91Ó°ÊÓ

A modulus function, also known as an absolute value function, is a fundamental concept in calculus and algebra. It is denoted by the symbol \(|x|\), and, in essence, it extracts the positive magnitude of a real number. For instance, the modulus of both -3 and 3 is 3. This function transforms negative inputs into positive outputs while keeping positive inputs unchanged. The equation of a modulus function could be written in the form \(|x-c|\) where \(c\) is some constant. Here, it measures the distance of \(x\) from \(c\) on a number line. In the exercise, we dealt with \(f(x) = |x - 2|\), which exhibits a distinct property: a sharp point or 'cusp', indicating potential non-differentiability.

To understand differentiability at a point, one needs to examine the concept of left-hand derivative. It represents the rate of change of the function as \(x\) approaches a particular point from the left. Mathematically, the left-hand derivative of a function \(f(x)\) at a point \(x = a\) is given by:

• \[ f'_-(a) = \lim_{{h \to 0^-}} \frac{{f(a+h) - f(a)}}{h} \]

For the modulus function \(f(x) = |x - 2|\), when approaching \(x = 2\) from the left ( ext{with} \(x < 2\)), the expression \(|x - 2|\) simplifies to \(-(x - 2)\). Therefore, the left-hand derivative evaluates to -1. This negative value implies the slope of the tangent moving left to the point is decreasing.

The right-hand derivative, analogous to its left-hand counterpart, gauges how a function behaves as it approaches a designated point from the right. It's defined by the formula:

• \[ f'_+(a) = \lim_{{h \to 0^+}} \frac{{f(a+h) - f(a)}}{h} \]

For \(f(x) = |x - 2|\), when we move to the right of \(x = 2\) ( ext{where} \(x > 2\)), the modulus presents as \(x - 2\). Consequently, the derivative for this segment becomes 1, indicating a positive slope. In essence, it portrays that as we move rightward from \(x = 2\), the function ascends uniformly.

A 'cusp' emerges in a function where there is an abrupt directional transition, typified by a sharp point or corner in the graph. At these points, the function often fails to be differentiable, as seen with \(f(x) = |x-2|\) at \(x=2\). Instead of a smooth curve, the graph displays a distinct V-shape at the cusp, where both the left-hand and right-hand derivatives yield different values (specifically, -1 and +1). This disparity accentuates the non-differentiability at cusps because for a function to be differentiable at a particular point, its left-hand and right-hand derivatives must coincide, which is evidently not the case here. Such unique characteristics make cusps an intriguing yet challenging feature in calculus.