// Numbas version: exam_results_page_options {"name": "F\u00f8lger og rekker", "metadata": {"description": "", "licence": "All rights reserved"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", "", "", "", "", "", ""], "questions": [{"name": "Taylor-polynomet", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}, {"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt en funksjon $f(x)=e^{\\var{k}x}$.

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2 .. 7#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["k"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Finn Taylor-polynomet av grad 4 til $f(x)$ i punktet $a=0$.

$p_nf(x)=$ [[0]] $+$ [[1]] $+$ [[2]] $+$ [[3]] $+$ [[4]]

Det er gitt en funksjon $f(x)=\\sin x$

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Grafen til $f(x)=\\sin x$ er tegnet i grønt, og grafen til Taylor-polynomet til $f$ av grad $n$ er tegnet i orange. For hver graf skal du avgjøre hva $n$ er.

", "

"], "matrix": [[0, 0, "1", 0, 0], ["1", 0, 0, 0, 0], [0, 0, 0, 0, "1"], [0, "1", 0, 0, 0], [0, 0, 0, "1", 0]], "layout": {"type": "all", "expression": ""}, "answers": ["Taylor-polynomet av grad 1", "Taylor-polynomet av grad 3", "Taylor-polynomet av grad 5", "Taylor-polynomet av grad 7", "Taylor-polynomet av grad 9"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Taylorpolynom av grad 2 og Hessematrisa", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}, {"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Gitt en funksjon $f(x,y)=e^{\\var{a}x}\\cdot \\sin(\\var{b}y)$ og et punkt $a=\\Big(0,\\dfrac{\\pi}{2}\\Big) $

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Taylorpolynomet av grad 2 blir da

$P_2(x,y)=$[[0]]$+$[[1]] $x+$[[2]]$(y-\\frac{\\pi}{2})+$[[3]]$x^2+$[[4]]$x(y-\\frac{\\pi}{2})+$[[5]]$(y-\\frac{\\pi}{2})^2$

Talylor-polynomet til $f$ av grad $n$ om punktet $a$ er

$T_nf(x)=\\sum\\limits_{k=0}^n \\frac{f^{(0)}(a)}{k!}(x-a)^k$ og restledd $R_nf(x)=f(x)-T_nf(x)$.

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

La $g(x)=e^x$ og $a=0$. Da

$T_0e^x=$ [[0]] og $T_0e^{0.2}=$ [[4]]

$T_1e^x=$ [[1]] og $T_1e^{0.2}=$ [[5]]

$T_2e^x=$ [[2]] og $T_2e^{0.2}=$ [[6]]

$T_3e^x=$ [[3]] og $T_3e^{0.2}=$ [[7]] (avrund til 4 desimaler)

Regn ut med kalkulator $e^{0.2}$ = [[0]] (avrund til 4 desimaler)

La $g(x)=e^x$ og $a=0$, som over. Da er

$R_0e^x=$ [[0]] (avrund til 4 desimaler).

$R_1e^x=$ [[1]] (avrund til 4 desimaler).

$R_2e^x=$ [[2]] (avrund til 4 desimaler).

$R_3e^x=$ [[3]] (avrund til 4 desimaler).

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.2214", "maxValue": "0.2214", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "0.2214", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": ".0214", "maxValue": ".0214", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": ".0214", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": ".0014", "maxValue": ".0014", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": ".0014", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": ".0001", "maxValue": ".0001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": ".0001", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

For hvilken naturlige tall $n$ er $\\lvert R_ne^x(.2) \\rvert $ mindre enn $10^{-6}=.000001$?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$n = 2$", "$n = 3$", "$n = 4$", "$n = 5$", "$n = 100$"], "matrix": [0, 0, "0", "1", "1"], "distractors": ["", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Taylorrekke til 1/(1-x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

La $f(x)=\\dfrac{1}{1-x}$

La oss finne Taylorrekka til $f(x)$ i $x=0$

$f(0)=$ [[0]]

$f'(x)=$ [[1]] $f'(0)=$ [[2]] $\\dfrac{f'(0)}{1!}=$ [[2]]

$f''(x)=$ [[3]] $f''(0)=$ [[4]] $\\dfrac{f''(0)}{2!}=$ [[5]]

$f'''(x)=$ [[6]] $f'''(0)=$ [[7]] $\\dfrac{f'''(0)}{3!}=$ [[8]]

$f^{(n)}(x)=$ [[9]] $f^{(n)}(0)=$ [[10]] $\\dfrac{f^{(n)}(0)}{n!}=$ [[11]]

$Tf(x)=\\sum\\limits_{n=0}^{\\infty}$ [[12]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(1-x)^-2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "2*(1-x)^-3", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "6*(1-x)^-4", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "6", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "n!*(1-x)^(-n-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "n!", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "1", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x^n", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "n", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Taylorrekke til f(x)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/formula.PNG", "/srv/numbas/media/question-resources/formula.PNG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Hvilke av følgende funksjoner $f(x)$ er lik sin Taylorrekke $Tf(x)$ for alle $x$

"], "matrix": ["0.5", "0.5", "0.5", "0.5", 0, 0], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Delsummer", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt en følge $\\{a_n\\}$ slik at $a_n=\\var{a}n+\\var{b}$ og rekken $\\sum\\limits_{n=0}^{\\infty}a_n$ er generert av følgen $\\{a_n\\}$.

", "advice": "", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 6#2)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(3 .. 7#2)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Skriv inn de delsummene

$S_0$=[[0]]

$S_1$=[[1]]

$S_2$=[[2]]

$S_3$=[[3]]

Det er gitt en rekke $\\sum\\limits_{n=0}^{\\infty}a_n$ , der $a_n\\geq0$ for alle $n$.

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Velg riktige påstander

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\sum\\limits_{n=0}^{\\infty}a_n$ konvergerer $\\Rightarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n=0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ konvergerer $\\Leftarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n=0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ konvergerer $\\Leftrightarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n=0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ divergerer $\\Rightarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n\\neq0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ divergerer $\\Leftarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n\\neq0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ divergerer $\\Leftrightarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n\\neq0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ er begrenset $\\Rightarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n=0$", "$\\sum\\limits_{n=0}^{\\infty}a_n$ er begrenset $\\Leftarrow$ $\\lim\\limits_{n\\rightarrow\\infty}a_n=0$"], "matrix": ["1", 0, 0, 0, "1", 0, "1", 0], "distractors": ["", "", "", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Geometriske rekker", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}, {"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt en geometrisk rekke $\\sum\\limits_{k=0}^{\\infty}a_0r^k$

Match påstandene med betingelsene på $r$

", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": false, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Den uendelig rekken konvergerer, og summen er lik $\\frac{a_0}{1-r}$", "Den uendelige rekken divergerer", "Summen av rekken er gitt ved $\\sum\\limits_{k=0}^na_0r^k=(n+1)a_0$"], "matrix": [[0, 0, "1"], [0, "1", 0], ["1", 0, 0]], "layout": {"type": "all", "expression": ""}, "answers": ["$r=1$", "$|r|\\geq1$", "$-1<r<1$"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Grense sammenligningstest", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}, {"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt en rekke $\\sum\\limits_{n=0}^{\\infty}\\frac{n+1}{2n^2+2}$.

Vi regner ut at $\\lim\\limits_{n\\rightarrow\\infty}\\dfrac{\\frac{n+1}{2n^2+2}}{\\frac{1}{n}}=\\lim\\limits_{n\\rightarrow\\infty}\\dfrac{n^2+n}{2n^2+2}=\\frac{1}{2}$

Videre vet vi at at rekken $\\sum\\limits_{n=0}^{\\infty}\\frac{1}{n}$ divergerer.

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Hva kan vi da si om rekken $\\sum\\limits_{n=0}^{\\infty}\\frac{n+1}{2n^2+2}$?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Den konvergerer", "Den divergerer", "Det er umulig å si noe om konvergensen av rekken $\\sum\\limits_{n=0}^{\\infty}\\frac{n+1}{2n^2+2}$"], "matrix": [0, "1", 0], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Konvergens av rekker og subrekker", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt to rekker $\\sum\\limits_{n=0}^{\\infty}a_n$ og $\\sum\\limits_{n=107}^{\\infty}a_n$

Velg riktige påstander

Hvilke av følgende funksjonsfølger konvergerer punktvis mot $e^x$ når $n\\rightarrow\\infty$?

", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$e^{x+n}$", "$T_n e^x$", "$\\frac{e^x}{xn}$", "$\\Big(1+\\frac{x}{n}\\Big)^n$", "$e^x+\\frac{x^2}{n}$", "$e^{x+\\frac{1}{n}}$"], "matrix": [0, "1", 0, "1", "1", "1"], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "punktvis konvergens og uniform konvergens", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

La $\\{f_n(x)\\}, f(x): A\\subseteq\\mathbb{R}\\rightarrow\\mathbb{R}$.

Velg rigtig påstand

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\{f_n(x)\\}$ konvergerer punktvis mot $f(x)$ på $A$ $\\Rightarrow$ $\\{f_n(x)\\}$ konvergerer uniformt mot $f(x)$ på $A$", "$\\{f_n(x)\\}$ konvergerer uniformt mot $f(x)$ på $A$ $\\Rightarrow$ $\\{f_n(x)\\}$ konvergerer punktvis mot $f(x)$ på $A$", "$\\{f_n(x)\\}$ konvergerer punktvis mot $f(x)$ på $A$ $\\Leftrightarrow$ $\\{f_n(x)\\}$ konvergerer uniformt mot $f(x)$ på $A$"], "matrix": [0, "1", 0], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Rekke 1/n", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ida Friestad Pedersen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/792/"}, {"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

For rekken $\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n}$ ser vi at $\\lim\\limits_{n\\rightarrow\\infty}a_n=\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n}=0$

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Hvilken påstand er da riktig?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Rekken $\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n}$ konvergerer siden $\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n}=0$", "Rekken $\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n}$ konvergerer og summen er lik 2.", "Selv om $\\lim\\limits_{n\\rightarrow\\infty}\\frac{1}{n}=0$ divergerer rekken $\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n}$ "], "matrix": [0, 0, "1"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Sammenligningstest", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt to positive rekker $\\sum\\limits_{n=0}^{\\infty}a_n$ og $\\sum\\limits_{n=0}^{\\infty}b_n$ slik at $a_n\\geq b_n$ for alle $n$.

Velg riktige påstander

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\sum\\limits_{n=0}^{\\infty}a_n$ konvergerer $\\Rightarrow$ $\\sum\\limits_{n=0}^{\\infty}b_n$ konvergerer", "$\\sum\\limits_{n=0}^{\\infty}a_n$ konvergerer $\\Leftarrow$ $\\sum\\limits_{n=0}^{\\infty}b_n$ konvergerer", "$\\sum\\limits_{n=0}^{\\infty}a_n$ konvergerer $\\Leftrightarrow$ $\\sum\\limits_{n=0}^{\\infty}b_n$ konvergerer", "$\\sum\\limits_{n=0}^{\\infty}a_n$ divergerer $\\Rightarrow$ $\\sum\\limits_{n=0}^{\\infty}b_n$ divergerer", "$\\sum\\limits_{n=0}^{\\infty}a_n$ divergerer $\\Leftarrow$ $\\sum\\limits_{n=0}^{\\infty}b_n$ divergerer", "$\\sum\\limits_{n=0}^{\\infty}a_n$ divergerer $\\Leftrightarrow$ $\\sum\\limits_{n=0}^{\\infty}b_n$ divergerer"], "matrix": ["1", 0, 0, 0, "1", 0], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Summen av geometriske rekker", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Det er gitt to geometriske rekker $\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{a}^n}$ og $\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{b}^n}$ med kvotienter $r=\\frac{1}{\\var{a}}$ og $r=\\frac{1}{\\var{b}}$ henholdsvis.

Bruk formelen $\\sum\\limits_{n=0}^{\\infty}r^na_0=\\frac{a_0}{1-r}$ for $|r|<1$ og finn

$\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{a}^n}$=[[0]]

$\\sum\\limits_{n=0}^{\\infty}\\frac{1}{\\var{b}^n}$=[[1]]

Finn videre

$\\sum\\limits_{n=0}^{\\infty}\\Big(\\frac{1}{\\var{a}^n}+\\frac{1}{\\var{b}^n}\\Big)=$[[0]]

$\\sum\\limits_{n=0}^{\\infty}\\Big(\\frac{1}{\\var{a}^n}-\\frac{1}{\\var{b}^n}\\Big)=$[[1]]

$\\sum\\limits_{n=0}^{\\infty}\\Big(\\frac{1}{\\var{a}^n}\\cdot\\frac{1}{\\var{b}^n}\\Big)=$ [[2]]