Plotting graphs using derivatives examples. Complete function study example online

If the problem requires a complete study of the function f (x) = x 2 4 x 2 - 1 with the construction of its graph, then we will consider this principle in detail.

To solve a problem of this type, you should use the properties and graphs of the main elementary functions. The research algorithm includes the following steps:

Finding the domain of definition

Since research is carried out on the domain of definition of the function, it is necessary to start with this step.

Example 1

The given example involves finding the zeros of the denominator in order to exclude them from the ODZ.

4 x 2 - 1 = 0 x = ± 1 2 ⇒ x ∈ - ∞ ; - 1 2 ∪ - 1 2 ; 1 2 ∪ 1 2 ; +∞

As a result, you can get roots, logarithms, and so on. Then the ODZ can be searched for a root of an even degree of type g (x) 4 by the inequality g (x) ≥ 0, for the logarithm log a g (x) by the inequality g (x) > 0.

Studying the boundaries of the ODZ and finding vertical asymptotes

There are vertical asymptotes at the boundaries of the function, when the one-sided limits at such points are infinite.

Example 2

For example, consider the border points equal to x = ± 1 2.

Then it is necessary to study the function to find the one-sided limit. Then we get that: lim x → - 1 2 - 0 f (x) = lim x → - 1 2 - 0 x 2 4 x 2 - 1 = = lim x → - 1 2 - 0 x 2 (2 x - 1 ) (2 x + 1) = 1 4 (- 2) · - 0 = + ∞ lim x → - 1 2 + 0 f (x) = lim x → - 1 2 + 0 x 2 4 x - 1 = = lim x → - 1 2 + 0 x 2 (2 x - 1) (2 x + 1) = 1 4 (- 2) (+ 0) = - ∞ lim x → 1 2 - 0 f (x) = lim x → 1 2 - 0 x 2 4 x 2 - 1 = = lim x → 1 2 - 0 x 2 (2 x - 1) (2 x + 1) = 1 4 (- 0) 2 = - ∞ lim x → 1 2 - 0 f (x) = lim x → 1 2 - 0 x 2 4 x 2 - 1 = = lim x → 1 2 - 0 x 2 (2 x - 1) (2 x + 1) = 1 4 ( + 0) 2 = + ∞

This shows that the one-sided limits are infinite, which means the straight lines x = ± 1 2 are the vertical asymptotes of the graph.

Study of a function and whether it is even or odd

When the condition y (- x) = y (x) is satisfied, the function is considered even. This suggests that the graph is located symmetrically with respect to Oy. When the condition y (- x) = - y (x) is satisfied, the function is considered odd. This means that the symmetry is relative to the origin of coordinates. If at least one inequality is not satisfied, we obtain a function of general form.

The equality y (- x) = y (x) indicates that the function is even. When constructing, it is necessary to take into account that there will be symmetry with respect to Oy.

To solve the inequality, intervals of increasing and decreasing are used with the conditions f " (x) ≥ 0 and f " (x) ≤ 0, respectively.

Definition 1

Stationary points- these are the points that turn the derivative to zero.

Critical points- these are internal points from the domain of definition where the derivative of the function is equal to zero or does not exist.

When making a decision, the following notes must be taken into account:

for existing intervals of increasing and decreasing inequalities of the form f " (x) > 0, critical points are not included in the solution;
points at which the function is defined without a finite derivative must be included in the intervals of increasing and decreasing (for example, y = x 3, where the point x = 0 makes the function defined, the derivative has the value of infinity at this point, y " = 1 3 x 2 3, y "(0) = 1 0 = ∞, x = 0 is included in the increasing interval);
To avoid disagreements, it is recommended to use mathematical literature recommended by the Ministry of Education.

Inclusion of critical points in intervals of increasing and decreasing if they satisfy the domain of definition of the function.

Definition 2

For determining the intervals of increase and decrease of a function, it is necessary to find:

derivative;
critical points;
divide the domain of definition into intervals using critical points;
determine the sign of the derivative on each of the intervals, where + is an increase, and - is a decrease.

Example 3

Find the derivative on the domain of definition f " (x) = x 2 " (4 x 2 - 1) - x 2 4 x 2 - 1 " (4 x 2 - 1) 2 = - 2 x (4 x 2 - 1) 2 .

Solution

To solve you need:

find stationary points, this example has x = 0;
find the zeros of the denominator, the example takes the value zero at x = ± 1 2.

We place points on the number axis to determine the derivative on each interval. To do this, it is enough to take any point from the interval and perform the calculation. If the result is positive, we depict + on the graph, which means the function is increasing, and - means it is decreasing.

For example, f " (- 1) = - 2 · (- 1) 4 - 1 2 - 1 2 = 2 9 > 0, which means that the first interval on the left has a + sign. Consider on the number line.

Answer:

the function increases on the interval - ∞; - 1 2 and (- 1 2 ; 0 ] ;
there is a decrease in the interval [ 0 ; 1 2) and 1 2 ; + ∞ .

In the diagram, using + and -, the positivity and negativity of the function are depicted, and the arrows indicate decrease and increase.

Extremum points of a function are points where the function is defined and through which the derivative changes sign.

Example 4

If we consider an example where x = 0, then the value of the function in it is equal to f (0) = 0 2 4 · 0 2 - 1 = 0. When the sign of the derivative changes from + to - and passes through the point x = 0, then the point with coordinates (0; 0) is considered the maximum point. When the sign changes from - to +, we obtain a minimum point.

Convexity and concavity are determined by solving inequalities of the form f "" (x) ≥ 0 and f "" (x) ≤ 0. Less commonly used is the name convexity down instead of concavity, and convexity upward instead of convexity.

Definition 3

For determining the intervals of concavity and convexity necessary:

find the second derivative;
find the zeros of the second derivative function;
divide the definition area into intervals with the appearing points;
determine the sign of the interval.

Example 5

Find the second derivative from the domain of definition.

Solution

f "" (x) = - 2 x (4 x 2 - 1) 2 " = = (- 2 x) " (4 x 2 - 1) 2 - - 2 x 4 x 2 - 1 2 " (4 x 2 - 1) 4 = 24 x 2 + 2 (4 x 2 - 1) 3

We find the zeros of the numerator and denominator, where in our example we have that the zeros of the denominator x = ± 1 2

Now you need to plot the points on the number line and determine the sign of the second derivative from each interval. We get that

Answer:

the function is convex from the interval - 1 2 ; 1 2 ;
the function is concave from the intervals - ∞ ; - 1 2 and 1 2; + ∞ .

Definition 4

Inflection point– this is a point of the form x 0 ; f (x 0) . When it has a tangent to the graph of the function, then when it passes through x 0 the function changes sign to the opposite.

In other words, this is a point through which the second derivative passes and changes sign, and at the points themselves it is equal to zero or does not exist. All points are considered to be the domain of the function.

In the example, it was clear that there are no inflection points, since the second derivative changes sign while passing through the points x = ± 1 2. They, in turn, are not included in the scope of definition.

Finding horizontal and oblique asymptotes

When defining a function at infinity, you need to look for horizontal and oblique asymptotes.

Definition 5

Oblique asymptotes are depicted using straight lines given by the equation y = k x + b, where k = lim x → ∞ f (x) x and b = lim x → ∞ f (x) - k x.

For k = 0 and b not equal to infinity, we find that the oblique asymptote becomes horizontal.

In other words, asymptotes are considered to be lines to which the graph of a function approaches at infinity. This facilitates quick construction of a function graph.

If there are no asymptotes, but the function is defined at both infinities, it is necessary to calculate the limit of the function at these infinities in order to understand how the graph of the function will behave.

Example 6

Let's consider as an example that

k = lim x → ∞ f (x) x = lim x → ∞ x 2 4 x 2 - 1 x = 0 b = lim x → ∞ (f (x) - k x) = lim x → ∞ x 2 4 x 2 - 1 = 1 4 ⇒ y = 1 4

is a horizontal asymptote. After examining the function, you can begin to construct it.

Calculating the value of a function at intermediate points

To make the graph more accurate, it is recommended to find several function values at intermediate points.

Example 7

From the example we considered, it is necessary to find the values of the function at the points x = - 2, x = - 1, x = - 3 4, x = - 1 4. Since the function is even, we get that the values coincide with the values at these points, that is, we get x = 2, x = 1, x = 3 4, x = 1 4.

Let's write and solve:

F (- 2) = f (2) = 2 2 4 2 2 - 1 = 4 15 ≈ 0, 27 f (- 1) - f (1) = 1 2 4 1 2 - 1 = 1 3 ≈ 0 , 33 f - 3 4 = f 3 4 = 3 4 2 4 3 4 2 - 1 = 9 20 = 0 , 45 f - 1 4 = f 1 4 = 1 4 2 4 1 4 2 - 1 = - 1 12 ≈ - 0.08

To determine the maxima and minima of the function, inflection points, and intermediate points, it is necessary to construct asymptotes. For convenient designation, intervals of increasing, decreasing, convexity, and concavity are recorded. Let's look at the picture below.

It is necessary to draw graph lines through the marked points, which will allow you to approach the asymptotes by following the arrows.

This concludes the full exploration of the function. There are cases of constructing some elementary functions for which geometric transformations are used.

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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 mathematics! Let's go!

Domain of definition

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 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:

Scope of definition.
Continuity.
Even or odd.
Periodicity.
Asymptotes.
Zeros.
Sign constancy.
Increasing and decreasing.
Extremes.
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 illustrative examples 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. Consequently, 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 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. If we talk in simple language, then the zero of the 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 value:

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, and 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.

One of the possible schemes for studying a function and constructing a graph is decomposed into the following stages of solving the problem: 1. Domain of definition of the function (O.O.F.). 2. Function break points, their nature. Vertical asymptotes. 3. Even, odd, periodicity of the function. 4. Points of intersection of the graph with the coordinate axes. 5. Behavior of the function at infinity. Horizontal and oblique asymptotes. 6. Intervals of monotonicity of a function, maximum and minimum points. 7. Directions of convexity of the curve. Inflection points. 8. Function graph. Example 1. Construct a graph of the function y = 1. (vereiora or curl of Maria Anieei). - the entire numerical axis. 2. There are no break points; there are no vertical asymptotes. 3. The function is even: , so its graph is symmetrical with respect to the Oy axis\ non-periodic. From the parity of the function it follows that it is enough to construct its graph on the half-line x ^ O, and then mirror it in the Oy axis. 4. At x = 0 we have Yx, so the graph of the function lies in the upper half-plane y > 0. Scheme for constructing the graph of the function Study of functions to an extremum using derivatives higher order Calculation of the roots of equations by the methods of chords and tangents that the graph has a horizontal asymptote y = O, there are no oblique asymptotes. So the function increases when and decreases when. Point x = 0 is critical. When x passes through the point x = 0, the derivative y"(x) changes sign from minus to plus. Therefore, the point x = 0 is the maximum point, y(Q) = I. This result is quite obvious: /(x) = T^ IV*. The second derivative vanishes at the points x =. We examine the point x = 4- (hereinafter, we have a symmetry consideration): when we obtain (the curve is convex upward). graph of the function. We summarize the results of the study in the table: Inflection point max In the table, the arrow “Y” indicates an increase in the function, the arrow “\” indicates its decrease. The graph of the function is shown in Fig. 33. Example 2. Construct a graph of the function (Newton’s trident). ) - the entire numerical axis, excluding point 2. We have that the line x = 0 is a vertical asymptote 3. The function is neither even nor odd [a function of general position], assuming that the graph of the function intersects the axis. Oh at the point (-1,0). There are no oblique and horizontal asymptotes, hence the critical point. The second derivative of the function at a point, so x = is the minimum point. The second derivative turns into uul at a point and changes its sign when passing through this point. Therefore, the point is the inflection point of the curve. For) we have e. the convexity of the curve is directed downward; for -I we have. the convexity of the curve is directed upward. The results of the study are summarized in a table: Does not exist Does not exist Inflection point Does not exist. The vertical asymptote of the derivative vanishes at x = e,/2. and when x passes through this point, y" changes sign. Consequently, is the abscissa of the inflection point of the curve. We summarize the results of the study in the table: Inflection point. The graph of the function is shown in Fig. 37. Example 4. Construct a graph of the function the entire numerical axis, excluding the point Point point discontinuity of the 2nd kind of function. Since Km . then the direct vertical asymptote of the graph of the function. Assuming y = 0, we have that the graph of the function intersects the Ox axis at the point. Therefore, the graph of the function has an oblique asymptote. From the condition. we obtain - the critical point. The second derivative of the function y" = D > 0 everywhere in the domain of definition, in particular, at the point - the minimum point of the function. 7. Since, everywhere in the domain of definition of the function, the convexity of its graph is directed downward. The results of the study are summarized in a table: Doesn't exist Doesn't exist Doesn't exist. x = 0 - vertical asymptote The graph of the function is shown in Fig. Example 5. Construct a graph of the function the entire number axis. 2. Continuous everywhere. There are no vertical asymptotes. 3. General provisions, non-periodic. 4. The function vanishes at 5. Thus, the graph of the function has a slanted asymptote. The derivative vanishes at the point and does not exist at. When x passes through the point) the derivative does not change sign, so there is no extremum at the point x = 0. When a point x passes through a point, the derivative) changes sign from “+” to So the function has a maximum. When x passes through the point x = 3 (x > I), the derivative y"(x) changes sign, i.e., at the point x = 3 the function has a minimum. 7. Finding the second derivative Scheme for constructing a graph of a function Study of functions to an extremum using derivatives higher order Calculation of roots of equations by chord and tangent methods The second derivative y"(x) does not exist at the point x = 0 and when x passes through the point x = 0 y" changes sign from + to so that the point (0,0) of the curve is a point There is no inflection with a vertical tangent. At the point x = 3 there is no inflection of the graph. Everywhere in the half-plane x > 0 the convexity of the curve is directed upward. The results of the study are summarized in the table: Does not exist Does not exist Does not exist Inflection point (0.0) with a vertical tangent The graph of the function is presented. in Fig. 39. §7. Study of functions on an extremum using derivatives of higher order. To find the points of maximum and minimum of functions, the Taylor formula can be used. Let the function /(x) have a derivative of the nth order in a certain neighborhood of the point xq. , continuous at the point x0. Let 0. Then if the number n is odd, then the function f(x) at the point x0 has no extremum; when n is even, then at the point x0 the function f(x) has a maximum if /(n)(x0)< 0, и минимум, если /. В силу определения точек максимума и минимума вопрос о том, имеет ли функция f(x) в точке х0 экстремум, сводится к тому, существует ли такое <5 >0, which is in the interval, the difference - /(x0) retains its sign. Using the Taylor formula as a condition, then from (1) we obtain 1condition f(n*(r) is continuous at a point and Φ Therefore, due to the stability of the name of a continuous function, there exists such that in the interval () does not change and coincides with the sign of f(n)( xo). Let's consider the possible cases: 1) n is an even number and / Then I therefore by virtue of (2). According to the definition, this means that the point r is the minimum point of the function /(r). 2) n - even and. Then we will have i together with this and Therefore, the point i will be in this case the maximum point of the function /(r). 3) n is an odd number, / - Then for x > x0 the sign > will coincide with the sign of /(n)(th), and for r th it will be the opposite. Therefore, no matter how small 0, the sign of the difference f(r) - f(r) will not be the same for all x e (r - 6, r + £). Consequently, in this case the function f(r) at the point th has no extremum. Example. Let us consider the functions A. It is easy to see that the point x = 0 is critical point both functions. For the function y = x4, the first of the nonzero derivatives at the point x = 0 is the 4th order derivative: Thus, here n = 4 is even and. Therefore, at the point x = 0 the function y = x4 has a minimum. For the function y = x), the first of the derivatives that are nonzero at the point x = 0 is the 3rd order derivative. So in this case n = 3 is odd, and at the point x = 0 the function y = x3 does not have an extremum. Comment. Using Taylor's formula, we can prove the following theorem, which expresses sufficient conditions for the inflection point. "Theorem 12. Let the function /(r) in some neighborhood of the point r0 have a derivative of the th order, continuous at the point xq. Let, but /(n)(*o) Φ 0. Then, if n is an odd number, then the point Mo(x0, f(xо)) is the inflection point of the graph of the function y = f(x). The simplest example is provided by the function §8. Calculating the roots of equations using the methods of chords and tangents. The problem is to find the real root of the equation. Let us assume that the following conditions are met: 1) the function f(x) is continuous on the interval [a, 6]; 2) the numbers f(a) and f(b) are opposite in sign: 3) on the interval [a, 6] there exist derivatives f"(x) and f "(x), maintaining a constant sign on this segment. From conditions 1) and 2) by virtue of the Bolzano-Cauchy theorem (p. 220) it follows that the function /(x) vanishes at least at one point £ € ( a, b), that is, equation (1) has at least one real root £ in the interval (a, 6) since, by virtue of condition 3), the derivative /"(x) on [a, b\ remains constant. sign, then f(x) is monotonic on [a, b] and therefore in the interval (a, b) equation (1) has only one real root. Consider a method for calculating the approximate value of this single real root £ € (a, 6) of the equation ( I) with any degree of accuracy. Four cases are possible (Fig. 40): 1) Fig. 40 For definiteness, let us take the case when f\x) > 0, f"(x) > 0 on the segment [a, 6) (Fig. 41). Let us connect the points A(a, /(a)) and B(b, f(b)) by chord A B. This is a segment of a straight line passing through points A and B, the equation of which is Point aj, at which the chord AB intersects the Ox axis, is located between ai (and is a better approximation to a. Assuming in (2) y = 0, we find From Fig. 41 it is easy to notice that the point a\ will always be located on the side from which the signs f(x) and f"(x) are opposite. Let us now draw a tangent to the curve y = f(x) in point B(b, f(b)), i.e. at that end of the arc ^AB at which f(x) and /"(i) have the same sign. This is an essential condition: without it, the point of intersection tangent to the Ox axis may not give an approximation to the desired root at all. The point b\, at which the tangent intersects the Ox axis, is located between £ and b on the same side as 6, and is a better approximation to than b. This tangent is determined by the equation Assuming y = 0 in (3), we find b\: Scheme for constructing a graph of a function Study of functions to an extremum using higher order derivatives Calculation of the roots of equations using the methods of chords and tangents Thus, we have Let the absolute error of the approximation C of the root £ be given in advance . For the absolute error of the approximate values of aj and 6, the root £, we can take the value |6i - ai|. If this error is greater than the permissible one, then, taking the segment as the original one, we find the following approximations of the root where. Continuing this process, we obtain two sequences of approximate values. Sequences (an) and (bn) are monotonic and limited and, therefore, have limits. Let It can be shown that if the conditions formulated above are met, 1 to the only root of the equation / Example. Find the root (equation r2 - 1 = 0 on the segment . Thus, all conditions are met to ensure the existence of a single root (equation x2 - 1 = 0 on the segment . . and the method should work. 8 in our case a = 0, b = 2. When n = I from (4) and (5) we find When n = 2 we obtain which gives an approximation to the exact value of the root (with absolute error Exercises Construct graphs of functions: Find the largest and smallest value functions on given segments: Explore the behavior of functions in the neighborhood given points using higher order derivatives: Answers

First, try to find the domain of the function:

Did you manage? Let's compare the answers:

Is everything right? Well done!

Now let's try to find the range of values of the function:

Found it? Let's compare:

Got it? Well done!

Let's work with graphs again, only now it's a little more complicated - find both the domain of definition of the function and the range of values of the function.

How to find both the domain and range of a function (advanced)

Here's what happened:

I think you've figured out the graphs. Now let’s try to find the domain of definition of a function in accordance with the formulas (if you don’t know how to do this, read the section about):

Did you manage? Let's check answers:

, since the radical expression must be greater than or equal to zero.
, since you cannot divide by zero and the radical expression cannot be negative.
, since, respectively, for all.
, since you cannot divide by zero.

However, we still have one more unanswered point...

I will repeat the definition once again and emphasize it:

Did you notice? The word "single" is a very, very important element of our definition. I'll try to explain it to you with my fingers.

Let's say we have a function defined by a straight line. . At, we substitute this value into our “rule” and get that. One value corresponds to one value. We can even make a table of the different values and graph this function to see for ourselves.

"Look! - you say, ““ occurs twice!” So maybe a parabola is not a function? No, it is!

The fact that “ ” appears twice is not a reason to accuse the parabola of ambiguity!

The fact is that, when calculating for, we received one game. And when calculating with, we received one game. So that's right, a parabola is a function. Look at the graph:

Got it? If not, here you go life example very far from mathematics!

Let’s say we have a group of applicants who met while submitting documents, each of whom told in a conversation where he lives:

Agree, it is quite possible for several guys to live in one city, but it is impossible for one person to live in several cities at the same time. This is like a logical representation of our “parabola” - Several different X's correspond to the same game.

Now let's come up with an example where the dependency is not a function. Let’s say these same guys told us what specialties they applied for:

Here we have a completely different situation: one person can easily submit documents for one or several directions. That is one element sets are put into correspondence several elements multitudes. Respectively, this is not a function.

Let's test your knowledge in practice.

Determine from the pictures what is a function and what is not:

Got it? And here it is answers:

The function is - B, E.
The function is not - A, B, D, D.

You ask why? Yes, here's why:

In all pictures except IN) And E) There are several for one!

I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the range of permissible values of an argument and the range of definition of a function. Let's move on to the next section - how to set a function?

Methods for specifying a function

What do you think the words mean? "set function"? That's right, this means explaining to everyone what function we are talking about in this case. And explain it in such a way that everyone understands you correctly and the function graphs drawn by people based on your explanation are the same.

How can this be done? How to set a function? The simplest method, which has already been used more than once in this article, is using the formula. We write a formula, and by substituting a value into it, we calculate the value. And as you remember, a formula is a law, a rule by which it becomes clear to us and to another person how an X turns into a Y.

Usually, this is exactly what they do - in tasks we see ready-made functions specified by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” baffles. Let's understand everything in order, and let's start with the analytical method.

Analytical method of specifying a function

The analytical method is to specify a function using a formula. This is the most universal, comprehensive and unambiguous method. If you have a formula, then you know absolutely everything about a function - you can make a table of values from it, you can build a graph, determine where the function increases and where it decreases, in general, study it in full.

Let's consider the function. What's the difference?

"What does it mean?" - you ask. I'll explain now.

Let me remind you that in the notation the expression in brackets is called an argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (the expression in brackets) is, we will write it instead in the expression.

In our example it will look like this:

Let's consider another task related to the analytical method of specifying a function, which you will have on the exam.

Find the value of the expression at.

I'm sure that at first you were scared when you saw such an expression, but there is absolutely nothing scary about it!

Everything is the same as in the previous example: whatever the argument (the expression in brackets) is, we will write it instead in the expression. For example, for a function.

What needs to be done in our example? Instead you need to write, and instead -:

shorten the resulting expression:

That's it!

Independent work

Now try to find the meaning of the following expressions yourself:

, If
, If

Did you manage? Let's compare our answers: We are used to the fact that the function has the form

Even in our examples, we define the function in exactly this way, but analytically it is possible to specify the function in an implicit form, for example.

Try building this function yourself.

Did you manage?

This is how I built it.

What equation did we finally derive?

Right! Linear, which means that the graph will be a straight line. Let's make a table to determine which points belong to our line:

This is exactly what we were talking about... One corresponds to several.

Let's try to draw what happened:

Is what we got a function?

That's right, no! Why? Try to answer this question with the help of a drawing. What did you get?

“Because one value corresponds to several values!”

What conclusion can we draw from this?

That's right, a function cannot always be expressed explicitly, and what is “disguised” as a function is not always a function!

Tabular method of specifying a function

As the name suggests, this method is a simple sign. Yes, yes. Like the one you and I have already made. For example:

Here you immediately noticed a pattern - the Y is three times larger than the X. And now the task to “think very carefully”: do you think that a function given in the form of a table is equivalent to a function?

Let's not talk for a long time, but let's draw!

So. We draw the function specified by the wallpaper in the following ways:

Do you see the difference? It's not all about the marked points! Take a closer look:

Have you seen it now? When we define a function in a tabular way, we display on the graph only those points that we have in the table and the line (as in our case) passes only through them. When we define a function analytically, we can take any points, and our function is not limited to them. This is the peculiarity. Remember!

Graphical method of constructing a function

The graphical method of constructing a function is no less convenient. We draw our function, and another interested person can find what y is equal to at a certain x and so on. Graphical and analytical methods are among the most common.

However, here you need to remember what we talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Do you remember? Just in case, I’ll copy here the definition of what a function is:

As a rule, people usually name exactly the three ways of specifying a function that we have discussed - analytical (using a formula), tabular and graphical, completely forgetting that a function can be described verbally. How is this? Yes, very simple!

Verbal description of the function

How to describe a function verbally? Let's take our recent example - . This function can be described as “every real value of x corresponds to its triple value.” That's it. Nothing complicated. You, of course, will object - “there are so complex functions, which are simply impossible to ask verbally!” Yes, there are such, but there are functions that are easier to describe verbally than to define with a formula. For example: “each natural value of x corresponds to the difference between the digits of which it consists, while the largest digit contained in the number’s notation is taken as the minuend.” Now let's look at how our verbal description of the function is implemented in practice:

The largest digit in a given number is, respectively, the minuend, then:

Main types of functions

Now let's move on to the most interesting part - let's look at the main types of functions with which you have worked/are working and will work in the course of school and college mathematics, that is, let's get to know them, so to speak, and give them brief description. Read more about each function in the corresponding section.

Linear function

A function of the form where, are real numbers.

The graph of this function is a straight line, so constructing a linear function comes down to finding the coordinates of two points.

The position of the straight line on the coordinate plane depends on the angular coefficient.

The scope of a function (aka the scope of valid argument values) is .

Range of values - .

Quadratic function

Function of the form, where

The graph of the function is a parabola; when the branches of the parabola are directed downwards, when the branches are directed upwards.

Many properties quadratic function depend on the value of the discriminant. The discriminant is calculated using the formula

The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:

Domain of definition

The range of values depends on the extremum of the given function (vertex point of the parabola) and the coefficient (direction of the branches of the parabola)

Inverse proportionality

The function given by the formula, where

The number is called the coefficient of inverse proportionality. Depending on the value, the branches of the hyperbola are in different squares:

Scope of definition - .

Range of values - .

SUMMARY AND BASIC FORMULAS

1. A function is a rule according to which each element of a set is associated with a single element of the set.

- this is a formula denoting a function, that is, the dependence of one variable on another;
- variable quantity, or, argument;
- dependent quantity - changes when the argument changes, that is, according to any specific formula reflecting the dependence of one quantity on another.

2. Valid argument values, or the domain of a function, is what is associated with the possibilities in which the function makes sense.

3. Function range- this is what values it takes, given acceptable values.

4. There are 4 ways to set a function:

analytical (using formulas);
tabular;
graphic
verbal description.

5. Main types of functions:

: , where, are real numbers;
: , Where;
: , Where.

Build function

We offer to your attention a service for constructing graphs of functions online, all rights to which belong to the company Desmos. Use the left column to enter functions. You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the window with the graph, you can hide both the left column and the virtual keyboard.

Benefits of online charting

Visual display of entered functions
Building very complex graphs
Construction of graphs specified implicitly (for example, ellipse x^2/9+y^2/16=1)
The ability to save charts and receive a link to them, which becomes available to everyone on the Internet
Control of scale, line color
Possibility of plotting graphs by points, using constants
Plotting several function graphs simultaneously
Plotting graphs in polar system coordinates (use r and θ(\theta))

With us it’s easy to build charts of varying complexity online. Construction is done instantly. The service is in demand for finding intersection points of functions, for depicting graphs for further moving them into a Word document as illustrations when solving problems, and for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this website page is Google Chrome. Correct operation is not guaranteed when using other browsers.