// Numbas version: finer_feedback_settings
{"name": "Ugur's copy of Laplace:  Inverse Laplace Completing the Square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Laplace:  Inverse Laplace Completing the Square", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p></p>\n<p>Determine the <strong>inverse Laplace Transform</strong> of the following using <strong>completion of the square</strong>.</p>\n<p></p>\n<p></p>\n<p style=\"text-align: center;\">\\(F(s)=\\dfrac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\\)</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: center;\"></p>", "advice": "<p>We have</p>\n<p>\\(F(s)=\\frac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\\).</p>\n<p>By completing the square (on the denominator) we get:</p>\n<p>\\(F(s)=\\frac{\\var{B}s+\\var{C}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}}\\).</p>\n<p>Re-grouping the numerator gives:</p>\n<p>\\(F(s)=\\frac{\\var{B}(s+\\var{b1})-\\simplify{{B}*{b1}-{C}}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} = \\frac{\\var{B}(s+\\var{b1})}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} - \\frac{\\simplify{{B}*{b1}-{C}}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} = \\frac{\\var{B}(s+\\var{b1})}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} &nbsp;- \\frac{\\simplify{{B}*{b1}-{C}}}{\\sqrt{\\simplify{{c1}-{b1}^2}}}\\cdot\\frac{\\sqrt{\\simplify{{c1}-{b1}^2}}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} \\)</p>\n<p>Now, the expression above is easy enough to hunt from Laplace Transform table:</p>\n<p>\\(f(t)=\\var{B}e^{-\\var{b1}t}\\cos\\left(\\sqrt{\\simplify{{c1}-{b1}^2}}t\\right)+\\frac{-\\simplify{{B}*{b1}-{C}}}{\\sqrt{\\simplify{{c1}-{b1}^2}}}e^{-\\var{b1}t}\\sin\\left(\\sqrt{\\simplify{{c1}-{b1}^2}}t\\right)\\)</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"C": {"name": "C", "group": "Ungrouped variables", "definition": "random(2 .. 14#1)", "description": "", "templateType": "randrange", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "{b1}^2+{f}^2", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(6 .. 12#1)", "description": "", "templateType": "randrange", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "a1", "B", "b1", "C", "c1", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write down the inverse Laplace transform</p>\n<p>\\(\\mathscr{L}^{-1}\\{F(s)\\}=\\)&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({B}*e^(-{b1}*t)*cos({f}*t)+({C}-{B}*{b1})/{f}*e^(-{b1}*t)*sin({f}*t))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}]}], "sortAnswers": true}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}