// Numbas version: finer_feedback_settings
{"name": "Tangent line approximation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Tangent line approximation", "tags": [], "metadata": {"description": "Use the tangent to get a linear approximation.<br>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<p><strong>a)</strong> The derivative of \\[ f(x) = \\sqrt[\\var{n}]{x}&nbsp; =&nbsp;x^{\\frac{1} {\\var{n}}} \\] is given by<br/>\\[ f'(x) = \\frac{\\mathrm{d}}{\\mathrm{d}x}(x^{\\frac{1} {\\var{n}}}) = {\\frac{1} {\\var{n}}} \\cdot x^{\\frac{\\var{1-n}} {\\var{n}}}&nbsp; .\\]</p>\n<p>The equation of the tangent to the graph of \\( f \\) at \\( ( a, f(a)) \\) is obtained by <br/>\\[ y - f(a) = f'(a) \\cdot (x-a) \\]</p>\n<p>For \\( a = \\var{a} \\) , this means <br/>\\[ y - \\var{m} = \\frac{1} {\\var{ricoN}} \\cdot \\left( x-\\var{a} \\right) \\]<br/>or equivalently<br/>\\[ y = \\var{m} + \\frac{1} {\\var{ricoN}} \\cdot \\left( x-\\var{a} \\right)&nbsp; = \\frac{1} {\\var{ricoN}} \\cdot x +&nbsp; \\frac {\\var{m*(n-1)}} {\\var{n}}\\]</p>\n<p></p>\n<p><strong>b)</strong> The linear approximation of&nbsp; \\( f(a+\\var{eps}) \\) is given by <br/>\\[ f(a) + f'(a) \\cdot \\var{eps} \\]<br/>When \\( a = \\var{a} \\)&nbsp; this leads to<br/>\\[ \\var{rklben} \\]<br/>which needs to be rounded up to&nbsp; \\( \\var{p+3} \\) decimals.</p>", "rulesets": {}, "extensions": [], "variables": {"rklcN": {"name": "rklcN", "group": "Ungrouped variables", "definition": "n", "description": "<p>cst rklvgl noemer</p>", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "<p>grondtal in a = m^n</p>", "templateType": "anything"}, "ricoN": {"name": "ricoN", "group": "Ungrouped variables", "definition": "(n*m^(n-1))", "description": "", "templateType": "anything"}, "rklbenadering": {"name": "rklbenadering", "group": "Ungrouped variables", "definition": "m+eps/(n*m^(n-1))", "description": "", "templateType": "anything"}, "rklcT": {"name": "rklcT", "group": "Ungrouped variables", "definition": "m*(n-1)", "description": "<p>teller constante rklvgl</p>", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "<p>decimaal om bij a op te tellen</p>", "templateType": "anything"}, "eps": {"name": "eps", "group": "Ungrouped variables", "definition": "10^(-p)", "description": "<p>eps</p>", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "m^n", "description": "<p>getal a</p>", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "<p>exponent in a = m^n</p>", "templateType": "anything"}, "rklben": {"name": "rklben", "group": "Ungrouped variables", "definition": "dpformat(m+eps/(n*m^(n-1)),12)", "description": "<p>rklben</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "n", "a", "p", "eps", "ricoN", "rklcT", "rklcN", "rklbenadering", "rklben"], "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, "prompt": "<h2>Consider the function \\( f: \\mathbb{R}^+&nbsp; \\to&nbsp; \\mathbb{R} : x \\mapsto&nbsp; \\sqrt[\\var{n}]{x}&nbsp; \\, \\, . \\)</h2>"}, {"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>Write an equation of the tangent line to the graph of&nbsp; \\( f \\) at the point \\( ( a, f(a)) \\) if \\( a = \\var{a} \\).</p>\n<p>Do this in the form \\(&nbsp; y = number1 \\cdot x+number2 \\). Use fractions, but no decimals.</p>\n<p>\\( y = \\) [[0]]&nbsp; \\( \\cdot x+ \\)&nbsp; [[1]]</p>\n<p></p>", "gaps": [{"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": "1/{ricoN}", "maxValue": "1/{ricoN}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "rklcT/rklcN", "maxValue": "rklcT/rklcN", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>Approximate&nbsp; \\( f(\\var{a+eps}) \\) up to \\( \\var{p+3} \\) decimals using this tangent line:</p>\n<p>&nbsp;\\( f(\\var{a+eps}) \\)&nbsp; approximately equals &nbsp; [[0]]</p>\n<p></p>", "gaps": [{"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": "rklbenadering", "maxValue": "rklbenadering", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "p+3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Paul Verheyen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3610/"}]}]}], "contributors": [{"name": "Paul Verheyen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3610/"}]}