Problem 16 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow \pi / 2} \cos (x-\pi) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit

We need to find the limit of the function \( \cos(x-\pi) \) as \( x \) approaches \( \frac{\pi}{2} \). This means we want to determine the behavior of \( \cos(x-\pi) \) right near \( x = \frac{\pi}{2} \).

Step 2: Rewrite the Function

The expression \( x-\pi \) can be rewritten as \( x - \frac{\pi}{2} - \frac{\pi}{2} \). Therefore, \( \cos(x-\pi) = \cos\left((x-\frac{\pi}{2}) - \frac{\pi}{2}\right) \). This allows us to analyze the cosine function near \( x = \frac{\pi}{2} \).

Step 3: Evaluate Numerically Using a Table

Create a table with values of \( x \) approaching \( \frac{\pi}{2} \) from the left and right, and calculate \( \cos(x-\pi) \) for each. For example, for \( x = \frac{\pi}{2} - 0.1 \), \( \cos((\frac{\pi}{2}-0.1) - \pi) \), which simplifies to \( \cos(-1.57 - 0.1) = \cos(-1.67) \). Do the same for values like \( \frac{\pi}{2}-0.01 \), \( \frac{\pi}{2}+0.01 \), etc.

Step 4: Analyze the Pattern

The pattern in the table shows that as \( x \) approaches \( \frac{\pi}{2} \), the value of \( \cos(x-\pi) \) approaches \( 0 \). This is due to \( \cos(-\frac{\pi}{2}) = 0 \), which is the form the function takes as \( x \) approaches \( \frac{\pi}{2} \).

Step 5: Use a Graph to Double Check

Plot the function \( \cos(x-\pi) \) and observe the graph near \( x = \frac{\pi}{2} \). The graph should show that as \( x \) approaches \( \frac{\pi}{2} \) from both sides, the function approaches 0. This visual confirmation supports our numerical findings.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that plays a significant role in various mathematical and real-world applications. It is represented as \( \cos(x) \), and oscillates between -1 and 1, corresponding to the angle \( x \). Understanding the behavior of the cosine function is vital, especially when dealing with limits, as seen in the given exercise.With the example of \( \cos(x - \pi) \), the expression \( x - \pi \) shifts the graph of \( \cos(x) \) by \( \pi \) units to the right. Given that the cosine function is periodic with a period of \( 2\pi \), this simply flips the graph, allowing us to analyze the function efficiently.To delve deeper:

• Cosine of 0 or multiples of \( 2\pi \) is 1.
• Cosine of \( \pi \) (and its odd multiples) is -1.
• Cosine of \( \pi/2 \) (and its odd multiples) is 0.

These values help us predict the behavior of \( \cos(x-\pi) \) as \( x \) nears specific points like \( \pi/2 \). By recognizing the transformation, students can better anticipate and confirm results obtained through calculation or graphical representation.

Numerical Evaluation

Numerical evaluation is a practical approach when calculating the limit of a function. Using values close to the point of interest, students can predict the behavior of the function as it approaches that point. This is especially useful when exploring the limit of \( \cos(x - \pi) \) as \( x \) approaches \( \frac{\pi}{2} \).To perform this:

• Choose values for \( x \) that are near \( \frac{\pi}{2} \) from both the lower and upper sides, such as \( \frac{\pi}{2} - 0.1 \), \( \frac{\pi}{2} + 0.1 \), etc.
• Substitute these values into the function \( \cos((x-\pi)) \).
• Observe the results to find a trend; as \( x \) closes in on \( \frac{\pi}{2} \), the cosine approaches 0.

Numerical evaluation not only provides a means to compute approximate values, but also helps in developing an intuitive understanding of the function's behavior at specific points. By interacting with actual numbers, students can visualize the limit and solidify their comprehension.

Graphical Analysis

Graphical analysis complements numerical evaluation by providing a visual method to investigate the behavior of functions. In our exercise, examining the function \( \cos(x - \pi) \) through a graph provides insight into how the function behaves as \( x \rightarrow \frac{\pi}{2} \).When graphing:

• Plot points for \( x \) values either side of \( \frac{\pi}{2} \).
• Use graphing tools or software to create a visual plot of \( \cos(x - \pi) \).
• Observe the graph for patterns or consistent behaviors as \( x \) gets closer to \( \frac{\pi}{2} \).

The graph should clearly illustrate how the function approaches zero as \( x \) nears \( \frac{\pi}{2} \). This graphical method not only augments the findings from numerical evaluation but also strengthens understanding by showcasing the behavior of the function in a continuous, unbroken format.In conclusion, combining numerical and graphical analyses can drasticall enhance comprehension of limits, making them crucial tools for evaluating the utility of trigonometric functions like the cosine function.