// Numbas version: exam_results_page_options {"name": "Test trinn 8 - 13", "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "

Etter at du har levert prøven, kan du se hvordan du har gjort det på prøven. du kan også se hvilke oppgaver du har feil på. 

", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true}, "timing": {"allowPause": true, "timeout": {"action": "warn", "message": "

Tiden er ute og du må levere nå.

"}, "timedwarning": {"action": "warn", "message": "

Du har 5 minutter igjen på prøven.

"}}, "allQuestions": true, "shuffleQuestions": false, "percentPass": "50", "duration": 5400, "pickQuestions": 0, "navigation": {"onleave": {"action": "warnifunattempted", "message": "

Du har ikke svart på alle oppgavene, vil du likevel sende svarene?

"}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "metadata": {"description": "

Her finner du oppgaver fra klassetrinn 8 til 13

", "licence": "All rights reserved"}, "type": "exam", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Br\u00f8kregning med variable 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Morten Brekke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/478/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "": []}, "parts": [{"stepsPenalty": "0", "prompt": "

Trekk sammen

\n

$\\displaystyle\\simplify{(x+{a})/({b}x)-({c}x-{d})/({ee}x)}=$[[0]]

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Utvid brøkene slik at de får en felles nevner. Deretter kan du addere eller subtrahere. 

\n
\n

(Husk å forkorte hvis det er mulig)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({ee-b*c}x+{ee*a+b*d})/({b*ee}x)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "0", "prompt": "

Regn ut

\n

$\\displaystyle{\\frac{\\frac{x}{\\var{j}}+\\frac{1}{\\var{l}}}{\\frac{\\var{g}x}{\\var{j*l}}+\\frac{1}{\\var{j}}}} =$[[0]]

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Finn fellesnevner for småbrøkene, og multipliser med denne over og under hovedbrøkstreken.

\n
\n

For eksempel: 

\n

\\[\\frac{\\frac{2x}{3}+\\frac{1}{5}}{\\frac{4x}{15}-\\frac{1}{3}}=\\frac{(\\frac{2x}{3}+\\frac{1}{5})\\cdot 15}{(\\frac{4x}{15}-\\frac{1}{3})\\cdot 15}=\\frac{2x\\cdot 5+1\\cdot3}{4x-1\\cdot 5}=\\frac{10x+3}{4x-5}\\]

\n

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({l}x+{j})/({g}x+{l})", "marks": "1", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

Regn ut og oppgi svaret ditt som en brøk eller et bokstavuttrykk. Bruk  /  som brøkstrek, for eksempel skal $\\frac{x-2}{3}$ skrives som (x-2)/3 og $\\frac{(x+1)}{(x-4)}$ som (x+1)/(x-4)

", "variable_groups": [{"variables": ["a", "b", "c", "d", "ee", "g", "j", "l"], "name": "numerical fractions"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..3)", "templateType": "anything", "group": "numerical fractions", "name": "a", "description": ""}, "c": {"definition": "random(2..5)", "templateType": "anything", "group": "numerical fractions", "name": "c", "description": ""}, "b": {"definition": "random(2..6 except a)", "templateType": "anything", "group": "numerical fractions", "name": "b", "description": ""}, "d": {"definition": "random(2..5)", "templateType": "anything", "group": "numerical fractions", "name": "d", "description": ""}, "g": {"definition": "random(3..7)", "templateType": "anything", "group": "numerical fractions", "name": "g", "description": ""}, "ee": {"definition": "random(2..5 except b)", "templateType": "anything", "group": "numerical fractions", "name": "ee", "description": ""}, "j": {"definition": "random(2..5)", "templateType": "anything", "group": "numerical fractions", "name": "j", "description": ""}, "l": {"definition": "random(2..5 except j)", "templateType": "anything", "group": "numerical fractions", "name": "l", "description": ""}}, "metadata": {"description": "

Add, subtract, multiply and divide algebraic fractions.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Potenser 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Morten Brekke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/478/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "": []}, "parts": [{"stepsPenalty": 0, "prompt": "

Regn ut

\n

$\\displaystyle{\\left(\\frac{y}{\\var{k}}\\right)^\\var{m+1}} =$[[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "y^{m+1}/{k}^{m+1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Husk potensregnereglene

\n

$(a\\cdot b)^n = a^n\\cdot b^n$

\n

$\\displaystyle{\\left(\\frac{a}{b}\\right)^n=\\frac{a^n}{b^n}}$

\n

Se eventuelt denne videoen for hjelp:

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "0", "prompt": "

Regn ut

\n

$\\displaystyle{\\left(\\var{a}x\\right)^\\var{n}} =$[[0]]

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}^{n}*x^{n}", "marks": "1", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Husk potensregnereglene

\n

$(a\\cdot b)^n = a^n\\cdot b^n$

\n

$\\displaystyle{\\left(\\frac{a}{b}\\right)^n=\\frac{a^n}{b^n}}$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Skriv så enkelt som mulig:

\n

$\\displaystyle{(\\var{b}y)^\\var{-m+1}\\cdot (-\\var{a}y)^\\var{k}} =$[[0]]

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{b}^{-m+1}*{-a}^{k}*y^{-m+k+1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Husk potensregnereglene

\n

$a^n\\cdot a^m=a^{n+m} $

\n

$\\displaystyle{\\frac{a^n}{a^m}=a^{n-m}}$

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Regn ut:

\n

$\\displaystyle{\\left(\\var{d}\\cdot a^2\\right)^\\var{l}} = $[[0]] 

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{d}^{l}*a^{2*l}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Husk potensregnereglene

\n

$a^n\\cdot a^m=a^{n+m} $

\n

$\\displaystyle{\\frac{a^n}{a^m}=a^{n-m}}$

\n

$\\displaystyle{\\left(a^n\\right)^m=a^{n\\cdot m}}$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Skriv så enkelt som mulig:

\n

$\\displaystyle{\\frac{\\left(y^\\var{n}z\\right)^{-\\var{m}}\\cdot \\left(y z^\\var{k}\\right)^\\var{m}}{\\left(y^{-\\var{n-1}}z\\right)^\\var{m+1}}} = $[[0]] 

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "y^{n-1}*z^{-2*m+k*m-1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "steps": [{"prompt": "

Husk potensregnereglene

\n

$a^n\\cdot a^m=a^{n+m} $

\n

$\\displaystyle{\\frac{a^n}{a^m}=a^{n-m}}$

\n

$\\displaystyle{\\left(a^n\\right)^m=a^{n\\cdot m}}$

\n

Se eventuelt denne filmsnutten for hjelp:

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "

NB! I denne oppgaven kan du uttrykke svaret ved potenser. For eksempel kan du skrive 3^(-2)*x^7 for å få $3^{-2}\\cdot x^7$ og x^2*y for å få $x^2\\cdot y$

", "variable_groups": [{"variables": ["a", "b", "d", "n", "m", "k", "l"], "name": "numerical fractions"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "numerical fractions", "name": "a", "description": ""}, "b": {"definition": "random(2..9)", "templateType": "anything", "group": "numerical fractions", "name": "b", "description": ""}, "d": {"definition": "random(2..5)", "templateType": "anything", "group": "numerical fractions", "name": "d", "description": ""}, "k": {"definition": "random(2..4 except m)", "templateType": "anything", "group": "numerical fractions", "name": "k", "description": ""}, "m": {"definition": "random(2..4)", "templateType": "anything", "group": "numerical fractions", "name": "m", "description": ""}, "l": {"definition": "random(2..4)", "templateType": "anything", "group": "numerical fractions", "name": "l", "description": ""}, "n": {"definition": "random(2..3 except a)", "templateType": "anything", "group": "numerical fractions", "name": "n", "description": ""}}, "metadata": {"description": "

Working with powers

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Vekstfart - grafisk og ved regning", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/rett_linje_stigningstall.png", "/srv/numbas/media/question-resources/rett_linje_stigningstall.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Morten Brekke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/478/"}], "functions": {"eqnline": {"definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n \n// The curve is described by the equation \n// y = x^2-(a+b)x+a*b\n\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox: [-6,10,6,-10],\n axis: false,\n showNavigation: false,\n grid: true\n});\n \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,1],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,1],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n// Draw the function\n\n\nboard.create('functiongraph',[function(x){ return (x-a)*(x-b);},-6,6]);\nboard.create('functiongraph',[function(x){ return (2*x1-a-b)*(x-x1)+y1; strokeColor: 'black'},-6,6]);\nboard.create('point',[x2,y2],{size:2});\n\nreturn div;", "type": "html", "language": "javascript", "parameters": [["a", "number"], ["b", "number"], ["x1", "number"], ["y1", "number"], ["x2", "number"], ["y2", "number"]]}}, "ungrouped_variables": ["a", "b", "c", "x1", "y1", "dist", "x2", "y2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

We know that the graph crosses the $x$-axis at both $(\\var{a},0)$ and $(\\var{b},0)$. Since this is a quadratic, we know our equations has two roots, and by the previous observation, they are at $\\var{a}$ and $\\var{b}$. Hence we can write our equation as $\\simplify{y=(x-{a})(x-{b})}$ which simplifies to $\\simplify{y=x^2-({a}+{b})x+({a}*{b})}$.

\n

\n

To find the coefficients of the turning point of the quadratic, we know the x-coordinate of the turning point will correspond to the solution to $dy/dx=0$. So we get $\\simplify{2x-({a}+{b})}=0$ hence $\\simplify{x=({a}+{b})/2}$. We substitute this value of x back into the equation of the quadratic to find the corresponding y-coordinate.

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

Finn grafisk vekstfarten til $f$ i punktet $(\\var{x1}, \\var{y1})$

\n

Vekstfarten er: [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "2*x1-(a+b)", "minValue": "2*x1-(a+b)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "steps": [{"prompt": "

Hint: Vekstfarten til funksjonen i punktet $(\\var{x1}, \\var{y1})$ er stigningstallet til tangenten i dette punktet. Dette stigningstallet kan vi finne ved grafisk avlesning. Se for eksempel på den rette linjen tegnet nedenfor (som ikke nødvendigvis svarer til tangenten i denne oppgaven):

\n

\n

Her er stigningstallet $a = \\frac{\\Delta x}{\\Delta y}=\\frac{2}{1}=2$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Finn ved regning vekstfarten til $f$ i punktet A med koordinatene $(\\var{x2}, \\var{y2})$

\n

Vekstfart: [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "2*x2-(a+b)", "minValue": "2*x2-(a+b)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "steps": [{"prompt": "

Hint: Vekstfarten i punktet $(\\var{x2}, \\var{y2})$ er gitt ved $f'(\\var{x2})$.

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "

{eqnline(a,b,x1,y1,x2,y2)}

\n

Figuren ovenfor viser grafen til funksjonen

\n

$\\displaystyle\\simplify[all, !Noleadingminus]{f(x)=x^2-{a+b}x+{a*b}}$

\n

sammen med tangenten til grafen i punktet $(\\var{x1}, \\var{y1})$ og punktet A med koordinater $(\\var{x2}, \\var{y2})$

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "variables": {"a": {"definition": "random(-3..3 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-2..2 except [0,a,-a])", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "dist": {"definition": "random(-2..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "dist", "description": ""}, "y2": {"definition": "(x2-a)*(x2-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "y2", "description": ""}, "x2": {"definition": "floor((a+b)/2)-dist", "templateType": "anything", "group": "Ungrouped variables", "name": "x2", "description": ""}, "y1": {"definition": "(x1-a)*(x1-b)", "templateType": "anything", "group": "Ungrouped variables", "name": "y1", "description": ""}, "x1": {"definition": "floor((a+b)/2)+dist", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Morten's copy of Stasjonaere punkter 2 (kap 7 Sinus)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Morten Brekke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/478/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "r1", "r2", "an", "d", "ap", "xmin", "xmax"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepsPenalty": 0, "prompt": "

$f(x)=\\simplify[all,!collectNumbers,!noleadingminus, fractionNumbers]{{a}x^3+{b}x^2-{c}x+{d}}$

\n

$f'(x)=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Bruk blant annet derivasjonsregelen $(x^n )'=n x^{n-1}$

\n

Videoen i denne lenken viser et eksempel på derivasjon av en polynomfunksjon.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3*{a}x^2+2*{b}x-{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Undersøk hvor  $f'(x)$ skifter fortegn, og bestem eventuelle topp- og bunnpunkter:

\n

Koordinatene til bunnpunktet er: ( [[0]] , [[2]] )

\n

Koordinatene til toppunktet er: ( [[1]] , [[3]] )

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Når vi deriverer får vi $\\displaystyle f'(x)=\\simplify[std]{{3*a}x^2+{2*b}x-{c}}$.

\n

For å finne de stasjonære punktene må vi løse  $\\displaystyle f'(x)=0$ for $x$, dvs likningen

\n

$\\simplify[std]{{3*a}x^2+{2*b}x-{c}=0}$.

\n

Denne likningen har to løsninger; $x=\\var{r1}$ og $x=-\\var{r2}$. For klassifisere de stasjonære punktene (avgjøre om de er toppunkt, bunnpunkt eller terrassepunkt) kan vi tegne fortegnslinja til den deriverte.  

\n

Dere kan finne mer hjelp i denne videosnutten:

\n

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "xmin", "minValue": "xmin", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "xmax", "minValue": "xmax", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "a*xmin^3+b*xmin^2-c*xmin+d", "minValue": "a*xmin^3+b*xmin^2-c*xmin+d", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "a*xmax^3+b*xmax^2-c*xmax+d", "minValue": "a*xmax^3+b*xmax^2-c*xmax+d", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

Her skal vi finne koordinatene til funksjonens stasjonære punkter, og avgjøre om de er toppunkter, bunnpunkter eller terrassepunkter.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-2..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "3*a*r1*r2", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "3*a*(r2-r1)/2", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "r1": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "r2": {"definition": "random(0..1#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "r2", "description": ""}, "an": {"definition": "if(a<0,1,0)", "templateType": "anything", "group": "Ungrouped variables", "name": "an", "description": ""}, "ap": {"definition": "if(a<0,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ap", "description": ""}, "xmax": {"definition": "an*r1-ap*r2", "templateType": "anything", "group": "Ungrouped variables", "name": "xmax", "description": ""}, "xmin": {"definition": "-an*r2+ap*r1", "templateType": "anything", "group": "Ungrouped variables", "name": "xmin", "description": ""}, "d": {"definition": "random(-10..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "

Finding the stationary points of a cubic with two turning points

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Morten Brekke", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/478/"}], "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/rett_linje_stigningstall.png", "/srv/numbas/media/question-resources/rett_linje_stigningstall.png"]]}