Question

Sketch the graph of \(y=\ln \cos x+\ln \sec x\) on \((-\pi / 2, \pi / 2)\), but think before you begin.

Short answer

The graph of the given function is a horizontal line at \(y=0\) between \(-\pi/2\) and \(\pi/2\), excluding the endpoints.

Step-by-step solution

Understand the Function

First, let's simplify the function. We have:\[y = \ln (\cos x) + \ln (\sec x)\]Using the property of logarithms \( \ln a + \ln b = \ln (a \cdot b) \), we can rewrite this as:\[y = \ln (\cos x \cdot \sec x)\]Since \(\sec x = \frac{1}{\cos x}\), we have:\[ y = \ln ((\cos x) \cdot (\frac{1}{\cos x})) = \ln(1) \]Thus, the function simplifies to \(y = 0\) everywhere in its domain.

Determine Domain of the Function

In the original function, both \(\ln \cos x\) and \(\ln \sec x\) must be defined. This requires: - \(\cos x > 0\), because the logarithm of a negative number or zero is undefined.\(\sec x = \frac{1}{\cos x}\)- Thus, \(x eq \pm \pi/2\) because \(\cos(\pi/2) = 0\). Within \((-\pi / 2, \pi / 2)\), \( \cos x > 0\) except at endpoints. Therefore, the domain is exactly \((-\pi / 2, \pi / 2)\).

Describe the Graph

Given the simplification, the graph of the function \(y = \ln \cos x + \ln \sec x\) during the interval \((-\pi/2, \pi/2)\) is simply a horizontal line at \(y = 0\). There are no breaks or bends in the line except approaching the endpoints at \(-\pi/2\) and \(\pi/2\), where the graph does not exist due to the undefined nature of the original logarithmic terms.

Sketch the Graph

To sketch the graph, plot a horizontal line along the x-axis (where \(y=0\)) from \(-\pi/2\) to \(\pi/2\). Keep in mind that the graph does not include the points \(-\pi/2\) and \(\pi/2\) because they are not part of the domain. Make sure to present these points as open circles or simply leave them unconnected to indicate that they do not belong to the graph.

Key concepts

Graph Sketching

Graph sketching is a fundamental skill in calculus where we create a visual representation of a given function. In the exercise provided, sketching the graph helps us visualize the behavior of the function over the specified domain.
To begin, we first simplified the original function, finding out the function actually equals zero throughout the interval \((-\pi/2, \pi/2)\). This means the graph is a straight horizontal line on the y-axis at \(y=0\).

• The domain, \((-\pi/2, \pi/2)\), shows where the function is defined.
• The endpoints are excluded, meaning we do not plot the exact points \(-\pi/2\) and \(\pi/2\). This is where open circles help demonstrate discontinuity.
• The function remains unchanged, indicating no additional twists or curves, emphasizing the simplicity of plotting a horizontal line.

In conclusion, understanding how to sketch graphs effectively starts with analyzing the function thoroughly, as done here, and concludes in sketching a clean and informative plot.

Logarithmic Functions

Logarithmic functions involve the logarithm operation, which is the inverse of exponentiation. They are critical for transforming and understanding the behavior of multiplicative relationships.
In our exercise, the function used two logarithmic terms: \(\ln(\cos x)\) and \(\ln(\sec x)\).

• Property Utilization: The logarithmic identity utilized here is \(\ln a + \ln b = \ln (a \cdot b)\). Applying it combined the terms effectively.
• Logarithmic Limitations: The domain considerations showed why \(\cos x\) must always be greater than zero. Logarithms aren’t defined for zero and negative values, which shapes the function’s domain.
• Simplification: The expression \(\ln(1)\) simplifies directly to zero, elucidating the result of the simplification process.

An understanding of these key properties allows for deeper comprehension of how logarithmic functions contribute to simplifying complex expressions like in our exercise.

Trigonometric Functions

Trigonometric functions are fundamental in relating angles to ratios of triangle sides. Here, the function involved calls for an understanding of cosine and secant.
- **Cosine (**\(\cos x\)**):** It represents the ratio of the adjacent side to the hypotenuse. For our function, \(\cos x > 0\) ensures that the output values are valid under the logarithmic operation.
- **Secant (**\(\sec x\ = \frac{1}{\cos x}\)**):** The secant is the reciprocal of cosine. Integrating \(\sec x\) in a logarithm enforces no zero or negative cosine values, maintaining the operation's validity.

• Domain Impact: Recognizing the reciprocal nature of secant underscores why \(x eq \pm \pi/2\), because \(\cos(\pm \pi/2)\) equals zero, leading to undefined terms.
• Coefficient Analysis: The algorithmic understanding helps rearrange these terms, which is crucial in simplifying \(\ln(\cos x \cdot \sec x)\) to \(\ln(1)\).

Trigonometric functions like cosine and secant provide elemental support to logarithmic function evaluations, making understanding them essential for calculus success.