Problem 4 - Calculus for Biology and Medicine

Question

Each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extremevalue theorem. With the help of a graphing calculator, graph each function and locate its global extrema. (Note that a function may assume a global extremum at more than one point.) \(f(x)=\cos \frac{x}{2}, 0 \leq x \leq 2 \pi\)

Step-by-Step Solution

Answer

Global maximum is 1 at \(x = 0\) and \(x = 2\pi\); global minimum is \(-1\) at \(x = \pi\).

Step 1: Understanding the Problem

We need to find the global extrema (maximum and minimum values) of the function \( f(x) = \cos \frac{x}{2} \) on the interval \( [0, 2\pi] \). We'll accomplish this by utilizing a graphing calculator to visualize the function's behavior on the given interval. Since the function is continuous on a closed interval, the Extreme Value Theorem assures us it will have both a maximum and minimum on \([0, 2\pi]\).

Step 2: Setting Up the Graphing Calculator

Turn on the graphing calculator and enter the function \( y = \cos \frac{x}{2} \). Ensure the calculator is set to radian mode, as the interval and function are given in terms of \( \pi \). Set the window settings so that the \( x \)-range is from \( 0 \) to \( 2\pi \). The \( y \)-range should encompass typical cosine values, which range between \(-1\) and \(1\).

Step 3: Graph the Function

Graph the function \( f(x) = \cos \frac{x}{2} \) using the graphing calculator. Observe the wave-like pattern due to the cosine function, which will oscillate between \(-1\) and \(1\). The graph provides a visual representation of the function, showing peaks and valleys within the interval \([0, 2\pi]\).

Step 4: Locating the Extrema

Using the graphing calculator's features, identify the highest points (maxima) and lowest points (minima) of the graph within the interval. The global maximum on this interval is found at the points where \( x = 0 \) and \( x = 2\pi \), with a value of \(1\). The global minimum occurs at \( x = \pi \), with a value of \(-1\).

Step 5: Conclusion

The global extrema of the function \( f(x) = \cos \frac{x}{2} \) over the interval \( [0, 2\pi] \) are a maximum value of \(1\) at \( x = 0 \) and \( x = 2\pi \), and a minimum value of \(-1\) at \( x = \pi \). The function achieves its extremum values multiple times, consistent with the properties of periodic functions.

Key Concepts

Understanding Global Extrema

Global extrema refer to the highest and lowest points a function attains on a specified interval. For a better grasp, think of it as standing on the tallest hill or in the deepest valley. When dealing with a continuous function over a closed interval, the Extreme Value Theorem assures that a global maximum and minimum must exist somewhere within the interval.

In our given problem, we looked at the function \(f(x) = \cos \frac{x}{2}\) over \([0, 2\pi]\). By finding the global maximum and minimum on this interval, we determine the global extrema. Each point where the function reaches its highest or lowest values is crucial for understanding the overall behavior of the function.

Here are some quick notes to remember:

• Global maximum is the peak value a function reaches.
• Global minimum is the lowest value it reaches.
• The Extreme Value Theorem states these values will exist for continuous functions over closed intervals.

Utilizing a Graphing Calculator

A graphing calculator is a powerful tool for visualizing functions. It can help identify the critical values of a function on a specific interval and confirm theoretical calculations visually.

For \(f(x) = \cos \frac{x}{2}\), using a graphing calculator:

• Enter the function in radian mode because the interval uses \(\pi\).
• Set the graph window to capture the full range of \(x\) from \(0\) to \(2\pi\) and \(y\) from -1 to 1, matching the cosine's typical values.
• Observe the graph to see the wave-like pattern common in cosine functions.

This visual method allows students to pinpoint approximate values for the function's peaks and valleys. It enhances understanding by turning abstract numbers into tangible, graphical insights, helping you to easily spot where the function hits its maximum and minimum values.

Importance of Continuous Functions

A continuous function, like \(f(x) = \cos \frac{x}{2}\), is essential to discussions involving the Extreme Value Theorem. Continuity ensures there are no breaks, jumps, or hollow points over the interval, which guarantees that extreme values are not only theoretical but actually present.

In practical terms, having a continuous function means:

• You can trace the function over any point in the interval without lifting your pencil.
• The theorem holds true only on closed intervals, where the boundaries are included.
• Graphing a continuous function gives a smooth curve or line, providing a clearer insight into changing behaviors over the interval.

The function \(f(x) = \cos \frac{x}{2}\) is particularly straightforward as it is periodic and standard within the closed interval, reinforcing this very concept.