// Numbas version: finer_feedback_settings
{"name": "Taylor series (three terms)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"answer": "{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "prompt": "\n        <p>Input the first three terms in the Taylor series in the form $a+b(x-\\var{c})+c(x-\\var{c})^2$ for suitable coefficients $a,\\;b$ and $c$.</p>\n        <p><br/>Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.</p>\n        ", "notallowed": {"message": "<p>Do not input factorials or decimals in the Taylor series.</p>", "showStrings": false, "partialCredit": 0, "strings": ["!", "."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 1e-06, "type": "jme", "answersimplification": "all,fractionNumbers,!collectNumbers", "marks": 4, "vsetrangepoints": 5}], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "tm1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tm0*b", "name": "tm1", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a=4 or a=9 or a=25 or a=36 or a=49,2,if(a=8 or a=27,3,if(a=32,5,if(a=16,random(2,4),random(2..5)))))", "name": "n", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,4,8,9,16,27,32,25,36,49)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "c", "description": ""}, "tm0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a^(1/n)", "name": "tm0", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*switch(a=1,random(2..9),a=4,random(3,5,7,9),a=8,random(1,3,5,7,9),a=9,random(1,2,4,5,7,8),a=16,random(1,3,5,7,9),a=32,random(1,3,5,7,9),a=25,random(1,2,4,6,7,9),a=27,random(1,2,4,5,7,8),a=36,random(1,5,7,9),random(1,2,3,4,5,8,9))", "name": "b", "description": ""}, "tm2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-(n-1)*tm1*b", "name": "tm2", "description": ""}}, "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "c", "n"], "rulesets": {}, "name": "Taylor series (three terms)", "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are asked to find the first 3 terms in the Taylor series at $x=\\var{c}$ for $f(x)=(\\simplify[all]{{a-b*c}+{b}*x})^{1/\\var{n}}$ i.e. up to terms in $x^2$.</p>", "tags": ["3 term Taylor series", "approximation", "approximations", "Calculus", "calculus", "checked2015", "function", "functions", "mas1601", "MAS1601", "series approximation", "series expansion", "Taylor series", "taylor series", "three term Taylor series"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                                        \t\t<p><strong>20/06/2012:</strong></p>\n                                        \t\t<p>Added tags.</p>\n                                        \t\t<p>Added !collectNumbers to some rules so that polynomials presented in standard order.</p>\n                                        \t\t<p>Added more explanation to prompt in question.</p>\n                                        \t\t<p>Also included ! and . in forbidden strings together with message.</p>\n                                        \t\t<p><strong>3/07/2012:<br /></strong></p>\n                                        \t\t<p>Added tags.</p>\n                                        \t\t<p><strong>9/07/2012:</strong></p>\n                                        \t\t<p>Improved display of first line of Advice.</p>\n                                        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the first 3 terms in the Taylor series at $x=c$ for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.</p>"}, "functions": {}, "advice": "<p>The first three terms in the Taylor series are given by $\\simplify[all]{a+b(x-{c})+c(x-{c})^2}$ where $\\displaystyle a=f(\\var{c}),\\;\\;b=f'(\\var{c}),\\;\\;c=\\frac{f''(\\var{c})}{2}$<br/>For this example, <br/>\\[\\begin{eqnarray*} f'(x)&amp;=&amp;\\simplify[all,fractionNumbers]{{b}/{n}*({a-b*c}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&amp;=&amp;\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a-b*c}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \\]<br/>and so we get:<br/>\\[\\begin{eqnarray*} a&amp;=&amp;f(\\var{c})=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&amp;=&amp;f'(\\var{c})=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&amp;=&amp;\\frac{f''(\\var{c})}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\\]<br/>Hence the first three terms of the Taylor series are:<br/>\\[\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2} \\]</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/"}]}