// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Laplace of constants and powers of t", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "parts": [{"marks": 1, "checkVariableNames": false, "vsetRange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "answer": "{a}/s", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkingAccuracy": 0.001, "type": "jme", "prompt": "<p>Find $L[\\var{a} ]$.</p>", "expectedVariableNames": [], "showPreview": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "checkVariableNames": false, "vsetRange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "answer": "{b}/s^2", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkingAccuracy": 0.001, "type": "jme", "prompt": "<p>Find $L[\\var{b}t ]$.</p>", "expectedVariableNames": [], "showPreview": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "checkVariableNames": false, "vsetRange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "answer": "{a}/s+{b}/s^2", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkingAccuracy": 0.001, "type": "jme", "prompt": "<p>Find $L[\\var{a}+\\var{b}t ]$.</p>", "expectedVariableNames": [], "showPreview": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "checkVariableNames": false, "vsetRange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "answer": "fact({a})/s^{a+1}", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkingAccuracy": 0.001, "type": "jme", "prompt": "<p>Find $L[t^\\var{a}]$.</p>", "expectedVariableNames": [], "showPreview": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "checkVariableNames": false, "vsetRange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "answer": "({a}*(fact({c})))/s^{c+1}+({b}*(fact({d})))/s^{d+1}", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkingAccuracy": 0.001, "type": "jme", "prompt": "<p>Find $L[\\var{a}t^\\var{c}+\\var{b}t^\\var{d}]$.</p>", "expectedVariableNames": [], "showPreview": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "name": "Simon's copy of Laplace of constants and powers of t", "statement": "<p>Use the following standard Laplace transform results in the questions below:</p>\n<p>$L[af(t)+bg(t)]=aL[f(t)]+bL[g(t)]$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Linearity)</p>\n<p>$L[k]=\\frac{k}{s}$</p>\n<p>$L[t]=\\frac{1}{s^2}$</p>\n<p>$L[t^n]=\\frac{n!}{s^{n+1}}$</p>\n<p></p>", "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "variables": {"a": {"definition": "random(1..9#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "a"}, "b": {"definition": "random(2..9#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "b"}, "c": {"definition": "random(2..5#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "c"}, "d": {"definition": "random(6..9#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "d"}}, "advice": "<p><strong>(a)</strong></p>\n<p>$L[k]=\\frac{k}{s}$</p>\n<p>Hence&nbsp;$L[\\var{a} ]=\\frac{\\var{a}}{s}$.</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p></p>\n<p>Note $L[\\var{b}t ]=\\var{b}L[t]$ by linearity</p>\n<p></p>\n<p>Now we use&nbsp;$L[t]=\\frac{1}{s^2}$ to obtain:</p>\n<p>$L[\\var{b}t ]=\\var{b}L[t]=\\var{b}\\frac{1}{s^2} =\\frac{\\var{b}}{s^2}$</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>Use linearity to obtain</p>\n<p>$L[\\var{a}+\\var{b}t ]=L[\\var{a}]+L[\\var{b}t ]=\\frac{\\var{a}}{s}+\\frac{\\var{b}}{s^2}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;using our results from parts&nbsp;<strong>(a)&nbsp;</strong>and&nbsp;<strong>(b)</strong></p>\n<p></p>\n<p><strong>(d)</strong></p>\n<p>We use&nbsp;$L[t^n]=\\frac{n!}{s^{n+1}}$</p>\n<p>Hence&nbsp;$L[t^\\var{a}]=\\frac{\\var{a}!}{s^{\\var{a}+1}}=\\simplify{{a}!/(s^({a}+1))}$.</p>\n<p></p>\n<p><strong>(e)</strong></p>\n<p>By linearity, $L[\\var{a}t^\\var{c}+\\var{b}t^\\var{d}]=\\var{a}L[t^\\var{c}]+\\var{b}L[t^\\var{d}]$.</p>\n<p>Now we use&nbsp;$L[t^n]=\\frac{n!}{s^{n+1}}$ to obtain</p>\n<p>$L[\\var{a}t^\\var{c}+\\var{b}t^\\var{d}]=\\var{a}L[t^\\var{c}]+\\var{b}L[t^\\var{d}]= \\var{a}\\times \\frac{\\var{c}!}{s^{\\var{c}+1}} +\\var{b}&nbsp;\\times \\frac{\\var{d}!}{s^{\\var{d}+1}}=\\frac{\\var{a}\\times\\var{c}!}{s^{\\var{c+1}}}+\\frac{\\var{b}&nbsp;\\times\\var{d}!}{s^{\\var{d+1}}}$</p>", "tags": [], "metadata": {"description": "<p>Laplace of constants and powers of t</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"js": "", "css": ""}, "rulesets": {}, "variable_groups": [], "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}