// Numbas version: exam_results_page_options
{"name": "Numbastest uke 43: Differensiallikninger", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Hvilke av likninger er differensiallikningene?", "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": "All rights reserved"}, "statement": "<p>Hvilke av likningene er differensiallikninger?</p>", "advice": "<p>En differensiallikning (f&oslash;rsteordens) er en likning som inkluderer en ukjent funksjon (f.eks. $y(x), y(t), y, x(y), x(t)$ osv.), den deriverte av denne funksjonen og variabel.</p>\n<p>$y'+y+y^2=x$ er en differensiallikning med $y(x)$ som en ukjente funksjon</p>\n<p>$y'=y$ er en differensiallikning med $y(x)$ som en ukjente funksjon</p>\n<p>$y'+5e^{xy}=\\sin(x)$ er en differensiallikning med $y(x)$ som en ukjente funksjon</p>\n<p>$x'+x=y$ er en differensiallikning med $x(y)$ som en ukjente funksjon</p>\n<p>$x^2+y^2=y'$ er en differensiallikning med $y(x)$ som en ukjente funksjon</p>\n<p>$\\cos(y'(x))=\\sin(x)$ er en differensiallikning med $y(x)$ som en ukjente funksjon</p>\n<p>$y'(t)=cos(t)$ er en differensiallikning med $y(t)$ som en ukjente funksjon</p>\n<p>$y(t)=cos(t)$ er ikke en differensiallikning fordi den ikke inkluderer noen deriverte&nbsp;</p>\n<p>$x^2+x=y$ er ikke en differensiallikning fordi den ikke inkluderer noen deriverte</p>\n<p>$z=x^2+y^2$ er ikke en differensiallikning fordi den ikke inkluderer noen deriverte</p>\n<p>$x^2+2x+1=0$ er ikke en differensiallikning fordi den ikke inkluderer noen deriverte</p>\n<p>$y'(x)+y(x)$ er ikke en likning</p>\n<p></p>", "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, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$y'+y+y^2=x$", "$y'=y$", "$y'+5e^{xy}=\\sin(x)$", "$y(t)=cos(t)$", "$y'(t)=cos(t)$", "$x'+x=y$", "$x^2+x=y$", "$x^2+y^2=y'$", "$z=x^2+y^2$", "$x^2+2x+1=0$", "$y'(x)+y(x)$", "$\\cos(y'(x))=\\sin(x)$"], "matrix": ["0.5", "0.5", "0.5", "0", "0.5", "0.5", 0, "0.5", 0, 0, 0, "0.5"], "distractors": ["", "", "", "", "", "", "", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Avgj\u00f8r ved regning hvilke av f\u00f8lgende funksjoner er l\u00f8sninger til difflikninger", "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": "All rights reserved"}, "statement": "<p>Avgj&oslash;r ved regning hvilke av f&oslash;lgende funksjoner er l&oslash;sninger til differensiallikningene i listen til venstre</p>", "advice": "<p>1) $\\var{out}(\\var{inp})=\\simplify{{a}e^({b}{inp})+{c}}$ er l&oslash;sningen til&nbsp; $\\var{out}'=\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}+{d}{out}}$ fordi</p>\n<p>VS: $\\var{out}'=\\simplify{{a}*{b}e^({b}{inp})}$</p>\n<p>HS: $\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}+{d}{out}}=\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}}+(\\var{d})\\cdot(\\simplify{{a}e^({b}{inp})+{c}})=\\simplify{{a}*{b}e^({b}{inp})}$</p>\n<p>2) $\\var{out}(\\var{inp})=\\simplify{{a}e^({b}{inp})+{c}}$ er l&oslash;sningen til&nbsp; $\\var{out}''=\\simplify{{a*(b^2)}e^({b}{inp})}$ fordi</p>\n<p>$\\var{out}''=(\\simplify{{a}*{b}e^({b}{inp})})'=\\simplify{{a*(b^2)}e^({b}{inp})}$</p>\n<p>3) $\\var{out}(\\var{inp})=\\simplify{{a}sin({b}{inp})}$ er l&oslash;sningen til&nbsp; $\\var{out}''=\\simplify{{-a*(b^2)}}\\sin(\\var{b}\\var{inp})$ fordi</p>\n<p>$\\var{out}'=\\simplify{{a*b}cos({b}{inp})}$</p>\n<p>$\\var{out}''=(\\simplify{{a*b}cos({b}{inp})})'=\\simplify{-{a*b^2}sin({b}{inp})}$</p>\n<p>4) $\\var{out}(\\var{inp})=\\simplify{{a}sin({b}{inp})}$ er l&oslash;sningen til $\\var{out}''=\\simplify{-{b^2}{out}}$ fordi</p>\n<p>VS: $\\var{out}''=\\simplify{{-a*(b^2)}}\\sin(\\var{b}\\var{inp})$</p>\n<p>HS: $\\simplify{-{b^2}{out}}=\\simplify{-{b^2}}\\cdot\\simplify{{a}sin({b}{inp})}=\\simplify{{-b^2*a}sin({b}{inp})}$&nbsp;</p>\n<p>5) $\\var{out}(\\var{inp})=\\simplify{{a}{inp}*e^({b}{inp})}$ er l&oslash;sningen til $\\dfrac{\\var{out}'+\\var{out}}{\\simplify{{a}e^({b}{inp})}}=\\simplify{{b+1}{inp}+1}$ fordi</p>\n<p>$\\var{out}'=\\simplify{{a}*e^({b}{inp})+{a*b}{inp}*e^({b}{inp})}=\\simplify{{a}*e^({b}{inp})*(1+{b}{inp})}$</p>\n<p>$\\var{out}'+\\var{out}=\\simplify{{a}*e^({b}{inp})*(1+{b}{inp})}+\\simplify{{a}{inp}*e^({b}{inp})}=\\simplify{{a}*e^({b}{inp})*(1+{b+1}{inp})}$</p>\n<p>$\\dfrac{\\var{out}'+\\var{out}}{\\simplify{{a}e^({b}{inp})}}=\\simplify{{b+1}{inp}+1}$</p>\n<p>6)$\\var{out}(\\var{inp})=\\simplify{{a}{inp}*e^({b}{inp})}$ er l&oslash;sningen til $\\var{out}''=\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\var{out}$ fordi</p>\n<p>$\\var{out}''=(\\simplify{{a}*e^({b}{inp})*(1+{b}{inp})})'=\\simplify{{a*b}*e^({b}{inp})*(2+{b}{inp})}=\\simplify{{2*a*b}*e^({b}{inp})+{a*b^2}{inp}*e^({b}{inp})}$</p>\n<p>$\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\var{out}=\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\cdot\\simplify{{a}{inp}*e^({b}{inp})}=\\simplify{{2*a*b}*e^({b}{inp})+{a*b^2}{inp}*e^({b}{inp})}$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..3 except 0 except b)", "description": "", "templateType": "anything", "can_override": false}, "inp": {"name": "inp", "group": "Ungrouped variables", "definition": "expression(random('x','r','s','t'))", "description": "", "templateType": "anything", "can_override": false}, "out": {"name": "out", "group": "Ungrouped variables", "definition": "expression(random('f','h','g','y'))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "inp", "out"], "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, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\var{out}'=\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}+{d}{out}}$", "$\\var{out}''=\\simplify{{a*(b^2)}e^({b}{inp})}$", "$\\var{out}''=\\simplify{{-a*(b^2)}}\\sin(\\var{b}\\var{inp})$", "$\\var{out}''=\\simplify{-{b^2}{out}}$", "$\\dfrac{\\var{out}'+\\var{out}}{\\simplify{{a}e^({b}{inp})}}=\\simplify{{b+1}{inp}+1}$", "$\\var{out}''=\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\var{out}$"], "matrix": [["1", "0", 0], ["1", "0", 0], [0, "1", 0], [0, "1", 0], [0, 0, "1"], [0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["$\\var{out}(\\var{inp})=\\simplify{{a}e^({b}{inp})+{c}}$", "$\\var{out}(\\var{inp})=\\simplify{{a}sin({b}{inp})}$", "$\\var{out}(\\var{inp})=\\simplify{{a}{inp}*e^({b}{inp})}$"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Differensiallikningen y'=ay", "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": "<p>Hint: For &aring; skrive flere eksponenter etter hverandre, sett en parentes rundt eksponentene:&nbsp; ^(...)</p>\n<p>Gitt differensiallikningen $y'(\\var{t})=\\var{a}y(\\var{t})$</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 15#1)", "description": "", "templateType": "randrange", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "expression(random('x','r','s','t'))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "t"], "variable_groups": [], "functions": {}, 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"absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}]}], "sortAnswers": false}, {"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": "<p>Finn l&oslash;sningen som oppfyller initialbetingelsen $y(\\frac{1}{\\var{a}})=e$</p>\n<p>$y(\\var{t})=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", 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