// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Laplace of e^(at)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Laplace transform of e^{at}</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["a", "c", "b", "d"], "rulesets": {}, "statement": "<p>Note that the Laplace transform of $e^{at}$ is $\\frac{1}{s-a}$</p>\n<p>$L[e^{at}]=\\frac{1}{s-a}$</p>\n<p></p>\n<p>Use this result to solve the following:</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Recall that&nbsp;$L[e^{at}]=\\frac{1}{s-a}$</p>\n<p></p>\n<p><strong>(a)</strong></p>\n<p>Using the above result, $L[e^{\\var{a}t}]=\\frac{1}{s-\\var{a}}$</p>\n<p></p>\n<p><strong>(b)</strong>&nbsp;&nbsp;</p>\n<p>Using the above result, &nbsp;$L[e^{\\var{b}t}]=\\frac{1}{s-(\\var{b})}=\\frac{1}{s+\\var{-b}}$</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>When you have the Laplace transform of two functions added together you just get the Laplace transform of each function and add the two answers.</p>\n<p>$L[f(t)+g(t)]=L[f(t)]+L[g(t)]$</p>\n<p></p>\n<p>Hence&nbsp; $L[e^{\\var{c}t}+e^{\\var{d}t}]=L[e^{\\var{c}t}]+L[e^{\\var{d}t}]=\\frac{1}{s-(\\var{c})}+\\frac{1}{s-\\var{d}}=\\simplify{1/(s-{c})+1/(s-{d})}$</p>", "extensions": [], "name": "Simon's copy of Laplace of e^(at)", "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"d": {"description": "", "templateType": "anything", "name": "d", "definition": "random (-9..9 except [0,1,-1])", "group": "Ungrouped variables"}, "a": {"description": "", "templateType": "anything", "name": "a", "definition": "random (2..9 except [0,1])", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "anything", "name": "b", "definition": "random (-9..-2)", "group": "Ungrouped variables"}, "c": {"description": "", "templateType": "anything", "name": "c", "definition": "random (-9..-2 except [0,1,-1])", "group": "Ungrouped variables"}}, "variable_groups": [], "parts": [{"expectedVariableNames": [], "answer": "1/(s-{a})", "showCorrectAnswer": true, "showPreview": true, "marks": 1, "scripts": {}, "type": "jme", "variableReplacements": [], "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkVariableNames": false, "showFeedbackIcon": true, "checkingType": "absdiff", "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Find the Laplace transform of $e^{\\var{a}t}$</p>", "customMarkingAlgorithm": "", "checkingAccuracy": 0.001}, {"expectedVariableNames": [], "answer": "1/(s-{b})", "showCorrectAnswer": true, "showPreview": true, "marks": 1, "scripts": {}, "type": "jme", "variableReplacements": [], "vsetRangePoints": 5, "unitTests": [], "failureRate": 1, "checkVariableNames": false, "showFeedbackIcon": true, "checkingType": "absdiff", "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Find the Laplace transform of $e^{\\var{b}t}$</p>", "customMarkingAlgorithm": "", "checkingAccuracy": 0.001}, {"answer": "1/(s-{c})+1/(s-{d})", "customMarkingAlgorithm": "", "variableReplacements": [], "checkingAccuracy": 0.001, "type": "jme", "vsetRangePoints": 5, "stepsPenalty": 0, "showFeedbackIcon": true, "checkingType": "absdiff", "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "steps": [{"variableReplacementStrategy": "originalfirst", "prompt": "<p>When you have the Laplace transform of two functions added together you just get the Laplace transform of each function and add the two answers.</p>\n<p>$L[f(t)+g(t)]=L[f(t)]+L[g(t)]$</p>", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "information", "variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": []}], "checkVariableNames": false, "prompt": "<p>Find the Laplace transform of $ { e^{ \\var{c} t}+e^{ \\var{d} t} }$</p>\n<p></p>", "showCorrectAnswer": true, "marks": 1, "scripts": {}, "unitTests": [], "expectedVariableNames": [], "showPreview": true, "variableReplacementStrategy": "originalfirst", "failureRate": 1}], "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/"}]}