// Numbas version: exam_results_page_options
{"name": "Adrian's copy of Maclaurin 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+{tm2}/{2*a^2*n^2}*x^2", "prompt": "<p>First 3 terms = ?<br/>Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.</p>", "showpreview": true, "scripts": {}, "checkingaccuracy": 1e-06, "showCorrectAnswer": true, "expectedvariablenames": [], "checkvariablenames": false, "vsetrange": [0, 1], "answersimplification": "all,fractionNumbers,!collectNumbers", "type": "jme", "marks": 4, "checkingtype": "absdiff", "vsetrangepoints": 5}], "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "n"], "rulesets": {}, "preamble": {"js": "", "css": ""}, "statement": "<p>You are asked to find the first 3 terms in the MacLaurin series for $f(x)=(\\simplify[all]{{a}+{b}*x})^{1/\\var{n}}$ i.e. up to terms in $x^2$.</p>", "variables": {"tm0": {"description": "", "group": "Ungrouped variables", "name": "tm0", "definition": "a^(1/n)", "templateType": "anything"}, "tm1": {"description": "", "group": "Ungrouped variables", "name": "tm1", "definition": "tm0*b", "templateType": "anything"}, "b": {"description": "", "group": "Ungrouped variables", "name": "b", "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))", "templateType": "anything"}, "a": {"description": "", "group": "Ungrouped variables", "name": "a", "definition": "random(1,4,8,9,16,27,32,25,36,49)", "templateType": "anything"}, "s1": {"description": "", "group": "Ungrouped variables", "name": "s1", "definition": "random(1,-1)", "templateType": "anything"}, "n": {"description": "", "group": "Ungrouped variables", "name": "n", "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)))))", "templateType": "anything"}, "tm2": {"description": "", "group": "Ungrouped variables", "name": "tm2", "definition": "-(n-1)*tm1*b", "templateType": "anything"}}, "variable_groups": [], "showQuestionGroupNames": false, "functions": {}, "type": "question", "tags": ["approximate", "approximations", "calculus", "Calculus", "checked2015", "first three terms in a maclaurin series", "First three terms in MacLaurin series", "functions", "maclaurin series", "MacLaurin series", "mas1601", "MAS1601"], "metadata": {"description": "<p>Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.</p>", "licence": "Creative Commons Attribution 4.0 International", "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><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 in Advice.</p>\n                \t\t<p>&nbsp;</p>\n                \t\t"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>The first three terms in the MacLaurin series are given by $a+bx+cx^2$ where $\\displaystyle a=f(0),\\;\\;b=f'(0),\\;\\;c=\\frac{f''(0)}{2}$<br/>For this example, <br/>\\[\\begin{eqnarray*} f'(x)&amp;=&amp;\\simplify[all,fractionNumbers]{{b}/{n}*({a}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&amp;=&amp;\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \\]<br/>and so we get:<br/>\\[\\begin{eqnarray*} a&amp;=&amp;f(0)=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&amp;=&amp;f'(0)=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&amp;=&amp;\\frac{f''(0)}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\\]<br/>Hence the first three terms of the MacLaurin series are:<br/>\\[\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2} \\]</p>", "name": "Adrian's copy of Maclaurin series (three terms)", "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}