// Numbas version: exam_results_page_options {"name": "Differentiation: second derivatives, core facts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: second derivatives, core facts", "tags": [], "metadata": {"description": "

Q1 is true/false question covering some core facts, notation and basic examples.  Q2 has two functions for which second derivative needs to be determined.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

This question deals with second derivatives.

\n

-----------------------------------

", "advice": "

Correct answers are not given for the first question, because you should read the information provided to determine what is correct.

\n

", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "part b", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "part b", "definition": "random(-5..5 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "change these", "definition": "random(10..18)+random(1..9)/10", "description": "", "templateType": "anything", "can_override": false}, "statements": {"name": "statements", "group": "do not change these", "definition": "map(if(rand[j]=1,\n statements_true[j],\n statements_false[j]),j,0..n-1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "change these", "definition": "length(statements_true)", "description": "", "templateType": "anything", "can_override": false}, "statements_true": {"name": "statements_true", "group": "change these", "definition": "[ \"The first derivative of a function is just the derivative of the function\",\n \"The second derivative of $f$ is the derivative of the derivative of $f$ \",\n \"The second derivative of $g$ can be denoted by $\\\\frac{d^2g}{dt^2}$\",\n \"The second derivative of $f$ can be denoted by $\\\\frac{d^2f}{dx^2}$\",\n \"From the video, if the second derivative is positive, then the original graph is curving upwards\",\n \"From the video, if the second derivative is negative, then the original graph is curving downwards\"]\n ", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "change these", "definition": "random(3..7)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "part b", "definition": "random(-6..-9)", "description": "", "templateType": "anything", "can_override": false}, "statements_false": {"name": "statements_false", "group": "change these", "definition": "[ \"The first derivative of a function is different from the derivative of the function\",\n \"The second derivative of $f$ is the derivative of $f$ \",\n \"The second derivative of $g$ is denoted by $\\\\frac{d^2g}{d^2t} $\",\n \"The second derivative of $g$ can be denoted by $\\\\frac{d^2g}{dx^2}$\",\n \"From the video, if the second derivative is positive, then the original graph is positive\",\n \"From the video, if the second derivative is negative, then the original graph is negative\"\n ]", "description": "", "templateType": "anything", "can_override": false}, "marks": {"name": "marks", "group": "do not change these", "definition": "matrix(map(if(rand[j]=1,[max_mark/n,-max_mark/3+max_mark/n],[-max_mark/3+max_mark/n,max_mark/n]),j,0..n-1))\n", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "change these", "definition": "random(6..9)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "part b", "definition": "random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "rand": {"name": "rand", "group": "do not change these", "definition": "repeat(random(0..1),n)", "description": "", "templateType": "anything", "can_override": false}, "max_mark": {"name": "max_mark", "group": "change these", "definition": "2", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "part b", "definition": "random(-2..-7)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "change these", "variables": ["statements_true", "statements_false", "max_mark", "n", "a", "b", "c"]}, {"name": "do not change these", "variables": ["rand", "statements", "marks"]}, {"name": "part b", "variables": ["a1", "n1", "b1", "a2", "b2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Which of the following are true and which are false?

\n

\n

In the following, $f(x) = \\sin(x)$ and $g(t) = \\cos(t)$.

", "minMarks": 0, "maxMarks": "0", "minAnswers": "{n}", "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": false, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": false, "markingMethod": "sum ticked cells", "choices": "{statements}", "matrix": "{marks}", "layout": {"type": "all", "expression": ""}, "answers": ["

True

", "

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": "

$f(x) = \\simplify{{a1}*x^{n1} + {b1}sin(x)}$.  What is $\\frac{d^2 f}{dx^2}$?

\n

[[0]]

\n

\n

\n

\n

$g(t) = \\simplify{{a2}ln(t) + {b2} cos(t)}$. What is $g''(t)$?

\n

[[1]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a1*n1*(n1-1)}*x^{n1-2} - {b1}sin(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, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{-a2}/t^2 - {b2} cos(t)", "answerSimplification": "all", "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": "t", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}, {"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": "Paul Howes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/632/"}, {"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/"}]}