{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":45280,"title":"La derivada numérica","description":"El concepto de pendiente o inclinación de una función recta es muy intuitivo para cualquier persona. Desde el punto de vista matemático, la pendiente indica cuánto se incrementa o decrementa la función _y = f(x)_ al incrementar el valor de la variable _x_. Matemáticamente,\r\n\r\n\u003c\u003chttps://www.calculadoraconversor.com/wp-content/uploads/2017/06/formula-pendiente-de-una-recta-que-pasa-por-dos-puntos.png\u003e\u003e\r\n\r\nSin embargo, ¿cuál es la pendiente en las funciones que no son rectas? Obviamente, la pendiente será distinta en cada punto de la función. Por lo tanto, podríamos obtener otra función _f'(x)_ que, para cada punto _x_, nos indicase el valor de la pendiente en la función _f(x)_ original. A esta función se le conoce como función derivada o derivada. La derivada se define matemáticamente como la pendiente de la recta tangente a la función _f(x)_ cuando nos aproximamos al punto _x_.\r\n\r\n\u003c\u003chttps://4.bp.blogspot.com/-98krAw0r0fg/WC7WQ_pV2DI/AAAAAAAAvAQ/zCbdPM8jf0QKkusoIC3KrzhHwhh-qAvlgCPcB/s1600/funcion%2Bderivada%2Bpor%2Bla%2Bdefinicion.png\u003e\u003e\r\n\r\nCuando tenemos funciones conocidas, es fácil calcular la derivada a partir de unas reglas de derivación. Pero, ¿y si no conocemos la función? ¿y si tenemos una señal ruidosa y aleatoria, o la mezcla de varias funciones desconocidas? En estos casos, podemos calcular la derivada (es decir, el límite) de forma aproximada, tomando un valor de _h_ pequeño. \r\n\r\nEn este problema, debes calcular el vector derivada _dv_ de un vector _v_ y un tamaño de paso _h_ como parámetros de entrada.  ","description_html":"\u003cp\u003eEl concepto de pendiente o inclinación de una función recta es muy intuitivo para cualquier persona. Desde el punto de vista matemático, la pendiente indica cuánto se incrementa o decrementa la función \u003ci\u003ey = f(x)\u003c/i\u003e al incrementar el valor de la variable \u003ci\u003ex\u003c/i\u003e. Matemáticamente,\u003c/p\u003e\u003cimg src = \"https://www.calculadoraconversor.com/wp-content/uploads/2017/06/formula-pendiente-de-una-recta-que-pasa-por-dos-puntos.png\"\u003e\u003cp\u003eSin embargo, ¿cuál es la pendiente en las funciones que no son rectas? Obviamente, la pendiente será distinta en cada punto de la función. Por lo tanto, podríamos obtener otra función \u003ci\u003ef'(x)\u003c/i\u003e que, para cada punto \u003ci\u003ex\u003c/i\u003e, nos indicase el valor de la pendiente en la función \u003ci\u003ef(x)\u003c/i\u003e original. A esta función se le conoce como función derivada o derivada. La derivada se define matemáticamente como la pendiente de la recta tangente a la función \u003ci\u003ef(x)\u003c/i\u003e cuando nos aproximamos al punto \u003ci\u003ex\u003c/i\u003e.\u003c/p\u003e\u003cimg src = \"https://4.bp.blogspot.com/-98krAw0r0fg/WC7WQ_pV2DI/AAAAAAAAvAQ/zCbdPM8jf0QKkusoIC3KrzhHwhh-qAvlgCPcB/s1600/funcion%2Bderivada%2Bpor%2Bla%2Bdefinicion.png\"\u003e\u003cp\u003eCuando tenemos funciones conocidas, es fácil calcular la derivada a partir de unas reglas de derivación. Pero, ¿y si no conocemos la función? ¿y si tenemos una señal ruidosa y aleatoria, o la mezcla de varias funciones desconocidas? En estos casos, podemos calcular la derivada (es decir, el límite) de forma aproximada, tomando un valor de \u003ci\u003eh\u003c/i\u003e pequeño.\u003c/p\u003e\u003cp\u003eEn este problema, debes calcular el vector derivada \u003ci\u003edv\u003c/i\u003e de un vector \u003ci\u003ev\u003c/i\u003e y un tamaño de paso \u003ci\u003eh\u003c/i\u003e como parámetros de entrada.\u003c/p\u003e","function_template":"function dv = your_fcn_name(v,h)\r\n  dv = v;\r\nend","test_suite":"%%\r\nh = 0.001;\r\nv = 0:h:50; % Recta\r\ny_correct = ones(1,length(v)-1);\r\nassert(isequal(round(your_fcn_name(v,h)),round(y_correct)))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":385299,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2020-01-27T17:54:20.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-01-27T17:23:49.000Z","updated_at":"2026-03-02T14:48:11.000Z","published_at":"2020-01-27T17:23:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.png\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.png\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEl concepto de pendiente o inclinación de una función recta es muy intuitivo para cualquier persona. Desde el punto de vista matemático, la pendiente indica cuánto se incrementa o decrementa la función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey = f(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e al incrementar el valor de la variable\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Matemáticamente,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSin embargo, ¿cuál es la pendiente en las funciones que no son rectas? Obviamente, la pendiente será distinta en cada punto de la función. Por lo tanto, podríamos obtener otra función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef'(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e que, para cada punto\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, nos indicase el valor de la pendiente en la función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e original. A esta función se le conoce como función derivada o derivada. La derivada se define matemáticamente como la pendiente de la recta tangente a la función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e cuando nos aproximamos al punto\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCuando tenemos funciones conocidas, es fácil calcular la derivada a partir de unas reglas de derivación. Pero, ¿y si no conocemos la función? ¿y si tenemos una señal ruidosa y aleatoria, o la mezcla de varias funciones desconocidas? En estos casos, podemos calcular la derivada (es decir, el límite) de forma aproximada, tomando un valor de\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e pequeño.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEn este problema, debes calcular el vector derivada\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003edv\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e de un vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e y un tamaño de paso\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e como parámetros de entrada.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\"},{\"partUri\":\"/media/image2.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\"}]}"},{"id":58941,"title":"Compute measurement uncertainty","description":"Suppose a variable  depends on  independent variables , , . If the independent variables have uncertainty , , etc. and the uncertainties are independent, then the uncertainty in  can be estimated with \r\n\r\nFor this problem, the relationship between  and  will be power laws of the form\r\n\r\nFor example, the relationship  would have c = 0.5 and a = [1 2]. \r\nWrite a function that takes a vector of values of , the coefficient , exponents , and a vector of uncertainties  and returns the absolute uncertainty , relative uncertainty , and the index of the variable that contributes the most to the uncertainty in . Identifying the largest contributor to the uncertainty tells the experimentalist which measurement to target first to improve the measurement.  ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 307.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 153.8px; transform-origin: 407px 153.8px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.8583px 8px; transform-origin: 61.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSuppose a variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.4583px 8px; transform-origin: 40.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e depends on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.3667px 8px; transform-origin: 72.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e independent variables \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" alt=\"x1\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" alt=\"x2\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"56.5\" height=\"20\" alt=\"x3,...,xn\" style=\"width: 56.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.092px 8px; transform-origin: 145.092px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. If the independent variables have uncertainty \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"20\" alt=\"Deltax1\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"20\" alt=\"Deltax2\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.267px 8px; transform-origin: 209.267px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, etc. and the uncertainties are independent, then the uncertainty in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.5667px 8px; transform-origin: 71.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e can be estimated with \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 50.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 25.4px; text-align: left; transform-origin: 384px 25.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"166\" height=\"51\" alt=\"Deltay = sqrt(sum((dy/dxj Deltaxj)^2))\" style=\"width: 166px; height: 51px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.258px 8px; transform-origin: 132.258px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor this problem, the relationship between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" alt=\"xj\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.7417px 8px; transform-origin: 93.7417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will be power laws of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 27px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13.5px; text-align: left; transform-origin: 384px 13.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"124.5\" height=\"27\" alt=\"y = c x1^a1 x2^a2 x3^a3 ...xn^an\" style=\"width: 124.5px; height: 27px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 34.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.4px; text-align: left; transform-origin: 384px 17.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1917px 8px; transform-origin: 92.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"75\" height=\"35\" alt=\"KE = (1/2)mv^2\" style=\"width: 75px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.2917px 8px; transform-origin: 39.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e would have \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.95px 8px; transform-origin: 26.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ec = 0.5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.65px 8px; transform-origin: 34.65px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ea = [1 2]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147.267px 8px; transform-origin: 147.267px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a vector of values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.875px 8px; transform-origin: 48.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the coefficient \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1167px 8px; transform-origin: 38.1167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, exponents \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.7333px 8px; transform-origin: 93.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and a vector of uncertainties \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"21.5\" height=\"18\" alt=\"Deltax\" style=\"width: 21.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and returns the absolute uncertainty \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"21.5\" height=\"18\" alt=\"Deltay\" style=\"width: 21.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.5667px 8px; transform-origin: 64.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, relative uncertainty \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37\" height=\"18.5\" alt=\"Deltay/y\" style=\"width: 37px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 190.308px 8px; transform-origin: 190.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the index of the variable that contributes the most to the uncertainty in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.225px 8px; transform-origin: 330.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Identifying the largest contributor to the uncertainty tells the experimentalist which measurement to target first to improve the measurement.  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [dy,dyrel,i1] = uncertainty1(x,c,a,dx)\r\n%  dy    = uncertainty in y\r\n%  dyrel = relative uncertainty in y (i.e., dy/y)\r\n%  i1    = index of the variable that contributes most to the uncertainty\r\n%  x     = values of the independent variables\r\n%  c     = coefficient of the relationship\r\n%  a     = vector of exponents\r\n%  dx    = vector of uncertainties\r\n\r\n   dy = sqrt(sum((dydx*dx).^2));\r\n   dyrel = dy/y;\r\n   i1 = find(max(dy));\r\nend","test_suite":"%% Volume of a cylinder: V = (pi/4)*D^2*H\r\nD = 75;         %  Diameter (mm)\r\ndD = 3;         %  Diameter uncertainty (mm)\r\nH = 75;         %  Height (mm)\r\ndH = 3;         %  Height uncertainty (mm)\r\n[dV,dVrel,i1] = uncertainty1([D H],pi/4,[2 1],[dD dH]);\r\ndV_correct = 2.9636e+04;\r\ndVrel_correct = 0.0894;\r\ni1_correct = 1;\r\nassert(abs(dV-dV_correct)\u003c1)\r\nassert(abs(dVrel-dVrel_correct)\u003c1e-4)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Hydraulic conductivity from constant-head permeameter test: K = VL/TAd\r\nV  = 7.54e-5;         %  Volume (m3)\r\ndV = 1e-6;            %  Volume uncertainty (m3)\r\nL  = 0.1;             %  Length (m)\r\ndL = 1e-3;            %  Length uncertainty (m)\r\nT  = 1000;            %  Time (sec)\r\ndT = 1;               %  Time uncertainty (sec)\r\nA  = 1.2657e-3;       %  Area (m2)\r\ndA = 6.3e-5;          %  Area uncertainty (m2)\r\nd  = 0.05;            %  Head difference (m)\r\ndd = 1e-3;            %  Head difference uncertainty (m)\r\n[dK,dKrel,i1] = uncertainty1([V L T A d],1,[1 1 -1 -1 -1],[dV dL dT dA dd]);\r\ndK_correct = 6.691e-6;\r\ndKrel_correct = 0.0562;\r\ni1_correct = 4;\r\nassert(abs(dK-dK_correct)\u003c1e-9)\r\nassert(abs(dKrel-dKrel_correct)\u003c1e-4)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Travel time: T = ne L^2/(K dh)\r\nne  = 0.1;            %  Effective porosity\r\ndne = 0.02;           %  Effective porosity uncertainty \r\nL   = 20;             %  Length (m)\r\ndL  = 0.3;            %  Length uncertainty (m)\r\nK   = 0.5;            %  Hydraulic conductivity (m/d)\r\ndK  = 0.08;           %  Hydraulic conductivity uncertainty (m/d)\r\ndh  = 1.5;            %  Head difference (m)\r\nddh = 0.2;            %  Head difference uncertainty (m)\r\n[dT,dTrel,i1] = uncertainty1([ne L K dh],1,[1 2 -1 -1],[dne dL dK ddh]);\r\ndT_correct = 15.48;\r\ndTrel_correct = 0.290;\r\ni1_correct = 1;\r\nassert(abs(dT-dT_correct)\u003c1e-2)\r\nassert(abs(dTrel-dTrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Travel time: T = ne L^2/(K dh)\r\nne  = 0.1;            %  Effective porosity\r\ndne = 0.01;           %  Effective porosity uncertainty \r\nL   = 20;             %  Length (m)\r\ndL  = 0.3;            %  Length uncertainty (m)\r\nK   = 0.5;            %  Hydraulic conductivity (m/d)\r\ndK  = 0.08;           %  Hydraulic conductivity uncertainty (m/d)\r\ndh  = 1.5;            %  Head difference (m)\r\nddh = 0.2;            %  Head difference uncertainty (m)\r\n[dT,dTrel,i1] = uncertainty1([ne L K dh],1,[1 2 -1 -1],[dne dL dK ddh]);\r\ndT_correct = 12.43;\r\ndTrel_correct = 0.233;\r\ni1_correct = 3;\r\nassert(abs(dT-dT_correct)\u003c1e-2)\r\nassert(abs(dTrel-dTrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Travel time: T = ne L^2/(K dh)\r\nne  = 0.1;            %  Effective porosity\r\ndne = 0.01;           %  Effective porosity uncertainty \r\nL   = 20;             %  Length (m)\r\ndL  = 1.7;            %  Length uncertainty (m)\r\nK   = 0.5;            %  Hydraulic conductivity (m/d)\r\ndK  = 0.08;           %  Hydraulic conductivity uncertainty (m/d)\r\ndh  = 1.5;            %  Head difference (m)\r\nddh = 0.2;            %  Head difference uncertainty (m)\r\n[dT,dTrel,i1] = uncertainty1([ne L K dh],1,[1 2 -1 -1],[dne dL dK ddh]);\r\ndT_correct = 15.3;\r\ndTrel_correct = 0.287;\r\ni1_correct = 2;\r\nassert(abs(dT-dT_correct)\u003c1e-2)\r\nassert(abs(dTrel-dTrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Discharge by Manning's equation: Q = (1/n) R^(2/3) S0^(1/2) A = (1/n) A^(5/3) S0^(1/2)/P^(2/3)\r\nn   = 0.015;          %  Manning roughness coefficient\r\ndn  = 0.002;          %  Manning roughness coefficient uncertainty \r\nA   = 0.02;           %  Area (m2)\r\ndA  = 2e-3;           %  Area uncertainty (m2)\r\nP   = 0.04;           %  Wetted perimeter (m)\r\ndP  = 8e-3;           %  Wetted perimeter uncertainty (m)\r\nS0  = 0.01;           %  Slope (-)\r\ndS0 = 7e-4;           %  Slope uncertainty (-)\r\n[dQ,dQrel,i1] = uncertainty1([n A P S0],1,[-1 5/3 -2/3 1/2],[dn dA dP dS0]);\r\ndQ_correct = 2.13e-2;\r\ndQrel_correct = 0.254;\r\ni1_correct = 2;\r\nassert(abs(dQ-dQ_correct)\u003c1e-4)\r\nassert(abs(dQrel-dQrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Discharge by Manning's equation: Q = (1/n) R^(2/3) S0^(1/2) A = (1/n) A^(5/3) S0^(1/2)/P^(2/3)\r\nn   = 0.015;          %  Manning roughness coefficient\r\ndn  = 0.002;          %  Manning roughness coefficient uncertainty \r\nA   = 0.02;           %  Area (m2)\r\ndA  = 1e-3;           %  Area uncertainty (m2)\r\nP   = 0.04;           %  Wetted perimeter (m)\r\ndP  = 4e-3;           %  Wetted perimeter uncertainty (m)\r\nS0  = 0.01;           %  Slope (-)\r\ndS0 = 7e-4;           %  Slope uncertainty (-)\r\n[dQ,dQrel,i1] = uncertainty1([n A P S0],1,[-1 5/3 -2/3 1/2],[dn dA dP dS0]);\r\ndQ_correct = 1.46e-2;\r\ndQrel_correct = 0.174;\r\ni1_correct = 1;\r\nassert(abs(dQ-dQ_correct)\u003c1e-4)\r\nassert(abs(dQrel-dQrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-12-30T14:25:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-09-02T03:35:52.000Z","updated_at":"2026-02-10T14:38:18.000Z","published_at":"2023-09-02T03:35:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose a variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e independent variables \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x3,...,xn\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_3,\\\\ldots, x_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. If the independent variables have uncertainty \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltax1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltax2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, etc. and the uncertainties are independent, then the uncertainty in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be estimated with \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltay = sqrt(sum((dy/dxj Deltaxj)^2))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta y = \\\\left[\\\\sum_{j = 1}^n \\\\left(\\\\frac{\\\\partial y}{\\\\partial x_j}\\\\Delta x_j\\\\right)^2\\\\right]^{1/2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, the relationship between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"xj\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_j\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will be power laws of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = c x1^a1 x2^a2 x3^a3 ...xn^an\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = c x_1^{a_1} x_2^{a_2} x_3^{a_3}\\\\cdots x_n^{a_n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"KE = (1/2)mv^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eKE = \\\\frac{1}{2} m v^2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e would have \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ec = 0.5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea = [1 2]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a vector of values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the coefficient \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"c\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, exponents \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and a vector of uncertainties \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltax\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and returns the absolute uncertainty \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltay\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, relative uncertainty \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltay/y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta y/y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the index of the variable that contributes the most to the uncertainty in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Identifying the largest contributor to the uncertainty tells the experimentalist which measurement to target first to improve the measurement.  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":45280,"title":"La derivada numérica","description":"El concepto de pendiente o inclinación de una función recta es muy intuitivo para cualquier persona. Desde el punto de vista matemático, la pendiente indica cuánto se incrementa o decrementa la función _y = f(x)_ al incrementar el valor de la variable _x_. Matemáticamente,\r\n\r\n\u003c\u003chttps://www.calculadoraconversor.com/wp-content/uploads/2017/06/formula-pendiente-de-una-recta-que-pasa-por-dos-puntos.png\u003e\u003e\r\n\r\nSin embargo, ¿cuál es la pendiente en las funciones que no son rectas? Obviamente, la pendiente será distinta en cada punto de la función. Por lo tanto, podríamos obtener otra función _f'(x)_ que, para cada punto _x_, nos indicase el valor de la pendiente en la función _f(x)_ original. A esta función se le conoce como función derivada o derivada. La derivada se define matemáticamente como la pendiente de la recta tangente a la función _f(x)_ cuando nos aproximamos al punto _x_.\r\n\r\n\u003c\u003chttps://4.bp.blogspot.com/-98krAw0r0fg/WC7WQ_pV2DI/AAAAAAAAvAQ/zCbdPM8jf0QKkusoIC3KrzhHwhh-qAvlgCPcB/s1600/funcion%2Bderivada%2Bpor%2Bla%2Bdefinicion.png\u003e\u003e\r\n\r\nCuando tenemos funciones conocidas, es fácil calcular la derivada a partir de unas reglas de derivación. Pero, ¿y si no conocemos la función? ¿y si tenemos una señal ruidosa y aleatoria, o la mezcla de varias funciones desconocidas? En estos casos, podemos calcular la derivada (es decir, el límite) de forma aproximada, tomando un valor de _h_ pequeño. \r\n\r\nEn este problema, debes calcular el vector derivada _dv_ de un vector _v_ y un tamaño de paso _h_ como parámetros de entrada.  ","description_html":"\u003cp\u003eEl concepto de pendiente o inclinación de una función recta es muy intuitivo para cualquier persona. Desde el punto de vista matemático, la pendiente indica cuánto se incrementa o decrementa la función \u003ci\u003ey = f(x)\u003c/i\u003e al incrementar el valor de la variable \u003ci\u003ex\u003c/i\u003e. Matemáticamente,\u003c/p\u003e\u003cimg src = \"https://www.calculadoraconversor.com/wp-content/uploads/2017/06/formula-pendiente-de-una-recta-que-pasa-por-dos-puntos.png\"\u003e\u003cp\u003eSin embargo, ¿cuál es la pendiente en las funciones que no son rectas? Obviamente, la pendiente será distinta en cada punto de la función. Por lo tanto, podríamos obtener otra función \u003ci\u003ef'(x)\u003c/i\u003e que, para cada punto \u003ci\u003ex\u003c/i\u003e, nos indicase el valor de la pendiente en la función \u003ci\u003ef(x)\u003c/i\u003e original. A esta función se le conoce como función derivada o derivada. La derivada se define matemáticamente como la pendiente de la recta tangente a la función \u003ci\u003ef(x)\u003c/i\u003e cuando nos aproximamos al punto \u003ci\u003ex\u003c/i\u003e.\u003c/p\u003e\u003cimg src = \"https://4.bp.blogspot.com/-98krAw0r0fg/WC7WQ_pV2DI/AAAAAAAAvAQ/zCbdPM8jf0QKkusoIC3KrzhHwhh-qAvlgCPcB/s1600/funcion%2Bderivada%2Bpor%2Bla%2Bdefinicion.png\"\u003e\u003cp\u003eCuando tenemos funciones conocidas, es fácil calcular la derivada a partir de unas reglas de derivación. Pero, ¿y si no conocemos la función? ¿y si tenemos una señal ruidosa y aleatoria, o la mezcla de varias funciones desconocidas? En estos casos, podemos calcular la derivada (es decir, el límite) de forma aproximada, tomando un valor de \u003ci\u003eh\u003c/i\u003e pequeño.\u003c/p\u003e\u003cp\u003eEn este problema, debes calcular el vector derivada \u003ci\u003edv\u003c/i\u003e de un vector \u003ci\u003ev\u003c/i\u003e y un tamaño de paso \u003ci\u003eh\u003c/i\u003e como parámetros de entrada.\u003c/p\u003e","function_template":"function dv = your_fcn_name(v,h)\r\n  dv = v;\r\nend","test_suite":"%%\r\nh = 0.001;\r\nv = 0:h:50; % Recta\r\ny_correct = ones(1,length(v)-1);\r\nassert(isequal(round(your_fcn_name(v,h)),round(y_correct)))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":385299,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2020-01-27T17:54:20.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-01-27T17:23:49.000Z","updated_at":"2026-03-02T14:48:11.000Z","published_at":"2020-01-27T17:23:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.png\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.png\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEl concepto de pendiente o inclinación de una función recta es muy intuitivo para cualquier persona. Desde el punto de vista matemático, la pendiente indica cuánto se incrementa o decrementa la función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey = f(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e al incrementar el valor de la variable\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Matemáticamente,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSin embargo, ¿cuál es la pendiente en las funciones que no son rectas? Obviamente, la pendiente será distinta en cada punto de la función. Por lo tanto, podríamos obtener otra función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef'(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e que, para cada punto\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, nos indicase el valor de la pendiente en la función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e original. A esta función se le conoce como función derivada o derivada. La derivada se define matemáticamente como la pendiente de la recta tangente a la función\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e cuando nos aproximamos al punto\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCuando tenemos funciones conocidas, es fácil calcular la derivada a partir de unas reglas de derivación. Pero, ¿y si no conocemos la función? ¿y si tenemos una señal ruidosa y aleatoria, o la mezcla de varias funciones desconocidas? En estos casos, podemos calcular la derivada (es decir, el límite) de forma aproximada, tomando un valor de\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e pequeño.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEn este problema, debes calcular el vector derivada\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003edv\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e de un vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e y un tamaño de paso\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e como parámetros de entrada.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\"},{\"partUri\":\"/media/image2.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\"}]}"},{"id":58941,"title":"Compute measurement uncertainty","description":"Suppose a variable  depends on  independent variables , , . If the independent variables have uncertainty , , etc. and the uncertainties are independent, then the uncertainty in  can be estimated with \r\n\r\nFor this problem, the relationship between  and  will be power laws of the form\r\n\r\nFor example, the relationship  would have c = 0.5 and a = [1 2]. \r\nWrite a function that takes a vector of values of , the coefficient , exponents , and a vector of uncertainties  and returns the absolute uncertainty , relative uncertainty , and the index of the variable that contributes the most to the uncertainty in . Identifying the largest contributor to the uncertainty tells the experimentalist which measurement to target first to improve the measurement.  ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 307.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 153.8px; transform-origin: 407px 153.8px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.8583px 8px; transform-origin: 61.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSuppose a variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.4583px 8px; transform-origin: 40.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e depends on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72.3667px 8px; transform-origin: 72.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e independent variables \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" alt=\"x1\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" alt=\"x2\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"56.5\" height=\"20\" alt=\"x3,...,xn\" style=\"width: 56.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.092px 8px; transform-origin: 145.092px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. If the independent variables have uncertainty \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAoCAYAAABTsMJyAAACuUlEQVRoQ+1XPUscURTd7SOSpLLWIgFBi/jRaCHEQLALmI9qC9FYiYWCVmKRgBYWKTSWFkHzD5L0Ek2VKhYqaGEqlRB/gOfAu3Bdd+bd93Zndllm4LDLzJt777nfUy610VVuIy6lgkyrRrOITBGZHDwQm2YXsO0T8DEHG80qYsiMQ/o34AboMGvK4WAMmZ+wa8jZ9ga/eznYaVIRSqYbUo+V5CP8f2rSlMOhUDKMwktn14NWi04ImUcw/hLYBv4BC47MAX6Hc3C8V0UImTVImwX6nVSdboO498urLeMDVjKMyhmwD7xwNjHlJt3/7+p+msmv8XAMGAWeAF8B3pNLHMYU7gFOQvhbybyH0HXgFUDDeQ0Ah0qZTzmbx3PgFFgFpCPKe3TOCMB2TzLMgq0syHBIsk6qO5du06ylGaNyRmPXnWV7fwhUAGkuz5TTjCJLpk8AUVxrpmijqPQxcGXQLs2ER9lAOoEJICitqvVY0ozep7KkecKodTnBIdH5g3dYN7waMnx9ZKQulqEwaQ9jPW06o0JWnM94Z9q9Z4kos2ADmAdqbh0+MlKUvZ70+Y/nMkTTiOvM0E5Ia+3cBdl4hHhiFNPIyOpiSR22VBmiluhIqxcHsFMuJtQas4MzTJpNFBmmwTuAQ9JXmLqgLTVAw34DMm8sW0Q0GfHcFyi0tls9RNMWUEaxD3jr6lDSR7KEaXXtoqGDFU1mCVI+AH+B84TwV99mx5PupKPDdCU4bFknc4C0Yd3auVmQxI563hAyut0audw7JqkjzYG1xEtvETo9+ZxnK454tcCoyOguE0tE3mOX4uc1VxdGuZah0jySnousKDL1Esjq/YJMVp6tV24RmXo9mNX7sphGbQBZGRUql+vMFCDDlQN5BfgB3Pnc8C2aoYqber4g01T3pygvIlNEJgcPtFWa3QKli5ApMjNG5gAAAABJRU5ErkJggg==\" width=\"25.5\" height=\"20\" alt=\"Deltax1\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"20\" alt=\"Deltax2\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.267px 8px; transform-origin: 209.267px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, etc. and the uncertainties are independent, then the uncertainty in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.5667px 8px; transform-origin: 71.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e can be estimated with \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 50.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 25.4px; text-align: left; transform-origin: 384px 25.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"166\" height=\"51\" alt=\"Deltay = sqrt(sum((dy/dxj Deltaxj)^2))\" style=\"width: 166px; height: 51px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.258px 8px; transform-origin: 132.258px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor this problem, the relationship between \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" alt=\"xj\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.7417px 8px; transform-origin: 93.7417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will be power laws of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 27px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13.5px; text-align: left; transform-origin: 384px 13.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"124.5\" height=\"27\" alt=\"y = c x1^a1 x2^a2 x3^a3 ...xn^an\" style=\"width: 124.5px; height: 27px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 34.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 17.4px; text-align: left; transform-origin: 384px 17.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.1917px 8px; transform-origin: 92.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"75\" height=\"35\" alt=\"KE = (1/2)mv^2\" style=\"width: 75px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.2917px 8px; transform-origin: 39.2917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e would have \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.95px 8px; transform-origin: 26.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ec = 0.5\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 34.65px 8px; transform-origin: 34.65px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ea = [1 2]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147.267px 8px; transform-origin: 147.267px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes a vector of values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.875px 8px; transform-origin: 48.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the coefficient \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1167px 8px; transform-origin: 38.1167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, exponents \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.7333px 8px; transform-origin: 93.7333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and a vector of uncertainties \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"21.5\" height=\"18\" alt=\"Deltax\" style=\"width: 21.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and returns the absolute uncertainty \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"21.5\" height=\"18\" alt=\"Deltay\" style=\"width: 21.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.5667px 8px; transform-origin: 64.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, relative uncertainty \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37\" height=\"18.5\" alt=\"Deltay/y\" style=\"width: 37px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 190.308px 8px; transform-origin: 190.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the index of the variable that contributes the most to the uncertainty in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.225px 8px; transform-origin: 330.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Identifying the largest contributor to the uncertainty tells the experimentalist which measurement to target first to improve the measurement.  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [dy,dyrel,i1] = uncertainty1(x,c,a,dx)\r\n%  dy    = uncertainty in y\r\n%  dyrel = relative uncertainty in y (i.e., dy/y)\r\n%  i1    = index of the variable that contributes most to the uncertainty\r\n%  x     = values of the independent variables\r\n%  c     = coefficient of the relationship\r\n%  a     = vector of exponents\r\n%  dx    = vector of uncertainties\r\n\r\n   dy = sqrt(sum((dydx*dx).^2));\r\n   dyrel = dy/y;\r\n   i1 = find(max(dy));\r\nend","test_suite":"%% Volume of a cylinder: V = (pi/4)*D^2*H\r\nD = 75;         %  Diameter (mm)\r\ndD = 3;         %  Diameter uncertainty (mm)\r\nH = 75;         %  Height (mm)\r\ndH = 3;         %  Height uncertainty (mm)\r\n[dV,dVrel,i1] = uncertainty1([D H],pi/4,[2 1],[dD dH]);\r\ndV_correct = 2.9636e+04;\r\ndVrel_correct = 0.0894;\r\ni1_correct = 1;\r\nassert(abs(dV-dV_correct)\u003c1)\r\nassert(abs(dVrel-dVrel_correct)\u003c1e-4)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Hydraulic conductivity from constant-head permeameter test: K = VL/TAd\r\nV  = 7.54e-5;         %  Volume (m3)\r\ndV = 1e-6;            %  Volume uncertainty (m3)\r\nL  = 0.1;             %  Length (m)\r\ndL = 1e-3;            %  Length uncertainty (m)\r\nT  = 1000;            %  Time (sec)\r\ndT = 1;               %  Time uncertainty (sec)\r\nA  = 1.2657e-3;       %  Area (m2)\r\ndA = 6.3e-5;          %  Area uncertainty (m2)\r\nd  = 0.05;            %  Head difference (m)\r\ndd = 1e-3;            %  Head difference uncertainty (m)\r\n[dK,dKrel,i1] = uncertainty1([V L T A d],1,[1 1 -1 -1 -1],[dV dL dT dA dd]);\r\ndK_correct = 6.691e-6;\r\ndKrel_correct = 0.0562;\r\ni1_correct = 4;\r\nassert(abs(dK-dK_correct)\u003c1e-9)\r\nassert(abs(dKrel-dKrel_correct)\u003c1e-4)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Travel time: T = ne L^2/(K dh)\r\nne  = 0.1;            %  Effective porosity\r\ndne = 0.02;           %  Effective porosity uncertainty \r\nL   = 20;             %  Length (m)\r\ndL  = 0.3;            %  Length uncertainty (m)\r\nK   = 0.5;            %  Hydraulic conductivity (m/d)\r\ndK  = 0.08;           %  Hydraulic conductivity uncertainty (m/d)\r\ndh  = 1.5;            %  Head difference (m)\r\nddh = 0.2;            %  Head difference uncertainty (m)\r\n[dT,dTrel,i1] = uncertainty1([ne L K dh],1,[1 2 -1 -1],[dne dL dK ddh]);\r\ndT_correct = 15.48;\r\ndTrel_correct = 0.290;\r\ni1_correct = 1;\r\nassert(abs(dT-dT_correct)\u003c1e-2)\r\nassert(abs(dTrel-dTrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Travel time: T = ne L^2/(K dh)\r\nne  = 0.1;            %  Effective porosity\r\ndne = 0.01;           %  Effective porosity uncertainty \r\nL   = 20;             %  Length (m)\r\ndL  = 0.3;            %  Length uncertainty (m)\r\nK   = 0.5;            %  Hydraulic conductivity (m/d)\r\ndK  = 0.08;           %  Hydraulic conductivity uncertainty (m/d)\r\ndh  = 1.5;            %  Head difference (m)\r\nddh = 0.2;            %  Head difference uncertainty (m)\r\n[dT,dTrel,i1] = uncertainty1([ne L K dh],1,[1 2 -1 -1],[dne dL dK ddh]);\r\ndT_correct = 12.43;\r\ndTrel_correct = 0.233;\r\ni1_correct = 3;\r\nassert(abs(dT-dT_correct)\u003c1e-2)\r\nassert(abs(dTrel-dTrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Travel time: T = ne L^2/(K dh)\r\nne  = 0.1;            %  Effective porosity\r\ndne = 0.01;           %  Effective porosity uncertainty \r\nL   = 20;             %  Length (m)\r\ndL  = 1.7;            %  Length uncertainty (m)\r\nK   = 0.5;            %  Hydraulic conductivity (m/d)\r\ndK  = 0.08;           %  Hydraulic conductivity uncertainty (m/d)\r\ndh  = 1.5;            %  Head difference (m)\r\nddh = 0.2;            %  Head difference uncertainty (m)\r\n[dT,dTrel,i1] = uncertainty1([ne L K dh],1,[1 2 -1 -1],[dne dL dK ddh]);\r\ndT_correct = 15.3;\r\ndTrel_correct = 0.287;\r\ni1_correct = 2;\r\nassert(abs(dT-dT_correct)\u003c1e-2)\r\nassert(abs(dTrel-dTrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Discharge by Manning's equation: Q = (1/n) R^(2/3) S0^(1/2) A = (1/n) A^(5/3) S0^(1/2)/P^(2/3)\r\nn   = 0.015;          %  Manning roughness coefficient\r\ndn  = 0.002;          %  Manning roughness coefficient uncertainty \r\nA   = 0.02;           %  Area (m2)\r\ndA  = 2e-3;           %  Area uncertainty (m2)\r\nP   = 0.04;           %  Wetted perimeter (m)\r\ndP  = 8e-3;           %  Wetted perimeter uncertainty (m)\r\nS0  = 0.01;           %  Slope (-)\r\ndS0 = 7e-4;           %  Slope uncertainty (-)\r\n[dQ,dQrel,i1] = uncertainty1([n A P S0],1,[-1 5/3 -2/3 1/2],[dn dA dP dS0]);\r\ndQ_correct = 2.13e-2;\r\ndQrel_correct = 0.254;\r\ni1_correct = 2;\r\nassert(abs(dQ-dQ_correct)\u003c1e-4)\r\nassert(abs(dQrel-dQrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n\r\n%% Discharge by Manning's equation: Q = (1/n) R^(2/3) S0^(1/2) A = (1/n) A^(5/3) S0^(1/2)/P^(2/3)\r\nn   = 0.015;          %  Manning roughness coefficient\r\ndn  = 0.002;          %  Manning roughness coefficient uncertainty \r\nA   = 0.02;           %  Area (m2)\r\ndA  = 1e-3;           %  Area uncertainty (m2)\r\nP   = 0.04;           %  Wetted perimeter (m)\r\ndP  = 4e-3;           %  Wetted perimeter uncertainty (m)\r\nS0  = 0.01;           %  Slope (-)\r\ndS0 = 7e-4;           %  Slope uncertainty (-)\r\n[dQ,dQrel,i1] = uncertainty1([n A P S0],1,[-1 5/3 -2/3 1/2],[dn dA dP dS0]);\r\ndQ_correct = 1.46e-2;\r\ndQrel_correct = 0.174;\r\ni1_correct = 1;\r\nassert(abs(dQ-dQ_correct)\u003c1e-4)\r\nassert(abs(dQrel-dQrel_correct)\u003c1e-3)\r\nassert(isequal(i1,i1_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2023-12-30T14:25:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-09-02T03:35:52.000Z","updated_at":"2026-02-10T14:38:18.000Z","published_at":"2023-09-02T03:35:52.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose a variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e independent variables \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x3,...,xn\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_3,\\\\ldots, x_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. If the independent variables have uncertainty \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltax1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltax2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, etc. and the uncertainties are independent, then the uncertainty in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be estimated with \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltay = sqrt(sum((dy/dxj Deltaxj)^2))\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta y = \\\\left[\\\\sum_{j = 1}^n \\\\left(\\\\frac{\\\\partial y}{\\\\partial x_j}\\\\Delta x_j\\\\right)^2\\\\right]^{1/2}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, the relationship between \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"xj\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_j\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will be power laws of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = c x1^a1 x2^a2 x3^a3 ...xn^an\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = c x_1^{a_1} x_2^{a_2} x_3^{a_3}\\\\cdots x_n^{a_n}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"KE = (1/2)mv^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eKE = \\\\frac{1}{2} m v^2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e would have \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ec = 0.5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea = [1 2]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes a vector of values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the coefficient \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"c\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, exponents \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and a vector of uncertainties \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltax\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and returns the absolute uncertainty \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltay\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, relative uncertainty \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Deltay/y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta y/y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the index of the variable that contributes the most to the uncertainty in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Identifying the largest contributor to the uncertainty tells the experimentalist which measurement to target first to improve the measurement.  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"derivatives\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"derivatives\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"derivatives\"","","\"","derivatives","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f7724f573a8\u003e":null,"#\u003cMathWorks::Search::Field:0x00007f7724f57308\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f7724f56a48\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f7724f57628\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f7724f57588\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f7724f574e8\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f7724f57448\u003e":"tag:\"derivatives\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f7724f57448\u003e":"tag:\"derivatives\""},"queried_facets":{}},"query_backend":{"connection":{"configuration":{"index_url":"http://index-op-v2/solr/","query_url":"http://search-op-v2/solr/","direct_access_index_urls":["http://index-op-v2/solr/"],"direct_access_query_urls":["http://search-op-v2/solr/"],"timeout":10,"vhost":"search","exchange":"search.topic","heartbeat":30,"pre_index_mode":false,"host":"rabbitmq-eks","port":5672,"username":"search","password":"J3bGPZzQ7asjJcCk","virtual_host":"search","indexer":"amqp","http_logging":"true","core":"cody"},"query_connection":{"uri":"http://search-op-v2/solr/cody/","proxy":null,"connection":{"parallel_manager":null,"headers":{"User-Agent":"Faraday v1.0.1"},"params":{},"options":{"params_encoder":"Faraday::FlatParamsEncoder","proxy":null,"bind":null,"timeout":null,"open_timeout":null,"read_timeout":null,"write_timeout":null,"boundary":null,"oauth":null,"context":null,"on_data":null},"ssl":{"verify":true,"ca_file":null,"ca_path":null,"verify_mode":null,"cert_store":null,"client_cert":null,"client_key":null,"certificate":null,"private_key":null,"verify_depth":null,"version":null,"min_version":null,"max_version":null},"default_parallel_manager":null,"builder":{"adapter":{"name":"Faraday::Adapter::NetHttp","args":[],"block":null},"handlers":[{"name":"Faraday::Response::RaiseError","args":[],"block":null}],"app":{"app":{"ssl_cert_store":{"verify_callback":null,"error":null,"error_string":null,"chain":null,"time":null},"app":{},"connection_options":{},"config_block":null}}},"url_prefix":"http://search-op-v2/solr/cody/","manual_proxy":false,"proxy":null},"update_format":"RSolr::JSON::Generator","update_path":"update","options":{"url":"http://search-op-v2/solr/cody"}}},"query":{"params":{"per_page":50,"term":"tag:\"derivatives\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"derivatives\"","","\"","derivatives","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007f7724f573a8\u003e":null,"#\u003cMathWorks::Search::Field:0x00007f7724f57308\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007f7724f56a48\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007f7724f57628\u003e":1,"#\u003cMathWorks::Search::Field:0x00007f7724f57588\u003e":50,"#\u003cMathWorks::Search::Field:0x00007f7724f574e8\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007f7724f57448\u003e":"tag:\"derivatives\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007f7724f57448\u003e":"tag:\"derivatives\""},"queried_facets":{}},"options":{"fields":["id","difficulty_rating"]},"join":" "},"results":[{"id":45280,"difficulty_rating":"easy"},{"id":58941,"difficulty_rating":"medium"}]}}