Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

Arc length may be negative if f(x) < 0 on part of the interval in question.

검증된 단계별 안내

Recall the definition of arc length for a function \(f(x)\) on an interval \([a,b]\): the arc length \(L\) is given by the integral \(L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\).

Note that the integrand \(\sqrt{1 + (f'(x))^2}\) is always non-negative because it is a square root of a sum of squares, which cannot be negative.

Since the integrand is non-negative and the limits of integration satisfy \(a < b\), the value of the integral (arc length) must be non-negative.

The value of \(f(x)\) itself (whether positive or negative) does not affect the sign of the arc length because the formula depends on \(f'(x)\), the derivative, and the square root expression, not directly on \(f(x)\).

Therefore, arc length cannot be negative even if \(f(x) < 0\) on part of the interval; the statement is false.

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.

Definition of Arc Length

Arc length measures the distance along a curve between two points and is always a non-negative quantity. It is calculated by integrating the square root of 1 plus the derivative squared, ensuring the result represents a length, which cannot be negative.

Role of the Function's Sign in Arc Length

The sign of the function f(x) does not affect the arc length because arc length depends on the magnitude of the derivative, not the function's value. Even if f(x) is negative, the length along the curve remains positive.

Integral of Absolute Values and Non-negativity

Arc length involves integrating the square root of the sum of squares, which is always non-negative. This ensures the integral accumulates positive values, preventing the arc length from being negative regardless of the function's behavior.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.Arc length may be negative if f(x) < 0 on part of the interval in question.