Name: accuracy and approximation in calculations
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Specialised Topic in Science

112.51.c.1.

112.51.c.2.C.

Accuracy, limits, and approximation

엔지니어링 계산에서 정확도, 한계 및 근사치는 값을 나타내는 데 사용됩니다.
차원이 다른 두 개의 정사각형 블록을 생각해 보십시오. 해당 영역의 정확한 값은 유효 숫자를 기준으로 가장 가까운 값으로 숫자를 반올림하여 얻을 수 있습니다.
밑변 반지름 r과 높이 h를 갖는 오른쪽 원형 원뿔을 고려하십시오. 원뿔이 꼭짓점에서 x 거리만큼 원형으로 슬라이스되어 슬라이스된 요소의 두께가 Δx를 갖도록 하면 그 부피를 결정할 수 있습니다.
Δv에서 dv로, Δx에서 dx로 한계를 적용하고 고차 차이를 무시하면 부피에 대한 표현을 얻을 수 있습니다.
O에서 작은 각도를 대체하는 호 AB를 고려하십시오. 호 길이는 직각 삼각형의 밑변에 근사화할 수 있습니다.
빗변의 길이가 통일성이면 호 길이는 sine theta와 같으며 대략 theta와 같습니다. 또한 코사인 세타는 거의 단위와 같습니다.
1도의 theta에 대해 sin theta와 tan theta의 근사값은 거의 같습니다.

<ol>
	<li>메리암, J.L.; Kraige, L.G. 및 Bolton, J.N. (2020). 엔지니어링 역학 \u2012 정역학. 뉴저지 호보켄: 존 와일리. 쪽 – 13-14.</li>
</ol>

정확도, 극한 및 근사치

정확도, 극한, 근사치는 여러 분야, 특히 공학 계산에서 흔히 사용됩니다. 이러한 개념은 주어진 값이 실제 값에 최대한 가깝도록 보장하는 데 필수적입니다.
정확도는 측정된 값이 참값 또는 실제 값에 얼마나 가까운 지를 의미합니다. 공학 역학에서는 결과가 정밀하고 정확한지 확인하기 위해 이론적 또는 실험적 분석 중에 반복적인 측정이 수행됩니다.
모든 솔루션의 정확도는 경험적 데이터와 다양한 계산 또는 측정을 통해 얻은 결과를 기반으로 합니다. 액체로 채워진 원통형 용기를 생각해 보십시오. 액체의 부피는 용기의 치수를 측정하고 적절한 공식을 사용하여 부피를 계산하고 결정할 수 있습니다. 그러나 실제로는 용기의 치수를 절대적인 정밀도로 측정하는 것이 불가능합니다.
근사치는 분석적으로 해결하기 어려운 복잡한 함수를 처리할 때 특히 유용합니다. 근사 정도는 측정에 사용된 유효 숫자의 수를 나타냅니다. 더 많은 유효 숫자를 사용할수록 결과가 더 정확해집니다. 근사치는 유용하지만 계산의 정확성에 영향을 미치는 오차가 발생할 수 있으므로 사용할 때는 주의해야 합니다. 즉, 근사치로 인해 발생하는 오차와 그 오차가 결과에 어떤 영향을 미칠 수 있는지 이해하는 것이 중요합니다.
예를 들어 상대적으로 작은 각도의 경우 일반적으로 15° 또는 0.26 라디안인 경우 근사치를 사용하여 사인, 코사인 및 탄젠트 함수를 단순화할 수 있습니다. 특히 각도가 15°보다 클 경우 이러한 근사의 정확도가 감소합니다. 또는 0.26 라디안. 그러나 15°보다 작거나 같은 각도의 경우; (0.26 라디안), 이러한 수식은 정확도가 높으며 정확한 값과 거의 동일한 값을 생성합니다.
극한의 개념은 엔지니어링 계산에서도 똑같이 중요합니다. 극한은 특히 독립 변수가 특정 값에 접근할 때 함수의 동작을 확인하는 데 사용됩니다. x가 특정 값에 가까워질수록 무한대에 접근하는 함수를 생각해 봅시다. 극한은 지정된 x 값을 초과하지 않고 도달할 수 있는 최대값을 정의합니다. 극한은 중요한 애플리케이션에서 시스템의 안정성과 성능을 결정하는 데 특히 유용합니다.

정확도, 한계 및 근사치는 많은 분야, 특히 엔지니어링 계산에서 일반적입니다. 이러한 개념은 다음을 보장하는 데 필수적입니다.

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

An Introduction to Statics

정역학 입문

KEY TERMS AND CONCEPTS

주요 용어 및 개념

accuracy and approximation in calculations

Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate.

The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured dimensions and a formula, but those dimensions cannot be measured with absolute precision in practice.

Approximations are useful when a function is too complex to solve exactly. The degree of approximation depends on the number of significant figures used in the measurements. More significant figures usually give a result that is closer to the true value.

Approximations can also introduce error, so the effect of the error must be understood. For small angles, usually less than or equal to 15° or 0.26 radians, sine, cosine, and tangent can be approximated with good accuracy. This accuracy becomes worse when the angle is greater than 15° or 0.26 radians.

Limits are another key idea in engineering calculations. A limit describes the behavior of a function as an independent variable approaches a specific value. For a function that grows toward infinity as x approaches a certain value, the limit helps define the maximum value that can be reached without passing that x value. Limits are useful for judging the stability and performance of systems in critical applications.

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112.51.c.1.F.

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112.51.c.2.A.

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Accuracy and Approximation in Calculations

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