Experiments in Periodicity

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1 Experiments in Periodicity Kaitlin Keon and Yasmine Sasannejad May 9, 2005 Introduction Periodic functions are prevalent in various aspects of mathematics and physics. They describe a variety of recurring phenomena. These types of functions let us analyze the repetition of various occurrences in the world around us. Everything from the rising of sun to the motion of sounds wave can be examined using periodic functions. Periodic functions are defined as functions whose values repeat at regular intervals, called periods. In this laboratory, we will examine periodic functions, specifically trigonometric functions, to understand their various properties, including area accumulation and periodicity. We will use the computer program CALCWIN to investigate the graphs and characteristics of periodic functions. It is important to first establish the language of describing periodic functions. For a function y = f(x) to be p-periodic, there must exist a positive real number p such that for every real number x, f(x + p) = f(x) where the smallest positive p is referred to as the period of f. For example, the trigonometric functions f(x) = sin(x) and f(x) = cos(x) are both 2Πperiodic. It is interesting to examine the periods for the rest of the basic trigonometric functions. The trigonometric functions f(x) = tan(x) and f(x) = cot(x) are Π-periodic. Also, f(x) = sec(x) and f(x) = csc(x) are 2Π-periodic. 1

2 Examples of Periodic Functions Let s examine different forms of periodic trigonometric functions to determine their periodicity. It is beneficial to add and multiply trigonometric functions to see its effect on the period. We first looked at the function f(x) = sin(x) + cos(x). Similar to sin(x) and cos(x), this function is also 2Π-periodic. The graph is shifted to lie in between cos(x) and sin(x). It also appears to be stretched vertically. Another example of a trigonometric function is f(x) = 2sin(x). This function is also 2Π-periodic but it is vertically stretched so that the maximum y-values are 2 and 2 instead of 1 and 1. Now let s examine the derivatives of these functions to see if they have comparable periods. The derivative of f(x) = sin(x) + cos(x) is f(x) = cos(x) sin(x). The graph of this function appears to be 2Π-periodic. Compared to f(x) = sin(x) + cos(x), f(x) = cos(x) sin(x) is shifted to the left by Π. For the next example, the derivative of 2sin(x) is 2cos(x). This 2 new function has a period of 2Π and is also shifted to the left by Π. It is 2 important to note that the derivatives of periodic functions have the same period. For these two examples, the derivatives were just shifted versions of the original functions. Area Accumulation In this section, we will use CALCWIN to look at area accumulation functions and try to determine their periodicity. Let s first define area accumulation function. By the fundamental theorem of calculus, a continuous function f always has an antiderivative, given by an area accumulation function. For any real number a, we define an area accumulation function A for f whose value A(x) is the signed area between the graph of f and the x-axis over the interval from a to x. For area accumulation, areas below the x-axis are assigned a negative value while those above the x-axis are considered positive. In this investigation, we will vary the value for a to see its effect on area accumulation functions. First, let s examine the area accumulation for the function f(x) = sin(x). By using CALCWIN, we can assess the area of the function by using rectangles to estimate the integral. For the x-values 8 to 12, the approximation of the area using 100 rectangles is If we increase the number of 2

3 rectangles the program uses, the area approximation becomes more accurate. Using 200 rectangles produces the area approximation of If we continue to increase the number of rectangles, the sum will become even lower. For 1,000 rectangles, the sum is estimated to be In actuality, the integral from 8 to 12 is It is also intriguing to look at the sum of the area in a specified section of the graph to see if the area is zero in the given period of the function. This would make sense because the two sections of the graph, one above the x-axis and one below, should cancel each other out to produce an area of zero. For the function sin(x), when the limits are from 0 to 6.28, the sum is which suggests that the period is 2Π (approximated as 6.28 on the computer program). Now, let s examine the area accumulations for the previous examples. For the function f(x) = sin(x) + cos(x), the integral from 8 to 12 is The integral from to with 5000 rectangles is This suggests that the period of this function is 2Π because = 6.27 which is very close to 2Π given that the computer program can t be and isn t entirely accurate. For the second example, the area accumulation for 2sin(x) from 8 to 12 with 5,000 rectangles is The sum of the area for the period between 0 and 6.28 with 5,000 rectangles is This evidence supports the claim that the period is 2Π. Table 1 below shows the area accumulation for f(x) = sin(x) over the interval from 8 to 12, beginning at x = 8, and going in increments of 1 for each step. Here, i represents the number of approximating rectangles being used for that specified interval (there are 10 rectangles per unit). We know the period of this function is 2π, so Table 1 is difficult to analyze accurately since we are looking at intervals not corresponding to the period. To solve this, let us look at another set of data, Table 2, which has been generated in the same manner, only this time the intervals correspond directly to the function s period, 2π. Also, this time we used 2000 rectangles for our approximation, meaning there are 100 rectangles per unit, which will give us much more accurate data. Even though our data is roughly approximated, we can see a pattern in Table 2. The area accumulation for the interval [0, 2π] is equivalent to that of the interval [ π, pi]. This trend as well as our Graph from Table 1 suggest that f(x) = sin(x) is p-periodic for p = 2π, as we already knew. 3

4 Table 1 x i Sum So Far, A(x) Table 2 x i Sum So Far, A(x) 2π π π π π We can also examine the area accumulation of a function by plotting its antiderivative, the integral. This is essentially the same as what we have done before, however now we will look at the area accumulation plotted as a numerical function in order to compare it visually with our original function. We have done this in Figure 1 (see attached), once again using CALCWIN. 4

5 The antiderivative is the function plotted on top. We know already that the antiderivative of sin(x) is cos(x). However, our plotted graph here is not exactly cos(x). The graph is actually equivalent to f(x) = cos(x), only shifted slightly downward. Why is this? This depends on the interval we choose to integrate. If the interval chosen is a multiple of the function s period, the total area accumulation should be zero. Our interval of [-8, 12] is not a multiple of our function s period, 2π, and therefore there will be extra area being counted which does not have its corresponding area of the opposite sign to cancel it out in the summation. So, if our interval were a multiple of 2π, such as [-6.28, 6.28], our area accumulation over the entire interval would be zero, and the plotted antiderivative would be exactly cos(x). A New Type of Function To further understand periodic functions, it is interesting to investigate the properties of the function f(x) = sin(x + cos(x)) to determine its period and other important characteristics. For this function, the addition of cos(x) shifts the graph so that there are fewer values above the x-axis than below the x-axis. Because of this fascinating property, we wanted to further investigate the functions area accumulation and periodicity. We will investigate how to move this graph back into alignment with the x-axis in the section entitled Finding a Constant, Q. There is an important definition that will be utilized in this section. We must define a function that is linearly p-periodic. We say f(x) is linearly p-periodic if and only if there exists a real constant M such that for all x, f(x + p) = f(x) + M The constant M is called the p-translation constant of f. By looking at the graph of f(x) = sin(x+cos(x)), we can conjecture that it is 2π-periodic. But how can we know for sure? We can prove it! Theorem: f(x) = sin(x + cos(x)) is 2π-periodic. Proof: For a function f(x) to be p-periodic, there must exist a p such that: f(x) = f(x + p) for every real number x. If f(x) = sin(x + cos(x)) is 2π-periodic, then, it is true that: 5

6 sin(x + cos(x)) = sin(x + 2π + cos(x + 2π)) for every real x. This implies that sin(x + 2π + cos(x + 2π)) sin(x + cos(x)) = 0 for every real x. Expanding the left side of this equation, we get the following 10 term expression, which must be equivalent to 0. Note: This expansion can be done using a standard electronic graphing utility. sin(2π)cos(x)cos(sin(2π)sin(x))cos(cos(2π)cos(x)) + cos(2π)sin(x)cos(sin(2π)sin(x))cos(cos(2π)cos(x)) cos(2π)cos(x)sin(sin(2π)sin(x))cos(cos(2π)cos(x)) + sin(2π)sin(x)sin(sin(2π)sin(x))cos(cos(2π)cos(x)) + cos(2π)cos(x)cos(sin(2π)sin(x))sin(cos(2π)cos(x)) sin(2π)sin(x)cos(sin(2π)sin(x))sin(cos(2π)cos(x)) + sin(2π)cos(x)sin(sin(2π)sin(x))sin(cos(2π)cos(x)) + cos(2π)sin(x)sin(sin(2π)sin(x))sin(cos(2π)cos(x)) sin(x)cos(cos(x)) cos(x)sin(cos(x)) = 0 Knowing that sin(2π)=0, it follows that any term with sin(2π) as a factor will cancel out, as well as those with sin(sin(2π)), since sin(0)=0. This reduces our equation considerably to become the following: cos(2π)sin(x)cos(sin(2π)sin(x))cos(cos(2π)cos(x)) + cos(2π)cos(x)cos(sin(2π)sin(x))sin(cos(2π)cos(x)) sin(x)cos(cos(x)) cos(x)sin(cos(x)) = 0 Noticing now that cos(2π)=1, and that cos(0)=1, we are left with: sin(x)cos(cos(x)) + cos(x)sin(cos(x)) sin(x)cos(cos(x)) cos(x)sin(cos(x)) = 0 Now it is easy to see that our like terms cancel and we are left with 0 = 0. Ensuring the Area Accumulation to be p-periodic If the area accumulation, A(x), of a function is not p-periodic, what effect does that have on our study? We might be interested to know a way to ensure that A(x) will be p-periodic. This can be done simply by subtracting a constant, Q, to our function f(x). Our new function will still be p-periodic of course, since the subtraction of a constant will only shift the graph vertically, 6

7 which does not have an effect on the period of the graph. Essentially, the subtraction of our Q will shift f(x) until it lies on the x-axis such that the positive area under the curve will balance the negative area under the curve, thereby ensuring that the total area under the curve f(x) will be 0. Finding a Constant, Q After some careful thought and manipulation of the data through trial and error, we determined the way to always find a value for Q that will satisfy: f 1 (x) = f(x) Q, where f 1 (x) has a p-periodic area accumulation function, A 1 (x). If A(x) represents the total area accumulation for f(x), then to find Q, we use the formula: Q = A(x) p. Note that in this equation, our p represents the period of f(x). Let s look at an example. Consider f(x) = sin(x + cos(x)) (This is an example we will look at in depth in the next section). The area accumulation function for this graph is linearly p-periodic, as can be seen in Figure 2 (see attached). We want A(x) to be p-periodic. When looking at the area of f(x) = sin(x + cos(x)) in Figure 3 (see attached), we see that the area above the x-axis is less than the area below. Since we already know the period for f(x) is 2π, and we know the value of A(x) over the periodic interval to be (calculated with CALCWIN), we can find Q: Q = π Now, our new formula, f 1 (x) will be: f 1 (x) = sin(x + cos(x)) We can now see the graph once it has been shifted by Q in the bottom half of Figure 4 (see attached). Also, we can see the area accumulation function of the shifted graph in the top half of Figure 4. Now, our value for A 1 (x) is approximated to be , which is considerably close to 0. This is illustrated in Figure 5 (see attached). 7

8 The process of manipulating the function by subtracting Q can be considered logically. The value of Q is the area of the function divided by the function s period. Since the period of a function is the x value that the function covers, when you divide by this value, you get the area per unit of x. Therefore, Q produces the area for each x value. When you subtract Q from the function, you shift the graph so that every value of x moves towards the x-axis. When this occurs, there is balance of the area above and below the x-axis and the accumulated area is zero. For any function, you can obtain Q and subtract it from the function so that the function is p-periodic. Conclusion Periodic functions exist in our every day lives. However, we rarely think of them mathematically as periodic functions. This laboratory investigated properties of periodic functions to obtain a better understanding of how they work. Specifically, trigonometric functions have a variety of interesting characteristics that make them a topic of intense and on-going research. We examined trigonometric functions using specific examples to try to understand how different values and manipulations of functions affect the area accumulation and periodicity. One interesting function that we investigated in depth was f(x) = sin(x + cos(x)). This function proved to be very fascinating as we tried to obtain a value for Q for which the function would be p-periodic. Once we found the value for Q for this specific function, we could generalize to other functions about how to obtain the value for Q. Through this laboratory, we learned a lot about the intricate aspects of periodic functions and their many properties. Further investigation would be beneficial to try to understand linking several functions of sines and cosines together such as f(x) = sin(x + cos(x + sin(x + cos(x + cos(x))))). If we had more time, we would be able to better generalize to this type of linking and understand how to obtain a value for Q for any function with linked functions of sine and cosine. 8

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