Describe The X-values At Which The Function Is Differentiable

Describing the x-values at which a Function is Differentiable

Differentiability, a cornerstone concept in calculus, signifies a function's smoothness at a specific point. A function is differentiable at a point if it has a derivative at that point, essentially meaning it has a well-defined tangent line. Understanding where a function is differentiable requires examining its properties and identifying potential points of non-differentiability. This article will delve into the criteria for differentiability, explore common scenarios where functions fail to be differentiable, and provide a systematic approach to determining the x-values at which a given function is differentiable.

Understanding Differentiability

Before we delve into identifying points of differentiability, let's solidify our understanding of the concept. A function f(x) is differentiable at a point x = a if the limit

lim (h→0) [(f(a + h) - f(a))/h]

exists. This limit represents the slope of the tangent line to the graph of f(x) at x = a, and it's also known as the derivative of f(x) at x = a, denoted as f'(a). Geometrically, this means the function has a well-defined tangent line at x = a, without any sharp corners or cusps.

Key Requirements for Differentiability:

• Continuity: A crucial prerequisite for differentiability is continuity. If a function is not continuous at a point, it cannot be differentiable at that point. A discontinuity creates a "break" in the graph, preventing the existence of a well-defined tangent line.

• Smoothness: Even if a function is continuous, it may still not be differentiable. The function must be "smooth" at the point in question. This means there should be no sharp corners, cusps, or vertical tangents. These features indicate an abrupt change in the slope, making the derivative undefined.

Common Scenarios of Non-Differentiability

Several situations can lead to a function being non-differentiable at certain points. Let's examine some common scenarios:

1. Discontinuities

As mentioned earlier, a discontinuity is a guaranteed point of non-differentiability. There are three main types of discontinuities:

• Removable Discontinuity: This occurs when the function has a "hole" at a point, but the limit of the function as x approaches that point exists. While the function value might be undefined at the point, the limit exists, and a single point could "fix" the discontinuity. However, even if you could redefine the function to fill the hole, the original function isn't differentiable there, due to the limit definition requiring existence of the function at the point.

• Jump Discontinuity: In a jump discontinuity, the function "jumps" from one value to another at a specific point. The limit from the left and the limit from the right differ, resulting in a clear break in the graph, making a tangent line impossible.

• Infinite Discontinuity (Vertical Asymptote): If a function has a vertical asymptote at a point, it's non-differentiable there. The function approaches positive or negative infinity as x approaches the point, preventing the existence of a finite derivative.

2. Sharp Corners (Cusps)

A sharp corner or cusp occurs when the left-hand derivative and the right-hand derivative at a point exist but are not equal. This implies that the slope of the tangent line approaches different values from the left and the right, leading to an undefined derivative. The absolute value function, |x|, is a classic example, exhibiting a cusp at x = 0.

3. Vertical Tangents

A vertical tangent occurs when the slope of the tangent line approaches infinity or negative infinity at a point. In such cases, the derivative is undefined, making the function non-differentiable at that point. The function f(x) = x^(1/3) has a vertical tangent at x = 0.

4. Non-smooth Points

Sometimes, a function might appear continuous but still lack differentiability at specific points. This often arises from functions involving piecewise definitions or absolute values, where the transition between pieces might not be smooth. Consider a function defined piecewise. If at the boundary points the derivative from the left and right do not match, the function is non-differentiable at those boundary points.

Identifying x-values of Differentiability: A Systematic Approach

Determining the x-values where a function is differentiable involves a step-by-step process:

1. Analyze the Function's Definition: Begin by carefully examining the function's definition. Identify any potential points of discontinuity, sharp corners, vertical tangents, or areas where the function might be defined piecewise.

2. Check for Continuity: Ensure the function is continuous at all points. Any discontinuity automatically renders the function non-differentiable at that point.

3. Compute the Derivative: If the function appears continuous, find its derivative using standard differentiation rules. This derivative represents the slope of the tangent line at each point where it exists.

4. Examine the Derivative for Undefined Points: The derivative might be undefined at certain points, even if the function is continuous. These points of undefined derivatives represent potential points of non-differentiability. Investigate these points further.

5. Evaluate One-Sided Derivatives: If the derivative is undefined at a point, calculate the one-sided derivatives (left-hand and right-hand derivatives) to check for sharp corners or cusps. If they are unequal, the function is non-differentiable at that point.

6. Consider Vertical Tangents: If the derivative approaches infinity or negative infinity at a point, the function has a vertical tangent and is non-differentiable at that point.

7. Analyze Piecewise Functions: Pay careful attention to the transition points of piecewise-defined functions. Check for continuity and compare left-hand and right-hand derivatives at these boundaries.

8. Confirm Differentiability: Based on the analysis above, you can definitively identify the x-values where the function is differentiable – these will be the points where the derivative exists and is finite, and the function is continuous.

Examples and Illustrations

Let's illustrate this systematic approach with some examples:

Example 1: f(x) = |x|

1. Definition: The absolute value function.

2. Continuity: Continuous everywhere.

3. Derivative: f'(x) = 1 for x > 0, f'(x) = -1 for x < 0, undefined at x = 0.

4. Undefined Derivative: The derivative is undefined at x = 0.

5. One-sided Derivatives: The left-hand derivative is -1, and the right-hand derivative is 1. They are unequal, indicating a cusp at x = 0.

6. Conclusion: f(x) = |x| is differentiable for all x ≠ 0.

Example 2: f(x) = x^(1/3)

1. Definition: Cube root function.

2. Continuity: Continuous everywhere.

3. Derivative: f'(x) = (1/3)x^(-2/3)

4. Undefined Derivative: The derivative is undefined at x = 0.

5. Limit of Derivative: As x approaches 0, the derivative approaches infinity, indicating a vertical tangent.

6. Conclusion: f(x) = x^(1/3) is differentiable for all x ≠ 0.

Example 3: Piecewise Function

Let's consider a piecewise function:

f(x) = x^2,  x ≤ 1
f(x) = 2x - 1, x > 1

1. Continuity: Check for continuity at x = 1. The limit from the left is 1, and the limit from the right is 1, and f(1) = 1. Therefore, continuous at x=1.

2. Derivative: f'(x) = 2x for x ≤ 1 and f'(x) = 2 for x > 1.

3. One-sided derivatives at x = 1: The left-hand derivative is 2(1) = 2 and the right-hand derivative is 2. They are equal.

4. Conclusion: This function is differentiable everywhere, including at x = 1.

Conclusion

Determining the x-values at which a function is differentiable is crucial for various applications in calculus and beyond. By systematically examining the function's continuity, derivative, and handling special cases such as cusps and vertical tangents, we can accurately identify the intervals where the function is differentiable. Remember that continuity is a necessary but not sufficient condition for differentiability. The function must also be smooth, devoid of sharp corners or vertical tangents, to possess a well-defined derivative at a given point. Through a careful understanding of these concepts and a systematic approach, one can effectively analyze the differentiability of diverse functions.