Problem 15 - Applied Calculus

Question

Most breeding birds in the northeast US migrate elsewhere during the winter. The number of bird species in an Ohio forest preserve oscillates between a high of 28 in June and a low of 10 in December. \(^{97}\) (a) Graph the number of bird species in this preserve as a function of \(t,\) the number of months since June. Include at least three years on your graph. (b) What are the amplitude and period of this function? (c) Find a formula for the number of bird species, \(B\), as a function of the number of months, \(t\) since June.

Step-by-Step Solution

Answer

Amplitude: 9, Period: 12 months. Formula: \( B(t) = 19 + 9 \cos\left(\frac{2\pi}{12}(t)\right) \).

Step 1: Analyzing the Periodicity

The number of bird species is said to oscillate between a high of 28 in June and a low of 10 in December. This indicates that the function is likely periodic, specifically sinusoidal, because it oscillates regularly and reaches its maximum and minimum values in predictable patterns.

Step 2: Determine the Amplitude

The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. Here, the amplitude is \(\frac{28 - 10}{2} = 9\).

Step 3: Calculate the Period

The period of the function is the time it takes to complete one full cycle. Since the species' count goes from maximum in June to minimum in December and reaches maximum again in June, the period is 12 months, which corresponds to one year.

Step 4: Define the Midline

The midline of the sinusoidal function is the average of the maximum and minimum values, calculated as \(\frac{28 + 10}{2} = 19\). This represents the vertical shift of the sinusoidal graph.

Step 5: Constructing the Function Formula

Using the information from the previous steps, the function describing the number of bird species, \( B \), as a function of months \( t \), is given by: \[ B(t) = 19 + 9 \cos\left(\frac{2\pi}{12}(t)\right) \]This formula reflects a cosine function shifted vertically and adjusted for amplitude and period.

Step 6: Graphing Over Three Years

To graph the function over three years, plot \( B(t) = 19 + 9 \cos\left(\frac{2\pi}{12}(t)\right) \) with \( t \) ranging from 0 to 36. This will show the oscillations between 28 and 10, repeating every 12 months from June to June.

Key Concepts

Amplitude

Amplitude is a fundamental concept in understanding sinusoidal functions. In the context of this exercise, amplitude helps us determine the fluctuation range of the bird species in the forest preserve. It represents how far the function goes above and below its average level, also known as the midline.

To calculate the amplitude of a sinusoidal function, use the formula:\[ \text{Amplitude} = \frac{\text{Max Value} - \text{Min Value}}{2} \]This formula essentially measures half the distance between the highest and lowest values the function attains. For this exercise, the maximum number of bird species is 28, and the minimum is 10. Plug in these values:\[ \text{Amplitude} = \frac{28 - 10}{2} = 9 \]This means that the number of bird species oscillates 9 units above and below the average value.

Periodicity

Periodicity in a function refers to the length of time it takes for the function to complete one full cycle, returning to its starting point. For sinusoidal functions like the one in this problem, the periodic nature is evident as the number of bird species repeats its pattern regularly.

To find the period, we need to identify the interval within which the function makes one complete oscillation. For the birds in the Ohio forest preserve, they reach their maximum in June, drop to a minimum in December, and then rise back to a maximum the following June. This cycle is 12 months long:

• June (maximum) to December (minimum)
• Back to June (maximum)

Thus, the period of the function is 12 months. This periodic behavior helps in predicting the number of species at any given month through the years.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as the cosine function used in this problem, provides a visual representation of periodic behavior. In this exercise, graphing the number of bird species allows us to see how the count changes over time.

When graphing a cosine function like \( B(t) = 19 + 9 \cos\left(\frac{2\pi}{12}t\right) \), follow these steps:

• Identify the amplitude from the formula, which is 9.
• Determine the midline or vertical shift, which we calculated as 19.
• The period is captured by the fraction \( \frac{2\pi}{12} \), showing the cycle completes every 12 months.

Plot the graph over a range of at least 36 months to display three full cycles, showing the fluctuation from a high of 28 species to a low of 10. By understanding these components, you can predict and visualize the changes in bird species throughout the years, enhancing your comprehension of trigonometric function behavior.