Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The graph of the function is a horizontal line at \(y = 0\) on \((-\pi/2, \pi/2)\).

Step 1: Simplify the Expression

First, simplify the expression given in the function. We have: \[ y = \ln(\cos x) + \ln(\sec x) \] Recall that \(\ln(a) + \ln(b) = \ln(ab)\). Therefore, \[ y = \ln(\cos x \cdot \sec x) \].

Step 2: Simplify Further Using Trigonometric Identity

Recognize that \(\sec x = \frac{1}{\cos x}\). Thus, substitute this into the expression:\[ \cos x \cdot \sec x = \cos x \cdot \frac{1}{\cos x} = 1 \].

Step 3: Conclusion of Simplification

The expression simplifies to:\[ y = \ln(1) \].Recall that \(\ln(1) = 0\). Therefore, \[ y = 0 \].

Step 4: Plot the Graph

Since the function simplifies to a constant, \(y=0\), on the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), the graph is a horizontal line coinciding with the x-axis between these limits.

Key Concepts

Trigonometric identities

In calculus, understanding trigonometric identities is crucial, as they allow us to simplify complex expressions involving trigonometric functions. Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variable involved. In this exercise, one key trigonometric identity that helps simplify the given function is \[ \sec x = \frac{1}{\cos x} \]. This identity is instrumental because it turns the product \( \cos x \cdot \sec x \) into \( \cos x \cdot \frac{1}{\cos x} \), which simplifies to 1. These identities are often used to transform complex expressions into simpler forms, facilitating easier graphing and deeper analysis of the function's behavior. Understanding and recognizing these identities is essential, so take the time to familiarize yourself with the basic trigonometric identities like sine, cosine, tangent, and their respective reciprocals.

Graph sketching

Graph sketching is an important skill in calculus, as it provides a visual representation of a function's behavior. When a function is simplified, as in this exercise where \( y = \ln(\cos x) + \ln(\sec x) \) transforms into \( y = 0 \), the graph becomes straightforward. Here, the task was simplified by analyzing and reducing the expression to a constant function, leading to the realization that the graph is simply a horizontal line.

• The constant graph \( y = 0 \) is a line along the x-axis.
• This stretches across the entire interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

To sketch a graph effectively, simplification is key, and understanding the function's intervals, intercepts, and general shape is part of the strategic approach. Begin by asking how the expression can be simplified or interpreted, looking for identities or constants that characterize its overall outline. This strategy takes less time and results in a clean and accurate representation of the function.

Logarithmic functions

Logarithmic functions, such as the natural logarithm \( \ln(x) \), are a crucial part of calculus, allowing the transformation of multiplicative relationships into additive ones. This characteristic is harnessed effectively in this exercise through the law of logarithms that states: \[ \ln(a) + \ln(b) = \ln(ab) \], thereby simplifying the task of graphing complex expressions.

• It helps turn products into simple terms like in \( y = \ln(\cos x \cdot \sec x) \).
• This further translates into \( y = \ln(1) \), which equals 0.

Logarithms are particularly useful in calculus for solving exponential equations and finding rates of change. The simplicity of \( \ln(1) = 0 \) also underscores the elegance of logarithmic transformations, demonstrating their power to distill even complicated trigonometric expressions into easily manageable forms. Embrace logarithmic functions as tools for simplification and clarity in mathematical analysis.