Problem 59 - Calculus: Early Transcendentals

Question

Determine whether \( f'(0) \) exists. \( f(x) = \left\\{ \begin{array}{ll} x \sin \frac{1}{x} & \mbox{if \) x \neq 0 \(}\\\ 0 & \mbox{if \) x = 0 \(} \end{array} \right.\)

Step-by-Step Solution

Answer

The derivative \( f'(0) \) does not exist.

Step 1: Understand the Piecewise Function

The function \( f(x) \) is defined differently for \( x eq 0 \) and \( x = 0 \). For \( x eq 0 \), \( f(x) = x \sin\frac{1}{x} \). For \( x = 0 \), \( f(x) = 0 \). Our goal is to determine if the derivative \( f'(0) \) exists.

Step 2: Recall the Definition of the Derivative

The derivative \( f'(0) \) is given by the limit definition: \[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}. \] We will calculate this limit to determine if \( f'(0) \) exists.

Step 3: Substitute Values into the Limit

Since \( f(0) = 0 \), the limit becomes \[ f'(0) = \lim_{h \to 0} \frac{f(h)}{h}. \] For \( h eq 0 \), \( f(h) = h \sin\frac{1}{h} \), so the limit becomes \[ \lim_{h \to 0} \sin\frac{1}{h}. \]

Step 4: Evaluate the Limit

Since \( \sin \) fluctuates between \(-1\) and \(1\), \( \sin\frac{1}{h} \) also fluctuates between \(-1\) and \(1\) as \( h \to 0 \). Consequently, the limit \[ \lim_{h \to 0} \sin\frac{1}{h} \] does not exist because the function \( \sin\frac{1}{h} \) does not approach a single value.

Step 5: Conclusion on the Existence of f'(0)

Since the limit \( \lim_{h \to 0} \sin\frac{1}{h} \) did not approach a single value, the derivative \( f'(0) \) does not exist.

Key Concepts

Understanding Piecewise Functions

Piecewise functions are mathematical expressions defined by different rules or formulae over different intervals of the domain.
In simpler terms, instead of one continuous rule that applies everywhere, a function can have multiple expressions depending on the input value.
This is helpful in describing situations where a single formula is not sufficient to represent the entire behavior of a function. In the context of our exercise, the function \( f(x) \) is piecewise because it is defined by different expressions for \( x eq 0 \) and \( x = 0 \):

• For \( x eq 0 \), the rule is \( f(x) = x \sin\left(\frac{1}{x}\right) \). This part shows how the function behaves for all non-zero values of \( x \).
• For \( x = 0 \), the value of the function is simply set to \( 0 \). This ensures the function is well-defined at that point.

This type of function can arise in practical scenarios such as physics or engineering problems where conditions change at certain threshold values.

Introduction to Derivatives

The concept of a derivative is a fundamental one in calculus, representing the rate at which a function changes as its input changes.
Essentially, it provides us with a tool to find instantaneous rates of change or slopes of tangent lines to curves at given points.For a given function \( f(x) \), its derivative, often denoted \( f'(x) \) or \( \frac{df}{dx} \), signifies how \( f \) changes with respect to \( x \).
This is crucial in understanding behaviors of functions, optimizing problems, and modelling real-world phenomena. Some key points about derivatives include:

• A derivative is the slope of the tangent line to the graph of the function at any point.
• If a function has a derivative at a particular point, it is said to be differentiable at that point.
• Not all functions are differentiable everywhere, which can lead to endpoints, sharp corners, or discontinuities.

In our exercise, the challenge was to find the derivative at \( x = 0 \) to see how the function behaves right at that point.

Limit Definition of Derivative

The limit definition of a derivative is the foundational principle for determining the derivative of a function at a particular point.
It is expressed as:\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]This equation essentially calculates the slope of the secant line through two points on the function as these points become infinitesimally close.
When the limit exists, it becomes the derivative, which is the instantaneous rate of change or the slope of the tangent line at \( x = a \).For our problem, we analyzed \( f'(0) \) using:\[ f'(0) = \lim_{h \to 0} \frac{f(h)}{h}. \]Since \( \sin \left(\frac{1}{h}\right) \) oscillates between \(-1\) and \(1\) without settling to any single value as \( h \to 0 \), the limit does not exist.
Hence, \( f'(0) \) is undefined. Understanding the limit definition helps grasp how derivatives are not simply calculated from finite differences but from an infinitely small approach.