b. Show that

f(x) = { x² sin(1/x), x ≠ 0
0, x = 0
is differentiable at x = 0 and find f′(0).

검증된 단계별 안내

First, understand the definition of differentiability at a point. A function f(x) is differentiable at x = a if the limit of the difference quotient exists: lim(x -> a) [f(x) - f(a)] / (x - a).

For the given function f(x), we need to check differentiability at x = 0. The function is defined as f(x) = x² sin(1/x) for x ≠ 0 and f(x) = 0 for x = 0.

Calculate the difference quotient for f(x) at x = 0: [f(x) - f(0)] / (x - 0) = [x² sin(1/x) - 0] / x = x sin(1/x).

Evaluate the limit of x sin(1/x) as x approaches 0. Use the fact that |sin(1/x)| ≤ 1, so |x sin(1/x)| ≤ |x|. As x approaches 0, |x| approaches 0, thus the limit of x sin(1/x) is 0.

Since the limit exists and is equal to 0, f(x) is differentiable at x = 0. Therefore, f'(0) = 0.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Differentiability

A function is differentiable at a point if it has a defined derivative at that point. This means the function must be continuous at the point, and the limit of the difference quotient must exist as the point is approached. For f(x) to be differentiable at x = 0, we need to check these conditions specifically at x = 0.

Limit Definition of Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero: f'(a) = lim (h -> 0) [(f(a+h) - f(a))/h]. For f(x) at x = 0, this involves evaluating the limit of [x² sin(1/x)]/x as x approaches 0, which simplifies to finding the behavior of x sin(1/x) as x approaches 0.

Squeeze Theorem

The Squeeze Theorem is used to find the limit of a function trapped between two other functions that have the same limit at a point. If -1 ≤ sin(1/x) ≤ 1, then -x ≤ x sin(1/x) ≤ x. As x approaches 0, both -x and x approach 0, allowing us to conclude that x sin(1/x) also approaches 0, which is crucial for proving differentiability at x = 0.

b. Show thatf(x) = { x² sin(1/x), x ≠ 0 0, x = 0is differentiable at x = 0 and find f′(0).