Problem 53 - Calculus an Applied Approach

Question

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=(x-3)^{2 / 3} $$

Step-by-Step Solution

Answer

The function \(y=(x-3)^{2/3}\) is differentiable for all \(x\) values except \(x=3\).

Step 1: Compute the Derivative

First, utilize the power rule for differentiation, which states \(d/dx [x^n] = n*x^{n-1}\) to calculate the derivative of \(y\). The derivative is \(y' = (2/3)*(x-3)^{-1/3}\). However, we need to be careful, as this derivative is undefined when \(x=3\).

Step 2: Analyze the Derivative

Next analyze the derivative function. A function is differentiable at a point if the derivative exists at that point and the function is continuous. At \(x = 3\), the derivative is undefined. Thus, the function **is not differentiable** at \(x = 3\).

Step 3: Distinguish the Domain of Differentiability

Finally, the function can be differentiated for all \(x\) values except at \(x = 3\) as highlighted in Step 2. This gives us the domain of differentiability.

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus. It is used to find the derivative of a function that has a variable raised to a power. The power rule states that if you have a function written as \( f(x) = x^n \), where \( n \) is a real number, the derivative \( f'(x) \) is \( n \times x^{n-1} \). This means that you bring down the exponent as a multiplier and then decrease the exponent by one.In a practical sense, the power rule allows you to quickly compute the derivative of simple terms in any polynomial function. For example, consider the term \( x^3 \). By the power rule, its derivative is \( 3x^{3-1} = 3x^2 \). This makes it easier to handle polynomial functions, as you can apply the rule to each term individually and then combine the results to find the derivative of the whole polynomial.

• Easy to apply for any power of \( x \).
• Works for both positive and negative powers, as well as fractional exponents.

Derivative

A derivative is a key concept in calculus that represents an instantaneous rate of change. This is similar to how speed measures the rate of change of distance over time. In mathematics, the derivative of a function at a given point represents the slope of the tangent line to the graph of the function at that point.Derivatives have widespread applications in various fields like physics, engineering, and economics because they provide a way to measure how one quantity changes concerning another. For a given function \( y = f(x) \), its derivative, usually denoted by \( f'(x) \), offers us insight into the function's behavior, such as finding local maxima, minima, or understanding the nature of the graph.

• Used to determine the rate of change of a quantity.
• Indicates the slope of the function at any given point.

Derivatives can also determine whether a function is increasing or decreasing at a particular point, essential for understanding critical points in graphing and optimization problems.

Continuous Function

A continuous function is one where small changes in the input result in small changes in the output. This is important in determining the differentiability of a function because a function must first be continuous to be differentiable. If a function has any gaps, jumps, or undefined points, it will not be derivable at those points.In practical terms, a continuous function means that you can draw its graph without lifting your pen from the paper. Thinking about our original problem with the function \( y=(x-3)^{2/3} \), the function is continuous everywhere except possibly at points where the base becomes zero or undefined. At \( x=3 \), the derivative is undefined as it involves division by zero, indicating a point of discontinuity which impacts differentiability.

• Continuity is needed to ensure differentiability at a point.
• Breaks or jumps in the graph indicate discontinuity and possible non-differentiability.

Understanding the behavior at and near these points helps to resolve issues of differentiability and helps in correctly formulating the domain where a function is differentiable.