// Numbas version: exam_results_page_options {"timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "metadata": {"description": "
description
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "feedback": {"intro": "", "showactualmark": true, "showanswerstate": true, "allowrevealanswer": true, "showtotalmark": true, "feedbackmessages": [], "advicethreshold": 0}, "showstudentname": true, "name": "Joshua's copy of Week 1 homework test", "navigation": {"onleave": {"action": "none", "message": ""}, "preventleave": true, "showresultspage": "oncompletion", "showfrontpage": true, "reverse": true, "browse": true, "allowregen": true}, "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 1, "name": "Group", "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": "Which combinations of domain and codomain make the formula $g(x) = 2x$ into a function?
", "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": ["Domain $\\left\\{0,1,2\\right\\}$ and codomain $\\left\\{0,2,4\\right\\}$
", "Domain $\\left\\{0,1,2\\right\\}$ and codomain $\\left\\{0,2\\right\\}$
", "Domain $\\left\\{0,1,2\\right\\}$ and codomain $\\left\\{0,2,4,5\\right\\}$
", "Domain $\\mathbb N$ and codomain $\\mathbb N$.
"], "type": "m_n_2", "displayType": "checkbox", "minMarks": 0}, {"maxAnswers": 0, "displayColumns": 0, "prompt": "Which of the following forumulas make $h:\\mathbb Z \\mapsto \\mathbb Z$ into a function?
", "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": ["$h(x) = -x$
", "$h(x) = \\pm x$
", "$h(x) = \\sqrt{x}$
", "$h(x) = x^2$
", "$h(x) = \\frac12$
", "$h(x) = 1$
"], "type": "m_n_2", "displayType": "checkbox", "minMarks": 0}], "statement": "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.
\nWe 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
\n$ f: \\left\\{-1,0,1\\right\\} \\mapsto \\left\\{0,1\\right\\}, f(x) = x^2.$
\nThe distinctive aspect of a function is that for any input, the formula determines exactly one output.
", "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": "A short question explaining the domain of a function.
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "type": "question"}, {"name": "Testing JSXGraph for a network diagram", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Joshua Capel", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1479/"}], "advice": "", "statement": "Goal: Make a diagram so students can click on the edges in a network diagram and create a tree.
", "metadata": {"licence": "None specified", "description": ""}, "rulesets": {}, "ungrouped_variables": ["a", "b"], "variablesTest": {"maxRuns": 100, "condition": "a<>b"}, "preamble": {"js": "", "css": ""}, "tags": [], "parts": [{"showCorrectAnswer": true, "marks": 0, "gaps": [{"showFeedbackIcon": true, "marks": 1, "variableReplacements": [], "type": "numberentry", "maxValue": "1", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "1", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerFraction": false, "scripts": {"validate": {"order": "instead", "script": "console.log('Validating');\nreturn true;"}, "mark": {"order": "instead", "script": "console.log('Marking');\n\nvar selectedEdges=[]\n\nvar totalVertices=Object.keys(this.PointDict).length;\nconsole.log(\"There are \"+totalVertices+\" in this graph\");\n\nfor (i=0;i{createNetwork()}
\nThis will be the information only part.
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\nEnter the value for \\(C\\) as an exact fraction. \\(C=\\) [[2]]
\nEnter the value for \\(D\\). \\(D=\\) [[3]]
"}], "statement": "The diagram below shows a typical mass-spring-damper system as might apply to the suspension of a car.
\n(Masses have mass M, springs with stiffness k and dampers having damping coefficient B).
\nThe associated variables are displacement x(t) and force F(t).
\n\n
Initially the mass is at a distance \\(\\var{i0}cm\\) from the equilibrium point and is moving at \\(\\var{i1}cm/s\\).
\nIf \\(B=\\simplify{2*{a1}}\\) and \\(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})\\)
\nthen from Newton's law we get the differential equation:
\n\n\\(\\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})\\) where \\(x(0)=\\var{i0} \\,\\, and \\,\\, x'(0)=\\var{i1}\\)
\n\nThe solution of the equation is given by:
\n\n\\(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})}\\)
\n.
", "rulesets": {}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "advice": "\\(\\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})\\) where \\(x(0)=\\var{i0} \\,\\, and \\,\\, x'(0)=\\var{i1}\\)
\nThe Laplace transform of this is given by:
\n\\(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}\\)
\nGathering only \\(X(s)\\) terms on the left hand side and factoring gives:
\n\\((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}\\)
\n\\((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}\\)
\n\\(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}\\)
\nSolving the first fraction gives:
\n\\(X(s)=\\frac{A}{s+\\var{d1}}+\\frac{B}{s+\\var{a1}}+\\frac{C}{(s+\\var{a1})^2}\\)
\n\\(\\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})\\)
\nLet \\(s=-\\var{d1}\\)
\n\\(\\simplify{{c1}+({i0}*-{d1}+{i1}+({a1}+{a1})*{i0})*(-{d1}+{d1})}=\\simplify{(-{d1}+{a1})(-{d1}+{a1})}A\\)
\n\\(A=\\simplify{({c1})/((-{d1}+{a1})(-{d1}+{a1}))}\\)
\nLet \\(s=-\\var{a1}\\)
\n\\(\\simplify{{c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1})}=\\simplify{(-{a1}+{d1})}C\\)
\n\\(C=\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1}))}\\)
\nCompare the coefficients of \\(s^2\\)
\n\\(\\var{i0}=A+B\\)
\n\\(B=\\simplify{{i0}-(({c1})/((-{d1}+{a1})(-{d1}+{a1})))}\\)
\n\\(B=\\simplify{({i0}*(-{d1}+{a1})*(-{d1}+{a1})-{c1})/((-{d1}+{a1})*(-{d1}+{a1}))}\\)
\nWe can find the inverse Laplace transform of the second fraction without breaking it down:
\n\\(\\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})}\\)
\n\\(D=\\var{b}\\)
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