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{"name": "Differentiation 19 - Finding Tangent and Normal", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');", "css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n"}, "ungrouped_variables": ["a", "b", "c", "d"], "advice": "<p>The differentiate of a function gives the gradient of the&nbsp;original function at the point $x$.</p>\n<p>The tangent to a line is the gradient of the curve at the point, and the normal is the reciprocal of this value (the perpendicular line intersecting the curve).</p>\n<p>The equation of a line can be found in many ways, but given a point and gradient for a straight line, two of the simple manipulations are:</p>\n<p>Using $y=mx+c$ and substituting in to find $c$</p>\n<p>OR</p>\n<p>Using $\\frac{y_2-y_1}{x_2-x_1}=m$ and rearranging.</p>", "metadata": {"description": "<p>Using differentiation to find the tangent and normal to a line at a given point</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>Find an equation for the tangent to the curve $\\simplify{y={a[0]}x^2+{b[0]}x+{c[0]}}$ at the point at which it crosses the y-axis.</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>\n<p>Therefore, the gradient of the tangent at $x=0$ is&nbsp;[[1]]</p>\n<p>When $x=0$, $y=$&nbsp;[[2]]</p>\n<p>Hence, the equation for the tangent&nbsp;is:</p>\n<p>So, $y=$&nbsp;[[3]]</p>", "unitTests": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": 1, "checkVariableNames": false, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "answer": "2*{a[0]}*x+{b[0]}", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}, {"customName": "", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "useCustomName": false, "allowFractions": false, "variableReplacements": [], "unitTests": [], "minValue": "{b[0]}", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{b[0]}"}, {"customName": "", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "type": "numberentry", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "useCustomName": false, "allowFractions": false, "variableReplacements": [], "unitTests": [], "minValue": "{c[0]}", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "marks": 1, "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": 0, "mustBeReduced": false, "maxValue": "{c[0]}"}, {"variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, "marks": "2", "checkVariableNames": false, "valuegenerators": [{"value": "", "name": "x"}], "customName": "", "answer": "{b[0]}*x+{c[0]}", "showPreview": true, "scripts": {}, "variableReplacements": [], "unitTests": [], "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answerSimplification": "all"}], "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true}, {"customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "useCustomName": false, "variableReplacements": [], "prompt": "<p>Find an equation for the normal&nbsp;to the curve $\\simplify{y={a[1]}x^2+{b[1]}x+{c[1]}}$ at the point where $x=\\var{d}$.</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>\n<p>Therefore, gradient of the curve at $x=\\var{d}$ is&nbsp;[[1]]</p>\n<p>Meaning the gradient of the normal at $x=\\var{d}$ is&nbsp;[[4]]</p>\n<p>When $x=\\var{d}$, $y=$&nbsp;[[2]]</p>\n<p>Hence, the equation for the normal is:</p>\n<p></p>\n<p>So, $y=$&nbsp;[[3]]</p>", "unitTests": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "customMarkingAlgorithm": "", "type": "jme", "showFeedbackIcon": true, "useCustomName": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "showCorrectAnswer": true, 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