Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\lim _{h \rightarrow 0} \frac{\tan (t+h)-\tan t}{h}\)

Step-by-Step Solution

Answer

The limit is \( \sec^2(t) \).

Step 1: Recognize the derivative definition

The given expression is the definition of the derivative of a function at a point. Specifically, this is the derivative of \( \tan(t) \) with respect to \( t \). Identify that \( f(t) = \tan(t) \), and you're finding \( f'(t) \) using the limit definition of a derivative.

Step 2: Apply the derivative formula for tangent

The derivative of the tangent function is known to be the square of the secant function. Specifically, \( (\tan(t))' = \sec^2(t) \). This derivative tells us how the tangent changes with respect to \( t \).

Step 3: Conclusion using known derivative

Since the form of the limit matches the derivative of \( \tan(t) \), the evaluated limit is simply the derivative \( \sec^2(t) \). Therefore, the limit \( \lim _{h \rightarrow 0} \frac{\tan (t+h)-\tan t}{h} = \sec^2(t) \).

Key Concepts

Derivative of Tangent

Understanding the derivative of the tangent function is essential in calculus, especially when you work with trigonometric expressions. The tangent function, denoted as \( \tan(t) \), is a periodic function with a specific pattern. Its derivative helps us understand how \( \tan(t) \) changes as \( t \) changes. The derivative of the tangent function is \( \sec^2(t) \). This tells us that the rate of change of the tangent function at any point \( t \) is equal to the square of the secant function evaluated at the same point.

• \( \tan(t) \): Represents a trigonometric function related to sine and cosine.
• \( \sec^2(t) \): The square of the secant, or \( \left( \frac{1}{\cos(t)} \right)^2 \).

Understanding this relationship is foundational, as it provides an analytical way to explore how tangent grows or decreases, particularly in regions where the function is increasing rapidly.

Limit Definition of Derivative

The limit definition of the derivative is a fundamental concept in calculus. It forms the backbone for finding derivatives of many basic and complex functions. In simple terms, the derivative of a function at a point gives the slope of the tangent line to the function's graph at that point. The limit definition, specifically, is written as:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}.\]In the exercise, the expression \( \lim _{h \to 0} \frac{\tan (t+h)-\tan t}{h} \) uses this definition to find the derivative of the tangent function.

• It evaluates the difference in \( \tan(t) \) as \( t \) shifts by a tiny amount \( h \).
• This small change \( h \) approaches zero, effectively measuring instantaneously how \( \tan(t) \) is changing.

This approach helps in understanding the behavior of functions at a micro level, providing insights into their instantaneous rate of change.

Secant Function Derivative

The secant function, represented as \( \sec(t) \), is the reciprocal of the cosine function, \( \sec(t) = \frac{1}{\cos(t)} \). In calculus, derivatives involving trigonometric functions like the secant are incredibly useful for modeling and solving real-world problems. The derivative of a secant function is a bit more complex than some other basic derivatives, but it plays a crucial role when dealing with derivatives of tangent functions.For the tangent function, the derivative being \( \sec^2(t) \) shows a direct relationship:

• The derivative of \( \sec(t) \) itself relates to tangent, being \( \sec(t) \cdot \tan(t) \).
• This relationship helps link tangent and secant derivatives intricately, simplifying analysis across trigonometric contexts.

Grasping these connections is key to managing and manipulating functions that involve trigonometric identities in differential calculus.