// Numbas version: exam_results_page_options {"questions": [], "duration": 0, "name": "Maclaurin and Taylor series", "showQuestionGroupNames": false, "allQuestions": true, "percentPass": 0, "feedback": {"showanswerstate": true, "advicethreshold": 0, "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true}, "shuffleQuestions": false, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Maclaurin series (three terms)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "tm1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tm0*b", "description": "", "name": "tm1"}, "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)))))", "description": "", "name": "n"}, "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))", "description": "", "name": "b"}, "tm0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a^(1/n)", "description": "", "name": "tm0"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,4,8,9,16,27,32,25,36,49)", "description": "", "name": "a"}, "tm2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-(n-1)*tm1*b", "description": "", "name": "tm2"}}, "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"answer": "{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2", "vsetrange": [0, 1], "checkingaccuracy": 1e-06, "showCorrectAnswer": true, "expectedvariablenames": [], "prompt": "

First 3 terms = ?
Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.

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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 $x^2$.

", "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"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

20/06/2012:

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Added !collectNumbers to some rules so that polynomials presented in standard order.

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3/07/2012:

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9/07/2012:

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Improved display of first line in Advice.

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Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The first three terms in the MacLaurin series are given by $a+bx+cx^2$ where $\\displaystyle a=f(0),\\;\\;b=f'(0),\\;\\;c=\\frac{f''(0)}{2}$
For this example,
\$\\begin{eqnarray*} f'(x)&=&\\simplify[all,fractionNumbers]{{b}/{n}*({a}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&=&\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \$
and so we get:
\$\\begin{eqnarray*} a&=&f(0)=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&=&f'(0)=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&=&\\frac{f''(0)}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\$
Hence the first three terms of the MacLaurin series are:
\$\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2} \$

"}, {"name": "Taylor series (three terms)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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

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 $c$.

\n

Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.

\n ", "notallowed": {"message": "

Do not input factorials or decimals in the Taylor series.

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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 $x^2$.

", "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

20/06/2012:

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Added !collectNumbers to some rules so that polynomials presented in standard order.

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Added more explanation to prompt in question.

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Also included ! and . in forbidden strings together with message.

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3/07/2012:

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9/07/2012:

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Improved display of first line of Advice.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

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 $x^2$.

The first three terms in the Taylor series are given by $\\simplify[all]{a+b(x-{c})+c(x-{c})^2}$ where $\\displaystyle a=f(\\var{c}),\\;\\;b=f'(\\var{c}),\\;\\;c=\\frac{f''(\\var{c})}{2}$
For this example,
\$\\begin{eqnarray*} f'(x)&=&\\simplify[all,fractionNumbers]{{b}/{n}*({a-b*c}+{b}x)^(-{n-1}/{n})}\\\\ f''(x)&=&\\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a-b*c}+{b}x)^(-{2*n-1}/{n})} \\end{eqnarray*} \$
and so we get:
\$\\begin{eqnarray*} a&=&f(\\var{c})=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\ b&=&f'(\\var{c})=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\ c&=&\\frac{f''(\\var{c})}{2}=\\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \\end{eqnarray*}\$
Hence the first three terms of the Taylor series are:
\$\\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*(x-{c})+{tm2}/{2*a^2*n^2}*(x-{c})^2} \$

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Find the first few terms of the Maclaurin and Taylor series of given functions.

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