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{"name": "Week 1 homework test", "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "The domain and codomain of a function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}], "functions": {}, "ungrouped_variables": ["x1", "y1"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"maxAnswers": 0, "displayColumns": 0, "prompt": "<p>Which combinations of domain and codomain make the formula $g(x) = 2x$ into a function?</p>", "matrix": ["1", "-1", "1", "1"], "shuffleChoices": false, "variableReplacements": [], "minAnswers": "0", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "$g(2)=4$ is not in the codomain", "", ""], "showCorrectAnswer": true, "scripts": {}, "warningType": "none", "marks": 0, "choices": ["<p>Domain $\\left\\{0,1,2\\right\\}$ and codomain $\\left\\{0,2,4\\right\\}$</p>", "<p>Domain $\\left\\{0,1,2\\right\\}$ and codomain $\\left\\{0,2\\right\\}$</p>", "<p>Domain $\\left\\{0,1,2\\right\\}$ and codomain $\\left\\{0,2,4,5\\right\\}$</p>", "<p>Domain $\\mathbb N$ and codomain $\\mathbb N$.</p>"], "type": "m_n_2", "displayType": "checkbox", "minMarks": 0}, {"maxAnswers": 0, "displayColumns": 0, "prompt": "<p>Which of the following forumulas make $h:\\mathbb Z \\mapsto \\mathbb Z$ into a function?</p>", "matrix": ["1", "-1", "-1", "1", "-1", "1"], "shuffleChoices": false, "variableReplacements": [], "minAnswers": 0, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "$h(1)=\\pm 1$ and so the input of 1 determines two possible outputs.", "$h(2) = \\sqrt(2)$ is not an element of the codomain", "", "$\\frac12$ is not in the codomain", ""], "showCorrectAnswer": true, "scripts": {}, "warningType": "none", "marks": 0, "choices": ["<p>$h(x) = -x$</p>", "<p>$h(x) = \\pm x$</p>", "<p>$h(x) = \\sqrt{x}$</p>", "<p>$h(x) = x^2$</p>", "<p>$h(x) = \\frac12$</p>", "<p>$h(x) = 1$</p>"], "type": "m_n_2", "displayType": "checkbox", "minMarks": 0}], "statement": "<p>Informally, a function tells you how to transform elements of one set into another. The parts of a function are the: name, domain, codomain and formula.</p>\n<ul>\n<li>The <strong>name</strong> of a function can be anything. This is usually is just a letter such as $f,g$ or $h$.</li>\n<li>The <strong>domain </strong>is the set of elements to be transformed, otherwise known as the inputs to the function.</li>\n<li>The <strong>codomain</strong> is the set containing the transformed elements, otherwise known as the outputs to the function.</li>\n<li>The <strong>formula</strong> is an expression which tells you exactly how to transform an element of the domain into an element of the codomain.</li>\n</ul>\n<p>We use the notation $f: X\\mapsto Y$ to mean that the function named $f$ has domain $X$ and codomain $Y$. The formula is expressed separately, for example</p>\n<p style=\"text-align: center;\">$ f: \\left\\{-1,0,1\\right\\} \\mapsto \\left\\{0,1\\right\\}, f(x) = x^2.$</p>\n<p>The distinctive aspect of a function is that for any input, the formula determines exactly one output.&nbsp;</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"y1": {"definition": "x1+1", "templateType": "anything", "group": "Ungrouped variables", "name": "y1", "description": ""}, "x1": {"definition": "random(-1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}}, "metadata": {"description": "<p>A short question explaining the domain of a function.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "type": "question"}, {"name": "Application 1: Differential equation with a repeated linear factor & a delta function", "extensions": [], "custom_part_types": [], "resources": [["question-resources/MSD.jpg", "/srv/numbas/media/question-resources/MSD.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "gaps": [{"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": true, "correctAnswerFraction": true, "maxValue": "{c1}/((-{d1}+{a1})(-{d1}+{a1}))", "variableReplacements": [], "minValue": "{c1}/((-{d1}+{a1})(-{d1}+{a1}))", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}, {"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": true, "correctAnswerFraction": true, "maxValue": "{i0}-{c1}/((-{d1}+{a1})(-{d1}+{a1}))", "variableReplacements": [], "minValue": "{i0}-{c1}/((-{d1}+{a1})(-{d1}+{a1}))", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}, {"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": true, "correctAnswerFraction": true, "maxValue": "({c1}+(-{i0}*{a1}+{i1}+2*{a1}*{i0})*(-{a1}+{d1}))/(-{a1}+{d1})", "variableReplacements": [], "minValue": "({c1}+(-{i0}*{a1}+{i1}+2*{a1}*{i0})*(-{a1}+{d1}))/(-{a1}+{d1})", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}, {"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": false, "correctAnswerFraction": false, "maxValue": "b", "variableReplacements": [], "minValue": "b", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}], "type": "gapfill", "variableReplacements": [], "prompt": "<p>Enter the value for \\(A\\) as an exact fraction.&nbsp; &nbsp; &nbsp; \\(A=\\)&nbsp;[[0]]</p>\n<p>Enter the value for \\(B\\) as an exact fraction.&nbsp; &nbsp; &nbsp; \\(B=\\)&nbsp;[[1]]</p>\n<p>Enter the value for \\(C\\) as an exact fraction.&nbsp; &nbsp; &nbsp; \\(C=\\)&nbsp;[[2]]</p>\n<p>Enter the value for \\(D\\).&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(D=\\)&nbsp;[[3]]</p>"}], "statement": "<p>The diagram below shows a typical mass-spring-damper system as might apply to the suspension of a car.</p>\n<p>(Masses have mass M, springs with stiffness k and dampers having damping coefficient B).</p>\n<p>The associated variables are displacement x(t) and force F(t).</p>\n<p style=\"padding-left: 90px;\">&nbsp;<img src=\"resources/question-resources/MSD.jpg\" width=\"422\" height=\"265\"/></p>\n<p>Initially the mass is at a distance \\(\\var{i0}cm\\) from the equilibrium point and is moving at \\(\\var{i1}cm/s\\).</p>\n<p>If \\(B=\\simplify{2*{a1}}\\)&nbsp; and&nbsp; \\(k=\\simplify{{a1}^2}\\) and the system is subjected to an external applied force \\(F(t)=\\var{c1}e^{-\\var{d1}t}+\\var{b}\\delta(t-\\var{f})\\)</p>\n<p>then from Newton's law we&nbsp;get the differential equation:</p>\n<p></p>\n<p>&nbsp; &nbsp; &nbsp;\\(\\frac{d^2x}{dt^2}+\\simplify{{a1}+{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}}x(t)=\\var{c1}e^{-\\var{d1}t}+\\var{b}\\delta(t-\\var{f})\\) &nbsp;where \\(x(0)=\\var{i0} \\,\\, and \\,\\, &nbsp;x'(0)=\\var{i1}\\)</p>\n<p></p>\n<p>The solution of the equation is given by:</p>\n<p></p>\n<p>&nbsp; &nbsp; &nbsp;\\(x(t)=Ae^{-\\var{d1}t}+Be^{-\\var{a1}t}+Cte^{-\\var{a1}t}+Du(t-\\var{f})(t-\\var{f})e^{-\\var{a1}(t-\\var{f})}\\)</p>\n<p>.</p>", "rulesets": {}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "advice": "<p>\\(\\frac{d^2x}{dt^2}+\\simplify{{a1}+{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}}x(t)=\\var{c1}e^{-\\var{d1}t}+\\var{b}\\delta(t-\\var{f})\\) &nbsp;where \\(x(0)=\\var{i0} \\,\\, and \\,\\, &nbsp;x'(0)=\\var{i1}\\)</p>\n<p>The Laplace transform of this is given by:</p>\n<p style=\"padding-left: 30px;\">\\(s^2X(s)-\\var{i0}s-\\var{i1}+\\simplify{{a1}+{a1}}(sX(s)-\\var{i0})+\\simplify{{a1}*{a1}}X(s)=\\frac{\\var{c1}}{s+\\var{d1}}+\\var{b}e^{-\\var{f}s}\\)</p>\n<p>Gathering only \\(X(s)\\) terms on the left hand side and factoring gives:</p>\n<p style=\"padding-left: 30px;\">\\((s^2+\\simplify{{a1}+{a1}}s+\\simplify{{a1}*{a1}})X(s)=\\frac{\\var{c1}}{s+\\var{d1}}+\\var{i0}s+\\simplify{{i1}+({a1}+{a1})*{i0}}+\\var{b}e^{-\\var{f}s}\\)</p>\n<p style=\"padding-left: 30px;\">\\((s^2+\\simplify{{a1}+{a1}}s+\\simplify{{a1}*{a1}})X(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{a1})*{i0})*(s+{d1})}}{s+\\var{d1}}+\\var{b}e^{-\\var{f}s}\\)</p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{a1})*{i0})*(s+{d1})}}{(s+\\var{d1})(s+\\var{a1})^2}+\\frac{\\var{b}}{(s+\\var{a1})^2}e^{-\\var{f}s}\\)</p>\n<p>Solving the first fraction gives:</p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{A}{s+\\var{d1}}+\\frac{B}{s+\\var{a1}}+\\frac{C}{(s+\\var{a1})^2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+({i0}s+{i1}+({a1}+{a1})*{i0})*(s+{d1})}=A(s+\\var{a1})(s+\\var{a1})+B(s+\\var{d1})(s+\\var{a1})+C(s+\\var{d1})\\)</p>\n<p>Let \\(s=-\\var{d1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+({i0}*-{d1}+{i1}+({a1}+{a1})*{i0})*(-{d1}+{d1})}=\\simplify{(-{d1}+{a1})(-{d1}+{a1})}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{({c1})/((-{d1}+{a1})(-{d1}+{a1}))}\\)</p>\n<p>Let \\(s=-\\var{a1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1})}=\\simplify{(-{a1}+{d1})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1}))}\\)</p>\n<p>Compare the coefficients of \\(s^2\\) &nbsp;&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(\\var{i0}=A+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{{i0}-(({c1})/((-{d1}+{a1})(-{d1}+{a1})))}\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{({i0}*(-{d1}+{a1})*(-{d1}+{a1})-{c1})/((-{d1}+{a1})*(-{d1}+{a1}))}\\)</p>\n<p>We can find the inverse Laplace transform of the second fraction&nbsp;without&nbsp;breaking it down:&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{\\var{b}}{(s+\\var{a1})^2}e^{-\\var{f}s}\\) changes to \\(\\var{b}u(t-\\var{f})(t-\\var{f})e^{-\\var{a1}(t-\\var{f})}\\)</p>\n<p style=\"padding-left: 30px;\">\\(D=\\var{b}\\)</p>", "tags": [], "ungrouped_variables": ["a1", "c1", "d1", "i0", "i1", "b", "f", "B_1"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solve a&nbsp;Differential equation&nbsp;with a repeated linear factor &amp; a delta function</p>"}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..12#1)"}, "B_1": {"name": "B_1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "{i0}-{c1}/((-{d1}+{a1})(-{d1}+{a1}))"}, "f": {"name": "f", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(1..9#1)"}, "a1": {"name": "a1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..5)"}, "d1": {"name": "d1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(12..16)"}, "i0": {"name": "i0", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..10)"}, "i1": {"name": "i1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..10) "}, "c1": {"name": "c1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..10)"}}, "variablesTest": {"maxRuns": "222", "condition": ""}, "type": "question"}]}], "duration": 0, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "metadata": {"description": "<p>description</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "showQuestionGroupNames": false, "feedback": {"showtotalmark": true, "advicethreshold": 0, "allowrevealanswer": true, "feedbackmessages": [], "showactualmark": true, "showanswerstate": true, "intro": ""}, "type": "exam", "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/MSD.jpg", "/srv/numbas/media/question-resources/MSD.jpg"]]}