// Numbas version: exam_results_page_options
{"name": "Function approximation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "n"}, "est": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a^(m/n)+h*m*a^(m/n-1)/n,5)", "description": "", "name": "est"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a^(1/n),0)", "description": "", "name": "c"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n=2,random(4,9,16,25,36),if(n=3,random(8,27,64),if(n=4,random(16,81),random(32,243))))", "description": "", "name": "a"}, "tr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((a+h)^(m/n),5)", "description": "", "name": "tr"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m-n", "description": "", "name": "p"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n=2 or n=4,random(1,3,5), if(n=3,random(1,2,4,5),random(1,2,3,4,6)))", "description": "", "name": "m"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(random(1,-1)*random(0.001..0.1#0.001),3)", "description": "", "name": "h"}}, "ungrouped_variables": ["a", "c", "est", "h", "s1", "tr", "m", "n", "p"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Function approximation", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "x^({m}/{n})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{a}", "minValue": "{a}", "correctAnswerFraction": false, "marks": 0.25, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{h}", "minValue": "{h}", "correctAnswerFraction": false, "marks": 0.25, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{est}", "minValue": "{est}", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{tr}", "minValue": "{tr}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n        <p>In this case $f(x) =\\;$ [[0]].</p>\n      <p>$a= \\;$[[1]] and $h=\\; $[[2]]<br/>Input your estimation to $5$ decimal places: [[3]]</p>\n      <p>True value is: [[4]] (input to 5 decimal places).</p>\n        ", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Use the approximation $f(a+h) \\approx f(a)+hf^{\\prime}(a)$ to estimate \\[\\var{a+h}^{\\frac{\\var{m}}{\\var{n}}} \\]for a suitable function $f(x)$.</p>", "tags": ["application of differentiation", "approximation", "approximation of the value of a function using the tangent", "approximations", "calculus", "Calculus", "checked2015", "equation of tangent", "first order approximation", "functions", "maclaurin series", "MacLaurin series", "mas1601", "MAS1601", "tangent equation"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                        \t\t<p><strong>3/07/2012<br /></strong></p>\n                        \t\t<p>Added tags</p>\n                        \t\t<p><strong>20/06/2012:</strong></p>\n                        \t\t<p><strong><br /></strong>Added tags.</p>\n                        \t\t<p>Got rid of request for 5dps for the function!</p>\n                        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Approximate $f(x)=(a+h)^{m/n}$ by $f(a)+hf^{\\prime}(a)$ to 5 decimal places and compare with true value.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>We have $f(x)=x^\\frac{\\var{m}}{\\var{n}}$ and $a= \\var{a}$, $h=\\var{h}$.<br/>Note that $\\var{a}^\\frac{1}{\\var{n}}=\\var{c}$ and so using the approximation :<br/>$f(a+h)\\approx f(a)+hf^{\\prime}(a)$ and $f^{\\prime}(x) = \\frac{\\var{m}}{\\var{n}}\\simplify[std]{x^({m-n}/{n})}$ <br/>we have:<br/>\\[\\simplify[std]{{a+h}^({m}/{n})}\\approx \\simplify[simplifyFractions]{{a}^({m}/{n})+{h}*({m}/{n})*{a}^({p}/{n})}=\\simplify[std,!sqrtSquare]{{c}^{m}+ {h}*({m}/{n})*{c}^{m-n}}=\\var{est}\\]<br/>to 5 decimal places.<br/>The true value to 5 decimal places is {tr}.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}