Problem 46 - Calculus - AP Edition

Question

Show that \(f(x)\) is continuous but not differentiable at the indicated point. Sketch the graph of \(f\) (a) \(f(x)=\sqrt[3]{x}, x=0\) (b) \(f(x)=\sqrt[3]{(x-2)^{2}}, x=2\)

Step-by-Step Solution

Answer

Both functions are continuous but not differentiable at the given points due to vertical tangent and cusp, respectively.

Step 1: Understand Continuity and Differentiability

First, review the definitions. A function \(f(x)\) is continuous at a point \(x=a\) if the limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(f(a)\). Differentiability requires that the derivative \(f'(a)\) exists at \(x=a\).

Step 2: Analyze Continuity for Part (a)

The function given is \(f(x) = \sqrt[3]{x}\). To check for continuity at \(x=0\), verify the limit: \(\lim_{x \to 0} \sqrt[3]{x} = \sqrt[3]{0} = 0\). Since \(f(0) = 0\), \(f(x)\) is continuous at \(x=0\).

Step 3: Analyze Differentiability for Part (a)

To check differentiability, calculate the derivative: \(f'(x) = \frac{d}{dx} \sqrt[3]{x} = \frac{1}{3}x^{-rac{2}{3}}\). Evaluate \(\lim_{x \to 0} f'(x)\): as \(x \to 0\), \(f'(x) \to \infty\). The derivative does not exist at \(x=0\), so \(f(x)\) is not differentiable at this point.

Step 4: Graph Sketch for Part (a)

Sketch the graph of \(f(x) = \sqrt[3]{x}\). The curve passes through the origin \((0, 0)\) with an inflection point at \(x=0\). The graph is smooth but the slope becomes vertical as \(x\) approaches 0.

Step 5: Analyze Continuity for Part (b)

The function given is \(f(x) = \sqrt[3]{(x-2)^2}\). Verify continuity at \(x=2\) by checking the limit: \(\lim_{x \to 2} \sqrt[3]{(x-2)^2} = \sqrt[3]{0} = 0\). Since \(f(2) = 0\), \(f(x)\) is continuous at \(x=2\).

Step 6: Analyze Differentiability for Part (b)

The derivative is required to check differentiability. Differentiating \(f(x)\) gives a complex expression where the derivative is undefined at \(x=2\) due to a discontinuous slope. Specifically, the slope is sharply changing, preventing differentiability at \(x=2\).

Step 7: Graph Sketch for Part (b)

Sketch the graph of \(f(x) = \sqrt[3]{(x-2)^2}\). It resembles a 'cusp' at \(x=2\) where the slope abruptly changes direction, indicating non-differentiability.

Key Concepts

Cubic Root Function

The cubic root function, denoted as \ \( f(x) = \sqrt[3]{x} \ \), is a unique mathematical function that is continuous for all real numbers. This function takes each input \( x \) and returns its cube root. Unlike square root functions, the cubic root can accept negative values, giving them negative outputs, as the cube of a negative number is still negative.
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With respect to continuity, this means the cubic root function remains unbroken, or continuous, at every point across its domain. For instance, at \( x = 0 \), the limit and the function value both equate to zero, confirming continuity. However, differentiability tells us more about the change in direction or slope of a function's graph. While continuous, not all points may have well-defined slopes, leading to points where the function might not be differentiable.
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Navigating from mere continuity to the existence of a derivative connects us to complexities involving calculus, particularly concerning whether a smooth tangent can be drawn at a point, which in the case of cubic roots at certain values like zero, may not be possible as the slope nears infinity.

Derivative

The derivative of a function measures how it changes, or its rate of change, at any given point. For the cubic root function, \ \( f(x) = \sqrt[3]{x} \ \), the derivative exists wherever the function maintains smoothness, but not necessarily everywhere. Calculating its derivative involves a general power rule approach resulting in \ \( f'(x) = \frac{1}{3}x^{-rac{2}{3}} \ \).
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This expression indicates that the slope of the slope, or second rate of change, dramatically increases as \( x \) approaches zero. Essentially, as you walk closer to the origin, the slope shoots higher, heading towards infinity, unfurling the complexity there: the function is not differentiable at \( x=0 \ \).
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Similarly, analyzing derivatives in terms of direction changes or sharp turns translates into discontinuous slope points. For a modified function like \ \( f(x) = \sqrt[3]{(x-2)^2} \ \), presences of abrupt directional slope changes can prevent a derivative being defined at \( x = 2 \), highlighting another aspect where the function isn't differentiable.

Graph Sketching

Graph sketching is a crucial skill that visually represents how a function behaves over its domain, unveiling insights into both continuity and differentiability. With a function like \ \( f(x) = \sqrt[3]{x} \ \), its graph crosses the origin and shows a vertical slope at zero, highlighting a point where differentiability fails, yet the path remains unbroken, showcasing continuity. Points like these, known as inflection points, are where the curvature alters direction but maintains a smooth path.
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Adding complexity, \ \( f(x) = \sqrt[3]{(x-2)^2} \ \) introduces a cusp - a sharp point at \( x=2 \ \). A cusp graphically represents where the function is continuous but changes direction abruptly without forming a tangent, making the derivative undefined at that point. Such points visually and analytically illustrate locations of non-differentiability on a graph.
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Proper graph sketching involves:

• Identifying points of continuity and differentiability.
• Choosing significant values and plotting.
• Highlighting areas of steep slope or abrupt directional changes, like cusps or vertical segments.

Understanding these principles equip you better to comprehend the deeper nuances of function behavior, often more revealing than simple equations alone.