A function is called even (odd) if for any and the equality 
.
The graph of an even function is symmetrical about the axis 
.
The graph of an odd function is symmetrical about the origin.
Example 6.2. Examine whether a function is even or odd
1)
;
2)
;
3)
.
Solution.
1) The function is defined when 
. We'll find 
.
Those. 
. Means, this function is even.
2) The function is defined when 
Those. 
. Thus, this function is odd.
3) the function is defined for , i.e. For
,
. Therefore the function is neither even nor odd. Let's call it a function of general form.
Function 
is called increasing (decreasing) on a certain interval if in this interval each larger value of the argument corresponds to a larger (smaller) value of the function.
Functions increasing (decreasing) over a certain interval are called monotonic.
If the function 
differentiable on the interval 
and has a positive (negative) derivative 
, then the function 
increases (decreases) over this interval.
Example 6.3. Find intervals of monotonicity of functions
1)
;
3)
.
Solution.
1) This function is defined on the entire number line. Let's find the derivative.
The derivative is equal to zero if 
And 
. The domain of definition is the number axis, divided by dots 
,
at intervals. Let us determine the sign of the derivative in each interval.
In the interval 
the derivative is negative, the function decreases on this interval.
In the interval 
the derivative is positive, therefore, the function increases over this interval.
2) This function is defined if 
or
.
We determine the sign of the quadratic trinomial in each interval.
Thus, the domain of definition of the function
Let's find the derivative 
,
, If 
, i.e. 
, But 
. Let us determine the sign of the derivative in the intervals 
.
In the interval 
the derivative is negative, therefore, the function decreases on the interval 
. In the interval 
the derivative is positive, the function increases over the interval 
.
Dot 
called the maximum (minimum) point of the function 
, if there is such a neighborhood of the point 
that's for everyone 
from this neighborhood the inequality holds 
.
The maximum and minimum points of a function are called extremum points.
If the function 
at the point 
has an extremum, then the derivative of the function at this point is equal to zero or does not exist (a necessary condition for the existence of an extremum).
The points at which the derivative is zero or does not exist are called critical.
5. Sufficient conditions for the existence of an extremum.Rule 1. If during the transition (from left to right) through the critical point 
derivative 
changes sign from “+” to “–”, then at the point 
function 
has a maximum; if from “–” to “+”, then the minimum; If 
does not change sign, then there is no extremum.
Rule 2. Let at the point 
first derivative of a function 
equal to zero 
, and the second derivative exists and is different from zero. If 
, That 
– maximum point, if 
, That 
– minimum point of the function.
Example 6.4. Explore the maximum and minimum functions:
1)
;
2)
;
3)
;
4)
.
Solution.
1) The function is defined and continuous on the interval 
.
Let's find the derivative 
and solve the equation 
, i.e. 
.From here 
– critical points.
Let us determine the sign of the derivative in the intervals , 
.
When passing through points 
And 
the derivative changes sign from “–” to “+”, therefore, according to rule 1 
– minimum points.
When passing through a point 
the derivative changes sign from “+” to “–”, so 
– maximum point.
,
.
2) The function is defined and continuous in the interval 
. Let's find the derivative 
.
Having solved the equation 
, we'll find 
And 
– critical points. If the denominator 
, i.e. 
, then the derivative does not exist. So, 
– third critical point. Let us determine the sign of the derivative in intervals.
Therefore, the function has a minimum at the point 
, maximum in points 
And 
.
3) A function is defined and continuous if 
, i.e. at 
.
Let's find the derivative 
.
Let's find critical points: 
Neighborhoods of points 
do not belong to the domain of definition, therefore they are not extrema. So, let's examine the critical points 
And 
.
4) The function is defined and continuous on the interval 
. Let's use rule 2. Find the derivative 
.
Let's find critical points:
Let's find the second derivative 
and determine its sign at the points 
At points 
function has a minimum.
At points 
the function has a maximum.
Function study.
1) D(y) – Definition domain: the set of all those values of the variable x. for which the algebraic expressions f(x) and g(x) make sense.
If a function is given by a formula, then the domain of definition consists of all values of the independent variable for which the formula makes sense.
2) Properties of the function: even/odd, periodicity:
Functions whose graphs are symmetrical with respect to changes in the sign of the argument are called odd and even.
An odd function is a function that changes its value to the opposite when the sign of the independent variable changes (symmetrical relative to the center of coordinates).
An even function is a function that does not change its value when the sign of the independent variable changes (symmetrical about the ordinate).
Neither even nor odd function (function general view) is a function that does not have symmetry. This category includes functions that do not fall under the previous 2 categories.
Functions that do not belong to any of the categories above are called neither even nor odd(or general functions).
Odd functions
Odd power where is an arbitrary integer.
Even functions
Even power where is an arbitrary integer.
A periodic function is a function that repeats its values after a certain regular interval of the argument, that is, it does not change its value when adding to the argument some fixed non-zero number (period of the function) throughout the entire domain of definition.
3) Zeros (roots) of a function are the points where it becomes zero.
Finding the intersection point of the graph with the axis Oy. To do this you need to calculate the value f(0). Find also the points of intersection of the graph with the axis Ox, why find the roots of the equation f(x) = 0 (or make sure there are no roots).
The points at which the graph intersects the axis are called zeros of the function. To find the zeros of a function, you need to solve the equation, that is, find those values of “x” at which the function becomes zero.
4) Intervals of constancy of signs, signs in them.
Intervals where the function f(x) maintains sign.
An interval of constant sign is an interval at each point of which the function is positive or negative.
ABOVE the x-axis.
BELOW the axle.
5) Continuity (points of discontinuity, nature of the discontinuity, asymptotes).
A continuous function is a function without “jumps”, that is, one in which small changes in the argument lead to small changes in the value of the function.
Removable Break PointsIf the limit of the function exists, but the function is not defined at this point, or the limit does not coincide with the value of the function at this point:
,
then the point is called removable break point functions (in complex analysis, a removable singular point).
If we “correct” the function at the point of removable discontinuity and put 
, then we get a function that is continuous at a given point. This operation on a function is called extending the function to continuous or redefinition of the function by continuity, which justifies the name of the point as a point removable rupture.
Discontinuity points of the first and second kind
If a function has a discontinuity at a given point (that is, the limit of the function at a given point is absent or does not coincide with the value of the function at a given point), then for numerical functions there are two possible options associated with the existence of numerical functions unilateral limits:
if both one-sided limits exist and are finite, then such a point is called a discontinuity point of the first kind. Removable discontinuity points are discontinuity points of the first kind;
if at least one of the one-sided limits does not exist or is not a finite value, then such a point is called a discontinuity point of the second kind.
Asymptote - straight, which has the property that the distance from a point on the curve to this straight tends to zero as the point moves away along the branch to infinity.
Vertical
Vertical asymptote - limit line 
.
As a rule, when determining the vertical asymptote, they look for not one limit, but two one-sided ones (left and right). This is done in order to determine how the function behaves as it approaches the vertical asymptote from different directions. For example:
HorizontalHorizontal asymptote - straight species, subject to the existence limit
.
Oblique asymptote - straight species, subject to the existence limits
Note: a function can have no more than two oblique (horizontal) asymptotes.
Note: if at least one of the two limits mentioned above does not exist (or is equal to ), then the oblique asymptote at (or ) does not exist.
if in item 2.), then , and the limit is found by the formula horizontal asymptote, 
.
6) Finding intervals of monotonicity. Find intervals of monotonicity of a function f(x)(that is, intervals of increasing and decreasing). This is done by examining the sign of the derivative f(x). To do this, find the derivative f(x) and solve the inequality f(x)0. On intervals where this inequality holds, the function f(x)increases. Where the reverse inequality holds f(x)0, function f(x) is decreasing.
Finding local extremum. Having found the intervals of monotonicity, we can immediately determine the local extremum points where an increase is replaced by a decrease, local maxima are located, and where a decrease is replaced by an increase, local minima are located. Calculate the value of the function at these points. If a function has critical points that are not local extremum points, then it is useful to calculate the value of the function at these points as well.
Finding the largest and smallest values of the function y = f(x) on a segment (continued)
|
1. Find the derivative of the function: f(x). 2. Find the points at which the derivative is zero: f(x)=0x 1, x 2 ,... 3. Determine the affiliation of points X 1 ,X 2 , … segment [ a; b]: let x 1a;b, A x 2a;b . |
How to insert mathematical formulas on a website?
If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted onto the site in the form of pictures that are automatically generated by Wolfram Alpha. In addition to simplicity, this universal method will help improve the visibility of the site in search engines. It has been working for a long time (and, I think, will work forever), but is already morally outdated.
If you constantly use mathematical formulas on your site, then I recommend that you use MathJax - a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX or ASCIIMathML markup.
There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your website, which will be automatically loaded from a remote server at the right time (list of servers); (2) download the MathJax script from a remote server to your server and connect it to all pages of your site. The second method - more complex and time-consuming - will speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method as it is simpler, faster and does not require technical skills. Follow my example, and in just 5 minutes you will be able to use all the features of MathJax on your site.
You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or on the documentation page:
One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.
The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.
Any fractal is constructed according to a certain rule, which is applied sequentially an unlimited number of times. Each such time is called an iteration.
The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. The result is a set consisting of the remaining 20 smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process endlessly, we get a Menger sponge.
Even function.
A function whose sign does not change when the sign changes is called even. x.
x equality holds f(–x) = f(x). Sign x does not affect the sign y.
Schedule even function symmetrical relative to the coordinate axis (Fig. 1).
Examples of an even function:
y=cos x
y = x 2
y = –x 2
y = x 4
y = x 6
y = x 2 + x
Explanation:
Let's take the function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect the sign y. The graph is symmetrical about the coordinate axis. This is an even function.
Odd function.
A function whose sign changes when the sign changes is called odd. x.
In other words, for any value x equality holds f(–x) = –f(x).
The graph of an odd function is symmetrical with respect to the origin (Fig. 2).
Examples of odd function:
y= sin x
y = x 3
y = –x 3
Explanation:
Let's take the function y = – x 3 .
All meanings at it will have a minus sign. That is a sign x influences the sign y. If the independent variable is positive number, then the function is positive if the independent variable is a negative number, then the function is negative: f(–x) = –f(x).
The graph of the function is symmetrical about the origin. This is an odd function.
Properties of even and odd functions:
NOTE:
Not all functions are even or odd. There are functions that do not obey such gradation. For example, the root function at = √X does not apply to either even or odd functions (Fig. 3). When listing the properties of such functions, an appropriate description should be given: neither even nor odd.
As you know, periodicity is the repetition of certain processes at a certain interval. Functions that describe these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.