Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(x=\pi / 3\) in the identity in Example 7 and define the functions \(f\) and \(g\) as follows. \(f(h)=\frac{\sin [(\pi / 3)+h]-\sin (\pi / 3)}{h}\) \(g(h)=\cos \frac{\pi}{3}\left(\frac{\sin h}{h}\right)-\sin \frac{\pi}{3}\left(\frac{1-\cos h}{h}\right)\) (a) What are the domains of the functions \(f\) and \(g ?\) (b) Use a graphing utility to complete the table. (c) Use the graphing utility to graph the functions \(f\) and \(g\). (d) Use the table and graph to make a conjecture about the values of the functions \(f\) and \(g\) as \(h \rightarrow 0^{+}\).

Step-by-Step Solution

Answer

The domains of the functions \(f\) and \(g\) are \((- \infty, 0) \cup (0, +\infty)\). The conjecture about the limit of the functions as \(h \rightarrow 0^{+}\) should result from the analysis of the graph and the table of values, which should be reasoned based on the definitions of \(f\), \(g\) and their behavior around point \(h = 0\).

Step 1: Determine the Domains of \(f\) and \(g\)

The domain of a function is the set of all permissible input values (in this case, \(h\)). For both functions \(f(h)\) and \(g(h)\), these will be all real numbers except for \(h = 0\), because division by zero is undefined. Therefore, the domain for both \(f\) and \(g\) is \((- \infty, 0) \cup (0, +\infty)\).

Step 2: Apply a Graphing Utility to Draw the Functions

As this part is tool dependent and we cannot provide the visualization of the graphs here. However, it is suggested to use a graphing calculator or a graphing tool like Desmos or GeoGebra to plot the functions \(f(h)\) and \(g(h)\) using their respective definitions given in the problem statement. Then, visualize the graph and complete table for \(f(h)\) and \(g(h)\) at different values of \(h\).

Step 3: Form a Conjecture

Look at the graph and table that were constructed using the graphing tool. Observe the behavior of the functions \(f(h)\) and \(g(h)\) as \(h\) approaches 0 from the right. Do they converge to a certain value or do they diverge? If possible, try also to reason the behavior based on their definitions. For \(f\), it resembles a difference quotient for the derivative of a sin function, and for \(g\), it resembles trigonometric identities. From these observations, a conjecture about the limiting behavior of the functions as \(h \rightarrow 0^{+}\) can be formed.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational for many areas of calculus, including solving problems involving oscillations like waves and pendulums. In this particular exercise, sin and cos functions are at play. These functions help describe and analyze periodic phenomena. They are continuously differentiable and have useful properties like symmetry and periodicity.

• Sine function (\( \sin \theta \)): It’s an odd function with a period of \(2\pi\), meaning its values repeat every \(2\pi\) units. Its graph is a wave that oscillates between -1 and 1.
• Cosine function (\( \cos \theta \)): Similarly, this is an even function with the same period as the sine function. It starts at 1 when \( \theta = 0 \) and also oscillates between -1 and 1.
• Involved angles: In this problem, \( \theta \) is set to \( \frac{\pi}{3} \), which is a special angle commonly appearing in trigonometric problems with known sine and cosine values.

Trigonometric functions are crucial here to understand the limits and differentiability of functions involving such angles. This understanding will be further illustrated as we delve into derivatives.

Derivatives

The derivative is a measure of how a function changes as its input changes. It is a key concept in calculus that uses limits to find the instantaneous rate of change (or slope) of a function. In the exercise, analyzing derivatives gives insight into how the functions \(f(h)\) and \(g(h)\) behave as \(h\) approaches zero.

When dealing with trigonometric functions like sine and cosine, derivatives allow us to determine how the output of these functions changes given an infinitesimally small increase in their input. For instance, the derivative of \( \sin(x) \) is \( \cos(x) \), revealing how the wave shape changes at any given point.

• First Principle of Derivatives: This involves the limit of the difference quotient to define the derivative, specifically \( \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \).
• Use in Problem: The function \(f(h)\) in the exercise embodies a trigonometric difference quotient indicative of deriving \( \sin(x) \) at \( \frac{\pi}{3} \).

Understanding derivatives in the context of trigonometric functions is essential when examining their limiting behavior and ensuring correct interpretations of their rates of change.

Difference Quotient

The difference quotient is an important concept in calculus, used in the definition of derivatives. It helps to compute the slope of the secant line between two points on the graph of the function. For a function \(f\), the difference quotient is expressed as \( \frac{f(x+h) - f(x)}{h} \).

• Role in the Problem: In our exercise, function \(f(h)\) is defined with a difference quotient format, where it's similar to examining the derivative of \( \sin(\theta) \).
• Limiting Behavior: As \(h\) tends to zero, the difference quotient approaches the derivative of the function, offering insights into how the function behaves infinitesimally around a point.

Interpreting the difference quotient allows us to transition from average rates of change to instantaneous rates, thereby understanding the derivative. This concept provides the foundation for analyzing the function's behavior as \(h\) approaches 0.

Function Domains

When talking about the domain of a function, we're referring to all the permissible input values for the function. Domains are critical because they determine where a function is defined and can successfully operate. In this exercise, the domains of the functions \(f\) and \(g\) are discussed.

• Restrictions on Domains: Both \(f(h)\) and \(g(h)\) are undefined at \(h = 0\), since this would result in division by zero. Thus, their domains exclude zero and are noted as \((- \infty, 0) \cup (0, + \infty)\).
• Importance in Calculus: Knowing the domain is vital for understanding the function's behavior and ensuring mathematical operations are performed within permissible conditions.

Understanding function domains is crucial when dealing with limits and derivatives, as computations often approach undefined areas like division by zero. This knowledge keeps our calculations honest and reliable.