Problem 103 - Calculus for Biology and Medicine

Question

Let $$ f(x)=-3 \cos \left(\frac{\pi x}{5}\right), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Step-by-Step Solution

Answer

Amplitude: 3; Period: 10.

Step 1: Identify the Form of the Function

The function given is in the form \( f(x) = a \cos(bx) \). For the function \( f(x)=-3 \cos \left(\frac{\pi x}{5}\right) \), we can identify the constants: \( a = -3 \) and \( b = \frac{\pi}{5} \).

Step 2: Determine the Amplitude

The amplitude of a cosine function \( a \cos(bx) \) is given by the absolute value of \( a \). Here, \( a = -3 \), so the amplitude is \( |a| = |-3| = 3 \).

Step 3: Calculate the Period

The period of a cosine function \( a \cos(bx) \) is calculated using the formula \( \frac{2\pi}{b} \). Substituting \( b = \frac{\pi}{5} \), the period is \( \frac{2\pi}{\frac{\pi}{5}} = \frac{2\pi \times 5}{\pi} = 10 \).

Key Concepts

Understanding Amplitude

In trigonometric functions like the cosine function, the amplitude is an important concept. The amplitude refers to the maximum height of the wave from the center line. It essentially tells us how far the peaks and troughs of the wave are from the horizontal axis. The amplitude is always a positive number, regardless of whether the function appears stretched or flipped.

In the form \(a \cos(bx)\), the amplitude is given by the absolute value of \(a\). Absolute value means you ignore any negative sign. For the exercise function, \(f(x) = -3 \cos\left(\frac{\pi x}{5}\right)\), the value of \(a\) is \(-3\), and its absolute value is \(3\). Thus, the amplitude is \(3\).

Understanding amplitude helps in sketching correct graphs and predicting the behavior of trigonometric functions. Once you determine the amplitude, you can visualize how high and low the cosine wave will go relative to the x-axis.

Determining the Period

The period of a trigonometric function like cosine refers to how long it takes for the function to complete one full cycle. A cycle is one complete repetition of the wave pattern, after which the pattern starts to repeat. Knowing the period is crucial to understand how frequently the wave repeats over a range of x-values.

The period for a function in the form \(a \cos(bx)\) can be calculated using the formula \(\frac{2\pi}{b}\). For the given function \(f(x) = -3 \cos\left(\frac{\pi x}{5}\right)\), identify that \( b = \frac{\pi}{5} \). Substituting \(b\) in the formula gives us the period:

• \(\frac{2\pi}{\frac{\pi}{5}} = \frac{2\pi \times 5}{\pi} = 10\)

The result, \(10\), tells us that the wave pattern of the function repeats every 10 units along the x-axis. Fathoming the period is essential for graphing these functions correctly and understanding the oscillatory nature of trigonometric waves.

Exploring the Cosine Function

The cosine function is one of the fundamental trigonometric functions and is part of the family that includes sine and tangent. It has a distinctive wave pattern that is symmetrical about the y-axis. In standard form, \(\cos(x)\), the wave oscillates between \(+1\) and \(-1\), producing consistent and smooth curves.

When the cosine function is modified by constants \(a\) and \(b\) in \(a \cos(bx)\), the dynamics of the wave change. The amplitude \(a\) affects the height, while \(b\) influences the period, determining how fast the wave completes its cycle. For instance, in our example \(-3 \cos\left(\frac{\pi x}{5}\right)\), the negative sign inverts the wave, flipping it about the x-axis.

• Amplitude tells the height of waves.
• Period tells the length needed for a full wave cycle.
• The negative sign flips the wave direction.

The cosine function is periodic and predictable, making it essential for modeling phenomena such as sound waves, light waves, and other cyclical patterns in nature.