![](\"resources/question-resources/HELM2_fig30_2k7DAa7.png\")
Exercises for HELM Book 2.7.1
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "2.7.1.1 Write down a polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Unmarked question, with advice.
\nWrite down an example of a polynomial of given degree and given variable.
\nWrite down a non-polynomial function.
\nExplain why a polynomial with a fractional index is not a polynomial.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "(a) Write down a polynomial expression of degree $\\var{d}$ with independent variable $\\var{latex(v)}$.
\n(b) Write down a function which is not a polynomial.
\n(c) Explain why $\\var{c_poly}$ is not a polynomial.
\n(d) Write down a polynomial expression of degree $0$ with independent variable $\\var{latex(v)}$.
", "advice": "(a) For example, $\\var{poly_a}$
\n(b) For example, $y=\\frac{1}{x}$
\n(c) Fractional indices such as $\\frac{1}{x}$, $\\sqrt{x}$, and $x^{\\frac{1}{2}}$ are not allowed in a polynomial.
\n(d) For example, $P(\\var{latex(v)}) = 13$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random('a','b','c','d','t','w','x','y','z')", "description": "the variable
", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "the degree
", "templateType": "anything", "can_override": false}, "c_poly": {"name": "c_poly", "group": "Ungrouped variables", "definition": "expression(\n [\"y=\"+random(1..10)+\"+x+x^(1/2)\",\n \"f(x) = \"+random(1..10)+\"-\"+random(1..10)+\"/x\",\n \"g(w)=\"+random(1..10)+\"*w^\"+random(1..4)+\"+\"+random(1..10)+\"*sqrt(w)-\"+random(1..10)\n ][idx]\n)", "description": "", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "poly_a": {"name": "poly_a", "group": "Ungrouped variables", "definition": "simplify(expression(\"f(\"+v+\")=\"+[\n string(random(1..10)), \n random(1..10)+\"*\"+v+\"-\"+\n weighted_random([ [0,0.5], [random(1..10),0.5] ]), \n random(1..10)+\"*\"+v+\"^2-\"+ \n weighted_random([ [0,0.5], [random(1..10),0.5] ])+\"*\"+v+\"-\"+\n weighted_random([ [0,0.5], [random(1..10),0.5 ] ]),\n random(1..10)+\"*\"+v+\"^3+\"+\n weighted_random([ [0,0.5], [random(1..10),0.5]])+\"*\"+v+\"^2-\"+ \n weighted_random([ [0,0.5], [random(1..10),0.5] ])+\"*\"+v+\"-\"+\n weighted_random([ [0,0.5], [random(1..10),0.5] ])\n ][d]\n ),\"all\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v", "d", "c_poly", "idx", "poly_a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "2.7.1.2 State the degree of the polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given an arbitrary polynomial, identify its degree.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "What is the degree of the polynomial $\\var{q3_expr}$?
", "advice": "Look for the largest index on the variable. This is the degree.
\nIf there are no indices, look for a variable. If there is a variable with no index, then this could be rewritten with an index of 1, and this is the degree.
\nIf there is no variable, and only a constant, then the degree is $0$.
\nSo the degree of $\\var{q3_expr}$ is $\\var{q3_deg}$.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q3_expr": {"name": "q3_expr", "group": "Ungrouped variables", "definition": "simplify(expression(string(simplify(\nexpression( \n q3_expr_bits[0] + \"+\" + q3_expr_bits[1] + \"+\" + q3_expr_bits[2] + \"+\" +\n q3_expr_bits[3] + \"+\" + q3_expr_bits[4] + \"+\" + q3_expr_bits[5] + \"+\" +\n q3_expr_bits[6] + \"+\" + q3_expr_bits[7] + \"+\" + q3_expr_bits[8] + \"+\" +\n q3_expr_bits[9] ),\n[\"all\"]))),[\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "q3_expr_bits": {"name": "q3_expr_bits", "group": "Ungrouped variables", "definition": "[q3_coeffs[0] + letter + \"^9\",\n q3_coeffs[1] + letter + \"^8\" , \n q3_coeffs[2] + letter + \"^7\" , \n q3_coeffs[3] + letter + \"^6\" , \n q3_coeffs[4] + letter + \"^5\" , \n q3_coeffs[5] + letter + \"^4\" , \n q3_coeffs[6] + letter + \"^3\" , \n q3_coeffs[7] + letter + \"^2\" , \n q3_coeffs[8] + letter , \n q3_coeffs[9] \n ]", "description": "", "templateType": "anything", "can_override": false}, "q3_deg": {"name": "q3_deg", "group": "Ungrouped variables", "definition": "if(q3_coeffs[0]<>0,9,\n if(q3_coeffs[1]<>0,8,\n if(q3_coeffs[2]<>0,7,\n if(q3_coeffs[3]<>0,6,\n if(q3_coeffs[4]<>0,5,\n if(q3_coeffs[5]<>0,4,\n if(q3_coeffs[6]<>0,3,\n if(q3_coeffs[7]<>0,2,\n if(q3_coeffs[8]<>0,1,0\n )))))))))", "description": "", "templateType": "anything", "can_override": false}, "q3_coeffs": {"name": "q3_coeffs", "group": "Ungrouped variables", "definition": "[weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n random(-9..9)\n ]", "description": "", "templateType": "anything", "can_override": false}, "letter": {"name": "letter", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','m','n','p','q','r','s','t','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q3_expr", "q3_expr_bits", "q3_deg", "q3_coeffs", "letter"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q3_deg", "maxValue": "q3_deg", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "2.7.1.3 One-to-one or Many-to-one", "extensions": [], "custom_part_types": [], "resources": [["question-resources/HELM2_fig30_2k7DAa7.png", "/srv/numbas/media/question-resources/HELM2_fig30_2k7DAa7.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "For the graphs in HELM Book 2 Figure 30, identify which are one-to-one, and which are many-to-one.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "
Which of the six function shown are one-to-one and which are many-to-one?
", "advice": "A one-to-one function has only a single input ($x$) value mapping to each output ($y$) value.
\nA many-to-one function has multiple input values mapping to the same output value.
", "rulesets": {}, "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, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": false, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$P1$", "$Q1$", "$P2$", "$Q2$", "$P3$", "$Q3$"], "matrix": [["1", 0], ["1", 0], [0, "1"], [0, "1"], ["1", 0], [0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["one-to-one", "many-to-one"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "menu", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "This quiz is a Numbas implementation of the Helping Engineers Learn Maths (HELM) booklet 2.7, \"Some Common Functions\" exercises.
\nQuestions generally have multiple versions, clicking the \"Try another question like this one\" button will generate a new version.
", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/HELM2_fig30_2k7DAa7.png", "/srv/numbas/media/question-resources/HELM2_fig30_2k7DAa7.png"]]}