// Numbas version: finer_feedback_settings
{"name": "SIT190 - Week 7 - Quiz - Short", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], []], "questions": [{"name": "Musa's copy of 3 Rate of change", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Picture_curve.png", "/srv/numbas/media/question-resources/Picture_curve.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "Musa Mammadov", "profile_url": 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"group": "Ungrouped variables", "definition": "y0+random(4..12)", "description": "", "templateType": "anything"}, "t2": {"name": "t2", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "km2": {"name": "km2", "group": "Ungrouped variables", "definition": "t2*v2", "description": "", "templateType": "anything"}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "random(55..89)", "description": "", "templateType": "anything"}, "t3": {"name": "t3", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "km3": {"name": "km3", "group": "Ungrouped variables", "definition": "t3*random(35..59)", "description": "", "templateType": "anything"}, "sec3": {"name": "sec3", "group": "Ungrouped variables", "definition": "random(10,12,15,20,30,40,45)", "description": "", "templateType": "anything"}, "sec4": {"name": "sec4", "group": "Ungrouped variables", "definition": "random(10,12,15,20,30,40,45)", "description": "", "templateType": "anything"}, "tem40": {"name": "tem40", "group": "Ungrouped variables", "definition": "random(10..38)", "description": "", "templateType": "anything"}, "tem41": {"name": "tem41", "group": "Ungrouped variables", "definition": "random(10..38)", "description": "", "templateType": "anything"}, "t4": {"name": "t4", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "", "templateType": "anything"}, "aa": {"name": "aa", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}, "bb": {"name": "bb", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "bb2": {"name": "bb2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "aa2": {"name": "aa2", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "x0", "y0", "x1", "y1", "t2", "km2", "v2", "t3", "km3", "sec3", "sec4", "tem40", "tem41", "t4", "aa", "bb", "bb2", "aa2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p><img src=\"resources/question-resources/Picture_curve.png\" width=\"340\" height=\"222\"/></p>\n<p>Calculate the average rate of change over an interval in a graph between points $(x0,y0) = (\\var{x0},\\var{y0})$ and $(x1,y1)= (\\var{x1},\\var{y1})$</p>\n<p>Average rate = [[0]]</p>", "gaps": 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"unitTests": [], "prompt": "<p>If it takes {t2} hours to drive a distance of {km2} km on a motorway, what would be your average speed in km/h?</p>\n<p>Average speed = [[0]] km/h</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{v2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 simple derivatives", "extensions": [], "custom_part_types": [], "resources": 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{"name": "x0", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything"}, "y0": {"name": "y0", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "x0+random(4..12)", "description": "", "templateType": "anything"}, "y1": {"name": "y1", "group": "Ungrouped variables", "definition": "y0+random(4..12)", "description": "", "templateType": "anything"}, "t2": {"name": "t2", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "km2": {"name": "km2", "group": "Ungrouped variables", "definition": "t2*v2", "description": "", "templateType": "anything"}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "random(55..89)", "description": "", "templateType": "anything"}, "t3": {"name": "t3", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "km3": {"name": "km3", "group": "Ungrouped variables", "definition": "t3*random(35..59)", "description": "", "templateType": "anything"}, "sec3": {"name": "sec3", "group": "Ungrouped variables", "definition": "random(10,12,15,20,30,40,45)", "description": "", "templateType": "anything"}, "sec4": {"name": "sec4", "group": "Ungrouped variables", "definition": "random(10,12,15,20,30,40,45)", "description": "", "templateType": "anything"}, "tem40": {"name": "tem40", "group": "Ungrouped variables", "definition": "random(10..38)", "description": "", "templateType": "anything"}, "tem41": {"name": "tem41", "group": "Ungrouped variables", "definition": "random(10..38)", "description": "", "templateType": "anything"}, "t4": {"name": "t4", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "", "templateType": "anything"}, "aa": {"name": "aa", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}, "bb": {"name": "bb", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "bb2": {"name": "bb2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "aa2": {"name": "aa2", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}, "nn": {"name": "nn", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "nn1": {"name": "nn1", "group": "Ungrouped variables", "definition": "random(-9..-1)", "description": "", "templateType": "anything"}, "nn2": {"name": "nn2", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything"}, "nn3": {"name": "nn3", "group": "Ungrouped variables", "definition": "random(2..9 except nn2)", "description": "", "templateType": "anything"}, "ann": {"name": "ann", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything"}, "ann1": {"name": "ann1", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}, "ann23": {"name": "ann23", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}, "cnn23": {"name": "cnn23", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "ann4": {"name": "ann4", "group": "Ungrouped variables", "definition": "random(-9..9 except [-1,0,1])", "description": "", "templateType": "anything"}, "bnn4": {"name": "bnn4", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "cnn4": {"name": "cnn4", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "nn4": {"name": "nn4", "group": "Ungrouped variables", "definition": "random(4..8)", "description": "", "templateType": "anything"}, "nn5": {"name": "nn5", "group": "Ungrouped variables", "definition": "random(2..3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "x0", "y0", "x1", "y1", "t2", "km2", "v2", "t3", "km3", "sec3", "sec4", "tem40", "tem41", "t4", "aa", "bb", "bb2", "aa2", "nn", "nn1", "nn2", "nn3", "ann", "ann1", "ann23", "cnn23", "ann4", "bnn4", "cnn4", "nn4", "nn5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Find derivative of $y=\\var{ann4}x^\\var{nn4}+\\var{bnn4}x^\\var{nn5} +\\var{cnn4}$</p>\n<p>&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{ann4*nn4}x^{nn4-1} +{bnn4*nn5}x^{nn5-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Differentiation - Trigonometric Functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "<p>More work on differentiation with&nbsp;trigonometric functions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Differentiate the following&nbsp;trigonometric functions using the chain rule.</strong></p>\n<p>Do not write out $dy/dx$; only input the differentiated right hand side of each equation.</p>", "advice": "<p><strong></strong>If you don't know how to differentiate trigonometric functions, please see&nbsp;<em>'Differentiation 4 - Trigonometric Functions'</em>.</p>\n<p><strong></strong></p>\n<p><strong>These questions use the chain rule.</strong></p>\n<p>The earlier questions are easy to do by inspection, e.g using Part a:</p>\n<p style=\"text-align: center;\">$y=sin(\\var{c[0]}x)$.</p>\n<p>We differentiate the term(s) inside the function, here the term is $\\var{c[0]}x$.</p>\n<p>Then we derive $sin$ of any function, giving us $cos$.</p>\n<p>Putting our results together, we get</p>\n<p>$\\var{c[0]}cos(\\var{c[0]}x)$.</p>", "rulesets": {}, "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..5]", "description": "<p>coefficients</p>", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "repeat(random(3..6),2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "p"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$y=-5\\cos(\\var{c[3]}x)+\\sin(\\var{c[4]}x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "5{c[3]}sin({c[3]}x)+{c[4]}cos({c[4]}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Differentiation - Exponentials and Logs", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "<p>Differentiating exponentials and Logs<br></p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Differentiate the following.</strong></p>\n<p>Do not write out $dy/dx$; only input the differentiated right hand side of each equation.</p>\n<p>Remember to enclose all single powers inside a bracket, for example, $e^{2x}$ is inputted as $e$^$(2x)$, or use $\\ln(2)$ instead of $\\ln2.$</p>", "advice": "<p><strong>The key fact to understand here is that the differentiate of $e^x$ is $e^x$.</strong></p>\n<p>This can be proven by looking at evaluating limits etc. but it is not necessary to do so at this stage.</p>\n<p>The basic steps to differentiate an exponential function are:</p>\n<p>Differentiate the power of $e$, for example in Part b,&nbsp;<strong></strong>$y=\\var{c[1]}e^{\\var{p[1]}x}$, you would differentiate $\\var{p[1]}x$.</p>\n<p>In this example, it is $\\var{p[1]}$.</p>\n<p>Then multiply the coefficient of $e$ by this result.</p>\n<p>Here, you would find $\\simplify{{c[1]}{p[1]}e^({p[1]}x)}$.</p>\n<p>This is your final answer for the derivative.</p>\n<p></p>\n<p>Remember, don't be confused if there is no coefficient. The fact the term is there means the coefficient must be $1$, but we don't tend to write it out as, for example $1x$, we just say $x$.</p>\n<p></p>\n<p>Basic formulas:</p>\n<p>$\\frac{d}{dx} e^x = e^x$</p>\n<p>$\\frac{d}{dx} e^{u(x)} = e^{u(x)}\\frac{d}{dx} u$</p>\n<p>$\\frac{d}{dx} a^x = a^x \\ln(a)$</p>\n<p>$\\frac{d}{dx} a^{u(x)} = a^{u(x)} \\ln(a) \\frac{d}{dx} u$</p>\n<p>$\\frac{d}{dx} \\ln(x) = \\frac{1}{x} ~~ (x&gt;0)$</p>\n<p>$\\frac{d}{dx} \\ln|x| = \\frac{1}{x} ~~ (x\\neq 0)$</p>\n<p>$\\frac{d}{dx} \\log_a(x) = \\frac{1}{x \\ln a} ~~ (a&gt;0, a \\neq 1)$</p>\n<p>$\\frac{d}{dx} x^x = x^x (1+\\ln x)$</p>", "rulesets": {}, "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "repeat(random(2..4),7)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..8),7)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(2..6),7)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "p", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$y=\\var{c[1]}e^{\\var{p[1]}x}$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({c[1]}*{p[1]})*e^({p[1]}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Calculating gradients - polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "<p>Calculating gradients - polynomials</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "repeat(random(2..9),17)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..9),17)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(-9..9 except [-1,0,1]),17)", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "repeat(random(1..9),17)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "b", "a", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Given $y=\\var{a[1]}x^2+\\var{b[1]}x+\\var{c[1]},$ first calculate the differential $y^\\prime (x) = \\frac{dy}{dx}$ and then gradient $y^\\prime (\\var{x[1]}) $ at point $x = \\var{x[1]}$</p>\n<p>$y^\\prime (x) =$&nbsp;[[0]]</p>\n<p>$y^\\prime (\\var{x[1]}) =$ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{2*a[1]}x+{b[1]}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{2*a[1]*x[1]+b[1]}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Number of roots and stationary points of a graph", "extensions": ["geogebra", "jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "<p>A graph (of a cubic) is given. The question is to determine the number of roots and number of stationary points the graph has. Non-calculator. Advice is given.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Finding the number of roots and turning points based on a graph.</p>", "advice": "<p>(i) Definition: A '<strong>root</strong>' of a function $f(x)$ is a value of $x$ which makes $f(x)=0$. &nbsp;Visually a root can be found be seeing when the $y$-coordinate of the graph is $0$, i.e., when the graph crosses the $x$-axis. Therefore, to count the roots, you need to count how many times the graph crosses the $x$-axis. &nbsp;In this question, the graph crosses the $x$-axis $\\var{num_roots}$ time(s), so there are $\\var{num_roots}$ roots.</p>\n<p>(ii) Definition: A '<strong>stationary point</strong>' of a function is a point on the graph where $f'(x)=0$. &nbsp;Remember that $f'$ tells us the gradient of $f$, so visually a stationary point is where the gradient of the curve is 0. &nbsp;In this question, there is/are $\\var{num_stat}$ place(s) where the gradient of the graph is $0$, so the answer is $\\var{num_stat}$.</p>\n<p>(iii) <span class=\"ILfuVd\"><span class=\"e24Kjd\">There are 3 <b>types</b> of <b>stationary points</b>: maximum points, minimum points and points of inflection. </span></span></p>\n<p><span class=\"ILfuVd\"><span class=\"e24Kjd\">Consider what happens to the gradient at a maximum point. It is positive just before the maximum point, zero at the maximum point, then negative just after the maximum point.</span></span></p>\n<p>Just before a minimum point the gradient is negative, at the minimum the gradient is zero and just after the minimum point it is positive.</p>\n<p></p>", "rulesets": {}, "variables": {"num_roots": {"name": "num_roots", "group": "Ungrouped variables", "definition": "if(num_stat = 2, random(1..3), 1)", "description": "<p>The number of roots.</p>", "templateType": "anything"}, "vshift": {"name": "vshift", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "<p>Random amount of vertifical shift for sake of variability.</p>", "templateType": "anything"}, "num_stat": {"name": "num_stat", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "<p>Number of stationary&nbsp;points</p>", "templateType": "anything"}, "hshift": {"name": "hshift", "group": "Ungrouped variables", "definition": "random(-2..2)", "description": "<p>Random amount of horizontal shift to create variability.</p>", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-1..1 except 0)", "description": "<p>Coefficient of x^3</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num_roots", "num_stat", "a", "hshift", "vshift"], "variable_groups": [], "functions": {"plotgraph": {"parameters": [["num_stat", "number"], ["num_roots", "number"], ["a", "number"], ["h", "number"], ["v", "number"]], "type": "html", "language": "javascript", "definition": "// This functions plots a cubic with a certain number of\n// stationary points and roots.\n// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n// Max and min x and y values for the axis.\nvar x_min = -6;\nvar x_max = 6;\nvar y_min = -10;\nvar y_max = 10;\n\n\n// First, make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n  '500px',\n  '600px',\n {\n   boundingBox: [x_min,y_max,x_max,y_min],\n   axis: false,\n   showNavigation: true,\n   grid: true\n }\n);\n\n\n\n\n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,1],{\n  drawLabels: true,\n  label: {offset: [-4, -10]},\n  minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,1],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n\n\n\n// Plot the function.\n\nswitch (num_stat) {\n  case 0:\n    board.create('functiongraph',\n             [function(x){ return a*(Math.pow(x+h,3)+2*(x+h)+v);},x_min,x_max]);\n    break;\n    \n  case 1:\n    board.create('functiongraph',\n             [function(x){ return a*(Math.pow(x+h,3)+v);},x_min,x_max]);\n    break;\n    \n  case 2:\n    switch (num_roots) {\n      case 1:\n        board.create('functiongraph',\n        [function(x){ return a*((x+2+h)*(x+h)*(x-2+h)+5);},x_min,x_max]);\n        break;\n        \n      case 2:\n        board.create('functiongraph',\n        [function(x){ return a*((x+1+h)*(x+1+h)*(x-2+h));},x_min,x_max]);\n        break;\n        break;\n        \n      case 3:\n        board.create('functiongraph',\n        [function(x){ return a*((x+2+h)*(x+h)*(x-2+h));},x_min,x_max]);\n        break;\n    }\n    \n        \n    break;\n}\n\n\n// num_stat\n\n\n\n\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>{plotgraph(num_stat,num_roots, a, hshift, vshift)}</p>\n<p>Above is the graph of some function $f$.</p>\n<p>How many roots does $~~f(x)=0~~$ have?&nbsp;[[0]]</p>\n<p>How many stationary points does $f(x)$ have? [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "num_roots", "maxValue": "num_roots", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "num_stat", "maxValue": "num_stat", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "type": "question"}, {"name": "Musa's copy of 3 Find coordinates of stationary points of polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Musa Mammadov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4417/"}], "tags": [], "metadata": {"description": "<p>Finding the coordinates and determining the nature of the stationary points on a polynomial function</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "variables": {"y12": {"name": "y12", "group": "Ungrouped variables", "definition": "2(x12^3)-3(x12+x22)*x12^2+6*x12*x22*x12+c02", "description": "", "templateType": "anything"}, "y03": {"name": "y03", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything"}, "y32": {"name": "y32", "group": "Ungrouped variables", "definition": "if(y12<y22,y12,y22)", "description": "", "templateType": "anything"}, "c03": {"name": "c03", "group": "Ungrouped variables", "definition": "y03-((x03^3)/3-((x13+x23)/2)*x03^2+(x13*x23)*x03)", "description": "", "templateType": "anything"}, "y13": {"name": "y13", "group": "Ungrouped variables", "definition": "(x13^3)/3-(x13+x23)*x13^2/2+x13*x23*x13+c03", "description": "", "templateType": "anything"}, "y23": {"name": "y23", "group": "Ungrouped variables", "definition": "(x23^3)/3-((x13+x23)*x23^2)/2+x13*x23*x23+c03", "description": "", "templateType": "anything"}, "y43": {"name": "y43", "group": "Ungrouped variables", "definition": "if(y13<y23,y23,y13)", "description": "", "templateType": "anything"}, "x03": {"name": "x03", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "c02": {"name": "c02", "group": "Ungrouped variables", "definition": "y02-((2x02^3)-3(x12+x22)*x02^2+6(x12*x22)*x02)", "description": "", "templateType": "anything"}, "x43": {"name": "x43", "group": "Ungrouped variables", "definition": "if(y13<y23,x23,x13)", "description": "", "templateType": "anything"}, "y0": {"name": "y0", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything"}, "y22": {"name": "y22", "group": "Ungrouped variables", "definition": "2(x22^3)-3(x12+x22)*x22^2+6x12*x22*x22+c02", "description": "", "templateType": "anything"}, "c0": {"name": "c0", "group": "Ungrouped variables", "definition": "y0-(2x0^3-3(x1+x2)*x0^2+6(x1*x2)*x0)", "description": "", "templateType": "anything"}, "x4": {"name": "x4", "group": "Ungrouped variables", "definition": "if(y1<y2,x2,x1)", "description": "", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-3..3 except x1)", "description": "", "templateType": "anything"}, "y02": {"name": "y02", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything"}, "x22": {"name": "x22", "group": "Ungrouped variables", "definition": "random(-3..3 except [x2,x12])", "description": "", "templateType": "anything"}, "x0": {"name": "x0", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "2x2^3-3(x1+x2)*x2^2+6*x1*x2*x2+c0", "description": "", "templateType": "anything"}, "y33": {"name": "y33", "group": "Ungrouped variables", "definition": "if(y13<y23,y13,y23)", "description": "", "templateType": "anything"}, 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"group": "Ungrouped variables", "definition": "2x1^3-3*(x1+x2)*x1^2+6*x1*x2*x1+c0", "description": "", "templateType": "anything"}, "x42": {"name": "x42", "group": "Ungrouped variables", "definition": "if(y12<y22,x22,x12)", "description": "", "templateType": "anything"}, "x02": {"name": "x02", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "y4": {"name": "y4", "group": "Ungrouped variables", "definition": "if(y1<y2,y2,y1)", "description": "", "templateType": "anything"}, "y42": {"name": "y42", "group": "Ungrouped variables", "definition": "if(y12<y22,y22,y12)", "description": "", "templateType": "anything"}, "x13": {"name": "x13", "group": "Ungrouped variables", "definition": "random(-3..3 except x1)", "description": "", "templateType": "anything"}, "x111": {"name": "x111", "group": "Ungrouped variables", "definition": "min({x32}, {x42})", "description": "", "templateType": "anything"}, "x222": {"name": "x222", "group": "Ungrouped variables", "definition": "max({x32}, {x42})", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["y1", "y0", "y3", "y2", "y4", "y02", "y03", "x1", "x43", "x42", "y42", "y43", "x03", "x02", "y22", "y23", "x23", "x22", "x2", "x3", "x0", "c0", "c02", "x4", "x32", "x33", "x13", "y33", "y32", "x12", "c03", "y13", "y12", "x111", "x222"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>For the following function:</p>\n<p>\\[ \\simplify{y = 2x^3-3{(x12+x22)}x^2+6{x12*x22}x+{c02}} \\]</p>\n<p>Determine the coordinates and the nature of the stationary points.</p>\n<p>First Derivative; $y^{\\prime}(x) =$ [[4]]</p>\n<p>Give values of x where stationary points occur: smallest-$x_1$ =[[6]],&nbsp;&nbsp;&nbsp; largest-$x_2$ = [[7]]</p>\n<p>Second Derivative is $~ y^{\\prime\\prime}(x) =$ [[5]]</p>\n<p>Values of Second derivative at stationary points:&nbsp;</p>\n<p>$y^{\\prime\\prime}(x_1) = $[[8]]</p>\n<p>$y^{\\prime\\prime}(x_2) = $[[9]]</p>\n<p></p>\n<p>Minimum point: $\\big($ [[0]] $ , $ [[1]] $\\big)$ and maximum point: $\\big($ [[2]] $ , $ [[3]] $\\big)$</p>\n<p>Enter fractions in their simplest form.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{x32}", "answerSimplification": "all,fractionNumbers", "showPreview": 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