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

Dans les calculs d’ingénierie, la précision, les limites et l’approximation sont utilisées pour représenter une valeur.
Considérons deux blocs carrés de dimensions différentes. Les valeurs précises de leurs surfaces peuvent être obtenues en arrondissant leurs nombres aux valeurs les plus proches basées sur des chiffres significatifs.
Considérons un cône circulaire droit ayant un rayon de base, r, et une hauteur, h. Si le cône est découpé circulairement à une distance x de son sommet, de sorte que l’élément tranché ait une épaisseur, Δx, alors son volume peut être déterminé.
En appliquant la limite de Δv à dv et de Δx à dx, et en négligeant les différentiels d’ordre supérieur, on obtient l’expression du volume.
Considérons un arc AB, sous-tendant un petit angle en O. La longueur de l’arc peut être approchée de la base d’un triangle rectangle.
Si la longueur de l’hypoténuse est unie, la longueur de l’arc est égale au sinus thêta, approximativement égal à thêta. De plus, le cosinus thêta est également approximativement égal à l’unité.
Pour thêta d’un degré, la valeur approximative de sin thêta et tan thêta est presque la même.

<ol>
	<li>Meriam, J.L. ; Kraige, L.G., et Bolton, J.N. (2020). Génie mécanique \u2012 Statique. Hoboken, New Jersey : John Wiley. pp – 13 et 14.</li>
</ol>

précision, limites et approximations

La précision, les limites et les approximations sont courantes dans de nombreux domaines, notamment dans les calculs d'ingénierie. Ces concepts sont impératifs pour s'assurer qu'une valeur donnée est aussi proche que possible de sa valeur réelle. La précision est définie comme la proximité de la valeur mesurée à la valeur vraie ou réelle. En mécanique de l'ingénierie, des mesures répétées sont prises lors d'analyses théoriques ou expérimentales pour garantir que le résultat est précis et exact. La précision de toute solution est basée sur les données empiriques et les résultats obtenus à partir de différents calculs ou mesures. Considérons un contenant cylindrique rempli d'un liquide. Le volume du liquide peut être déterminé en mesurant les dimensions du contenant et en calculant le volume à l'aide d'une formule appropriée. Cependant, en pratique, il est impossible de mesurer les dimensions du contenant avec une précision absolue. Les approximations sont particulièrement utiles lorsqu'il s'agit de fonctions complexes difficiles à résoudre analytiquement. Le degré d'approximation fait référence au nombre de chiffres significatifs utilisés dans les mesures. Plus on utilise de chiffres significatifs, plus le résultat est précis. Bien que les approximations soient utiles, il faut être prudent lors de leur utilisation, car elles peuvent introduire des erreurs qui affectent la précision du calcul. Cela signifie qu'il est essentiel de comprendre l'erreur introduite par l'approximation et comment elle peut affecter le résultat. Par exemple, pour un angle relativement petit, généralement inférieur ou égal à 15° ou 0,26 radians, des approximations peuvent être utilisées pour simplifier les fonctions sinus, cosinus et tangente. Notamment, la précision de ces approximations diminue lorsque l'angle est supérieur à 15° ou 0,26 radians. Cependant, pour des angles inférieurs ou égaux à 15° (0,26 radians), ces formules ont une bonne précision et produisent des valeurs presque égales à la valeur exacte. Le concept de limites est tout aussi important dans les calculs d'ingénierie. Les limites sont utilisées pour déterminer le comportement d'une fonction, notamment lorsque la variable indépendante approche une valeur spécifique. Considérons une fonction qui tend vers l'infini lorsque x approche d'une certaine valeur. La limite définit la valeur maximale qui peut être atteinte sans dépasser la valeur spécifiée de x. Les limites sont particulièrement utiles pour déterminer la stabilité et les performances des systèmes dans des applications critiques.

Précision, limites et approximations

La précision, les limites et les approximations sont courantes dans de nombreux domaines, en particulier dans les calculs d’ingénierie. Ces concepts sont impératifs pour garantir qu’un ...

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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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accuracy, limits, and approximation

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...

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Accuracy and Approximation in Calculations

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