Repeating its values at some regular interval of the argument, that is, not changing its value when some fixed non-zero number is added to the argument ( period functions) over the entire domain of definition.
More formally, the function is said to be periodic with period T ≠ 0 (\displaystyle T\neq 0), if for each point x (\displaystyle x) from its point definition area x + T (\displaystyle x+T) and x − T (\displaystyle x-T) also belong to its domain of definition, and for them the equality f (x) = f (x + T) = f (x − T) (\displaystyle f(x)=f(x+T)=f(x-T)).
Based on the definition, the equality also holds for a periodic function f (x) = f (x + n T) (\displaystyle f(x)=f(x+nT)), where n (\displaystyle n)- any integer.
However, if a set of periods ( T , T > 0 , T ∈ R ) (\displaystyle \(T,T>0,T\in \mathbb (R) \)) there is a smallest value, it is called main (or main) period functions.
Examples
Sin (x + 2 π) = sin x , cos (x + 2 π) = cos x , ∀ x ∈ R . (\displaystyle \sin(x+2\pi)=\sin x,\;\cos(x+2\pi)=\cos x,\quad \forall x\in \mathbb (R) .)
- The Dirichlet function is periodic; its period is any non-zero rational number. It also does not have a main period.
Some features of periodic functions
and T 2 (\displaystyle T_(2))(However, this number will simply be a period). For example, the function f (x) = sin (2 x) − sin (3 x) (\displaystyle f(x)=\sin(2x)-\sin(3x)) the main period is 2 π (\displaystyle 2\pi ), at the function g (x) = sin (3 x) (\displaystyle g(x)=\sin(3x)) period is 2 π / 3 (\displaystyle 2\pi /3), and their sum f (x) + g (x) = sin (2 x) (\displaystyle f(x)+g(x)=\sin(2x)) the main period is obviously equal to π (\displaystyle \pi ).- The sum of two functions with incommensurable periods is not always a non-periodic function.
Studying the phenomena of nature, solving technical problems, we are faced with periodic processes that can be described by functions of a special kind.
A function y = f(x) with domain D is called periodic if there exists at least one number T > 0 such that the following two conditions are satisfied:
1) the points x + T, x − T belong to the domain D for any x ∈ D;
2) for each x from D we have the relation
f(x) = f(x + T) = f(x − T).
The number T is called the period of the function f(x). In other words, a periodic function is a function whose values are repeated after a certain interval. For example, the function y = sin x is periodic (Fig. 1) with a period of 2π.
Note that if the number T is the period of the function f(x), then the number 2T will also be its period, as well as 3T, and 4T, etc., i.e., a periodic function has infinitely many different periods. If among them there is the smallest (not equal to zero), then all other periods of the function are multiples of this number. Note that not every periodic function has such a smallest positive period; for example, the function f(x)=1 has no such period. It is also important to keep in mind that, for example, the sum of two periodic functions having the same smallest positive period T 0 does not necessarily have the same positive period. So, the sum of functions f(x) = sin x and g(x) = −sin x does not have the smallest positive period at all, and the sum of functions f(x) = sin x + sin 2x and g(x) = −sin x, whose least periods are 2π has the smallest positive period equal to π.
If the ratio of the periods of two functions f(x) and g(x) is a rational number, then the sum and product of these functions will also be periodic functions. If the ratio of the periods of the everywhere defined and continuous functions f and g is an irrational number, then the functions f + g and fg will already be non-periodic functions. So, for example, the functions cos x sin √2 x and cosj √2 x + sin x are non-periodic, although the functions sin x and cos x are periodic with a period of 2π, the functions sin √2 x and cos √2 x are periodic with a period of √2 π .
Note that if f(x) is a periodic function with period T, then the complex function (if, of course, it makes sense) F(f(x)) is also a periodic function, and the number T will serve as its period. For example, the functions y \u003d sin 2 x, y \u003d √ (cos x) (Fig. 2.3) are periodic functions (here: F 1 (z) \u003d z 2 and F 2 (z) \u003d √z). However, one should not think that if the function f(x) has the smallest positive period T 0 , then the function F(f(x)) will have the same smallest positive period; for example, the function y \u003d sin 2 x has the smallest positive period, which is 2 times less than the function f (x) \u003d sin x (Fig. 2).
It is easy to show that if the function f is periodic with period T, is defined and differentiable at every point of the real line, then the function f "(x) (derivative) is also a periodic function with period T, however, the antiderivative function F (x) (see Integral calculus) for f(x) will be a periodic function only if
F(T) − F(0) = T o ∫ f(x) dx = 0.
Purpose: to generalize and systematize students' knowledge on the topic "Periodicity of functions"; to form skills in applying the properties of a periodic function, finding the smallest positive period of a function, plotting periodic functions; promote interest in the study of mathematics; cultivate observation, accuracy.
Equipment: computer, multimedia projector, task cards, slides, clocks, ornament tables, folk craft elements
“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. Presentation of the topic and objectives of the lesson.
II. Checking homework.
We check the homework according to the samples, discuss the most difficult points.
III. Generalization and systematization of knowledge.
1. Oral frontal work.
Questions of theory.
1) Form the definition of the period of the function
2) What is 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) Use the circle to 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 the angle of 540º is one of the periods of the function y= cos(2x)
3. Prove that the 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 meet with the words PERIOD, PERIODICITY?
Students' answers: A period in music is a construction in which a more or less complete musical thought is stated. The geological period is part of an era and is divided into epochs with a period of 35 to 90 million years.
The half-life of a radioactive substance. Periodic fraction. Periodicals are printed publications that appear on strictly defined dates. Periodic system of Mendeleev.
6. The figures show parts of the graphs of periodic functions. Define 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 met with the construction of repeating elements?
Students answer: Elements of ornaments, folk art.
IV. Collective problem solving.
(Problem solving on slides.)
Let us consider one of the ways to study a function for periodicity.
This method bypasses the difficulties associated with proving that one or another period is the smallest, and also there is no need to touch on questions about arithmetic operations on periodic functions and about 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: Let's 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 x=-0.25 we get
(T)=0<=>T=n, n ∈ Z
We have obtained that all periods of the considered function (if they exist) are among integers. Choose among these numbers the smallest positive number. This is 1 . Let's check if it is actually 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.
Task 2. Show that the function f(x)=cos 2 (x) is periodic and find its main period.
Task 3. Find the main period of the function
f(x)=sin(1.5x)+5cos(0.75x)
Assume the T-period of the function, then for any X the ratio
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(1.5Т)+5cos(0.75Т)=5 |
Adding, we get:
10cos(0.75T)=10
2π n, n € Z
Let's choose from all numbers "suspicious" for the period the smallest positive one 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)
Hence, is the main period of the function f.
Task 4. Check if the function f(x)=sin(x) is periodic
Let T be the period of the function f. Then for any x
sin|x+T|=sin|x|
If x=0, then sin|T|=sin0, sin|T|=0 T=π n, n ∈ Z.
Suppose. That for some n the number π n is a period
considered function π n>0. Then sin|π n+x|=sin|x|
This implies that n must be both even and odd at the same time, which is impossible. Therefore, this function is not periodic.
Task 5. Check if the function is periodic
f(x)= 
Let T be the period f, then
, hence sinT=0, T=π n, n € Z. Let us assume that for some n the number π n is indeed the period of the given function. Then the number 2π n will also be a period
Since the numerators are equal, so are their denominators, so
Hence, the function f is not periodic.
Group work.
Tasks for group 1.
Tasks for group 2.
Check if the function f is periodic and find its main period (if it exists).
f(x)=cos(2x)+2sin(2x)
Tasks for group 3.
At the end of the work, the groups present their solutions.
VI. Summing up the lesson.
Reflection.
The teacher gives students cards with drawings and offers to paint over part of the first drawing in accordance with the extent to which, as it seems to them, they have mastered the methods of studying the function for periodicity, and in part of the second drawing, in accordance with their contribution to the work in the lesson.
VII. Homework
one). Check if function f is periodic and find its main 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 the beginning of analysis with in-depth study.
- Mathematics. Preparation for the exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
- Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.
Argument x, then it is called periodic if there is a 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 values of the argument, and therefore any number can be considered its period.
Usually interested in the smallest non-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.
With regard to simple, basic functions, the only way to establish their periodicity or non-periodicity is through calculations. But for complex functions, there are already some 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 the point x is equal to the tangent of the tangent of the graph of its antiderivative at this point to the x-axis, and since the antiderivative is periodically repeated, the derivative must also be repeated. For example, the derivative of the function sin(x) is cos(x), and it is periodic. Taking the derivative of cos(x) gives you -sin(x). Periodicity remains unchanged.
However, the reverse 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 - also a periodic function, and its period is T/k. For example sin(2x) is a periodic function and its period is π. Visually, this can be represented as follows: by multiplying x by some number, you seem to compress the graph of the function horizontally exactly as 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 dividing T1/T2 is a 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 LCM (12, 15) = 60.
This can be visualized as follows: the functions come with different “step widths”, but if the ratio of their widths is rational, then sooner or later (or rather, 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 irrational, then the total function will not be periodic at all. For example, let F1(x) = x mod 2 (the remainder of x divided by 2) and F2(x) = sin(x). T1 here will be equal to 2, and T2 is equal to 2π. The ratio of periods is equal to π - an irrational number. Therefore, the function sin(x) + x mod 2 is not periodic.
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Slides captions:
Algebra and the beginning of analysis, grade 10 (profile level) A.G. Mordkovich, P.E. Semenov Teacher Volkova S.E.
Definition 1 A function y = f (x), x ∈ X is said to have period T if for any x ∈ X the equality f (x - T) = f (x) = f (x + T) is true. If a function with a period T is defined at a point x, then it is also defined at the points x + T, x - T. Any function has a period equal to zero at T = 0, we get f (x - 0) = f (x) = f ( x + 0) .
Definition 2 A function that has a non-zero period T is called periodic. If a function y = f (x), x ∈ X, has a period T, then any multiple of T (i.e., a number of the form kT, k ∈ Z) is also its period.
Proof Let 2T be the period of the function. Then f(x) = f(x + T) = f((x + T) + T) = f(x + 2T), f(x) = f(x - T) = f((x - T) -T) = f(x - 2T). Similarly, it is proved that f(x) = f(x + 3 T) = f(x - 3 T), f(x) = f(x + 4 T) = f(x - 4 T), etc. So f(x - kT) = f(x) = f(x + kT)
The smallest period among the positive periods of a periodic function is called the main period of this function.
Features of the graph of a periodic function If T is the main period of the function y \u003d f (x), then it is enough: to build a branch of the graph on one of the intervals of length T, to perform a parallel transfer of this branch along the x axis by ±T, ±2T, ±3T, etc. . Usually choose a gap with ends at points
Properties of Periodic Functions 1. If f(x) is a periodic function with period T, then the function g(x) = A f(kx + b), where k > 0, is also periodic with period T 1 = T/k. 2. Let the function f 1 (x) and f 2 (x) be defined on the entire real axis and be periodic with periods T 1 > 0 and T 2 >0. Then, for T 1 /T 2 ∈ Q, the function f(x) = f(x) + f 2 (x) is a periodic function with period T equal to the least common multiple of the numbers T 1 and T 2 .
Examples 1. The periodic function y = f(x) is defined for all real numbers. Its period is 3 and f(0) =4 . Find the value of the expression 2f(3) - f(-3). Decision. T \u003d 3, f (3) \u003d f (0 + 3) \u003d 4, f (-3) \u003d f (0–3) \u003d 4, f (0) \u003d 4. Substituting the obtained values into the expression 2f (3) - f(-3) , we get 8 - 4 =4 . Answer: 4.
Examples 2. The periodic function y = f(x) is defined for all real numbers. Its period is 5, and f(-1) = 1. Find f(-12) if 2f(3) - 5f(9) = 9. Solution T = 5 F(-1) = 1 f(9) = f(-1 +2T) = 1⇨ 5f(9) = 5 2f(3) = 9 + 5f(9) = 14 ⇨f(3)= 7 F(-12) = f(3 – 3T) = f (3) = 7 Answer: 7.
References A.G. Mordkovich, P.V. Semyonov. Algebra and the beginnings of analysis (profile level), Grade 10 A.G. Mordkovich, P.V. Semyonov. Algebra and beginnings of analysis (profile level), Grade 10. Methodological guide for the teacher
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