91Ó°ÊÓ

Continuity is a foundational concept in calculus and plays a significant role when analyzing functions, particularly when discussing zeros within an interval. A function is said to be continuous on an interval if, intuitively, it can be drawn without lifting the pen from the paper across that range. For the function \( r(\theta) = \tan \theta - \cot \theta - \theta \) on the interval \((0, \pi/2)\), continuity is assured since both the tangent and cotangent functions, as well as the linear function \(\theta\), are continuous in this domain.

• \(\tan \theta\) and \(\cot \theta\) are defined and continuous between \(0\) and \(\pi/2\), excluding the endpoints.
• The combination of continuous functions results in \(r(\theta)\) being continuous.
• This continuity is crucial for applying the Intermediate Value Theorem (IVT) later.

Being able to assert that a function is continuous over an interval allows us to make powerful conclusions about its behavior and access useful theorems in calculus, such as the IVT.

Analyzing the derivative of a function gives insights into its behavior, such as whether it is increasing or decreasing over a particular interval. For \( r(\theta) = \tan \theta - \cot \theta - \theta \), the derivative is computed as \[ r'(\theta) = \sec^2 \theta + \csc^2 \theta - 1. \]This derivative helps in understanding the function's monotonicity.

• Both \(\sec^2 \theta\) and \(\csc^2 \theta\) are positive for \(\theta\) in \((0, \pi/2)\).
• Thus, \( r'(\theta) > 0 \) over the interval, showing that the function is strictly increasing.

When a function's derivative is positive like this over interval \((0, \pi/2)\), it implies that the function is moving upwards, with no local minima or maxima, confirming that it cannot loop back and have more than one zero in the interval.

Understanding a function's behavior as it approaches the boundaries of its domain is essential for analyzing its properties over an interval. For \( r(\theta) = \tan \theta - \cot \theta - \theta \), examining its limits as \(\theta\) approaches 0 and \(\pi/2\) provides crucial insights.

• As \( \theta \to 0^+ \), \( \tan \theta \to 0 \), \( \cot \theta \to \infty \), thus \( r(\theta) \to -\infty \).
• As \( \theta \to \pi/2^- \), \( \tan \theta \to \infty \), \( \cot \theta \to 0 \), leading \( r(\theta) \to \infty \).

These boundary behaviors are critical because they set up the conditions for using the Intermediate Value Theorem. Seeing \( r(\theta) \) transition from \(-\infty\) to \(\infty\) indicates that at some point in \((0, \pi/2)\), the function must cross the x-axis, meaning there is at least one zero in the interval. This information complements our derivative analysis by confirming that if there’s one zero, it’s unique due to the function's strict increase.