Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = (x − 2)²ᐟ³.

a. Does f′(2) exist?

검증된 단계별 안내

To determine if f'(2) exists, we need to check if the function f(x) = (x - 2)^(2/3) is differentiable at x = 2. Differentiability at a point requires the function to be continuous at that point and the derivative to exist.

First, check the continuity of f(x) at x = 2. Since f(x) is defined for all real numbers and involves a power of (x - 2), it is continuous at x = 2.

Next, consider the definition of the derivative: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit as x approaches 2.

Substitute f(x) = (x - 2)^(2/3) into the derivative definition: f'(2) = lim (h -> 0) [((2 + h - 2)^(2/3) - (2 - 2)^(2/3)) / h] = lim (h -> 0) [h^(2/3) / h].

Simplify the expression: h^(2/3) / h = h^(-1/3). As h approaches 0, h^(-1/3) becomes undefined because it involves division by zero. Therefore, the limit does not exist, and f'(2) does not exist.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For a derivative to exist at a point, the function must be continuous and smooth (differentiable) at that point.

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit of the function as it approaches the point from both sides exists, and the limit equals the function's value at that point. If a function is not continuous at a point, it cannot have a derivative there.

Cusp

A cusp is a point on the graph of a function where the direction of the curve changes abruptly, creating a sharp point. At a cusp, the function is continuous, but the derivative does not exist because the slope of the tangent line is not well-defined, as it approaches different values from the left and right.

Theory and ExamplesIn Exercises 51 and 52, give reasons for your answers.Let f(x) = (x − 2)²ᐟ³.a. Does f′(2) exist?