// Numbas version: exam_results_page_options {"name": "Repetisjon Laplace", "metadata": {"description": "Eksempler fra forelesningen, de fleste av den enkle sorten.
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Calculating particular harmonic components of a Fourier series expansion.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

En funsksjon $f(t)$ er gitt ved:

\$f(t)=\\left[ \\begin{array}{cc}\\,\\,\\var{a}&\\,\\,-\\var{L}<t<-\\simplify{{L}/2}\\\\\\,\\,\\var{b}&\\,\\,-\\simplify{{L}/2}<t<\\simplify{{L}/2}\\\\\\,\\,\\var{c}&\\,\\,\\simplify{{L}/2}<t<\\var{L}\\end{array}\\right] \\,\\,\\,\\, \\text{der} f(t+\\simplify{2*{L}})=f(t)\$

", "advice": "

We derived the formulae for a trigonometric Fourier series of a periodic function having period \$2L\$

\$2L=\\simplify{2*{L}}\\implies L=\\var{L}\$

We can either evaluate \$a_0=\\frac{1}{L}\\int_{-L}^{L}f(t)dt\$ or use the shortcut:

\$\\frac{a_0}{2}=\$ the average value of the wave over one complete cycle \$=\\frac{Area}{Base}\$

\$\\frac{a_0}{2}=\\frac{\\var{a}*\\simplify{{L}/2}+\\var{b}*\\var{L}+\\var{c}*\\simplify{{L}/2}}{\\simplify{2*{L}}}=\\simplify{({a}+2*{b}+{c})/4}\$

The formula for the Fourier coefficient \$a_k\$ is given by: \$a_k=\\frac{1}{L}\\int_{-L}^{L}f(t)cos\\left(\\frac{{k}\\pi}{L}t\\right)dt\$

\$a_k=\\frac{1}{\\var{L}}\\left(\\int_{-\\var{L}}^{-\\frac{\\var{L}}{2}}\\var{a}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}\\var{b}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{\\frac{\\var{L}}{2}}^{\\var{L}}\\var{c}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt\\right)\$

\$a_k=\\frac{1}{\\var{L}}\\left(\\frac{\\var{a}*\\var{L}}{{k}\\pi}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\var{L}}^{-\\frac{\\var{L}}{2}}+\\frac{\\var{b}*\\var{L}}{{k}\\pi}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}+\\frac{\\var{c}*\\var{L}}{{k}\\pi}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{\\frac{\\var{L}}{2}}^{\\var{L}}\\right)\$

\$a_k=\\frac{\\var{a}}{{k}\\pi}sin(-\\frac{{k}\\pi}{2})-\\frac{\\var{a}}{{k}\\pi}sin(-{k}\\pi)+\\frac{\\var{b}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})-\\frac{\\var{b}}{{k}\\pi}sin(-\\frac{{k}\\pi}{2})+\\frac{\\var{c}}{{k}\\pi}sin({k}\\pi)-\\frac{\\var{c}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})\$

\$a_k=-\\frac{\\var{a}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})-\\frac{\\var{c}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})\$

\$a_k=\\frac{\\simplify{-{a}+2{b}-{c}}}{{k}\\pi}sin(\\frac{{k}\\pi}{2})\$

\$a_\\var{k}=\\simplify{(-{a}+2{b}-{c})/({k}*pi)sin({k}*pi/2)}\$

The formula for the Fourier coefficient \$b_k\$ is given by: \$b_k=\\frac{1}{L}\\int_{-L}^{L}f(t)sin\\left(\\frac{{k}\\pi}{L}t\\right)dt\$

\$b_k=\\frac{1}{\\var{L}}\\left(\\int_{-\\var{L}}^{-\\frac{\\var{L}}{2}}\\var{a}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}\\var{b}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt+\\int_{\\frac{\\var{L}}{2}}^{\\var{L}}\\var{c}sin\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)dt\\right)\$

\$b_k=\\frac{1}{\\var{L}}\\left(-\\frac{\\var{a}*\\var{L}}{{k}\\pi}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\var{L}}^{-\\frac{\\var{L}}{2}}-\\frac{\\var{b}*\\var{L}}{{k}\\pi}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{-\\frac{\\var{L}}{2}}^{\\frac{\\var{L}}{2}}-\\frac{\\var{c}*\\var{L}}{{k}\\pi}cos\\left(\\frac{{k}\\pi}{\\var{L}}t\\right)|_{\\frac{\\var{L}}{2}}^{\\var{L}}\\right)\$

\$b_k=-\\frac{\\var{a}}{{k}\\pi}cos(-\\frac{{k}\\pi}{2})+\\frac{\\var{a}}{{k}\\pi}cos(-{k}\\pi)-\\frac{\\var{b}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}cos(-\\frac{{k}\\pi}{2})-\\frac{\\var{c}}{{k}\\pi}cos({k}\\pi)+\\frac{\\var{c}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})\$

\$cos(-\\theta)=cos(\\theta)\$

\$b_k=-\\frac{\\var{a}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{a}}{{k}\\pi}cos({k}\\pi)-\\frac{\\var{b}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{b}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})-\\frac{\\var{c}}{{k}\\pi}cos({k}\\pi)+\\frac{\\var{c}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})\$

\$b_k=\\frac{\\var{c}-\\var{a}}{{k}\\pi}cos(\\frac{{k}\\pi}{2})+\\frac{\\var{a}-\\var{c}}{{k}\\pi}cos({k}\\pi)\$

\$b_\\var{k}=\\simplify{({c}-{a})/({k}*pi)cos({k}*pi/2)}+\\simplify{(-{c}+{a})/({k}*pi)cos({k}*pi)}\$

Recall the amplitude of the \$\\var{k}\$th harmonic component is given by \$\\sqrt{(a_\\var{k})^2+(b_\\var{k})^2}\$

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Finn verdien på det første leddet i til Fourier serien \$\\frac{a_{0}}{2}\$. Angi svaret ditt med tre desimaler.

\$\\frac{a_{0}}{2}\$ = [[0]]

Finn et utrykk for Fourier koefissienten \$ a_{k}\$ og så regn ut \$a_{\\var{k}}\$. Angi svaret ditt med tre desimaler.

\$ a_{\\var{k}}\$ = [[0]]

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Finn et utrykk for Fourier koefissienten \$ b_{k}\$ og så regn ut \$b_{\\var{k}}\$. Angi svaret ditt med tre desimaler.

\$ b_{\\var{k}}\$ = [[0]]

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Regn ut amplituden til den \$\\var{k}\$te harmoniske komponenten. Angi svaret ditt med tre desimaler.

Amplituden = [[0]]

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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Finn løsningen x(t) til følgende likninger. Du har lov til å bruke en tabell. Skriv inn bare løsningen uten x(t)=... .

", "advice": "

See 'show steps'.

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", "templateType": "anything"}, "ac": {"name": "ac", "group": "dependent variables", "definition": "1/(a+1)/(b+1)", "description": "", "templateType": "anything"}, "bc": {"name": "bc", "group": "dependent variables", "definition": "1/(a+1)/(a-b)", "description": "", "templateType": "anything"}, "cc": {"name": "cc", "group": "dependent variables", "definition": "-ac-bc", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c", "h"], "variable_groups": [{"name": "dependent variables", "variables": ["d", "pro", "c1", "cb", "db", "ac", "bc", "cc"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$x'+(\\var{c})x=(\\var{d}) e^{\\var{b}t}$ gitt at $x(0)=\\var{a}$.

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$L\\{t\\}=\\frac{1}{s^2}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "{a}*e^({b}*t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$x''+(\\var{cb})x'+(\\var{db})x = e^{-t}$ gitt at $x(0)=0$ og $x'(0)=0$.

Laplacetransformer diff-likningen og skriv et utrykk for $X(s)$ som en brøk med 1 i telleren. Skriv bare brøken uten $X(s)$. Du skal delbrøkoppspalte svaret i neste oppgavedel.

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "1/(s+1)/(s-{a})/(s-{b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": "0", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$x''+(\\var{cb})x'+(\\var{db})x = e^{-t}$ gitt at $x(0)=0$ og $x'(0)=0$.

Delbrøkoppspalt brøken fra deloppgave b).

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$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "{ac}/(s+1)+{bc}/(s-{a})+{cc}/(s-{b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": "5", "vsetRange": ["0", "1"], "checkVariableNames": false, "maxlength": {"length": "0", "partialCredit": "0", "message": ""}, "minlength": {"length": "0", "partialCredit": "0", "message": ""}, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": "0", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$x''+(\\var{cb})x'+(\\var{db})x = e^{-t}$ gitt at $x(0)=0$ og $x'(0)=0$.

Ta $L^{-1}$ av det delbrøkoppspaltete utrykket og skriv løsningen av difflikningen uten $x(t)=$.

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "{ac}*e^(-t)+{bc}*e^({a}t)+{cc}*e^({b}t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": "5", "vsetRange": ["0", "1"], "checkVariableNames": false, "maxlength": {"length": "0", "partialCredit": "0", "message": ""}, "minlength": {"length": "0", "partialCredit": "0", "message": ""}, "valuegenerators": [{"name": "t", "value": ""}]}]}, {"name": "Laplace av eksponential og trigonometriske funksjoner", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Mathias Sandulescu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3503/"}, {"name": "Kari Myklevoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4114/"}], "tags": [], "metadata": {"description": "

Laplace of constants and powers of t

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Regn ut følgende Laplace transformerte. Du har lov til å bruke en tabell.

", "advice": "

See 'show steps'.

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Regn ut $L\\{e^{\\var{a}t} \\}$.

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "1/(s-{a})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\sin(\\var{b}t) \\}$

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

"}], "answer": "{b}/(s^2+{b}^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\cos(\\var{c}t) \\}$.

$L\\{\\cos(at)\\}=\\frac{s}{s^2+a^2}$

"}], "answer": "s/(s^2+{c}^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\sin(\\frac{\\var{c}t}{\\var{a}}) \\}$.

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

"}], "answer": "{c}/{a}/(s^2+({c}/{a}^2))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{t\\cos(\\var{c}t) \\}$

$L\\{t\\cos(at)\\}=\\frac{s^2-a^2}{(s^2+a^2)^2}$

"}], "answer": "(s^2-{c}^2)/(s^2+{c}^2)^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{e^{\\var{b}t}\\sin(\\var{c}t) \\}$

$L\\{e^{-at}\\sin(bt)\\}=\\frac{b}{(s+a)^2+b^2}$

"}], "answer": "({c})/((s-{b})^2+({c})^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{ \\frac{\\cos(\\var{a}t)} {e^{\\var{b}t}} \\}$ der s>0.

[[0]]

$L\\{\\cos(at)\\}=\\frac{s}{s^2+a^2}$

"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(s+{b})/({a}^2+(s+{b})^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}], "sortAnswers": false}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{ {\\cosh(\\var{d}t)}\\}$ der s>0.

$L\\{\\cosh(at)\\}=\\frac{s}{s^2-a^2}$

"}], "answer": "s/(s^2-{a}^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}]}, {"name": "Laplace linearitet og tidsskifte", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Mathias Sandulescu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3503/"}, {"name": "Kari Myklevoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4114/"}], "tags": [], "metadata": {"description": "

Laplace of constants and powers of t

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Regn ut følgende Laplace transformerte. Du har lov til å bruke en tabell.

", "advice": "

See 'show steps'.

", "rulesets": {}, "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-2..2 except 0)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(0..9 except 0)", "description": "", "templateType": "anything"}, "bxc": {"name": "bxc", "group": "Ungrouped variables", "definition": "random(-9..9 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0..9 except 0)", "description": "", "templateType": "anything"}, "csq": {"name": "csq", "group": "Ungrouped variables", "definition": "c*c", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(0..3 except 0)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(0..3 except 0)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["bxc", "b", "c", "d", "a", "csq", "c1", "b1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{({\\var{a})t^2+({\\var{b})}} \\}$.

$L\\{t\\}=\\frac{1}{s^2}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "2*{a}/(s^3)+{b}/(s)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": "0", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{ \\var{a}e^{\\var{c}t} +(\\var{b})\\sin(t) \\}$.

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

"}], "answer": "{a}/(s-{c})+{b}/(s^2+1^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": "5", "vsetRange": ["0", "1"], "checkVariableNames": false, "maxlength": {"length": "0", "partialCredit": "0", "message": ""}, "minlength": {"length": "0", "partialCredit": "0", "message": ""}, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{1/\\var{a}\\sin(\\var{a}t) +\\var{b} \\cos(\\var{c}t) \\}$.

$L\\{\\cos(at)\\}=\\frac{s}{s^2+a^2}$

"}], "answer": "1/(s^2+{a}^2)+({b})*s/(s^2+{csq})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Gitt at $F(s)= \\frac{(s+\\var{b1})(s+\\var{c1})}{s^2+2s+1}$.

Regn ut $L\\{ e^{-t}f(t)\\}$

$L\\{f(t)\\}=F(s) \\rightarrow L\\{e^{-at} f(t)\\}=F(s+a)$

"}], "answer": "(s+1+{b1})(s+1+{c1})/((s+1)^2+2(s+1)+1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Gitt at $F(s)= \\frac{(s+\\var{b1})(s+\\var{c1})}{s^2+2s+1}$.

Regn ut $L\\{ e^{-\\var{a}t}f(t)\\}$

$L\\{f(t)\\}=F(s) \\rightarrow L\\{e^{-at} f(t)\\}=F(s+a)$

"}], "answer": "(s+{a}+{b1})(s+{a}+{c1})/((s+{a})^2+2(s+{a})+1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": "0", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Gitt at $F(s)= \\frac{\\var{c}}{s^2+3}$.

Regn ut $L\\{u(t-\\var{d})f(t-\\var{d})\\}$

$L\\{f(t)\\}=F(s) \\rightarrow L\\{u(t-d) f(t-d)\\}=e^{-sd}F(s)$

"}], "answer": "e^(-{d}s){c}/(s^2+3)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": "5", "vsetRange": ["0", "1"], "checkVariableNames": false, "maxlength": {"length": "0", "partialCredit": "0", "message": ""}, "minlength": {"length": "0", "partialCredit": "0", "message": ""}, "valuegenerators": [{"name": "s", "value": ""}]}]}, {"name": "Laplace av integraler og deriverte", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Mathias Sandulescu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3503/"}, {"name": "Kari Myklevoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4114/"}], "tags": [], "metadata": {"description": "

Laplace of constants and powers of t

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Finn Laplace transformerte til følgende uttrykk gitt at $L\\{y(t)\\}=Y$, $y(0)=\\var{a}$ og $y'(0)=\\var{b}$. Du har lov til å bruke en tabell.

", "advice": "

See 'show steps'.

", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9..9 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-3..3 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(0..9 except 0)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{2y'(t)\\}$.

$L\\{y'(t)\\}=sY-y(0) $

"}], "answer": "2(s*Y-{a})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\var{c}y''-y'\\}$.

$L\\{y(t)\\}=Y$, $y(0)=\\var{a}$ og $y'(0)=\\var{b}$

$L\\{y'(t)\\}=sY-y(0) $

$L\\{y''(t)\\}=s^2 Y-s y(0)-y'(0) $

"}], "answer": "{c}(s^2*Y-s*{a}-{b})-(s*Y-{a})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{y''+\\var{b}y'+\\var{c}y\\}$.

$L\\{y(t)\\}=Y$, $y(0)=\\var{a}$ og $y'(0)=\\var{b}$

$L\\{y'(t)\\}=sY-y(0) $

$L\\{y''(t)\\}=s^2 Y-s y(0)-y'(0) $

"}], "answer": "(s^2*Y-s*{a}-{b})+{b}(s*Y-{a})+{c}*Y", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}, {"name": "y", "value": ""}]}]}, {"name": "Invers Laplace transformasjon", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Mathias Sandulescu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3503/"}, {"name": "Kari Myklevoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4114/"}], "tags": [], "metadata": {"description": "
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Regn ut den invers Laplace transformerte $L^{-1}$ til følgende utrykk. Du har lov til å bruke en tabell.

", "advice": "

See 'show steps'.

", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-3..3 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-2..2 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(0..9 except 0)", "description": "", "templateType": "anything"}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "2*random(-9..9 except 0)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9..9 except 0 except -1 except 1)", "description": "", "templateType": "anything"}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "2*random(-9..9 except 0)", "description": "", "templateType": "anything"}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "p2/2", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c", "d", "p1", "p2", "p3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": "0", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L^{-1}\\{\\frac {\\var{p1}}{s}\\}$.

$L\\{k\\}=\\frac{k}{s}$

$L\\{t\\}=\\frac{1}{s^2}$

$L\\{t^n\\}=\\frac{n!}{s^{n+1}}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{\\cos(at)\\}=\\frac{s}{s^2+a^2}$

"}], "answer": "{p1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": "5", "vsetRange": ["0", "1"], "checkVariableNames": false, "maxlength": {"length": "0", "partialCredit": "0", "message": ""}, "minlength": {"length": "0", "partialCredit": "0", "message": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L^{-1}\\{\\frac{\\var{p2}}{s^3}\\}$.

$L\\{k\\}=\\frac{k}{s}$

$L\\{t\\}=\\frac{1}{s^2}$

$L\\{t^n\\}=\\frac{n!}{s^{n+1}}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{\\cos(at)\\}=\\frac{s}{s^2+a^2}$

"}], "answer": "{p3}*t^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L^{-1}\\{\\frac{s+ (\\var{a})}{s^2+1}\\}$.

$L\\{k\\}=\\frac{k}{s}$

$L\\{t\\}=\\frac{1}{s^2}$

$L\\{t^n\\}=\\frac{n!}{s^{n+1}}$

$L\\{e^{at}\\}=\\frac{1}{s-a}$

$L\\{\\sin(at)\\}=\\frac{a}{s^2+a^2}$

$L\\{\\cos(at)\\}=\\frac{s}{s^2+a^2}$

"}], "answer": "cos(t)+{a}sin(t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "t", "value": ""}]}]}, {"name": "Laplace av konstanter og potenser av t", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Mathias Sandulescu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3503/"}, {"name": "Kari Myklevoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4114/"}], "tags": [], "metadata": {"description": "

Laplace of constants and powers of t

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Regn ut følgende Laplace transformerte. Du har lov til å bruke en tabell.

", "advice": "

See 'show steps'.

", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 9#1)", "description": "", "templateType": "randrange"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "", "templateType": "randrange"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(6 .. 9#1)", "description": "", "templateType": "randrange"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\var{a} \\}$.

$L\\{k\\}=\\frac{k}{s}$

"}], "answer": "{a}/s", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\var{b}t \\}$.

$L\\{t\\}=\\frac{1}{s^2}$

"}], "answer": "{b}/s^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\var{a}+\\var{b}t \\}$.

$L\\{k\\}=\\frac{k}{s}$

$L\\{t\\}=\\frac{1}{s^2}$

"}], "answer": "{a}/s+{b}/s^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{t^\\var{a}\\}$.

$L\\{t^n\\}=\\frac{n!}{s^{n+1}}$

"}], "answer": "fact({a})/s^{a+1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Regn ut $L\\{\\var{a}t^\\var{c}+(\\var{b})t^\\var{d}\\}$.

$L\\{t^n\\}=\\frac{n!}{s^{n+1}}$

"}], "answer": "({a}*(fact({c})))/s^{c+1}+({b}*(fact({d})))/s^{d+1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "s", "value": ""}]}]}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": false, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "

Hei,

dette er et utvalg av øvinger som jeg laget til forelesningen.

Det er ikke ment til å være et representativt utvalg av det som kommer på eksamen.

", "feedbackmessages": []}, "contributors": [{"name": "Mathias Sandulescu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3503/"}], "extensions": [], "custom_part_types": [], "resources": []}