Problem 54 - Calculus and its applications

Question

Let \(g(x)=\sqrt[3]{x}\). A student graphs this function, and the graph appears to be continuous for all real numbers \(x\). The student concludes that \(g\) is differentiable for all \(x\), which is false. Identify the error, and explain why the conclusion is false. What is the correct conclusion regarding the differentiability of \(g ?\)

Step-by-Step Solution

Answer

\( g(x) \) is not differentiable at \( x = 0 \). It is differentiable for \( x \neq 0 \).

Step 1: Understanding the Problem

We have a function \( g(x) = \sqrt[3]{x} \) which is the cube root of \( x \). The student noted that the graph of this function is continuous for all real numbers \( x \), and concluded that it is differentiable everywhere as well.

Step 2: Reviewing Differentiability vs. Continuity

A function being continuous at a point means it doesn't have any breaks, jumps, or holes there. Differentiability, however, requires the function to have a well-defined tangent at that point, meaning the derivative exists. Each differentiable function is continuous, but the converse is not always true.

Step 3: Define the Derivative of the Function

The differentiability of \( g(x) = \sqrt[3]{x} \) can be checked by computing its derivative. The derivative of \( g(x) \), using the power rule, is \( g'(x) = \frac{1}{3}x^{-\frac{2}{3}} = \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x^2}} \).

Step 4: Identifying Points of Concern

Examine \( g'(x) = \frac{1}{3\sqrt[3]{x^2}} \). Notice that the denominator is \( \sqrt[3]{x^2} \). When \( x = 0 \), \( \sqrt[3]{0^2} = 0 \), making the denominator zero and thus \( g'(0) \) is undefined. Thus, \( g(x) \) is not differentiable at \( x = 0 \).

Step 5: Conclusion on Differentiability

Although \( g(x) = \sqrt[3]{x} \) is continuous for all \( x \), it is not differentiable at \( x = 0 \) due to the undefined derivative at that point. The correct conclusion is that \( g(x) \) is differentiable for all \( x eq 0 \).

Key Concepts

Continuity in Mathematical Functions

The concept of continuity in mathematical functions is fundamental in understanding various aspects of calculus. When we say a function is continuous at a point, we mean that there are no breaks, jumps, or holes in the graph of the function at that point. Essentially, you could draw the graph without lifting your pencil. In formal terms, a function \( f(x) \) is continuous at a point \( x = a \) if the following condition holds:

• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \).

This continuity means when you graph \( g(x) = \sqrt[3]{x} \), it appears smooth and unbroken over its entire domain of real numbers. However, continuity doesn't automatically mean differentiability, which leads us to the next concept.

Understanding Differentiable Functions

Differentiable functions are a subset of continuous functions where not only is the function smooth and unbroken, but it also has a defined tangent at every point within its domain. When a function is differentiable, you can calculate a tangent line slope, or derivative, at each point. The concept of differentiability is more stringent than continuity, requiring the derivative to exist.For our function \( g(x) = \sqrt[3]{x} \), the student thought it was differentiable everywhere because it was continuous. However, by computing the derivative \( g'(x) = \frac{1}{3} x^{-rac{2}{3}} \), we see it becomes problematic at \( x = 0 \). Here:

• The derivative involves the term \( x^{-rac{2}{3}} \), creating a division by zero at \( x = 0 \).
• This makes the derivative undefined at \( x = 0 \), demonstrating that \( g(x) \) is not differentiable at that point.

Thus, while a function can be continuous everywhere, like \( g(x) \) is, it may not be differentiable at certain points.

The Role of Calculus Education

Calculus education plays a critical role in understanding these nuanced mathematical concepts, like continuity and differentiability. Through calculus, students learn to distinguish between these properties. By studying calculus, students gain insight into concepts such as:

• How to formally determine the continuity of a function using limits.
• Recognizing that a function being continuous does not imply it is differentiable.
• Calculating derivatives and understanding where they fail to exist.

These insights are vital to avoid common misconceptions like assuming that a continuous graph automatically signals a function’s differentiability—illustrated by the function \( g(x) = \sqrt[3]{x} \). Understanding this distinction is crucial in higher mathematics and demonstrates why calculus education remains a cornerstone of studying more advanced mathematical topics.