// Numbas version: exam_results_page_options {"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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Avgj\u00f8r ved regning hvilke av f\u00f8lgende funksjoner er l\u00f8sninger til difflikninger", "tags": [], "metadata": {"description": "", "licence": "All rights reserved"}, "statement": "

Avgjør ved regning hvilke av følgende funksjoner er løsninger til differensiallikningene i listen til venstre

", "advice": "

1) $\\var{out}(\\var{inp})=\\simplify{{a}e^({b}{inp})+{c}}$ er løsningen til  $\\var{out}'=\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}+{d}{out}}$ fordi

\n

VS: $\\var{out}'=\\simplify{{a}*{b}e^({b}{inp})}$

\n

HS: $\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}+{d}{out}}=\\simplify{{a*b-a*d}e^({b}*{inp})-{c*d}}+(\\var{d})\\cdot(\\simplify{{a}e^({b}{inp})+{c}})=\\simplify{{a}*{b}e^({b}{inp})}$

\n

2) $\\var{out}(\\var{inp})=\\simplify{{a}e^({b}{inp})+{c}}$ er løsningen til  $\\var{out}''=\\simplify{{a*(b^2)}e^({b}{inp})}$ fordi

\n

$\\var{out}''=(\\simplify{{a}*{b}e^({b}{inp})})'=\\simplify{{a*(b^2)}e^({b}{inp})}$

\n

3) $\\var{out}(\\var{inp})=\\simplify{{a}sin({b}{inp})}$ er løsningen til  $\\var{out}''=\\simplify{{-a*(b^2)}}\\sin(\\var{b}\\var{inp})$ fordi

\n

$\\var{out}'=\\simplify{{a*b}cos({b}{inp})}$

\n

$\\var{out}''=(\\simplify{{a*b}cos({b}{inp})})'=\\simplify{-{a*b^2}sin({b}{inp})}$

\n

4) $\\var{out}(\\var{inp})=\\simplify{{a}sin({b}{inp})}$ er løsningen til $\\var{out}''=\\simplify{-{b^2}{out}}$ fordi

\n

VS: $\\var{out}''=\\simplify{{-a*(b^2)}}\\sin(\\var{b}\\var{inp})$

\n

HS: $\\simplify{-{b^2}{out}}=\\simplify{-{b^2}}\\cdot\\simplify{{a}sin({b}{inp})}=\\simplify{{-b^2*a}sin({b}{inp})}$ 

\n

5) $\\var{out}(\\var{inp})=\\simplify{{a}{inp}*e^({b}{inp})}$ er løsningen til $\\dfrac{\\var{out}'+\\var{out}}{\\simplify{{a}e^({b}{inp})}}=\\simplify{{b+1}{inp}+1}$ fordi

\n

$\\var{out}'=\\simplify{{a}*e^({b}{inp})+{a*b}{inp}*e^({b}{inp})}=\\simplify{{a}*e^({b}{inp})*(1+{b}{inp})}$

\n

$\\var{out}'+\\var{out}=\\simplify{{a}*e^({b}{inp})*(1+{b}{inp})}+\\simplify{{a}{inp}*e^({b}{inp})}=\\simplify{{a}*e^({b}{inp})*(1+{b+1}{inp})}$

\n

$\\dfrac{\\var{out}'+\\var{out}}{\\simplify{{a}e^({b}{inp})}}=\\simplify{{b+1}{inp}+1}$

\n

6)$\\var{out}(\\var{inp})=\\simplify{{a}{inp}*e^({b}{inp})}$ er løsningen til $\\var{out}''=\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\var{out}$ fordi

\n

$\\var{out}''=(\\simplify{{a}*e^({b}{inp})*(1+{b}{inp})})'=\\simplify{{a*b}*e^({b}{inp})*(2+{b}{inp})}=\\simplify{{2*a*b}*e^({b}{inp})+{a*b^2}{inp}*e^({b}{inp})}$

\n

$\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\var{out}=\\dfrac{\\simplify{{b^2}{inp}+{2b}}}{\\var{inp}}\\cdot\\simplify{{a}{inp}*e^({b}{inp})}=\\simplify{{2*a*b}*e^({b}{inp})+{a*b^2}{inp}*e^({b}{inp})}$

", "rulesets": {}, "extensions": [], "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", "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}]}]}], "contributors": [{"name": "Elena Malyutina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7213/"}]}