Problem 81 - Calculus Single Variable

Question

A function \(f\) and a point \(c\) are given. Graph the function $$ \phi(x)=\frac{f(x)-f(c)}{x-c} $$ in an appropriate viewing window centered about the line \(x=c .\) Use the graph of \(\phi\) to decide whether or not \(f^{\prime}(c)\) exists. Explain the reason for your answer, and, if your answer is that \(f^{\prime}(c)\) exists, use the graph of \(\phi\) to approximate the value of \(f^{\prime}(c)\). $$ f(x)=\sqrt[3]{\cos (x)}, \quad \mathrm{c}=\pi / 2 $$

Step-by-Step Solution

Answer

Graph \(\phi(x)\) around \(x = \pi/2\). If the values approach the same limit, then \(f'(\pi/2)\) exists, approximated by that limit.

Step 1: Identify the Function and the Point

We have the function \(f(x) = \sqrt[3]{\cos(x)}\) and the point \(c = \pi/2\). Our goal is to evaluate the behavior of the function around the point \(x = \pi/2\) to determine if the derivative \(f'(\pi/2)\) exists.

Step 2: Set up the Difference Quotient

The function \(\phi(x)\) is defined as \(\phi(x) = \frac{f(x) - f(c)}{x-c}\). Substitute \(f(x) = \sqrt[3]{\cos(x)}\) and \(c = \pi/2\) to write \(\phi(x) = \frac{\sqrt[3]{\cos(x)} - \sqrt[3]{\cos(\pi/2)}}{x - \pi/2}\). Since \(\cos(\pi/2) = 0\), \(f(c) = 0\) and the function simplifies to \(\phi(x) = \frac{\sqrt[3]{\cos(x)}}{x - \pi/2}\).

Step 3: Determine Indeterminate Form at \(c\)

At \(x = \pi/2\), \(\phi(x)\) becomes an indeterminate form \(\frac{0}{0}\). To understand its behavior, we will analyze its graph around this point to determine the existence of the limit.

Step 4: Graph the Function \(\phi(x)\)

Graph \(\phi(x)\) with \(x\) values slightly less and more than \(\pi/2\) to observe the behavior. If the graph approaches the same value from both sides, the derivative exists at that value.

Step 5: Analyze the Graph

Observing the graph, check if \(\phi(x)\) approaches a common value from both sides. If it does, that is the limit \(f'(\pi/2)\). If \(\phi(x)\) diverges or does not stabilize, then \(f'(\pi/2)\) does not exist.

Step 6: Conclusion and Approximation

Based on the behavior of \(\phi(x)\) in the graph around \(\pi/2\), we decide if \(f'(\pi/2)\) exists. If it exists, approximate the value based on where \(\phi(x)\) stabilizes near \(x = \pi/2\).

Key Concepts

Difference Quotient

The difference quotient is a powerful tool used in calculus to explore the rate of change of a function as it approaches a specific point. It represents the average rate of change of the function over a small interval. Mathematically, the difference quotient for a function \(f(x)\) at a point \(c\) is expressed as:

• \( \phi(x) = \frac{f(x) - f(c)}{x - c} \)

This expression helps in approximating the derivative, provided the limit exists as \(x\) approaches \(c\).
For the function \(f(x) = \sqrt[3]{\cos(x)}\), and \(c = \pi/2\), the difference quotient becomes \( \phi(x) = \frac{\sqrt[3]{\cos(x)}}{x - \pi/2} \). This setup is necessary for understanding how \(f(x)\) behaves around the point \(c\).
Using the difference quotient is the first step in determining if a function is differentiable at a particular point.

Indeterminate Form

An indeterminate form is a mathematical expression that does not have a well-defined value without further analysis or modification. Common examples include \(\frac{0}{0}\). When exploring derivatives, such forms often appear when evaluating a difference quotient directly at the point \(c\).
For instance, substituting \(x = \pi/2\) into \( \phi(x) = \frac{\sqrt[3]{\cos(x)}}{x - \pi/2} \) results in an indeterminate form of \(\frac{0}{0}\). This indicates that further steps, such as graphical analysis or algebraic manipulation, are required to understand the behavior of the function at that point.
Resolving an indeterminate form often involves calculating limits or using L'Hôpital's Rule if applicable, but here we rely on graphical insights due to the complexity of the cube root and cosine function involved.

Graphical Analysis

Graphical analysis provides a visual approach to understanding the behavior of a function around critical points. By graphing \( \phi(x) = \frac{\sqrt[3]{\cos(x)}}{x - \pi/2} \), one can investigate how the function behaves as \(x\) approaches \(\pi/2\). When graphing, we look for how \( \phi(x) \) behaves:

• Does it approach a specific value from both sides of \(\pi/2\)?
• Does it diverge or oscillate?

If the function \( \phi(x) \) approaches the same value from both directions, it suggests the existence of the derivative \(f'(c)\). This graphical check is crucial because it visualizes information that algebra might not always clarify, particularly in complex functions like \( \sqrt[3]{\cos(x)} \).
By examining the graph, one can approximate the derivative, further solidifying the conceptual understanding of function behavior at that point.

Function Behavior

Understanding the behavior of a function involves considering how it changes and stabilizes around particular points. This is especially relevant when determining the existence of derivatives. For the function \(f(x) = \sqrt[3]{\cos(x)}\) and the point \(c = \pi/2\), we are particularly interested in how \( \phi(x) = \frac{\sqrt[3]{\cos(x)}}{x - \pi/2} \) behaves as \(x\) nears \(\pi/2\).
To assess whether a derivative exists, we observe if the difference quotient \( \phi(x) \) stabilizes around a certain value. If the graph shows that \( \phi(x) \) approaches a constant value from both directions, then the function's behavior suggests that the derivative exists, and we can estimate \(f'(\pi/2)\) by what \( \phi(x) \) converges to.
In conclusion, examining function behavior is not only about finding specific numbers but understanding how changes in the input affect output consistently at critical points like \(c = \pi/2\).