Problem 8 - Brief Applied Calculus

Question

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=x^{4 / 5}\)

Step-by-Step Solution

Answer

The function is not differentiable at \( x = 0 \) because the derivative at this point does not exist.

Step 1: Recall the Definition of the Derivative

To determine the differentiability of a function at a point, we need to compute the derivative using the limit definition. For a function \( f(x) \), the derivative at \( x = c \) is given by: \[f'(c) = \lim_{{h \to 0}} \frac{{f(c+h) - f(c)}}{h}\]For this problem, we want to check whether \( f(x) = x^{4/5} \) is differentiable at \( x = 0 \).

Step 2: Substitute in the Derivative Definition

Substituting the given function and the point \( x = 0 \) into the definition, we get: \[\lim_{{h \to 0}} \frac{{(0+h)^{4/5} - 0^{4/5}}}{h} = \lim_{{h \to 0}} \frac{{h^{4/5}}}{h} = \lim_{{h \to 0}} h^{-1/5}\]

Step 3: Evaluate the Limit

Next, we need to evaluate the limit \( \lim_{{h \to 0}} h^{-1/5} \). As \( h \) approaches zero, \( h^{-1/5} \) approaches infinity. This indicates that the limit does not exist because it becomes unbounded as \( h \) gets closer to zero.

Step 4: Conclude the Differentiability at the Point

Since the limit \( \lim_{{h \to 0}} h^{-1/5} \) does not exist, the derivative \( f'(0) \) also does not exist. Therefore, the function \( f(x) = x^{4/5} \) is not differentiable at \( x = 0 \).

Key Concepts

Function Limits

When we talk about function limits, we are considering how a function behaves as the input approaches a particular point. In simpler words, it tells us what value the function is getting close to, as the input edge nearer to some value.
A limit is denoted by \( \lim_{{x \to c}} f(x) \), it represents the value that \( f(x) \) approaches as \( x \) gets arbitrarily close to \( c \).To better understand, imagine you're walking towards a finish line. The closer you get to the finish line, the more predictable it becomes where you will end up.

• Limits help us predict this finish without actually reaching it just yet.
• They deal with the trend, not the value at the point itself.

In the given exercise, we are interested in the limit of the derivative definition. Specifically, we want to know what happens to the function \( h^{-1/5} \) as \( h \) goes close to zero. Unfortunately, this limit doesn't settle on a finite number, indicating a point where the function misbehaves, making it critical in determining differentiability.

Derivative Definition

The derivative is a fundamental concept in calculus. It represents the rate at which a function is changing at any given point. Think of it as measuring how fast or slow something happens.
Using the definition, derivatives help us see how the value of a function changes as its input changes, symbolized as \( f'(x) \), the derivative of \( f \).Mathematically, the derivative at a point \( x = c \) can be found using the limit definition:

• \( f'(c) = \lim_{{h \to 0}} \frac{{f(c+h) - f(c)}}{h} \)

This formula tells us to look at the average rate of change over small intervals approaching zero. In our exercise, this involved substituting the function and checking the limit for differentiability.

• If the limit results in a real number, the function is differentiable at that point.
• Conversely, if the limit doesn't exist or isn't finite, the function is not differentiable there.

The concept is powerfully predictive but hinges heavily on the limit's existence.

Non-Differentiable Functions

Not all functions have nice, neat derivatives at every point. Some functions have sharp corners or vertical tangents where derivatives do not exist. These points of trouble indicate non-differentiability.
Knowing whether a function is non-differentiable at a point tells us about the function's geometry:

• If the limit for the derivative doesn't yield a real number, expect non-differentiability.
• Sharp edges or abrupt changes in direction lead to such behavior.

In our example problem, \( f(x) = x^{4/5} \) was tested for differentiability at \( x = 0 \). As the limit approached infinity, it confirmed a hitch in the smoothness of the function at that point.Non-differentiable points need cautious handling because they mark where the algebraic niceties we've come to expect break down, often telling valuable stories about the nature of the functions involved. These quirks in functions can have significant implications in fields like optimization and physics, where exact slopes and smooth curves are vital.