In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x − 1, x ≥ 0
x² + 2x + 7, x < 0

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First, check the continuity of the function at x = 0. Evaluate the left-hand limit and the right-hand limit of f(x) as x approaches 0. For x ≥ 0, f(x) = 2x - 1, so the right-hand limit as x approaches 0 is f(0) = 2(0) - 1 = -1. For x < 0, f(x) = x² + 2x + 7, so the left-hand limit as x approaches 0 is f(0) = 0² + 2(0) + 7 = 7.

Since the left-hand limit (7) and the right-hand limit (-1) are not equal, the function is not continuous at x = 0. A function must be continuous at a point to be differentiable there.

Even though the function is not continuous at x = 0, let's check the derivatives from both sides for completeness. For x ≥ 0, the derivative of f(x) = 2x - 1 is f'(x) = 2.

For x < 0, the derivative of f(x) = x² + 2x + 7 is f'(x) = 2x + 2. Evaluate this derivative as x approaches 0 from the left, which gives f'(0) = 2(0) + 2 = 2.

Since the function is not continuous at x = 0, it cannot be differentiable at that point, even though the derivatives from both sides are equal. Differentiability requires continuity at the point in question.

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주요 개념

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Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two distinct formulas: one for x greater than or equal to 0 and another for x less than 0. Understanding how to evaluate and analyze these functions at specific points is crucial for determining properties like continuity and differentiability.

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. To check differentiability at x = 0, we need to ensure that the left-hand derivative (from x < 0) and the right-hand derivative (from x ≥ 0) are equal. If they differ, the function is not differentiable at that point.

Continuity

Continuity at a point means that the function's value at that point matches the limit of the function as it approaches that point from both sides. For f(x) to be differentiable at x = 0, it must first be continuous there. This involves checking that the values of the two pieces of the function agree at x = 0, ensuring no jumps or breaks in the graph.

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.f(x) = { 2x − 1, x ≥ 0 x² + 2x + 7, x < 0