Instructions
Please note that period ical does not always have the smallest positive period. So, for example, as period and constant functions can be absolutely any number, and it may not have the smallest positive period A. There are also non-permanent period ical functions, which do not have the least positive period A. However, in most cases the smallest positive period at period there are still ichical ones.
Least period sine is equal to 2?. Consider this example functions y=sin(x). Let T be arbitrary period ohm sine, in this case sin(a+T)=sin(a) for any value of a. If a=?/2, it turns out that sin(T+?/2)=sin(?/2)=1. However, sin(x)=1 only if x=?/2+2?n, where n is an integer. It follows that T=2?n, and therefore the smallest positive value is 2?n 2?.
Least positive period cosine is also equal to 2?. Consider the proof of this with an example functions y=cos(x). If T is arbitrary period om cosine, then cos(a+T)=cos(a). In the event that a=0, cos(T)=cos(0)=1. In view of this, the smallest positive value of T at which cos(x) = 1 is 2?.
Considering the fact that 2? – period sine and cosine, it will also be period ohm cotangent, as well as tangent, but not minimal, since, like , the smallest positive period tangent and cotangent are equal?. You can verify this by considering the following: the points corresponding to (x) and (x+?) on the trigonometric circle have diametrically opposite locations. The distance from point (x) to point (x+2?) corresponds to half a circle. By definition of tangent and cotangent tg(x+?)=tgx, and ctg(x+?)=ctgx, which means the smallest positive period cotangent and ?.
note
Do not confuse the functions y=cos(x) and y=sin(x) - having the same period, these functions are represented differently.
Helpful advice
For greater clarity, draw a trigonometric function for which the smallest positive period is calculated.
Sources:
- Handbook of mathematics, school mathematics, higher mathematics
A periodic function is a function that repeats its values after some non-zero period. The period of a function is a number that, when added to a function argument, does not change the value of the function.
You will need
- Knowledge of elementary mathematics and principles of analysis.
Instructions
Video on the topic
note
All trigonometric functions are periodic, and all polynomial functions with a degree greater than 2 are aperiodic.
Helpful advice
The period of a function consisting of two periodic functions is the Least Common Multiple of the periods of these functions.
If we consider points on a circle, then points x, x + 2π, x + 4π, etc. coincide with each other. Thus, trigonometric functions on a straight line periodically repeat their meaning. If the period is known functions, you can build a function on this period and repeat it on others.
Instructions
Let the function f(x) = sin^2(10x) be given. Consider sin^2(10x) = sin^2(10(x+T)). Use the formula for reduction: sin^2(x) = (1 - cos 2x)/2. Then you get 1 - cos 20x = 1 - cos 20(x+T) or cos 20x = cos (20x+20T). Knowing that the period of the cosine is 2π, 20T = 2π. This means T = π/10. T is the smallest period, and the function will be repeated after 2T, and after 3T, and to the side along the axis: -T, -2T, etc.
Helpful advice
Use formulas to reduce the degree of a function. If you already know the periods of any functions, try to reduce the existing function to the known ones.
A function whose values are repeated after a certain number is called periodic. That is, no matter how many periods you add to the value of x, the function will be equal to the same number. Any study of periodic functions begins with a search for the smallest period, so as not to do unnecessary work: it is enough to study all the properties on an interval equal to the period.
Instructions
As a result, you will get a certain identity, from which try to select the minimum period. For example, if we get the equality sin(2T)=0.5, therefore, 2T=P/6, that is, T=P/12.
If the equality turns out to be true only when T = 0 or the parameter T depends on x (for example, the equality 2T = x is obtained), assume that the function is not periodic.
To find out the shortest period functions containing only one trigonometric expression, use . If the expression contains sin or cos, the period for functions will be 2P, and for the functions tg, ctg set the smallest period P. Please note that the function should not be raised to any power, and the variable under the sign functions must not be multiplied by a number other than 1.
If cos or sin is inside functions raised to an even power, reduce the period 2P by half. Graphically you can see it like this: functions, below the x-axis, will be reflected symmetrically upward, so the function will repeat twice as often.
To find the smallest period functions given that the angle x is multiplied by any number, proceed as follows: determine the standard period of this functions(for example, for cos it is 2P). Then split it before the variable. This will be the required shortest period. The decrease in the period is clearly visible on the graph: it is exactly as many times as the angle under the trigonometric sign is multiplied by functions.
If your expression has two periodic functions multiplied by each other, find the smallest period for each separately. Then determine the least common factor for them. For example, for periods P and 2/3P, the smallest common factor will be 3P (it has no remainder on both P and 2/3P).
Calculating the average salary of employees is necessary for calculating temporary disability benefits and paying for business trips. The average earnings of specialists are calculated based on the time actually worked and depend on the salary, allowances, and bonuses specified in the staffing table.
Minimum Positive period functions in trigonometry it is denoted f. It is characterized by the smallest value of the positive number T, that is, a smaller value of T will no longer be period ohm functions .
You will need
- – mathematical reference book.
Instructions
1. Please note that period ical function does not invariably have a minimum correct period. So, for example, as period and continuous functions there can be any number unconditionally, which means it may not have the smallest positive period A. There are also non-permanent period ical functions, which do not have the smallest correct period A. However, in most cases the minimum is correct period at period There are still some ical functions.
2. Minimum period sine is equal to 2?. See the example for proof of this. functions y=sin(x). Let T be arbitrary period ohm sine, in this case sin(a+T)=sin(a) for any value of a. If a=?/2, it turns out that sin(T+?/2)=sin(?/2)=1. However, sin(x)=1 only in the case when x=?/2+2?n, where n is an integer. It follows that T=2?n, which means that the smallest positive value of 2?n is 2?.
3. Minimum correct period cosine is also equal to 2?. See the example for proof of this. functions y=cos(x). If T is arbitrary period om cosine, then cos(a+T)=cos(a). In the event that a=0, cos(T)=cos(0)=1. In view of this, the smallest positive value of T at which cos(x) = 1 is 2?.
4. Considering the fact that 2? – period sine and cosine, the same value will be period ohm cotangent, as well as tangent, however, not minimal, because, as is well known, the minimal is correct period tangent and cotangent are equal?. You can verify this by looking at the following example: the points corresponding to the numbers (x) and (x+?) on the trigonometric circle have diametrically opposite locations. The distance from point (x) to point (x+2?) corresponds to half a circle. By definition of tangent and cotangent tg(x+?)=tgx, and ctg(x+?)=ctgx, which means the minimum is correct period cotangent and tangent are equal?.
A periodic function is a function that repeats its values after some non-zero period. The period of a function is a number that, when added to the argument of a function, does not change the value of the function.
You will need
- Knowledge of elementary mathematics and basic review.
Instructions
1. Let us denote the period of the function f(x) by the number K. Our task is to discover this value of K. To do this, imagine that the function f(x), using the definition of a periodic function, we equate f(x+K)=f(x).
2. We solve the resulting equation regarding the unknown K, as if x were a constant. Depending on the value of K, there will be several options.
3. If K>0 – then this is the period of your function. If K=0 – then the function f(x) is not periodic. If the solution to the equation f(x+K)=f(x) does not exist for any K not equal zero, then such a function is called aperiodic and it also has no period.
Video on the topic
Note!
All trigonometric functions are periodic, and all polynomial functions with a degree greater than 2 are aperiodic.
Helpful advice
The period of a function consisting of 2 periodic functions is the least universal multiple of the periods of these functions.
If we consider points on a circle, then points x, x + 2π, x + 4π, etc. coincide with each other. Thus, trigonometric functions on a straight line periodically repeat their meaning. If the period is famous functions, it is possible to construct a function on this period and repeat it on others.
Instructions
1. The period is a number T such that f(x) = f(x+T). In order to find the period, solve the corresponding equation, substituting x and x+T as an argument. In this case, the previously known periods for functions are used. For the sine and cosine functions the period is 2π, and for the tangent and cotangent functions it is π.
2. Let the function f(x) = sin^2(10x) be given. Consider the expression sin^2(10x) = sin^2(10(x+T)). Use the formula to reduce the degree: sin^2(x) = (1 – cos 2x)/2. Then you get 1 – cos 20x = 1 – cos 20(x+T) or cos 20x = cos (20x+20T). Knowing that the period of the cosine is 2π, 20T = 2π. This means T = π/10. T is the minimum correct period, and the function will be repeated after 2T, and after 3T, and in the other direction along the axis: -T, -2T, etc.
Helpful advice
Use formulas to reduce the degree of a function. If you already know the periods of some functions, try to reduce the existing function to the famous ones.
A function whose values are repeated after a certain number is called periodic. That is, no matter how many periods you add to the value of x, the function will be equal to the same number. Any search for periodic functions begins with a search for the smallest period, so as not to perform unnecessary work: it is enough to study all the properties on an interval equal to the period.
Instructions
1. Use the definition periodic functions. All x values in functions replace with (x+T), where T is the minimum period functions. Solve the resulting equation, considering T to be an unknown number.
2. As a result, you will get a certain identity, from it try to select the smallest period. Let's say, if we get the equality sin(2T)=0.5, therefore, 2T=P/6, that is, T=P/12.
3. If the equality turns out to be correct only when T = 0 or the parameter T depends on x (say, the equality 2T = x is obtained), conclude that the function is not periodic.
4. In order to find out the minimum period functions containing only one trigonometric expression, use the rule. If the expression contains sin or cos, the period for functions will be 2P, and for the functions tg, ctg set the minimum period P. Please note that the function should not be raised to any power, and the variable under the sign functions should not be multiplied by a number other than 1.
5. If cos or sin is inside functions built to an even power, reduce the period 2P by half. Graphically you can see it like this: graph functions, located below the x axis, will be symmetrically reflected upward, and consequently the function will be repeated twice as often.
6. In order to find the minimum period functions given that the angle x is multiplied by any number, proceed as follows: determine the typical period of this functions(let's say for cos it's 2P). After that, divide it by the factor in front of the variable. This will be the desired minimum period. The decrease in the period is clearly visible on the graph: it is compressed exactly as many times as the angle under the trigonometric sign is multiplied by functions .
7. Please note that if x is preceded by a fractional number less than 1, the period increases, that is, the graph, on the contrary, stretches.
8. If your expression has two periodic functions multiplied by each other, find the minimum period for each separately. After this, determine the minimum universal factor for them. Let's say, for periods P and 2/3P, the minimum universal factor will be 3P (it is divisible without a remainder by both P and 2/3P).
Calculation of the average salary of employees is needed to calculate temporary disability benefits and pay for business trips. The average earnings of experts are calculated based on the actual time worked and depend on the salary, allowances, and bonuses specified in the staffing table.
You will need
- – staffing table;
- - calculator;
- – right;
- - production calendar;
- – time sheet or work completion certificate.
Instructions
1. In order to calculate the average salary of an employee, first determine the period for which you need to calculate it. As usual, this period is 12 calendar months. But if an employee works at the enterprise for less than a year, for example, 10 months, then you need to find the average earnings for the time that the expert performs his work function.
2. Now determine the amount of wages that were actually accrued to him for the billing period. To do this, use payslips according to which the employee was given all the payments due to him. If it is unthinkable to use these documents, then multiply the monthly salary, bonuses, and allowances by 12 (or the number of months that the employee has been working at the enterprise, if he has been employed by the company for less than a year).
3. Calculate your average daily earnings. To do this, divide the amount of wages for the billing period by the average number of days in a month (currently it is 29.4). Divide the resulting total by 12.
4. After this, determine the number of hours actually worked. To do this, use a time sheet. This document must be filled out by a timekeeper, personnel officer or other employee whose job description specifies this.
5. Multiply the number of hours actually worked by the average daily earnings. The amount received is the average salary of the expert for the year. Divide the total by 12. This will be your average monthly income. This calculation is used for employees whose wages depend on the actual time worked.
6. When an employee is paid piecework, then multiply the tariff rate (indicated in the staffing table and determined by the employment contract) by the number of products produced (use a work completion certificate or another document in which this is recorded).
Note!
Do not confuse the functions y=cos(x) and y=sin(x) - having an identical period, these functions are depicted differently.
Helpful advice
For greater clarity, draw a trigonometric function for which the minimum correct period is calculated.
At your request!
7. Find the smallest positive period of the function: y=2cos(0.2x+1).
Let's apply the rule: if the function f is periodic and has a period T, then the function y=Af(kx+b) where A, k and b are constant, and k≠0 is also periodic, and its period is T o = T: |k|. For us, T=2π is the smallest positive period of the cosine function, k=0.2. We find T o = 2π:0.2=20π:2=10π.
9. The distance from the point equidistant from the vertices of the square to its plane is 9 dm. Find the distance from this point to the sides of the square if the side of the square is 8 dm.
10. Solve the equation: 10=|5x+5x 2 |.
Since |10|=10 and |-10|=10, then 2 cases are possible: 1) 5x 2 +5x=10 and 2) 5x 2 +5x=-10. Divide each of the equalities by 5 and solve the resulting quadratic equations:
1) x 2 +x-2=0, roots according to Vieta’s theorem x 1 =-2, x 2 =1. 2) x 2 +x+2=0. The discriminant is negative - there are no roots.
11. Solve the equation:
To the right side of the equality we apply the main logarithmic identity:
We get equality:
We solve the quadratic equation x 2 -3x-4=0 and find the roots: x 1 =-1, x 2 =4.
13. Solve the equation and find the sum of its roots on the indicated interval.
22. Solve inequality:
Then the inequality will take the form: tgt< 2. Построим графики уравнений: y=tgt и y=2. Выберем промежуток значений переменной t, при которых график y=tgt лежит ниже прямой у=2.
24. Line y= a x+b is perpendicular to the straight line y=2x+3 and passes through the point C(4; 5). Make up its equation. Directy=k 1 x+b 1 and y=k 2 x+b 2 are mutually perpendicular if the condition k 1 ∙k 2 =-1 is met. It follows that A·2=-1. The desired straight line will look like: y=(-1/2) x+b. We will find the value of b if in the equation of our straight line instead X And at Let's substitute the coordinates of point C.
5=(-1/2) 4+b ⇒ 5=-2+b ⇒ b=7. Then we get the equation: y=(-1/2)x+7.
25. Four fishermen A, B, C and D boasted about their catch:
1. D caught more than C;
2. The sum of catches A and B is equal to the sum of catches C and D;
3. A and D together caught less than B and C together. Record the fishermen's catch in descending order.
We have: 1)
D>C; 2)
A+B=C+D; 3)
A+D 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.” 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 y=sin(x) = sin(x+360º) tg(x+π n)=tgx, n € Z sin(x+2π n)=sinx, n € Z 5) How to plot a periodic function? Oral exercises. 1) Prove the following relations a) sin(740º) = sin(20º) 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 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. A geological period is 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. Mendeleev's periodic system. 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 smallest, and also eliminates the need to touch upon questions about arithmetic operations on periodic functions and 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) Let's put x=-0.25 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 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. Therefore, this function is not periodic. Task 5. Check if the function is periodic f(x)= Let T be the period of f, then Since the numerators are equal, their denominators are 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/
A.N. Kolmogorov
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=cos(x) = cos(x+360º)
y=tg(x) = tg(x+18 0º)
y=ctg(x) = ctg(x+180º)
ctg(x+π n)=ctgx, n € Z
cos(x+2π n)=cosx, n € Z
b) cos(54º) = cos(-1026º)
c) sin(-1000º) = sin(80º)
b) ctg530º
c) sin1268º
d) cos(-7363º)
(x+T+0.25)=(x+0.25)
sin(1.5T)+5cos(0.75T)=5
, 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
