraph 4 that x=1 is the vertical asymptote of the curve.

There are no horizontal asymptotes.

x + 1 = y is the asymptote of the slope of this curve. There are no other asymptotes.

8. Taking into account the conducted studies, we build a graph (see the figure above).

The study of the function is carried out according to a clear scheme and requires the student to have solid knowledge of the basic mathematical concepts such as domain of definition and values, function continuity, 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 complete research and plotting of the 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 make 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.