Problem 55970. Interpolated Value Between Two Points

Created by Ben Placek

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Given two points, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>(x_1,y_1)</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>(x_2,y_2)</w:t></w:r></w:customXml><w:r><w:t>, a new location x, and a method </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>\"linear\"</w:t></w:r><w:r><w:t> or </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>\"cubic\"</w:t></w:r><w:r><w:t>, return the value of a linear or cubic interpolant through the two points at the new location x, using the formulas below.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Linear: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>y = (\\Delta y)t + y_1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Cubic: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>y = -2(\\Delta y)t^3 + 3(\\Delta y)t^2 + y_1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where  </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\Delta y = y_2 - y_1</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>t = \\frac{x - x_1}{x_2 - x_1}</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>(This formulation for the cubic interpolant has zero derivative at each interpolation point.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x_2 \\le x_1</w:t></w:r></w:customXml><w:r><w:t>, your function should produce an error with the message \"x2 must be greater than x1\".</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x \\notin [x_1,x_2]</w:t></w:r></w:customXml><w:r><w:t>, your function should produce a warning with the message \"x is outside the interval [x1,x2]\" (but still return the value of the interpolant).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the method is given as anything other than </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>\"linear\"</w:t></w:r><w:r><w:t> or </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>\"cubic\"</w:t></w:r><w:r><w:t>, your function should produce an error with the message \"method must be either \"linear\" or \"cubic\"\".</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div 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; "><div style="block-size: 348.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 174.25px; transform-origin: 407px 174.25px; vertical-align: baseline; "><div style="block-size: 43px; 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 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 56px 8px; transform-origin: 56px 8px; unicode-bidi: normal; "><span style="">Given two points, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 44.5px; height: 20px;" width="44.5" height="20"></span><span 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 44.5px; height: 20px;" width="44.5" height="20"></span><span 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: 103.5px 8px; transform-origin: 103.5px 8px; unicode-bidi: normal; "><span style="">, a new location x, and a method </span></span><span 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 32px 8.5px; transform-origin: 32px 8.5px; ">"linear"</span></span><span 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: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; "><span style=""> or </span></span><span 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: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 28px 8.5px; transform-origin: 28px 8.5px; ">"cubic"</span></span><span 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: 89.5px 8px; transform-origin: 89.5px 8px; unicode-bidi: normal; "><span style="">, return the value of a linear or cubic interpolant through the two points at the new location x, using the formulas below.</span></span></div><div 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; "><span 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: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "><span style="">Linear: </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 93px; height: 20px;" width="93" height="20"></span></div><div 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; "><span 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: 22px 8px; transform-origin: 22px 8px; unicode-bidi: normal; "><span style="">Cubic: </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 180px; height: 21px;" width="180" height="21"></span></div><div style="block-size: 40px; 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 20px; text-align: left; transform-origin: 384px 20px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; "><span style="">where  </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 80.5px; height: 20px;" width="80.5" height="20"></span><span 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" style="width: 66.5px; height: 40px;" width="66.5" height="40"></span><span 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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "><span style="">.</span></span></div><div style="block-size: 21px; 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 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 279.5px 8px; transform-origin: 279.5px 8px; unicode-bidi: normal; "><span style="">(This formulation for the cubic interpolant has zero derivative at each interpolation point.)</span></span></div><div 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; "><span 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: 5.5px 8px; transform-origin: 5.5px 8px; unicode-bidi: normal; "><span style="">If </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" style="width: 45px; height: 20px;" width="45" height="20"></span><span 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: 273px 8px; transform-origin: 273px 8px; unicode-bidi: normal; "><span style="">, your function should produce an error with the message "x2 must be greater than x1".</span></span></div><div style="block-size: 43px; 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 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 5.5px 8px; transform-origin: 5.5px 8px; unicode-bidi: normal; "><span style="">If </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIoAAAAoCAYAAAAhU2KBAAAFXElEQVR4Xu1bua9OQRT3/gGCTmkpREFBJEKjsLbWULzOUmgIiSUKS0JoFJZE8aJBEJU9oSBiK0iIwhJR6ETwB/D7xT0yxtw7Z5Z7833vziQn733fPTP3nDO/OXPOmflGJpRWLKCwwIiCp7AUC0woQCkgUFmgT0DZXWORx/j+kcpa44dpOlRZU6POVXz/0X7WJ6D8gvIvQF8tI4zh86XxgwGVJnPBdczBuRzfLXEtnL4BxWkElWn7wcTFNNRAmQoFZoN+gl5FzlmtESLHG4/dhh4opzEr20AbE7aJAhQ/tIcaKAy8PlS00BFj+NX/w1GA4rfUUANFvMlW6HnOr2stRwGK33hDA5TFli4T8fkG6BtoM+hH9fxthGcpQBlyoGyA/DtB8/16/OWYh/9Cg9oCFL+BB9KjMJO5bQHErHPwOcFDb/LM0nE7Pv9XFPLYoQClA6AwoJwGcrl8bhehaasNkj0Yw64I5opNxDyhQMmts3+amjm6kCfKo7B6twW0DDSj0uEo/u6r/ufza9Uzrnqufu0qpydhFZDZzCjILqFz7JcgepgFqRau+muA0qbOMWp0LU8UUIjg7yCWvN9XgODEMkWdVE3iGfxdD5pSkcYY9EAPK8bV+HvT0UmAVPdc8x6bRwOUtnSOkZd9upYnCiimcjxQk7MBepfLoJOgmDOSI+i3F3QHtMJhQQFSTm/C12iA0pbOsUDpWp5koJhegBN4FyRbUKgRnqMDtyluY7ccnffjO25LjFt4sutqXaTHOXUOtZGLvwt5koEiK5J/GY9wq4ltFCa1dZUei6ypOqfqK/3blicLUCRuSN0SRFnGO4x9zEZPwua6DmDydZUeh+q8CkKeAu0AuWKvVMBo5WGosBYktSnGkgdA9hULW54sQGE8wsCVq2uW4qV1RhFlubUcN5jaik3M1Rh6zUCrM2VfCWLsxZYzCDftqJGHZQV6/AsgVralmKmZt2SgME0+a0icYggJZm3PJEZIGbtpxYYGsyE6M41lpbiNbE100sgjF5I2GQuZNaunICYhvvOyJKBITYMv4nbBZtZTmibH9UzG4zO5NpDrhDgXUGJ1bgsoWnm45bwG2dueLE7fvEUDRdAoe65rj6SrOwHSFts4mTKOFOp24TveN/EhPhSUJr/Wo6To7AMKt6hFoM+ge8aqb9IrRR4ZV8obPvsGAYUuju0J6DzoCkhiCbuewgnmZIemylwhD0CTq/6iUEh1NxQ0TUDJpXMTUMRritysRfFA1NVyySNjy3GIL1tUA8XM1fkSWxlz23A9D5k8Gumi0YHbGj2XXCVoGitnHSWnzk1AsW1H/bid2544pzxiQ2aXrH0xW2xqaqDQzR0CEf33QWZWIi+gV1na8DwELLZRtH19K8M1Tp0Rcurs23rEdnIy7gJKTnloBy7IwyCe2fnCAzVQtBOVi0+2Mub5nyoAasbuqo6ikcXk8QFFeLlAxkAzQ18QyC/xzSj6aX67NJBAoRLvKsXbjE3EttpgNnAu/mHXAoWlAMZoKVc7NXLyuOQgSFv8G0igiDfxpWwag2h4BgUoDCxjEgCNjiYP38PaTggYBw4oXXsTGpBGYHr4xrL4F3z27d3aSfJ5FMlyYk7dtTKQrw6M3PLmgHhJjL+TshuvfwzUD8B4JsIM53og4kOMZfPWHUbaRwkp75CT8baqyxrZWFxjTYpxH+8TSeMdIh7BMKjlbUW5E2SPOVBA0Sg8TDxMfdeB5KyHxxOMDVg69x3E5dTTLjnYYzfVbhrl6NNvj3NOSO/GKkDp3ZTHKVyAEme33vUqQOndlMcpXIASZ7fe9SpA6d2UxylcgBJnt971+g3YZMc4OO7j9AAAAABJRU5ErkJggg==" style="width: 69px; height: 20px;" width="69" height="20"></span><span 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: 335.5px 8px; transform-origin: 335.5px 8px; unicode-bidi: normal; "><span style="">, your function should produce a warning with the message "x is outside the interval [x1,x2]" (but still return the value of the interpolant).</span></span></div><div style="block-size: 42.5px; 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 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 142px 8px; transform-origin: 142px 8px; unicode-bidi: normal; "><span style="">If the method is given as anything other than </span></span><span 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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 32px 8.5px; transform-origin: 32px 8.5px; ">"linear"</span></span><span 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: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; "><span style=""> or </span></span><span 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: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 28px 8.5px; transform-origin: 28px 8.5px; ">"cubic"</span></span><span 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: 151.5px 8px; transform-origin: 151.5px 8px; unicode-bidi: normal; "><span style="">, your function should produce an error with the message "method must be either "linear" or "cubic"".</span></span></div><div style="block-size: 21px; 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 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; "><span style=""></span></span></div></div></div>

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Last Solution submitted on Jan 14, 2023

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goc3 on 21 Oct 2022

This problem is not necessarily easy because it tests two disparate things. One of those things, warnings and errors, is rarely presented or tested on Cody. It would be helpful to have more problems that test warning and error usage.

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Problem 55970. Interpolated Value Between Two Points

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Problem 55970. Interpolated Value Between Two Points

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