Problem 51 - Calculus Early Transcendentals

Question

\(51-52\) Determine whether \(\mathrm{f}^{\prime}(0)\) exists. \(f(x)=\left\\{\begin{array}{ll}{x \sin \frac{1}{x}} & {\text { if } x \neq 0} \\\ {0} & {\text { if } x=0}\end{array}\right.\)

Step-by-Step Solution

Answer

\( f'(0) \) does not exist because the limit \( \lim_{x \to 0} \sin \frac{1}{x} \) does not exist.

Step 1: Understand the Function

The function is defined piecewise: \( f(x) = x \sin \frac{1}{x} \) for \( x eq 0 \), and \( f(x) = 0 \) when \( x = 0 \). Our goal is to find if \( f'(0) \) exists.

Step 2: Define the Derivative at Zero

For \( f'(0) \) to exist, the limit \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} \) must exist. Here, \( f(0) = 0 \). Thus, the limit simplifies to \( \lim_{x \to 0} \frac{x \sin \frac{1}{x} - 0}{x} \), which is \( \lim_{x \to 0} \sin \frac{1}{x} \).

Step 3: Analyze the Sine Function

The function \( \sin \frac{1}{x} \) oscillates between -1 and 1 as \( x \to 0 \). Hence, \( \sin \frac{1}{x} \) does not approach a single value and does not have a limit as \( x \) approaches 0.

Step 4: Conclusion on the Derivative

Since \( \lim_{x \to 0} \sin \frac{1}{x} \) does not exist, the limit \( \lim_{x \to 0} \frac{x \sin \frac{1}{x}}{x} \) does not exist. Therefore, \( f'(0) \) does not exist.

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions that have different rules for different ranges of their input values. These are particularly useful in modeling scenarios where a function behaves differently based on certain conditions.
Consider a function defined in two parts: like in our example, it operates as one function when \( x eq 0 \) and as another when \( x = 0 \).

• Allows for accurately representing real-world situations.
• Can handle functions that have discontinuities or abrupt changes.
• In our example, we have a divide at \( x = 0 \), making it crucial to understand each piece separately.

Understanding how to work with piecewise functions is fundamental in calculus, especially when determining limits and derivatives.

Derivatives

Derivatives provide a mathematical way to express how a function changes at any given point. Simply put, it is the rate of change or the slope of the function at a particular point.
In our context, we're interested in finding \( f'(0) \), the derivative of \( f(x) \) at \( x = 0 \). This requires understanding the specific behavior of the function near this point.

• The derivative at a point is defined as \( \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \).
• Key to finding the derivative, especially for piecewise functions, is checking if this limit converges to a single value.
• Understanding whether this limit exists or not helps determine the differentiability at that point.

Derivatives are central in calculus because they reveal dynamics hidden within functions.

Limits

In calculus, the concept of limits helps us understand the behavior of functions as they approach certain points, whether finite or infinite.
Here, the limit \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} \) is essential to determine if the derivative \( f'(0) \) exists.

• Limits describe what a function approaches as the input gets very close to some value.
• For derivatives, it involves analyzing the difference between the function value and the function value at a chosen point relative to \( x \).
• If the limit results in a particular number, the function is continuous and hence differentiable at that point.

Grasping limits allows you to explore where behaviors like continuities, discontinuities, and asymptotes occur.

Sine Function

The sine function, \( \sin(x) \), is a periodic function that oscillates between -1 and 1. It plays a fundamental role in trigonometry, especially in describing oscillatory behaviors.
In our problem, it's critical because \( f(x) = x \sin \frac{1}{x} \) involves the sine function with a reciprocal input, \( \frac{1}{x} \).

• When \( x \) gets close to 0, \( \frac{1}{x} \) grows large, causing \( \sin \frac{1}{x} \) to oscillate rapidly.
• Understanding this oscillation helps in analyzing: when the input changes just slightly, how does the output react?
• For \( \sin \) when involving certain transformations, predicting exact behaviors can be tricky.

The sine function's oscillatory nature is key in studying periodic phenomenon.

Oscillation

Oscillation refers to the repeated back-and-forth movement or fluctuation of a function's output within certain bounds. This is a vital concept when analyzing trigonometric functions like sine.
Our function uses an oscillating part \( \sin \frac{1}{x} \), which doesn't settle as \( x \to 0 \).

• Indicates that a function is not heading towards a particular value.
• For derivatives, sustained oscillation around a point means no single slope captures the function's behavior there.
• Here, oscillation prevents the limit for the derivative at 0 from existing, meaning \( f'(0) \) is undefined.

Recognizing oscillation in functions can enlighten discussions around behavior over infinite, minute changes.