Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.
"}, "extensions": [], "variables": {"a": {"definition": "random(1,4,8,9,16,27,32,25,36,49)", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s1": {"definition": "random(1,-1)", "name": "s1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "tm2": {"definition": "-(n-1)*tm1*b", "name": "tm2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "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))", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "tm0": {"definition": "a^(1/n)", "name": "tm0", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "tm1": {"definition": "tm0*b", "name": "tm1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "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)))))", "name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "ungrouped_variables": ["a", "tm0", "tm2", "b", "tm1", "s1", "n"], "advice": "The general formula for the Maclaurin Series for a function $f(x)$ is
\n$f(x) = f(0) + f'(0)x + \\frac{f''(0)}{2!}x^2 + \\dots + \\frac{f^{n}(0)}{n!}x^n + \\dots$
\nFor this example,
\\[\\begin{eqnarray*} f'(x)&=&\\simplify[all,fractionNumbers]{{b}/{n}*({a}+{b}x)^(-{n-1}/{n})}\\\\[10pt] 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*} f(0)&=\\simplify[all]{{a}^(1/{n})={tm0}}\\\\[10pt] f'(0)&=\\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\\\[10pt] f''(0)&=\\simplify[all,fractionNumbers]{{tm2}/{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} \\]
For further information see Chapter 2 - Series notes.
", "functions": {}, "tags": [], "statement": "Consider the function $f(x)=(\\simplify[all]{{a}+{b}*x})^{1/\\var{n}}$.
", "name": "Maclaurin series (first three terms)", "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "parts": [{"unitTests": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "variableReplacements": [], "checkingAccuracy": 0.001, "expectedVariableNames": [], "customMarkingAlgorithm": "", "type": "jme", "marks": 4, "failureRate": 1, "checkVariableNames": false, "showCorrectAnswer": true, "answerSimplification": "all,fractionNumbers,!collectNumbers", "showPreview": true, "vsetRange": [0, 1], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "answer": "{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2", "prompt": "Find the first 3 terms in the Maclaurin series for $f(x)$, i.e. up to terms in $x^2$.
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Input coefficients as fractions, not as decimals. Also do not use factorials in your answer. For example, input 6 rather than 3!.