Problem 102 - Calculus for Biology and Medicine

Question

Let $$ f(x)=7 \cos (2 x), \quad x \in \mathbf{R} $$ Find the amplitude and the period of \(f(x)\).

Step-by-Step Solution

Answer

Amplitude is 7, and period is \(\pi\).

Step 1: Understand the General Form of a Cosine Function

Recall that the general form of a cosine function is \( g(x) = A \cos(Bx + C) + D \), where \( A \) determines the amplitude, \( B \) affects the period, \( C \) is a phase shift, and \( D \) is a vertical shift. For our function \( f(x) = 7 \cos(2x) \), we compare to this general form.

Step 2: Determine the Amplitude

In the function \( f(x) = 7 \cos(2x) \), the coefficient of cosine, \( A = 7 \), represents the amplitude. Therefore, the amplitude of \( f(x) \) is \( 7 \).

Step 3: Calculate the Period

The period of a cosine function is given by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the cosine function. In \( f(x) = 7 \cos(2x) \), \( B = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \).

Key Concepts

Understanding Amplitude in Trigonometric Functions

Amplitude is a fundamental concept in trigonometry. It defines how tall or how high the wave of a function reaches from its central position. In simple terms, amplitude measures the maximum distance the wave goes above or below its midpoint or equilibrium line. Just think of waves in the ocean. The amplitude of such a wave would be the height from the calm sea level to the peak.

For cosine functions, the amplitude is represented by the coefficient in front of the cosine part in the equation.

• If the equation is in the form of \( f(x) = A \cos(Bx + C) + D \), then \( A \) is the amplitude.
• This value indicates how much the function stretches or compresses vertically.

In the given exercise, our function is \( f(x) = 7 \cos(2x) \).

• The amplitude here is straightforward as it's the number 7, making the function stretch from -7 to 7 in its range.

Learning to identify the amplitude can help you understand the wave's behavior and its bounds in oscillating motion.

The Concept of Period in Waves

When we chat about periods in trigonometry, we're really talking about how long it takes for a function to repeat its pattern. Imagine the hand of a clock going around; it repeats its cycle every 12 hours. Similarly, a trigonometric function like a cosine wave repeats its pattern over and over. This repeating length is what we call the period.

Here's how you determine the period of a cosine function:

• The general formula for period is \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the cosine.

Looking at the exercise function \( f(x) = 7 \cos(2x) \):

• We see \( B = 2 \), and by plugging it into our formula, we calculate the period as \( \frac{2\pi}{2} = \pi \).

This means the cosine wave of our function completes one cycle or wave every \( \pi \) units along the x-axis.

Knowing the period of a function is vital for predicting its behavior, just like knowing how long it takes for day to turn to night helps us plan our day.

Exploring the Cosine Function

The cosine function is one of the fundamental building blocks in trigonometry. It's often discussed alongside its sibling, the sine function, but has its own unique behavior and characteristics. At its heart, the cosine function is all about cycles – it's repetitive and periodic by nature.

Key traits of the cosine function include:

• Its wave-like graph, which oscillates smoothly above and below an equilibrium line.
• Starting at its maximum value when the angle is zero (unlike the sine function which starts at zero).

The general equation of the cosine function is \( f(x) = A \cos(Bx + C) + D \).

• Here, \( A \) as mentioned affects the amplitude or height of the wave.
• \( B \) influences the period, determining how frequently the cycle repeats.
• \( C \) and \( D \) can shift the wave horizontally and vertically, though they aren't present in our current example.

For our specific function \( f(x) = 7 \cos(2x) \):

• It's stripped down to its essentials with no \( C \) or \( D \), so we focus purely on its amplitude and period.

Understanding the cosine function enables us to model and interpret many real-world phenomena, from the swinging of a pendulum to the cyclical behavior of tides.