Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(g(x)=\tan x \sec ^{2} x ; \quad[0, \pi / 4]\)

Step-by-Step Solution

Answer

The function \(g(x)\) ranges from 0 to \(2\sqrt{2}\) on \([0, \pi/4]\).

Step 1: Understanding the Function

The function given is composed of trigonometric terms: \(g(x) = \tan x \sec^2 x\). We need to analyze the behavior of this function in the interval \([0, \pi/4]\). Remember, \(\sec x = \frac{1}{\cos x}\), hence \(\sec^2 x = \frac{1}{\cos^2 x}\).

Step 2: Simplifying the Expression

Break down the function: \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec^2 x = \frac{1}{\cos^2 x}\). Therefore, \(g(x) = \tan x \cdot \sec^2 x = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos^2 x} = \frac{\sin x}{\cos^3 x}\).

Step 3: Finding the Interval Values

Evaluate \(g(x)\) at the boundaries of the interval: 1. At \(x = 0\), \(g(0) = \frac{0}{1^3} = 0\).2. At \(x = \frac{\pi}{4}\), \(\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), so \(g\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}/2}{(\sqrt{2}/2)^3} = 2\sqrt{2}\).

Step 4: Analyzing the Behavior within the Interval

The function \(g(x) = \frac{\sin x}{\cos^3 x}\) increases over the interval \([0, \pi/4]\), since \(\sin x\) slowly increases and \(\cos x\) slowly decreases. Hence, \(g(x)\) starts at 0 and ends at \(2\sqrt{2}\).

Step 5: Conclusion

In conclusion, the function \(g(x)\) on the interval \([0, \pi/4]\) changes from 0 to \(2\sqrt{2}\). This is useful information if we need to analyze any integrals or further properties of \(g(x)\) in this domain.

Key Concepts

Function Analysis

The core of analyzing a function, especially when dealing with trigonometric functions, is to break it down into its individual components. Here, the function in question is \(g(x) = \tan x \sec^2 x\). This function comprises two distinct trigonometric entities. Understanding these can make analyzing their combination much simpler.

• Firstly, the tangent function, denoted as \(\tan x\), is defined as \(\frac{\sin x}{\cos x}\). This relationship tells us how \(\tan x\) behaves depending on the sine and cosine values for any given angle \(x\).
• Next, the secant squared function, noted as \(\sec^2 x\), is equivalent to \(\frac{1}{\cos^2 x}\). Secant itself, \(\sec x\), is the reciprocal of cosine.

By examining these elemental components, we can simplify the function to \(g(x) = \frac{\sin x}{\cos^3 x}\). This expression helps not only to visualize the function's behavior in a given interval but also indicates how both sine and cosine values change the result dynamically.

Trigonometric Identities

Trigonometric identities are powerful tools in simplifying and understanding trigonometric functions. In our function \(g(x) = \tan x \sec^2 x\), these identities help deconstruct and reorganize the expression for deeper analysis.

• One key identity applied here is \(\tan x = \frac{\sin x}{\cos x}\). This identity transforms the tangent function into a ratio of sine and cosine, revealing its nature and potential zero points.
• Another important identity involves the secant function: \(\sec x = \frac{1}{\cos x}\). Consequently, \(\sec^2 x = \frac{1}{\cos^2 x}\), which not only shows secant's relationship to cosine but also highlights the secant's positive increase as cosine values decrease.

Utilizing these identities, we can rewrite and simplify complex functions into manageable expressions. In the case of \(g(x)\), understanding these identities allowed us to frame it as \(\frac{\sin x}{\cos^3 x}\), facilitating the analysis within a specified interval.

Interval Evaluation

Evaluating a function within an interval involves examining how the function behaves at the boundaries and within that span of values. The focus for \(g(x) = \frac{\sin x}{\cos^3 x}\) is within the interval \([0, \pi/4]\). Here's how you perform such an evaluation:

• Check the function at the boundary points first. At \(x = 0\), we find that \(g(0) = \frac{0}{1^3} = 0\). This reveals that at the start of the interval, the function value is 0.
• At the other boundary, \(x = \pi/4\), both sine and cosine equal \(\frac{\sqrt{2}}{2}\). When substituting this into the function, \(g\left(\frac{\pi}{4}\right)\) computes to \(2\sqrt{2}\).
• To further understand the function within the interval, note its increasing nature: as sine increases and cosine decreases, the overall value of \(g(x)\) escalates.

Understanding this evaluation is crucial for predicting further behaviors of the function such as integration, solving equations, or real-life application scenarios.