Problem 7 - Brief Applied Calculus

Question

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=x^{2 / 5}\)

Step-by-Step Solution

Answer

The limit does not exist, so the function is not differentiable at \( x = 0 \).

Step 1: Write the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x = a \) is given by \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \). To determine if \( f(x) = x^{2/5} \) is differentiable at \( x = 0 \), we need to compute \( f'(0) \).

Step 2: Set Up the Limit for the Derivative at x = 0

Substitute \( a = 0 \) into the derivative formula: \( f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{h^{2/5} - 0^{2/5}}{h} = \lim_{h \to 0} \frac{h^{2/5}}{h} \).

Step 3: Simplify the Expression

Simplify the expression \( \frac{h^{2/5}}{h} = \frac{1}{h^{3/5}} \). Now the limit becomes \( \lim_{h \to 0} \frac{1}{h^{3/5}} \).

Step 4: Evaluate the Limit

As \( h \to 0 \), \( h^{3/5} \to 0 \), making \( \frac{1}{h^{3/5}} \) approach infinity. Thus, the limit does not exist.

Key Concepts

Understanding the Derivative Definition

The derivative is a fundamental concept in calculus, representing the rate at which a function changes at any given point. It's essentially the slope of the tangent line to the curve of the function at a specific point. To find the derivative of a function at a point, we use the following formula:

\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]This formula evaluates the function's behavior as the change in the input (denoted by \( h \)) approaches zero. It helps us understand how the function is behaving in very small neighborhoods around the point \( a \). Whether the derivative exists at a point can reveal much about the function's smoothness and continuous trend at that location.

When we apply this formula, we are looking for whether this limit exists and results in a finite number. If it does, then the function is differentiable at that point. If the limit diverges or does not exist, as it happens for some functions, such as our given function at \( x=0 \), the function is not differentiable at that point.

Demystifying Limit Evaluation for Derivatives

Evaluating limits is crucial in determining differentiability. In our exercise, we replaced \( a \) with 0 in the definition of the derivative to find:

\[f'(0) = \lim_{h \to 0} \frac{h^{2/5} - 0^{2/5}}{h} = \lim_{h \to 0} \frac{h^{2/5}}{h} = \lim_{h \to 0} \frac{1}{h^{3/5}} \]Here, it simplified to \( \frac{1}{h^{3/5}} \) as the exponents were subtracted in the fraction.

To evaluate this limit, we need to consider the behavior of \( h \) as it tends to zero. The expression \( h^{3/5} \) also tends towards zero, leading the entire limit expression, \( \frac{1}{h^{3/5}} \), to head toward infinity. This divergence indicates that the limit does not exist. Without a finite limit, the derivative cannot be computed, reflecting that the function isn’t differentiable at \( x = 0 \).This methodical approach to limit evaluation showcases its role in assessing if a function is smoothly continuous and differentiable.

Exploring Non-Differentiable Functions

Differentiability signifies a function's smoothness at a point, but not all functions are differentiable everywhere. Some functions might appear continuous but can have points where derivatives do not exist.

A classic example is the function \( f(x) = x^{2/5} \) at \( x = 0 \). It is continuous at that point, meaning there's no hole or jump, but it's not differentiable there. Why?

• The graph of \( f(x) \) has a sharp corner at \( x = 0 \), disrupting a smooth transition. Differentiability requires that the curve at that point aligns smoothly with a single tangent line, which isn't possible at a cusp or kink.
• The behavior of \( \frac{1}{h^{3/5}} \) as \( h \to 0 \) shows that the 'rate of change' heads to infinity, signs of a point where the function's slope isn't defined well.

Even with a function being continuous (no breaks in the curve), a sharp turn or a vertical tangent can prevent differentiability.
Understanding these key differences between continuity and differentiability helps in grasping why certain functions are not differentiable at specific points.