Y x find the period of the function. Periodicity of functions y = sin x, y = cos x - Knowledge Hypermarket
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 value of the argument, and therefore any number can be considered its period.
Usually you are interested in the smallest non-zero period of a 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.
For simple, basic functions, the only way to determine whether they are periodic or non-periodic is through calculation. 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 the antiderivative repeats periodically, the derivative must also repeat. For example, the derivative of the function 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 as follows: by multiplying x by some number, you seem to compress the graph of the function 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 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 equal to LCM (12, 15) = 60.
This can be visually represented as follows: 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 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 - an irrational number. Therefore, the function sin(x) + x mod 2 is not periodic.
The video lesson “Periodicity of functions y = sin x, y = cos x” reveals the concept of periodicity of a function, considers a description of examples of solving problems in which the concept of periodicity of a function is used. This video lesson is a visual aid for explaining the topic to students. Also, this manual can become an independent part of the lesson, freeing up the teacher to conduct individual work with students.
Visibility in presenting this topic is very important. To represent the behavior of a function, plotting it, it must be visualized. It is not always possible to make constructions using a blackboard and chalk in such a way that they are understandable to all students. In the video tutorial, it is possible to highlight parts of the drawing with color when constructing, and make transformations using animation. Thus, the constructions become more understandable to most students. Also, the video lesson features contribute to better memorization of the material.
The demonstration begins by introducing the topic of the lesson, as well as reminding students of material learned in previous lessons. In particular, the list of properties that were identified in the functions y = sin x, as well as y = cos x, is summarized. Among the properties of the functions under consideration, the domain of definition, range of values, parity (oddness), other features are noted - boundedness, monotonicity, continuity, points of least (greatest) value. Students are informed that in this lesson another property of a function is studied - periodicity.
The definition of a periodic function y=f(x), where xϵX, in which the condition f(x-Т)= f(x)= f(x+Т) for some Т≠0 is presented. Otherwise, the number T is called the period of the function.
For the sine and cosine functions under consideration, the fulfillment of the condition is checked using reduction formulas. It is obvious that the form of the identity sin(x-2π)=sinx=sin(x+2π) corresponds to the form of the expression defining the condition of periodicity of the function. The same equality can be noted for the cosine cos (x-2π)= cos x= cos (x+2π). This means that these trigonometric functions are periodic.
It is further noted how the property of periodicity helps to build graphs of periodic functions. The function y = sin x is considered. A coordinate plane is constructed on the screen, on which abscissas from -6π to 8π are marked with a step of π. A part of the sine graph is plotted on the plane, represented by one wave on the segment. The figure demonstrates how the graph of a function is formed over the entire definition domain by shifting the constructed fragment, resulting in a long sinusoid.
A graph of the function y = cos x is constructed using the property of its periodicity. To do this, a coordinate plane is constructed in the figure, on which a fragment of the graph is depicted. It is noted that such a fragment is usually constructed on the segment [-π/2;3π/2]. Similar to the graph of the sine function, the construction of the cosine graph is performed by shifting the fragment. As a result of the construction, a long sinusoid is formed.
Graphing a periodic function has features that can be used. Therefore they are given in a generalized form. It is noted that to construct a graph of such a function, a branch of the graph is first constructed on a certain interval of length T. Then it is necessary to shift the constructed branch to the right and left by T, 2T, 3T, etc. At the same time, another feature of the period is pointed out - for any integer k≠0, the number kT is also the period of the function. However, T is called the main period, since it is the smallest of all. For the trigonometric functions sine and cosine, the basic period is 2π. However, the periods are also 4π, 6π, etc.
Next, it is proposed to consider finding the main period of the function y = cos 5x. The solution begins with the assumption that T is the period of the function. This means that the condition f(x-T)= f(x)= f(x+T) must be met. In this identity, f(x)= cos 5x, and f(x+T)=cos 5(x+T)= cos (5x+5T). In this case, cos (5x+5T)= cos 5x, therefore 5T=2πn. Now you can find T=2π/5. The problem is solved.
In the second problem, you need to find the main period of the function y=sin(2x/7). It is assumed that the main period of the T function for a given function is f(x)= sin(2x/7), and after a period f(x+T)=sin(2x/7)(x+T)= sin(2x/7 +(2/7)T). after reduction we get (2/7)Т=2πn. However, we need to find the main period, so we take the smallest value (2/7)T=2π, from which we find T=7π. The problem is solved.
At the end of the demonstration, the results of the examples are summarized to form a rule for determining the basic period of the function. It is noted that for the functions y=sinkx and y=coskx the main periods are 2π/k.
The video lesson “Periodicity of functions y = sin x, y = cos x” can be used in a traditional mathematics lesson to increase the effectiveness of the lesson. It is also recommended that this material be used by a teacher providing distance learning to increase the clarity of the explanation. The video can be recommended to a struggling student to deepen their understanding of the topic.
TEXT DECODING:
“Periodicity of functions y = cos x, y = sin x.”
To construct graphs of the functions y = sin x and y = cos x, the properties of the functions were used:
1 area of definition,
2 value area,
3 even or odd,
4 monotony,
5 limitation,
6 continuity,
7 highest and lowest value.
Today we will study another property: the periodicity of a function.
DEFINITION. The function y = f (x), where x ϵ X (the Greek is equal to ef of x, where x belongs to the set x), is called periodic if there is a non-zero number T such that for any x from the set X the double equality holds: f (x - T)= f (x) = f (x + T)(eff from x minus te is equal to ef from x and equal to ef from x plus te). The number T that satisfies this double equality is called the period of the function
And since sine and cosine are defined on the entire number line and for any x the equalities sin(x - 2π)= sin x= sin(x+ 2π) are satisfied (sine of x minus two pi is equal to sine of x and equal to sine of x plus two pi ) And
cos (x- 2π)= cos x = cos (x+ 2π) (the cosine of x minus two pi is equal to the cosine of x and equal to the cosine of x plus two pi), then sine and cosine are periodic functions with a period of 2π.
Periodicity allows you to quickly build a graph of a function. Indeed, in order to construct a graph of the function y = sin x, it is enough to plot one wave (most often on a segment (from zero to two pi), and then by shifting the constructed part of the graph along the x-axis to the right and left by 2π, then by 4π and so on to get a sine wave.
(show right and left shift by 2π, 4π)
Similarly for the graph of the function
y = cos x, but we build one wave most often on the segment [; ] (from minus pi over two to three pi over two).
Let us summarize the above and draw a conclusion: to construct a graph of a periodic function with a period T, you first need to construct a branch (or wave, or part) of the graph on any interval of length T (most often this is an interval with ends at points 0 and T or - and (minus te by two and te by two), and then move this branch along the x(x) axis to the right and left by T, 2T, 3T, etc.
Obviously, if a function is periodic with period T, then for any integer k0 (ka not equal to zero) a number of the form kT (ka te) is also the period of this function. Usually they try to isolate the smallest positive period, which is called the main period.
As the period of the functions y = cos x, y = sin x, one could take - 4π, 4π, - 6π, 6π, etc. (minus four pi, four pi, minus six pi, six pi, and so on). But the number 2π is the main period of both functions.
Let's look at examples.
EXAMPLE 1. Find the main period of the function y = cos5x (the y is equal to the cosine of five x).
Solution. Let T be the main period of the function y = cos5x. Let's put
f (x) = cos5x, then f (x + T) = cos5(x + T) = cos (5x + 5T) (eff of x plus te is equal to the cosine of five multiplied by the sum of x and te is equal to the cosine of the sum of five x and five te).
cos (5x + 5T) = cos5x. Hence 5T = 2πn (five te equals two pi en), but according to the condition you need to find the main period, which means 5T = 2π. We get T=
(the period of this function is two pi divided by five).
Answer: T=.
EXAMPLE 2. Find the main period of the function y = sin (the y is equal to the sine of the quotient of two x by seven).
Solution. Let T be the main period of the function y = sin. Let's put
f (x) = sin, then f (x + T) = sin (x + T) = sin (x + T) (ef of x plus te is equal to the sine of the product of two sevenths and the sum of x and te is equal to the sine of the sum of two sevenths x and two sevenths te).
For the number T to be the period of the function, the identity must be satisfied
sin (x + T) = sin. Hence T= 2πn (two sevenths te is equal to two pi en), but according to the condition you need to find the main period, which means T= 2π. We get T=7
(the period of this function is seven pi).
Answer: T=7.
Summarizing the results obtained in the examples, we can conclude: the main period of the functions y = sin kx or y = cos kx (y is equal to sine ka x or y is equal to cosine ka x) is equal to (two pi divided by ka).
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+180º)
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. 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)
(x+T+0.25)=(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 is 1. Let's check whether it will actually be 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 relation 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. Therefore, 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 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/
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Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. There is an interesting article on this subject, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.
A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), upon magnification we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.
Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes that are as complex in their details as in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will appear as a whole, either exactly, or perhaps with a slight deformation."
Trigonometric functions are periodic, that is, they repeat after a certain period. As a result, it is enough to study the function on this interval and extend the discovered properties to all other periods.
Instructions1. If you are given a primitive expression in which there is only one trigonometric function (sin, cos, tg, ctg, sec, cosec), and the angle inside the function is not multiplied by any number, and it itself is not raised to any degree - use the definition. For expressions containing sin, cos, sec, cosec, boldly set the period to 2P, and if the equation contains tg, ctg, then P. Let’s say, for the function y=2 sinx+5, the period will be equal to 2P.
2. If the angle x under the sign of a trigonometric function is multiplied by some number, then, in order to find the period of this function, divide the typical period by this number. Let's say you are given a function y = sin 5x. The typical period for a sine is 2P; dividing it by 5, you get 2P/5 - this is the desired period of this expression.
3. To find the period of a trigonometric function raised to a power, estimate the parity of the power. For an even degree, reduce the typical period by half. Let's say, if you are given the function y = 3 cos^2x, then the typical period 2P will decrease by 2 times, so the period will be equal to P. Please note that the functions tg, ctg are periodic to P to every degree.
4. If you are given an equation containing the product or quotient of two trigonometric functions, first find the period for all of them separately. After this, find the minimum number that would contain the integer of both periods. Let's say the function y=tgx*cos5x is given. For tangent the period is P, for cosine 5x the period is 2P/5. The minimum number in which both of these periods can be accommodated is 2P, thus the desired period is 2P.
5. If you find it difficult to do as suggested or doubt the result, try to do it as defined. Take T as the period of the function; it is larger than zero. Substitute the expression (x + T) instead of x into the equation and solve the resulting equality as if T were a parameter or a number. As a result, you will discover the value of the trigonometric function and be able to find the smallest period. Let's say, as a result of the relief, you get the identity sin (T/2) = 0. The minimum value of T at which it is performed is 2P, this will be the result of the task.
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.
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 to 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.
Trigonometric equations are equations that contain trigonometric functions of an unknown argument (for example: 5sinx-3cosx =7). In order to learn how to solve them, you need to know some ways to do this.
1. The solution of such equations consists of 2 stages. The first is reforming the equation to acquire its simplest form. The simplest trigonometric equations are: Sinx=a; Cosx=a, etc.
2. The second is the solution of the simplest trigonometric equation obtained. There are basic ways to solve equations of this type: Solving algebraically. This method is famously known from school, from an algebra course. Otherwise called the method of variable replacement and substitution. Using reduction formulas, we transform, make a substitution, and then find the roots.
3. Factoring the equation. First, we move all the terms to the left and factor them.
4. Reducing the equation to a homogeneous one. Equations are called homogeneous equations if all terms are of the same degree and the sine and cosine of the same angle. In order to solve it, you should: first transfer all its terms from the right side to the left side; move all universal factors out of brackets; equate factors and brackets to zero; equated brackets give a homogeneous equation of a lower degree, which should be divided by cos (or sin) to the highest degree; solve the resulting algebraic equation regarding tan.
5. The next method is to move to a half angle. Say, solve the equation: 3 sin x – 5 cos x = 7. Let’s move on to the half angle: 6 sin (x / 2) · cos (x / 2) – 5 cos? (x / 2) + 5 sin ? (x / 2) = 7 sin ? (x / 2) + 7 cos ? (x/ 2) , after which we reduce all terms into one part (preferably the right side) and solve the equation.
6. Entry of auxiliary angle. When we replace the integer value cos(a) or sin(a). The sign “a” is an auxiliary angle.
7. Method of reforming a product into a sum. Here you need to apply the appropriate formulas. Let's say given: 2 sin x · sin 3x = cos 4x. Solve it by transforming the left side into a sum, that is: cos 4x – cos 8x = cos 4x ,cos 8x = 0 ,8x = p / 2 + pk ,x = p / 16 + pk / 8.
8. The final method is called multifunctional substitution. We transform the expression and make a change, say Cos(x/2)=u, and then solve the equation with the parameter u. When purchasing the total, we convert the value to the opposite.
Video on the topic
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 value. If the period of a function is known, it is possible to construct the function on this period and repeat it on others.
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, they use the already well-known periods for functions. 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 known ones.
Examining a function for evenness and oddness helps to build a graph of the function and understand the nature of its behavior. For this research, you need to compare this function written for the argument “x” and for the argument “-x”.
1. Write down the function you want to study in the form y=y(x).
2. Replace the argument of the function with “-x”. Substitute this argument into a functional expression.
3. Simplify the expression.
4. Thus, you have the same function written for arguments “x” and “-x”. Look at these two entries. If y(-x)=y(x), then it is an even function. If y(-x)=-y(x), then it is an odd function. If it is impossible to say about a function that y (-x)=y(x) or y(-x)=-y(x), then by the property of parity this is a function of universal form. That is, it is neither even nor odd.
5. Write down your findings. Now you can use them in constructing a graph of a function or in a future analytical study of the properties of a function.
6. It is also possible to talk about the evenness and oddness of a function in the case when the graph of the function is already given. Let's say the graph served as the result of a physical experiment. If the graph of a function is symmetrical about the ordinate axis, then y(x) is an even function. If the graph of a function is symmetrical about the abscissa axis, then x(y) is an even function. x(y) is a function inverse to the function y(x). If the graph of a function is symmetrical about the origin (0,0), then y(x) is an odd function. The inverse function x(y) will also be odd.
7. It is important to remember that the idea of evenness and oddness of a function has a direct connection with the domain of definition of the function. If, say, an even or odd function does not exist at x=5, then it does not exist at x=-5, which cannot be said about a function of a universal form. When establishing even and odd parity, pay attention to the domain of the function.
8. Finding a function for evenness and oddness correlates with finding a set of function values. To find the set of values of an even function, it is enough to look at half of the function, to the right or to the left of zero. If at x>0 the even function y(x) takes values from A to B, then it will take the same values and at x0 the odd function y(x) takes on the range of values from A to B, then at x sin^2 ? + cos^2 ? = 1. The third and fourth identities are obtained by dividing, respectively, by b^2 and a^2: a^2/b^2 + 1 = c^2/b^2 => tg^2 ? + 1 = 1/cos^2 ?;1 + b^2/a^2 = c^2/a^2 => 1 + 1/tg^2 ? = 1/sin^ ? or 1 + ctg^2 ? = 1/sin^2 ?. The fifth and sixth main identities are proven by determining the sum of the acute angles of a right triangle, which is equal to 90° or?/2. More difficult trigonometric identities: formulas for adding arguments, double and triple angles, reducing the degree, reforming the sum or products of functions, as well as formulas for trigonometric substitution, namely expressions of basic trigonometric functions in terms of tan half angle: sin ?= (2*tg ?/2)/(1 + tan^2 ?/2);cos ? = (1 – tg^2 ?/2)/(1 = tg^2 ?/2);tg ? = (2*tg ?/2)/(1 – tg^2 ?/2).
The need to find the minimum value of a mathematical function is of actual interest in solving applied problems, say, in economics. Minimizing losses is of great importance for business activities.
1. In order to find the minimum value of the function, it is necessary to determine at what value of the argument x0 the inequality y(x0) will be satisfied? y(x), where x? x0. As usual, this problem is solved on a certain interval or in each range of values of the function, if one is not specified. One aspect of the solution is finding fixed points.
2. A stationary point is the value of the argument at which the derivative of the function becomes zero. According to Fermat's theorem, if a differentiable function takes an extreme value at some point (in this case, a local minimum), then this point is stationary.
3. The function often takes its minimum value precisely at this point, but it cannot be determined invariably. Moreover, it is not always possible to say with precision what the minimum of the function is equal to or whether it takes an infinitely small value. Then, as usual, they find the limit to which it tends as it decreases.
4. In order to determine the minimum value of a function, it is necessary to perform a sequence of actions consisting of four stages: finding the domain of definition of the function, acquiring fixed points, reviewing the values of the function at these points and at the ends of the interval, finding the minimum.
5. It turns out that let some function y(x) be given on an interval with boundaries at points A and B. Find the domain of its definition and find out whether the interval is its subset.
6. Calculate the derivative of the function. Equate the resulting expression to zero and find the roots of the equation. Check whether these stationary points fall within the gap. If not, then they are not taken into account at a further stage.
7. Examine the gap for the type of boundaries: open, closed, compound or immeasurable. This determines how you search for the minimum value. Let's say the segment [A, B] is a closed interval. Plug them into the function and calculate the values. Do the same with a stationary point. Select the lowest total.
8. With open and immeasurable intervals the situation is somewhat more difficult. Here you will have to look for one-sided limits that do not invariably give an unambiguous result. Say, for an interval with one closed and one punctured boundary [A, B), one should find a function at x = A and a one-sided limit lim y at x? B-0.
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