Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

검증된 단계별 안내

Recall the formula for the arc length of a curve \(y = f(x)\) on the interval \([a, b]\): \[L = \int_a^b \sqrt{1 + \left( f'(x) \right)^2} \, dx\]

Identify the function given: \[f(x) = \ln \left| \tanh \left( \frac{x}{2} \right) \right|\] and the interval: \[[a, b] = [\ln 2, \ln 8]\]

Find the derivative \(f'(x)\) using the chain rule and the derivative of \(\ln |u|\) which is \(\frac{u'}{u}\): Let \[u = \tanh \left( \frac{x}{2} \right)\] then \[f'(x) = \frac{u'}{u}\] Calculate \(u'\) by differentiating \(\tanh \left( \frac{x}{2} \right)\) with respect to \(x\).

Substitute \(f'(x)\) into the arc length formula: \[L = \int_{\ln 2}^{\ln 8} \sqrt{1 + \left( f'(x) \right)^2} \, dx\] Simplify the expression under the square root as much as possible to make the integral easier to evaluate.

Evaluate the integral either by direct integration or by using a suitable substitution if needed, to find the arc length over the given interval.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

15m

주요 개념

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

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \). This formula calculates the length of the curve by summing infinitesimal line segments along the curve.

Derivative of the Function

To apply the arc length formula, you need the derivative f'(x). For f(x) = ln|tanh(x/2)|, use the chain rule and derivatives of hyperbolic functions to find f'(x), which is essential for evaluating the integrand \( \sqrt{1 + (f'(x))^2} \).

Properties of Hyperbolic Functions

Understanding hyperbolic functions like tanh(x) and their derivatives is crucial. For example, \( \frac{d}{dx} \tanh(x) = \text{sech}^2(x) \). These properties help simplify the derivative and the integrand in the arc length integral.