Derivatives Of F(x) = |x-3| Left, Right, And Differentiability

#introduction Absolute value functions are a fascinating topic in calculus, often presenting unique challenges when it comes to differentiation. These functions, characterized by their sharp corners, provide excellent examples for understanding the concept of derivatives from both the left and right, and the implications for differentiability. In this article, we will delve deep into the function f(x) = |x-3|, exploring its derivatives at the critical point x = 3. We will calculate the left-hand and right-hand derivatives, discuss their significance, and ultimately determine whether the function is differentiable at this point. This exploration is crucial for anyone studying calculus, as it highlights the importance of limits and the conditions necessary for a function to be differentiable. Understanding the behavior of absolute value functions near their vertices is a fundamental skill, applicable in various mathematical and real-world contexts.

Understanding the Function f(x) = |x-3|

Before we dive into the derivatives, let's first ensure we have a solid grasp of the function f(x) = |x-3|. The absolute value function, denoted by |x|, essentially returns the magnitude of a number, regardless of its sign. This means that |x| is equal to x if x is positive or zero, and -x if x is negative. The function f(x) = |x-3| is a transformation of the basic absolute value function, shifted 3 units to the right along the x-axis. This shift is crucial because it places the vertex, or the point where the function changes direction, at x = 3. To fully understand this function, it's helpful to express it as a piecewise function:

f(x) = x - 3, if x ≥ 3 f(x) = -(x - 3), if x < 3

This piecewise definition clearly shows the two different linear segments that make up the absolute value function. To the right of x = 3, the function behaves like a line with a slope of 1, while to the left, it acts like a line with a slope of -1. This change in slope at x = 3 is what creates the sharp corner and the potential issue with differentiability. Understanding this piecewise nature is essential for correctly calculating the derivatives from the left and right. The graph of f(x) = |x-3| further illustrates this point, with a distinct V-shape, the tip of which lies at the point (3, 0). This visual representation is a powerful tool for grasping the function's behavior and predicting its differentiability.

Calculating the Derivative from the Left at x = 3

The derivative of a function at a point represents the instantaneous rate of change of the function at that point. To find the derivative from the left at x = 3, we need to consider the limit of the difference quotient as x approaches 3 from the left side. This is often denoted as f'(3-). Mathematically, the left-hand derivative is defined as:

f'(3-) = lim (h→0-) [f(3 + h) - f(3)] / h

where h approaches 0 from the negative side (i.e., h is negative). Since we are approaching from the left, we use the piecewise definition of f(x) that applies when x < 3: f(x) = -(x - 3). Substituting this into the limit definition, we get:

f'(3-) = lim (h→0-) [-(3 + h - 3) - (-(3 - 3))] / h f'(3-) = lim (h→0-) [-h] / h f'(3-) = lim (h→0-) -1

Thus, the derivative from the left at x = 3 is -1. This means that as we approach x = 3 from the left, the function is decreasing at a constant rate of 1 unit for every 1 unit increase in x. The negative sign indicates the decreasing nature of the function on this interval. This result is consistent with our understanding of the function's piecewise nature, where the left side is a line with a slope of -1. The limit definition rigorously confirms this intuition, providing a precise value for the instantaneous rate of change.

Calculating the Derivative from the Right at x = 3

Now, let's calculate the derivative from the right at x = 3, denoted as f'(3+). This involves finding the limit of the difference quotient as x approaches 3 from the right side. The right-hand derivative is defined as:

f'(3+) = lim (h→0+) [f(3 + h) - f(3)] / h

where h approaches 0 from the positive side (i.e., h is positive). When approaching from the right, we use the piecewise definition of f(x) that applies when x ≥ 3: f(x) = x - 3. Substituting this into the limit definition, we get:

f'(3+) = lim (h→0+) [(3 + h - 3) - (3 - 3)] / h f'(3+) = lim (h→0+) [h] / h f'(3+) = lim (h→0+) 1

Therefore, the derivative from the right at x = 3 is 1. This indicates that as we approach x = 3 from the right, the function is increasing at a constant rate of 1 unit for every 1 unit increase in x. The positive sign here reflects the increasing nature of the function on the right side of x = 3. Similar to the left-hand derivative, this result aligns perfectly with the piecewise nature of the function, where the right side is a line with a slope of 1. The limit calculation provides a rigorous confirmation of this behavior.

Differentiability at x = 3: A Critical Analysis

A fundamental concept in calculus is that for a function to be differentiable at a point, the derivative from the left and the derivative from the right must exist and be equal. In our case, we have calculated:

f'(3-) = -1 f'(3+) = 1

Clearly, the left-hand derivative and the right-hand derivative at x = 3 are not equal. This leads us to a crucial conclusion: f(x) = |x-3| is not differentiable at x = 3. This non-differentiability is a direct consequence of the sharp corner or vertex at x = 3. At this point, the function's slope changes abruptly, making it impossible to define a single, unique tangent line. The inequality of the left-hand and right-hand derivatives is the mathematical manifestation of this geometric observation. This example vividly illustrates a key principle in calculus: the existence of a derivative at a point requires the function to be