Problem 9 - Applied Calculus

Question

The following table shows values of a periodic function \(f(x) .\) The maximum value attained by the function is 5 (a) What is the amplitude of this function? (b) What is the period of this function? (c) Find a formula for this periodic function. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\\\\hline f(x) & 5 & 0 & -5 & 0 & 5 & 0 & -5 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) Amplitude is 5. (b) Period is 8. (c) Formula: \(f(x) = 5 \cos\left(\frac{\pi}{4}x\right)\).

Step 1: Identify the Amplitude

The amplitude of a periodic function is half the distance between its maximum and minimum values. For this function, the maximum value is given as 5 and from the table, the minimum value is -5. Therefore, the amplitude is:\[ \text{Amplitude} = \frac{5 - (-5)}{2} = \frac{10}{2} = 5 \]

Step 2: Determine the Period

The period of a periodic function is the distance over which the function's values start to repeat. Observing the table, the function values repeat every 8 units (from 0 to 8, then from 8 to 16), indicating that the period is:\[ \text{Period} = 8 \]

Step 3: Formulate the Function

A standard sinusoidal function can be represented as \(f(x) = A \sin(Bx + C) + D\) or \(f(x) = A \cos(Bx + C) + D\). Given that \(f(0)=5\) and it hits maximum value when \(x=0\), we can use the cosine function. Since the maximum amplitude is at \(x=0\) and decreases to zero at \(x=2\), we can write:\[ f(x) = 5 \cos\left(\frac{\pi}{4}x\right) \]Here, \(A = 5\), the period factor \(B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{8} = \frac{\pi}{4}\), and there is no vertical shift or phase shift, so \(C = 0\) and \(D=0\).

Key Concepts

Amplitude

In the context of sinusoidal functions, the amplitude signifies the peak deviation of the function from its mean value. Think of it as how high or low the waves go. In practical terms, the amplitude is half the distance between the highest point of the function (the maximum value) and the lowest point (the minimum value).
For the given function where the maximum value is 5 and the minimum value is -5, the calculation of the amplitude becomes quite simple. You can use the formula:

• Amplitude = \( \frac{\text{Maximum value} - \text{Minimum value}}{2} \)
• Amplitude = \( \frac{5 - (-5)}{2} = \frac{10}{2} = 5 \)

Understanding amplitude helps to visualize how much a sinusoidal function deviates from its central axis. Here, an amplitude of 5 means that the peaks of the sine or cosine waves are 5 units away from the central value of the function, which is often zero if there is no vertical shift.

Period

The period of a periodic function such as a sinusoidal function represents the interval over which its pattern repeats. It's like knowing the length of one complete cycle of a wave. Determining the period helps to define how quickly or slowly the wave oscillates.
In our exercise, by examining the function's values in the table, the pattern shows that it repeats every 8 units. This is interpreted as the period. You can compute it as follows:

• Notice that starting at \(x = 0\), the function goes through a full cycle by \(x = 8\).
• Therefore, the period is 8.

This period tells us that every 8 units along the x-axis, the function returns to its starting position, making it essential in determining the frequency and structure of the sinusoidal model.

Sinusoidal Functions

Sinusoidal functions form the foundation of trigonometry and describe waves through sine and cosine functions. These functions are crucial for modeling periodic phenomena like sound waves, light waves, and seasonal changes.
In our given exercise, we describe the periodic function using the cosine form as:

• General formula: \( f(x) = A \cos(Bx + C) + D \)
• Given formula: \( f(x) = 5 \cos\left(\frac{\pi}{4}x\right) \)

Here's how the components fit in:

• Amplitude \(A\): This was computed as 5, indicating the wave's height above and below the mean.
• Frequency factor \(B\): Derived from the period. Since \(B = \frac{2\pi}{\text{Period}}\), and the period is 8, \(B = \frac{\pi}{4}\).
• Phase Shift \(C\): In our function, there is no horizontal translation so \(C = 0\).
• Vertical Shift \(D\): The function has not been lifted or lowered, thus \(D = 0\).

Understanding the structure of sinusoidal functions allows for effective modeling and prediction of natural cyclic patterns, making it an invaluable tool in mathematics and physics.