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{"name": "Laplace and Inverse Laplace Transforms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Laplace and Inverse Laplace Transforms", "tags": ["eg-247", "inverse laplace transform", "laplace transform", "week 2"], "metadata": {"description": "<p>A simple test of definitions, properties and transform tables. Useful for retrieval practice.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>This quiz is designed to help you to revise the Laplace transform properties and tables.</p>", "advice": "<p>For an explanation of the results, please review the notes.&nbsp;</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "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>Which of these integrals represents the Laplace and Inverse Laplace transformations?</p>", "minMarks": 0, "maxMarks": "2", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "checkbox", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<em>\\[\\frac{1}{2\\pi j}\\int_{\\sigma-j\\omega}^{\\sigma+j\\omega}\\,F(s)\\,e^{st}\\,ds\\]</em>", "<em>\\[\\int_{0}^{\\infty}\\,f(t)\\,e^{-st}\\,dt\\]</em>", "\\[\\int_{-\\infty}^{t}\\,f(\\tau)\\,g(t-\\tau)\\,d\\tau\\]", "\\[\\int_{-j\\omega}^{+j\\omega}\\,f(t)\\,e^{-j\\omega t}\\,dt\\]"], "matrix": [["0", "1"], ["1", "0"], [0, 0], [0, 0]], "layout": {"type": "all", "expression": ""}, "answers": ["Laplace Transform", "Inverse Laplace Transform"]}, {"type": "m_n_x", "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>Match the Laplace transformation to the time-domain operator.</p>", "minMarks": 0, "maxMarks": "5", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "checkbox", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<em>\\[\\int_{-\\infty}^{t}\\,f(\\tau)\\,d\\tau\\]</em>", "<em>\\[\\lim_{t\\rightarrow 0}\\,f(t)\\]</em>", "\\[\\frac{d}{dt}\\,f(t)\\]", "\\[f(t + nT)\\]", "\\[\\int_0^t\\,f_1(\\tau)f_2(t-\\tau)\\,d\\tau\\]"], "matrix": [["1", "0", 0, 0, 0], ["0", "0", 0, "1", 0], [0, "1", 0, 0, 0], [0, 0, "1", 0, 0], [0, 0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["\\[\\frac{F(s)}{s}+\\frac{f(0^-)}{s}\\]", "\\[sF(s) - f(0^-)\\]", "\\[\\frac{\\int_{0}^{T}\\,f(t)\\,e^{-sT}}{1-e^{-sT}}\\]", "\\[\\lim_{s\\rightarrow \\infty}\\,sF(s)\\]", "\\[F_1(s)\\,F_2(s)\\]"]}, {"type": "m_n_x", "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>Match each of these mathematical properties to the associated Laplace transform property.</p>", "minMarks": 0, "maxMarks": "4", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["\\[c_1f_1(t) + c_2f_2(t)+ \\cdots + c_nf_n(t)\\Leftrightarrow c_1F_1(s) + c_2F_2(s) + \\cdots + c_nF_n(s)\\]", "\\[f(t - a)\\,u_0(t - a) \\Leftrightarrow e^{-as} F(s)\\]", "\\[e^{-at}\\, f(t) \\Leftrightarrow F(s + a)\\]", "\\[f(at) \\Leftrightarrow (1/a) \\, F (s/a)\\]"], "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["Linearity", "Time&nbsp;shifting", "Frequency shifting", "Time&nbsp;scaling"]}, {"type": "1_n_2", "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>What property is this?</p>\n<p>\\[\\lim_{t\\rightarrow \\infty}\\,f(t)\\Leftrightarrow \\lim_{s\\rightarrow 0}\\,sF(s)\\]</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Convolution in the time domain", "Initial value theorem", "Final value theorem", "Differentiation in the time domain", "Integration in the time domain"], "matrix": [0, 0, "1", 0, 0], "distractors": ["Not this one!", "Close: but look at the limits!", "Correct!", "$f(t)$ is not a differential equation!!", "$f(t)$ is not an integral"]}, {"type": "m_n_x", "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>Match the elementary signal to its Laplace transform</p>", "minMarks": 0, "maxMarks": "7", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["\\[1\\]", "\\[\\frac{1}{s}\\]", "\\[\\frac{1}{s^2}\\]", "\\[\\frac{1}{s&nbsp;+ a}\\]", "\\[\\frac{\\omega}{(s+a)^2+\\omega^2}\\]", "\\[e^{-as}\\]", "\\[\\frac{1-e^{-as}}{s}\\]"], "matrix": [["1", 0, 0, 0, 0, 0, 0], [0, "1", 0, 0, 0, 0, 0], [0, 0, "1", 0, 0, 0, 0], [0, 0, 0, "1", 0, 0, 0], [0, 0, 0, 0, "1", 0, 0], [0, 0, 0, 0, 0, "1", 0], [0, 0, 0, 0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["Dirac delta (unit impulse): $\\delta(t)$", "Unit step: $u_0(t)$", "Unit ramp: $u_1(t) = t u_0(t)$", "Exponential decay: $e^{-at}u_0(t)$", "Damped sinusoid: $e^{-at}\\,\\sin(\\omega t)u_0(t)$", "Sampling function: $\\delta(t-a)$", "Gating function: $u_0(t)-u_0(t-a)$"]}, {"type": "m_n_x", "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>Choose the correct Inverse Laplace transform $f(t)$ of the the following transforms $F(s)$</p>", "minMarks": 0, "maxMarks": "6", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["\\[1\\]", "\\[\\frac{1}{s}\\]", "\\[\\frac{1}{s+a}\\]", "\\[\\frac{\\omega}{(s + a)^2 + \\omega^2}\\]", "\\[\\frac{s+a}{(s + a)^2 + \\omega^2}\\]", "\\[s\\]"], "matrix": [["1", 0, 0, 0, 0, 0], [0, "1", 0, 0, 0, 0], [0, 0, "1", 0, 0, 0], [0, 0, 0, "1", 0, 0], [0, 0, 0, 0, "1", 0], [0, 0, 0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["\\[\\delta(t)\\]", "\\[u_0(t)\\]", "\\[e^{-at}u_0(t)\\]", "\\[e^{-at}\\sin\\omega t\\,u_0(t)\\]", "\\[e^{-at}\\cos\\omega t\\,u_0(t)\\]", "\\[\\delta'(t)\\]"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Chris Jobling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4013/"}]}]}], "contributors": [{"name": "Chris Jobling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4013/"}]}