Average Value Of G(x) = 2cos(x) On [-π/2, Π/2] Calculation

In this comprehensive article, we will delve into the process of finding the average value of a function, specifically focusing on the function g(x) = 2cos(x) over the interval [-π/2, π/2]. The concept of average value is a fundamental one in calculus and has wide applications in various fields such as physics, engineering, and economics. Understanding how to calculate the average value of a function provides valuable insights into its overall behavior and characteristics within a given interval. We will explore the theoretical background, the step-by-step calculation, and the significance of the result. By the end of this article, you will have a solid grasp of how to determine the average value of a function and appreciate its practical implications.

In calculus, the average value of a function over an interval provides a measure of the function's typical value within that interval. It's akin to finding the average height of a curve over a specified range. To achieve this, we use the definite integral, a powerful tool that allows us to compute the area under a curve. The formula for the average value, often denoted as g_ave, involves integrating the function over the interval and then dividing by the length of the interval. This process effectively smooths out the function's fluctuations, giving us a single value that represents the function's central tendency over the given domain. The average value is not simply the midpoint between the function's maximum and minimum values; rather, it's a weighted average that takes into account all the function's values within the interval. This concept is crucial for understanding the overall behavior of a function and is used extensively in various applications where a representative value over a range is needed.

Before we dive into the specifics of our problem, let's briefly recap the core concepts we'll be using. The definite integral, denoted by ∫[a, b] f(x) dx, represents the area under the curve of the function f(x) between the limits a and b. The Fundamental Theorem of Calculus provides the link between integration and differentiation, allowing us to evaluate definite integrals using antiderivatives. An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). Finding antiderivatives is a key step in evaluating definite integrals. Trigonometric functions, such as cosine, have well-defined antiderivatives, which we will utilize in our calculation. The cosine function, cos(x), is periodic and symmetric about the y-axis, properties that will play a role in our analysis. Understanding these foundational concepts will enable us to approach the problem of finding the average value of g(x) = 2cos(x) with confidence and clarity. Now, let's move on to the specific problem at hand and see how these concepts come together to provide us with the solution.

Problem Statement

Our task is to find the average value, denoted as g_ave, of the function g(x) = 2cos(x) on the interval [-π/2, π/2]. This problem requires us to apply the formula for the average value of a function, which involves calculating a definite integral and dividing by the length of the interval. The function g(x) = 2cos(x) is a trigonometric function, specifically a cosine function scaled by a factor of 2. The interval [-π/2, π/2] represents a range of x-values over which we want to determine the average value of the function. This interval is centered around the y-axis and spans one full period of the cosine function. The average value, g_ave, will be a single numerical value that represents the mean height of the curve of g(x) over this interval. This value provides a sense of the function's typical behavior within the specified domain. To solve this problem, we will first set up the integral expression for the average value, then evaluate the integral using the antiderivative of the cosine function, and finally divide by the length of the interval to obtain the final result. Let's proceed with the calculation step-by-step to arrive at the solution.

The interval [-π/2, π/2] is particularly interesting because it represents a symmetric range around the y-axis. The cosine function, cos(x), is an even function, meaning that cos(x) = cos(-x). This symmetry implies that the area under the curve of cos(x) from 0 to π/2 is the same as the area under the curve from -π/2 to 0. The scaling factor of 2 in g(x) = 2cos(x) simply stretches the cosine function vertically, doubling its amplitude. This vertical stretch will affect the average value, making it twice the average value of the standard cosine function over the same interval. The problem's setup allows us to leverage the symmetry of the cosine function to simplify our calculations. We can anticipate that the average value will be positive since the cosine function is positive over most of the interval [-π/2, π/2]. Understanding these characteristics of the function and the interval will help us interpret the result we obtain and ensure that it makes sense in the context of the problem. Now, let's move on to the detailed calculation of the average value.

Before we begin the calculation, it's important to understand why we're interested in the average value. The average value provides a single number that summarizes the overall behavior of a function over an interval. It's a statistical measure that gives us a sense of the function's typical value within the specified domain. In the case of g(x) = 2cos(x), the average value tells us the average height of the curve over the interval [-π/2, π/2]. This can be useful in various applications, such as determining the average power of an alternating current or the average temperature over a period of time. The average value is not necessarily the same as the function's value at the midpoint of the interval; it's a weighted average that takes into account all the function's values within the interval. The formula for the average value ensures that we're considering the entire function's behavior, not just its value at a single point. By finding the average value, we gain a more comprehensive understanding of the function's characteristics and its overall contribution within the specified range. Now that we've highlighted the significance of the average value, let's proceed with the step-by-step calculation to determine its precise value for g(x) = 2cos(x) on the interval [-π/2, π/2]. We'll start by setting up the integral expression and then evaluate it using the appropriate techniques of calculus.

Formula for Average Value

The formula for the average value g_ave of a function g(x) on the interval [a, b] is given by:

g_ave = (1 / (b - a)) ∫[a, b] g(x) dx

This formula is the cornerstone of our calculation. It states that the average value is equal to the integral of the function over the interval, divided by the length of the interval. The integral, ∫[a, b] g(x) dx, represents the area under the curve of g(x) between the limits a and b. The term (1 / (b - a)) acts as a scaling factor, normalizing the area by the length of the interval. This normalization ensures that we obtain a representative average value, rather than just the total area. The formula is derived from the concept of dividing the area under the curve into infinitesimally small rectangles and then taking the average height of these rectangles. The definite integral provides a way to sum up these infinitesimal areas, and dividing by the interval length gives us the average height, which is the average value of the function. The formula is applicable to any continuous function over a closed interval and is a fundamental tool in calculus for analyzing function behavior. Understanding this formula is crucial for solving our problem and for applying the concept of average value in other contexts. Now, let's apply this formula to our specific problem by substituting the given function and interval into the equation.

The formula highlights the importance of both the integral and the interval length in determining the average value. The integral captures the cumulative effect of the function's values over the interval, while the interval length provides the scale over which we are averaging. If the function is constant, the average value is simply the constant value itself. However, if the function varies, the average value represents a weighted average that takes into account all the function's fluctuations. The formula is a powerful tool because it allows us to summarize the behavior of a function over an interval with a single number. This is particularly useful when dealing with complex functions that may have many peaks and valleys. The average value provides a sense of the function's overall trend and its typical magnitude within the specified range. The formula is also versatile and can be applied to a wide range of functions, including trigonometric, exponential, and polynomial functions. To use the formula effectively, it's essential to be proficient in calculating definite integrals, which often involves finding antiderivatives and applying the Fundamental Theorem of Calculus. Now that we have a firm grasp of the formula, let's proceed with applying it to our problem and see how it helps us find the average value of g(x) = 2cos(x) on the interval [-π/2, π/2]. We will substitute the function and the interval limits into the formula and then evaluate the resulting integral to obtain the solution.

Before we apply the formula, let's consider the components involved. The function g(x) = 2cos(x) is a trigonometric function, which we know has a periodic nature and well-defined antiderivatives. The interval [-π/2, π/2] is a symmetric interval centered around the y-axis, which can sometimes simplify calculations due to symmetry properties. The formula itself, g_ave = (1 / (b - a)) ∫[a, b] g(x) dx, tells us that we need to calculate the definite integral of g(x) over the interval [a, b], and then divide the result by the length of the interval, (b - a). In our case, a = -π/2 and b = π/2, so the interval length is π/2 - (-π/2) = π. The integral part of the formula, ∫[a, b] g(x) dx, involves finding the antiderivative of g(x) and then evaluating it at the limits of integration. The antiderivative of 2cos(x) is 2sin(x), which we will use in our calculation. Once we have the definite integral, we will divide it by the interval length, π, to obtain the average value, g_ave. Understanding these components and their roles in the formula will help us approach the calculation in a systematic and organized manner. Now, let's proceed with the substitution and evaluation steps to find the final answer.

Calculation

To calculate the average value, we first substitute the given function g(x) = 2cos(x) and the interval [-π/2, π/2] into the formula:

g_ave = (1 / (π/2 - (-π/2))) ∫[-π/2, π/2] 2cos(x) dx

Simplifying the denominator, we get:

g_ave = (1 / π) ∫[-π/2, π/2] 2cos(x) dx

Next, we find the antiderivative of 2cos(x), which is 2sin(x). Now we evaluate the definite integral:

∫[-π/2, π/2] 2cos(x) dx = 2sin(x) |[-π/2, π/2] = 2sin(π/2) - 2sin(-π/2)

Since sin(π/2) = 1 and sin(-π/2) = -1, we have:

2sin(π/2) - 2sin(-π/2) = 2(1) - 2(-1) = 2 + 2 = 4

Now we substitute this result back into the formula for g_ave:

g_ave = (1 / π) * 4 = 4/π

Therefore, the average value of the function g(x) = 2cos(x) on the interval [-π/2, π/2] is 4/π. This completes the calculation, and we have successfully found the average value using the formula and the properties of the cosine function.

Each step in the calculation is crucial for arriving at the correct answer. The initial substitution ensures that we are applying the formula correctly with the given function and interval. Simplifying the denominator makes the subsequent calculations easier. Finding the correct antiderivative is a key step, as it allows us to evaluate the definite integral. Evaluating the antiderivative at the limits of integration and subtracting gives us the value of the definite integral, which represents the net area under the curve. Finally, dividing by the interval length gives us the average value. It's important to pay attention to the signs and values of the trigonometric functions at the limits of integration, as these can significantly affect the result. In this case, the symmetry of the sine function, sin(-x) = -sin(x), plays a role in simplifying the calculation. The final result, 4/π, is a positive number, which is consistent with our expectation that the average value of 2cos(x) over the interval [-π/2, π/2] should be positive. Now that we have the numerical value, let's discuss its significance and what it tells us about the function's behavior over the given interval.

During the calculation process, we utilized the fundamental theorem of calculus, which connects the concepts of differentiation and integration. This theorem allows us to evaluate definite integrals by finding the antiderivative of the function and then evaluating it at the limits of integration. The antiderivative of 2cos(x) is 2sin(x), which we obtained by recalling the derivative of sin(x) is cos(x). The evaluation of the antiderivative at the limits π/2 and -π/2 involved using the known values of the sine function at these angles. The sine function oscillates between -1 and 1, and its values at π/2 and -π/2 are 1 and -1, respectively. These values are essential for calculating the definite integral. The algebraic manipulation of these values led us to the result of 4 for the definite integral. This value represents the net area under the curve of 2cos(x) over the interval [-π/2, π/2]. The final step of dividing by the interval length, π, gave us the average value, 4/π. This number represents the average height of the curve over the specified interval. Now that we have the average value, let's interpret its meaning and discuss its implications in the context of the function and the interval.

The result of the calculation, g_ave = 4/π, is an approximate value of 1.273. This means that, on average, the function g(x) = 2cos(x) has a value of approximately 1.273 over the interval [-π/2, π/2]. Graphically, this can be visualized as a horizontal line at y = 1.273 that cuts across the curve of 2cos(x), such that the area under the curve above the line is equal to the area below the curve above the line. The average value provides a representative single value for the function's behavior over the interval. It's not necessarily the midpoint between the maximum and minimum values of the function, but rather a weighted average that takes into account all the function's values within the interval. In this case, the maximum value of g(x) is 2, which occurs at x = 0, and the minimum value is 0, which occurs at x = ±π/2. The average value, 1.273, is closer to the maximum value than the minimum value, which is consistent with the fact that the cosine function is positive over most of the interval [-π/2, π/2]. This result provides a concise summary of the function's behavior and can be useful in various applications where a representative value over a range is needed. Now, let's summarize the entire process and highlight the key takeaways from this analysis.

Result

The average value g_ave of the function g(x) = 2cos(x) on the interval [-π/2, π/2] is:

g_ave = 4/π

This result signifies the mean height of the curve g(x) = 2cos(x) over the specified interval. It's a single numerical value that summarizes the overall behavior of the function within the given domain. The average value, 4/π, is a positive number, which aligns with the fact that the cosine function is positive over most of the interval [-π/2, π/2]. The result is obtained by applying the formula for the average value, which involves calculating the definite integral of the function and dividing by the length of the interval. The integral represents the net area under the curve, and dividing by the interval length normalizes this area to provide a representative average value. The result, 4/π, provides a concise summary of the function's behavior and can be used in various applications where a single representative value is needed. This completes our analysis of finding the average value of g(x) = 2cos(x) on the interval [-π/2, π/2]. We have successfully applied the formula, performed the calculation, and interpreted the result in the context of the function and the interval.

The result g_ave = 4/π is a specific value that we obtained through a rigorous mathematical process. It's not just an arbitrary number; it's a precise representation of the average height of the curve g(x) = 2cos(x) over the interval [-π/2, π/2]. The result is expressed as a fraction involving π, which is a common occurrence when dealing with trigonometric functions and integrals. The presence of π indicates the connection between the function's periodic nature and its average value. The numerical approximation of 4/π is approximately 1.273, which provides a more intuitive understanding of the magnitude of the average value. This value lies between the function's minimum value of 0 and its maximum value of 2 within the interval, which is expected for an average value. The result highlights the power of calculus in providing quantitative measures of function behavior. By using the definite integral and the average value formula, we were able to condense the function's behavior over a continuous interval into a single number. This single number can then be used for comparison, analysis, and application in various contexts. Now, let's reflect on the key concepts and steps involved in this problem and summarize the main takeaways from this analysis.

Interpreting the result, g_ave = 4/π, in a broader context, we can say that the average value provides a kind of