One-Sided Derivatives

Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

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Identify the piecewise function f(x) based on the graph: f(x) = x^2 for x < 0 and f(x) = x for x ≥ 0.

To find the left-hand derivative at P(0,0), compute the limit as x approaches 0 from the left: lim (x -> 0-) [f(x) - f(0)] / (x - 0). Since f(x) = x^2 for x < 0, this becomes lim (x -> 0-) [x^2 - 0] / x.

Simplify the left-hand limit expression: lim (x -> 0-) x^2 / x = lim (x -> 0-) x.

To find the right-hand derivative at P(0,0), compute the limit as x approaches 0 from the right: lim (x -> 0+) [f(x) - f(0)] / (x - 0). Since f(x) = x for x ≥ 0, this becomes lim (x -> 0+) [x - 0] / x.

Simplify the right-hand limit expression: lim (x -> 0+) x / x = lim (x -> 0+) 1.

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주요 개념

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One-Sided Derivatives

One-sided derivatives are limits that assess the behavior of a function as it approaches a specific point from one side only. The right-hand derivative considers the limit as the input approaches the point from the right, while the left-hand derivative considers the limit from the left. If these two limits exist but are not equal, the function is not differentiable at that point.

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which means the function must be continuous and have a consistent slope from both sides. If the left-hand and right-hand derivatives at a point are not equal, the function is not differentiable there. This concept is crucial for understanding the smoothness and behavior of functions at specific points.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for defining derivatives, as the derivative itself is the limit of the average rate of change of the function as the interval approaches zero. Understanding limits allows for the analysis of function behavior near points of interest, particularly in determining continuity and differentiability.

One-Sided DerivativesCompute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.