Analyzing F(x) = Sin(π/2 * X) On [0, 4]: A Detailed Discussion
Let's dive into a comprehensive analysis of the function f(x) = sin(π/2 * x) over the interval [0, 4]. This function combines trigonometric principles with a linear transformation, resulting in a wave-like behavior that's both interesting and predictable. We'll explore its key characteristics, such as its periodicity, amplitude, critical points, and concavity, all within the specified interval. So, buckle up, math enthusiasts, as we unravel the intricacies of this sinusoidal function!
Understanding the Basics of f(x) = sin(π/2 * x)
To get started, let's break down the function itself. The core of our function is the sine function, sin(x), which oscillates between -1 and 1. The argument of the sine function, in this case, is (π/2 * x). This linear transformation affects the period of the sine wave. Remember, the general form for a sinusoidal function is f(x) = A * sin(B(x - C)) + D, where:
- A represents the amplitude.
- B affects the period.
- C represents the horizontal shift.
- D represents the vertical shift.
In our case, A = 1, B = π/2, C = 0, and D = 0. The amplitude of our function is 1, meaning it oscillates between -1 and 1. Now, let's talk about the period. The period of the standard sine function, sin(x), is 2π. However, the presence of π/2 inside the sine function modifies this. The period of sin(B * x) is given by 2π / |B|. So, for our function, the period is 2π / (π/2) = 4. This means the function completes one full cycle over an interval of length 4.
Considering our interval is [0, 4], we'll observe exactly one full cycle of the sine wave. This makes our analysis a bit more manageable and allows us to focus on the key features within this single period. Understanding the period is crucial because it helps us predict the function's behavior and identify its critical points, such as maxima, minima, and zeros.
Key Characteristics within the Interval [0, 4]
Now that we understand the basics, let's delve into the key characteristics of f(x) = sin(π/2 * x) within the interval [0, 4]. This involves identifying critical points, intervals of increase and decrease, and concavity.
1. Intercepts (Zeros)
The intercepts, or zeros, of a function are the points where the function crosses the x-axis, meaning f(x) = 0. For our function, we need to find the values of x in the interval [0, 4] where sin(π/2 * x) = 0. We know that sin(θ) = 0 when θ is an integer multiple of π (i.e., 0, π, 2π, etc.). So, we need to solve for x in the equation:
π/2 * x = n * π, where n is an integer.
Dividing both sides by π/2, we get:
x = 2n
Now, we need to find the integer values of n that give us x values within our interval [0, 4]. Let's test a few values:
- If n = 0, then x = 2 * 0 = 0
- If n = 1, then x = 2 * 1 = 2
- If n = 2, then x = 2 * 2 = 4
So, the intercepts of our function within the interval [0, 4] are x = 0, x = 2, and x = 4. These are the points where the graph of the function crosses the x-axis. They also serve as crucial reference points for understanding the function's overall shape and behavior.
2. Maxima and Minima
Maxima and minima are the points where the function reaches its highest and lowest values, respectively. For f(x) = sin(π/2 * x), the maximum value is 1, and the minimum value is -1 (remember the amplitude is 1). To find where these occur, we need to find the values of x where sin(π/2 * x) = 1 and sin(π/2 * x) = -1. Let's start with the maximum:
sin(π/2 * x) = 1
We know that sin(θ) = 1 when θ = π/2 + 2πk, where k is an integer. So:
π/2 * x = π/2 + 2πk
Dividing both sides by π/2, we get:
x = 1 + 4k
Now, let's find the value of k that gives us an x within our interval [0, 4]:
- If k = 0, then x = 1
This gives us the maximum point at (1, 1).
Now, let's find the minimum:
sin(π/2 * x) = -1
We know that sin(θ) = -1 when θ = 3π/2 + 2πk, where k is an integer. So:
π/2 * x = 3π/2 + 2πk
Dividing both sides by π/2, we get:
x = 3 + 4k
Now, let's find the value of k that gives us an x within our interval [0, 4]:
- If k = 0, then x = 3
This gives us the minimum point at (3, -1).
Therefore, within the interval [0, 4], the function has a maximum at (1, 1) and a minimum at (3, -1). These points are crucial for sketching the graph and understanding the function's oscillating behavior.
3. Intervals of Increase and Decrease
To determine the intervals where the function is increasing or decreasing, we need to analyze its first derivative. Let's find the derivative of f(x) = sin(π/2 * x) using the chain rule:
f'(x) = (π/2) * cos(π/2 * x)
The function is increasing when f'(x) > 0 and decreasing when f'(x) < 0. So, we need to analyze the sign of cos(π/2 * x).
- f'(x) > 0 when cos(π/2 * x) > 0. Cosine is positive in the first and fourth quadrants. Within our interval, this corresponds to the intervals where 0 ≤ π/2 * x < π/2 and 3π/2 < π/2 * x ≤ 2π. Solving for x, we get 0 ≤ x < 1 and 3 < x ≤ 4. Thus, the function is increasing on the intervals [0, 1) and (3, 4].
- f'(x) < 0 when cos(π/2 * x) < 0. Cosine is negative in the second and third quadrants. Within our interval, this corresponds to the interval where π/2 < π/2 * x < 3π/2. Solving for x, we get 1 < x < 3. Thus, the function is decreasing on the interval (1, 3).
In summary, the function f(x) = sin(π/2 * x) is increasing on [0, 1) and (3, 4], and decreasing on (1, 3) within the interval [0, 4]. This confirms our earlier findings regarding the maximum and minimum points.
4. Concavity and Inflection Points
Concavity describes the curvature of the function. To determine concavity, we need to analyze the second derivative. Let's find the second derivative of f(x) = sin(π/2 * x):
First, we have f'(x) = (π/2) * cos(π/2 * x). Now, differentiating again:
f''(x) = (π/2) * (-sin(π/2 * x)) * (π/2) = -(π^2 / 4) * sin(π/2 * x)
- Concave Up: The function is concave up when f''(x) > 0. This means -(π^2 / 4) * sin(π/2 * x) > 0, which simplifies to sin(π/2 * x) < 0. We know sine is negative in the third and fourth quadrants. Within our interval, this corresponds to π < π/2 * x < 2π. Solving for x, we get 2 < x < 4. Thus, the function is concave up on the interval (2, 4).
- Concave Down: The function is concave down when f''(x) < 0. This means -(π^2 / 4) * sin(π/2 * x) < 0, which simplifies to sin(π/2 * x) > 0. We know sine is positive in the first and second quadrants. Within our interval, this corresponds to 0 < π/2 * x < π. Solving for x, we get 0 < x < 2. Thus, the function is concave down on the interval (0, 2).
Inflection points are where the concavity changes. These occur when f''(x) = 0 or where f''(x) is undefined. In our case, f''(x) = -(π^2 / 4) * sin(π/2 * x). We need to find where this equals zero:
-(π^2 / 4) * sin(π/2 * x) = 0
This is true when sin(π/2 * x) = 0. We already found these points when determining the intercepts: x = 0, x = 2, and x = 4. However, inflection points only occur where the concavity changes. At x = 0 and x = 4, we are at the endpoints of our interval, so they don't represent a change in concavity within the interval. At x = 2, the concavity changes from down to up. Therefore, the only inflection point within the interval (0, 4) is at x = 2, which corresponds to the point (2, 0).
Putting It All Together: Sketching the Graph
Now that we've analyzed the intercepts, maxima, minima, intervals of increase and decrease, and concavity, we have all the pieces we need to sketch the graph of f(x) = sin(π/2 * x) on the interval [0, 4]. Here's a quick recap of our findings:
- Intercepts: x = 0, x = 2, x = 4
- Maximum: (1, 1)
- Minimum: (3, -1)
- Increasing: [0, 1) and (3, 4]
- Decreasing: (1, 3)
- Concave Down: (0, 2)
- Concave Up: (2, 4)
- Inflection Point: (2, 0)
Using this information, we can sketch a smooth, continuous curve that starts at (0, 0), increases to a maximum at (1, 1), decreases to a minimum at (3, -1), and returns to (4, 0). The curve is concave down from 0 to 2 and concave up from 2 to 4, with an inflection point at (2, 0). Guys, you've got the full picture now!
Conclusion
In this detailed discussion, we've thoroughly analyzed the function f(x) = sin(π/2 * x) on the interval [0, 4]. We've explored its periodicity, amplitude, critical points, intervals of increase and decrease, concavity, and inflection points. By understanding these characteristics, we can accurately sketch the graph of the function and gain a deeper appreciation for its behavior. This example demonstrates the power of calculus and trigonometric principles in understanding and analyzing mathematical functions. Keep exploring, guys, and you'll uncover even more fascinating mathematical landscapes!
