What is called the period of a function? Periodic function
FUNCTIONS AND LIMITS IX
§ 207. Periodic functions
Function at =f (x ) is called periodic if there is a number T =/= 0, such that for all values X from the domain of definition of this function.
f (x + T) = f (x ) .
The number T in this case is called the period of the function.
Periodic ones are, for example, trigonometric functions at = sin X And at =cos X . Their period is 2 π . An example of a periodic non-trigonometric function is the function at = {X ), which for each number X matches its fractional part*.
* For the fractional part of a number, see Chapter VIII, § 187.
For example, (3.56) = 0.56; (2.01) = 0.01, etc. If to an arbitrary number X add 1, then only the integer part of this number will change; the fractional part will remain the same. Hence, ( X + 1} = {X ) and therefore the function at = {X ) is periodic with period 1.
From equality f (x + T) = f (x ) it follows that all values of the function at =f (x ) repeat with period T. This is reflected in graphic representation periodic functions. So, for example, in the interval the sinusoid has the same shape as in the intervals, etc. (Fig. 282).
Figure 283 shows the graph of the function at = {X ). Function frequency at = {X ) determines that its graph in the interval has the same shape as in the intervals, etc.
If T is the period of the function f (x ), then 2T, 3T, 4T, etc. are also periods of this function.
Really,
f (x + 2T) = f [(x + T) + T] = f (x + T) = f (x ),
f (x + 3T) = f [(x + 2T) + T] = f (x + 2T) = f (x )
etc. In addition, the period of the function f (x ) can be considered any of the numbers: - T, - 2T, - 3T, etc. In fact,
f (x - T) = f [(x - T) + T] = f (x ),
f (x - 2T) = f [(x - 2T) + 2T] = f (x )
etc. So, if the number T is the period of the function f (x ), then for any integer P number P T is also the period of this function. That's why every periodic function has an infinite number of periods . For example, the period of the function at = sin x you can count any of the numbers: 2 π , 4π , 6π , - 2π , - 4π , and the period of the function at = {X ) - any of the numbers: 1, 2, 3, - 1, - 2, - 3, etc.
Talking about the period of a function at = f (x ), usually meaning the smallest positive period. So, we say that the period of the function at = sin X is the number 2 π , period of the function at = tg X - number π , period of the function ( X ) - number 1, etc.
However, it should be borne in mind that a periodic function may not have the smallest positive period.
For example, for the function f (x ) = 3 (Fig. 284) any real number is a period. But among the positive real numbers there is no smallest. Therefore the function f (x ) = 3, having an infinite number of periods, does not have the smallest positive period.
Exercises
For each of these functions (No. 1613-1621), find the smallest positive period:
1613. at = sin 2 X . 1619. at = sin (3 X - π / 4).
1614. at =cos x / 2 . 1620. at = sin 2 X
1615. at = tan 3 X . 1621. at = sin 4 X + cos 4 X .
1616. at =cos(1 - 2 X ).
1617. at = sin X cos X .
1618. at =ctg x / 3 ;
1622. Prove that the sum and product of two functions periodic with the same period T are functions periodic with period T.
1623*. Prove that the function at = sin X + {X ), which is the sum of two periodic functions at = sin X And at = {X ), itself is not periodic.
Doesn't this contradict the result of the previous problem?
1624. How to complete the graph of a function at = f (x ), periodic with period T, if it is specified only in the interval ?
Material for preparing for the algebra colloquium.
1. Definition of a function.
Function- variable dependence at from variable x, at which each value X corresponds single meaning variable at.
2. Definition of an increasing function.
Increasing function (not decreasing)- if for any values x 1 And x 2, such that x 1< х 2 , the inequality ( if a larger value of the argument from this interval corresponds to a larger value of the function).
To Using the graph of the function, determine the intervals of increase in the function, you need to move from left to right along the function graph line to highlight the intervals of the argument values X, where the graph is going up.
Decreasing function (not increasing)- if for any x 1 And x 2, such that x 1< х 2 , the inequality ( the larger value of the argument from this interval corresponds to the smaller value of the function.
To Using the graph of the function, determine the intervals of decrease in the function, you need to move from left to right along the graph line of the function to highlight the intervals of the argument values X, where the graph is going down.
Definition of even function, odd function, function general view.
The function is called even, if the following two conditions are met:
1. if (if X - X
2. for anyone X .
The function is called odd, if the following two conditions are met:
1. If the domain of definition of the function is symmetrical relative to the op-amp axis(If X belongs to the domain of definition of the function, then - X also belongs to the domain of definition of the function);
2. for anyone X from the domain of definition of the function the equality 
.
The function is called general function if these conditions are not met.
4. What conditions does the graph of an even and odd function have?
Property. Graph of an even function symmetrical about the OY axis.
Property. Graph of an odd function symmetrical about the origin.
Definition of a periodic function.
The function is called periodic, if there is a positive number T such that 
.
The smallest number with this property is called the period of the function.
6. List the main properties of the function y= sin x:
1) Function domain- all the values that the independent variable takes X.
The domain of this function is the set of all real numbers. Since instead of X into the equation y=sin(x) we can put any number. D (sin x) = R.
2) Function range- all the values that the dependent variable takes at.
The range of values of this function is the segment [-1;1]. E (sin x) = [-1;1].
3)
Function called periodic if there is a positive number T such that 
. The smallest number with this property is called the period of the function.
The function y=sin(x) is periodic, with a period of 2π.
4) The function y=sin(x) is odd. Recall that the graph of an odd function is symmetrical about the origin.
5) The function y=sin(x) takes:
Value equal to 0 when x =
Lowest value, equal to -1, for x= - ;
Positive values on the interval (0,π) and on the intervals obtained by shifting this interval by ;
Negative values on the interval () and on the intervals obtained by shifting this interval by ;
6) Function y=sin(x):
- increases on the interval [ - ; ], and on segments obtained by shifting this segment by ;
Decreases on the segment [; ], and on segments obtained by shifting this segment by ;
7. List the main properties of the function y= cos x:
1) The domain of this function is the set of all real numbers. D(cos) = R.
2) The range of values of this function is the segment [-1;1]. E (cos)=[-1;1].
3) The function y = cos (x) is periodic, with a period of 2.
4) The function y=cos(x) is even. Let me remind you that the graph of an odd function is symmetrical about the OY axis.
5) The function y=cos(x) takes:
A value equal to 0 when x = ;
Highest value, equal to 1, for x = ;
The smallest value is -1 at x = ;
Positive values on the interval () and on the intervals obtained by shifting this interval by ;
Negative values on the interval ( ; ) and on intervals obtained by shifting this interval by ;
6) Function y=cos(x):
- increases on the segment [ ;2 ], and on the segments obtained by shifting this segment by ;
Decreases on the segment , and on the segments obtained by shifting this segment by ;
Graphs of functions y=cos(x) and y= sin (x)
8. List the main properties of the function y= tan x:
1) The domain of this function is the set of all real numbers except .
Goal: summarize and systematize students’ knowledge on the topic “Periodicity of Functions”; develop skills in applying the properties of a periodic function, finding the smallest positive period of a function, constructing graphs of periodic functions; promote interest in studying mathematics; cultivate observation and accuracy.
Equipment: computer, multimedia projector, task cards, slides, clocks, tables of ornaments, elements of folk crafts
“Mathematics is what people use to control nature and themselves.”
A.N. Kolmogorov
During the classes
I. Organizational stage.
Checking students' readiness for the lesson. Report the topic and objectives of the lesson.
II. Checking homework.
We check homework using samples and discuss the most difficult points.
III. Generalization and systematization of knowledge.
1. Oral frontal work.
Theory issues.
1) Form a definition of the period of the function
2) Name the smallest positive period of the functions y=sin(x), y=cos(x)
3). What is the smallest positive period of the functions y=tg(x), y=ctg(x)
4) Using a circle, prove the correctness of the relations:
y=sin(x) = sin(x+360º)
y=cos(x) = cos(x+360º)
y=tg(x) = tg(x+18 0º)
y=ctg(x) = ctg(x+180º)
tg(x+π n)=tgx, n € Z
ctg(x+π n)=ctgx, n € Z
sin(x+2π n)=sinx, n € Z
cos(x+2π n)=cosx, n € Z
5) How to plot a periodic function?
Oral exercises.
1) Prove the following relations
a) sin(740º) = sin(20º)
b) cos(54º) = cos(-1026º)
c) sin(-1000º) = sin(80º)
2. Prove that an angle of 540º is one of the periods of the function y= cos(2x)
3. Prove that an angle of 360º is one of the periods of the function y=tg(x)
4. Transform these expressions so that the angles included in them do not exceed 90º in absolute value.
a) tg375º
b) ctg530º
c) sin1268º
d) cos(-7363º)
5. Where did you come across the words PERIOD, PERIODICITY?
Student answers: A period in music is a structure in which a more or less complete musical thought is presented. Geological period- part of an era and is divided into epochs with a period from 35 to 90 million years.
Half-life of a radioactive substance. Periodic fraction. Periodicals are printed publications that appear within strictly defined deadlines. Periodic table Mendeleev.
6. The figures show parts of the graphs of periodic functions. Determine the period of the function. Determine the period of the function.
Answer: T=2; T=2; T=4; T=8.
7. Where in your life have you encountered the construction of repeating elements?
Student answer: Elements of ornaments, folk art.
IV. Collective problem solving.
(Solving problems on slides.)
Let's consider one of the ways to study a function for periodicity.
This method avoids the difficulties associated with proving that a particular period is the shortest, and also eliminates the need to touch upon questions about arithmetic operations over periodic functions and on the periodicity of a complex function. The reasoning is based only on the definition of a periodic function and on the following fact: if T is the period of the function, then nT(n?0) is its period.
Problem 1. Find the smallest positive period of the function f(x)=1+3(x+q>5)
Solution: Assume that the T-period of this function. Then f(x+T)=f(x) for all x € D(f), i.e.
1+3(x+T+0.25)=1+3(x+0.25)
(x+T+0.25)=(x+0.25)
Let's put x=-0.25 and we get
(T)=0<=>T=n, n € Z
We have obtained that all periods of the function in question (if they exist) are among the integers. Let's choose the smallest positive number among these numbers. This 1 . Let's check whether it will actually be a period 1 .
f(x+1) =3(x+1+0.25)+1
Since (T+1)=(T) for any T, then f(x+1)=3((x+0.25)+1)+1=3(x+0.25)+1=f(x ), i.e. 1 – period f. Since 1 is the smallest of all positive integers, then T=1.
Problem 2. Show that the function f(x)=cos 2 (x) is periodic and find its main period.
Problem 3. Find the main period of the function
f(x)=sin(1.5x)+5cos(0.75x)
Let us assume the T-period of the function, then for any X the ratio is valid
sin1.5(x+T)+5cos0.75(x+T)=sin(1.5x)+5cos(0.75x)
If x=0, then
sin(1.5T)+5cos(0.75T)=sin0+5cos0
sin(1.5T)+5cos(0.75T)=5
If x=-T, then
sin0+5cos0=sin(-1.5T)+5cos0.75(-T)
5= – sin(1.5T)+5cos(0.75T)
| sin(1.5T)+5cos(0.75T)=5 – sin(1.5T)+5cos(0.75T)=5 |
Adding it up, we get:
10cos(0.75T)=10
2π n, n € Z
Let us choose the smallest positive number from all the “suspicious” numbers for the period and check whether it is a period for f. This number
f(x+)=sin(1.5x+4π )+5cos(0.75x+2π )= sin(1.5x)+5cos(0.75x)=f(x)
This means that this is the main period of the function f.
Problem 4. Let’s check whether the function f(x)=sin(x) is periodic
Let T be the period of the function f. Then for any x
sin|x+Т|=sin|x|
If x=0, then sin|Т|=sin0, sin|Т|=0 Т=π n, n € Z.
Let's assume. That for some n the number π n is the period
the function under consideration π n>0. Then sin|π n+x|=sin|x|
This implies that n must be both an even and an odd number, but this is impossible. That's why this function is not periodic.
Task 5. Check if the function is periodic
f(x)= 
Let T be the period of f, then
, hence sinT=0, Т=π n, n € Z. Let us assume that for some n the number π n is indeed the period of this function. Then the number 2π n will be the period
Since the numerators are equal, their denominators are also equal, therefore
This means that the function f is not periodic.
Work in groups.
Tasks for group 1.
Tasks for group 2.
Check if the function f is periodic and find its fundamental period (if it exists).
f(x)=cos(2x)+2sin(2x)
Tasks for group 3.
At the end of their work, the groups present their solutions.
VI. Summing up the lesson.
Reflection.
The teacher gives students cards with drawings and asks them to color part of the first drawing in accordance with the extent to which they think they have mastered the methods of studying a function for periodicity, and in part of the second drawing - in accordance with their contribution to the work in the lesson.
VII. Homework
1). Check if the function f is periodic and find its fundamental period (if it exists)
b). f(x)=x 2 -2x+4
c). f(x)=2tg(3x+5)
2). The function y=f(x) has a period T=2 and f(x)=x 2 +2x for x € [-2; 0]. Find the value of the expression -2f(-3)-4f(3.5)
Literature/
- Mordkovich A.G. Algebra and beginnings of analysis with in-depth study.
- Mathematics. Preparation for the Unified State Exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
- Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.
The number T such that for any x F(x + T) = F(x). This number T is called the period of the function.
There may be several periods. For example, the function F = const takes the same value for any value of the argument, and therefore any number can be considered its period.
Usually interested in the smallest equal to zero period of the function. For brevity, it is simply called a period.
A classic example of periodic functions is trigonometric: sine, cosine and tangent. Their period is the same and equal to 2π, that is, sin(x) = sin(x + 2π) = sin(x + 4π) and so on. However, of course, trigonometric functions are not the only periodic ones.
Relatively simple, basic functions the only way establish their periodicity or non-periodicity - calculations. But for complex functions there are already several simple rules.
If F(x) is with period T, and a derivative is defined for it, then this derivative f(x) = F′(x) is also a periodic function with period T. After all, the value of the derivative at point x is equal to the tangent of the tangent angle of the graph of its antiderivative at this point to the x-axis, and since it is repeated periodically, it must be repeated. For example, the derivative of functions sin(x) is equal to cos(x), and it is periodic. Taking the derivative of cos(x) gives you –sin(x). The frequency remains unchanged.
However, the opposite is not always true. Thus, the function f(x) = const is periodic, but its antiderivative F(x) = const*x + C is not.
If F(x) is a periodic function with period T, then G(x) = a*F(kx + b), where a, b, and k are constants and k is not equal to zero - is also a periodic function, and its period is T/k. For example, sin(2x) is a periodic function, and its period is π. This can be visually represented this way: by multiplying x by some number, you seem to compress the functions horizontally exactly that many times
If F1(x) and F2(x) are periodic functions, and their periods are equal to T1 and T2, respectively, then the sum of these functions can also be periodic. However, its period will not be a simple sum of periods T1 and T2. If the result of division T1/T2 is rational number, then the sum of the functions is periodic, and its period is equal to the least common multiple (LCM) of the periods T1 and T2. For example, if the period of the first function is 12, and the period of the second is 15, then the period of their sum will be equal to LCM (12, 15) = 60.
This can be visually represented as follows: the functions come with different “step widths,” but if the ratio of their widths is rational, then sooner or (more precisely, through the LCM of steps), they will become equal again, and their sum will begin a new period.
However, if the ratio of periods is , then the total function will not be periodic at all. For example, let F1(x) = x mod 2 (the remainder when x is divided by 2), and F2(x) = sin(x). T1 here will be equal to 2, and T2 will be equal to 2π. The ratio of periods is equal to π - irrational number. Therefore, the function sin(x) + x mod 2 is not periodic.
Sources:
- Theoretical information about functions
Many mathematical functions have one feature that makes them easier to construct: periodicity, that is, the repeatability of the graph on a coordinate grid at regular intervals.
Instructions
The most famous periodic functions in mathematics are sine and cosine. These functions have a wave-like and fundamental period equal to 2P. Also a special case of a periodic function is f(x)=const. Any number is suitable for position x; this function does not have a main period, since it is a straight line.
In general, a function is periodic if there is an integer N that is non-zero and satisfies the rule f(x)=f(x+N), thus ensuring repeatability. The period of a function is the smallest number N, but not zero. That is, for example, the function sin x is equal to the function sin (x+2ПN), where N=±1, ±2, etc.
Sometimes with a function it can
