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{"name": "Differentiation: Equation of tangent. Chain rule.", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Equation of tangent. Chain rule.", "tags": [], "metadata": {"description": "<p>A function $f(x) = cln(ax^2+bx) -x$ is sketched and tangent is also drawn. The equation of the tangent line is asked for and $x$-coordinate for horizontal tangent is asked for. Calculator.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>This is a calculator question.</p>\n<p>--------------------</p>", "advice": "<p>Remember to always check your answer is reasonable! You can do basic estimates from the diagram (e.g. is the gradient positive or negative, etc.)</p>\n<p></p>\n<p>(a)&nbsp;See&nbsp;?? for connection between differentiation and tangents. See&nbsp;?? about how to differentiate.</p>\n<p></p>\n<p>(b) To help you spot any errors, half-way through your calculations, you should end up having to solve the following quadratic equation:</p>\n<p>$\\simplify{{a'}x^2+{b'}x+{c'}=0}$.</p>\n<p>You can do this either by completing the square or by using the quadratic formula.</p>", "rulesets": {}, "extensions": ["jsxgraph"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"absfx": {"name": "absfx", "group": "part a", "definition": "abs(fx)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "part a", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "x0": {"name": "x0", "group": "part a", "definition": "(-b'+sqrt(b'*b'-4*a'*c'))/(2*a')", "description": "", "templateType": "anything", "can_override": false}, "a'": {"name": "a'", "group": "part a", "definition": "a", "description": "", "templateType": "anything", "can_override": false}, "fx": {"name": "fx", "group": "part a", "definition": "c*ln(a*x^2+b*x)-x", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "part a", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "y_int": {"name": "y_int", "group": "part a", "definition": "fx-x*m", "description": "", "templateType": "anything", "can_override": false}, "fx0": {"name": "fx0", "group": "part a", "definition": "c*ln(a*(x0)^2+b*x0)-x0", "description": "", "templateType": "anything", "can_override": false}, "b'": {"name": "b'", "group": "part a", "definition": "b-2*a*c", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "part a", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "part a", "definition": "c*(2*a*x+b)/(a*x^2+b*x)-1", "description": "", "templateType": "anything", "can_override": false}, "c'": {"name": "c'", "group": "part a", "definition": "-c*b", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "part a", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "and(m<>0,(fx0<8))", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "part a", "variables": ["a", "b", "c", "x", "fx", "m", "absfx", "y_int", "x0", "fx0", "a'", "b'", "c'"]}], "functions": {"plot": {"parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["x0", "number"], ["y0", "number"], ["m", "number"]], "type": "html", "language": "javascript", "definition": "// It creates the board, sets it up, then returns an\n// HTML div tag containing the board.\n\n\n\n// Max and min x and y values for the axis.\nvar x_min = -1.1;\nvar x_max = 9;\nvar y_min = -5;\nvar y_max = 10;\n\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: false,\n   grid: false,\n   axis:false,\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\n\n// create the x-axis and y-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 });\n\n\n\n\n// Plot the function.\n    board.create('functiongraph',\n             [function(x){ return c*Math.log(a*x*x+b*x)-x},0,x_max],\n                {strokeWidth:2});\n\n//Plot the tangent.\n    board.create('functiongraph',\n             [function(x){ return y0+(x-x0)*(m)},x_min,x_max]);\n\n// Plot coordinates.\n    board.create('circle',[[x0,y0],0.1],{color:'red'});\n\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "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>The curve with equation $y = \\simplify{{c}ln({a}x^2+{b}x) -x}$ is sketched below.</p>\n<p>{plot(a,b,c,x,fx,m)}</p>\n<p>In addition, the tangent to the curve at $x=\\var{x}$ has been drawn.</p>\n<p></p>\n<p></p>\n<p>(a) What is the equation of the tangent? Give all coefficients to 3 s.f.</p>\n<p>$y= $&nbsp; [[0]]</p>\n<p></p>\n<p></p>\n<p>(b) When is the gradient of the curve equal to 0? Give your answer to 3 s.f.</p>\n<p>When $x=$&nbsp;[[1]]&nbsp;</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x*{siground(m,3)}+{siground(y_int,3)}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "siground(x0,3)", "maxValue": "siground(x0,3)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Doug Satterford", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3185/"}]}]}], "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Doug Satterford", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3185/"}]}