// Numbas version: finer_feedback_settings {"showQuestionGroupNames": false, "feedback": {"intro": "", "feedbackmessages": [], "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "showtotalmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "metadata": {"description": "

Critical points, absolute minimum, local maximum and minimum points, increasing and decreasing, concavity

rebel

rebelmaths

", "licence": "Creative Commons Attribution 4.0 International"}, "navigation": {"allowregen": true, "reverse": true, "browse": true, "onleave": {"action": "none", "message": ""}, "showfrontpage": true, "showresultspage": "oncompletion", "preventleave": true}, "percentPass": "60", "duration": 0, "timing": {"allowPause": true, "timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}}, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-shuffled", "questions": [{"name": "Applications of differentiation ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers"]}, "advice": "

Parts A and B

Here, the question takes you throught the stages needed to find the solution. The reason we differentiate is that the derivative of a function tells us its gradient at a given point, and we want to find where the function has gradient zero because when the gradient is zero we either have a maximum or a minimum point.

Part C

The first part of this question is similar to parts A and B. The tricky bit is the second part! You need to work out the value of $t$ that produces the maximum piont but that is not the final answer - you need to use that value of $t$ to find the maximum height, which you do by substituting $t$ into the original function to find $y$.

", "statement": "", "ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "variables": {"g": {"definition": "random(-10 .. 10#1)", "group": "Ungrouped variables", "name": "g", "description": "", "templateType": "randrange"}, "w": {"definition": "random(2 .. 5#0.1)", "group": "Ungrouped variables", "name": "w", "description": "", "templateType": "randrange"}, "t": {"definition": "random(0 .. 1#0.1)", "group": "Ungrouped variables", "name": "t", "description": "", "templateType": "randrange"}, "c": {"definition": "random(2..7)", "group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything"}, "b": {"definition": "random(2..5)", "group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything"}, "f": {"definition": "random(-10 .. 10#1)", "group": "Ungrouped variables", "name": "f", "description": "", "templateType": "randrange"}, "z": {"definition": "random(20 .. 30#0.5)", "group": "Ungrouped variables", "name": "z", "description": "", "templateType": "randrange"}, "a": {"definition": "random(0 .. 10#0.5)", "group": "Ungrouped variables", "name": "a", "description": "", "templateType": "randrange"}, "h": {"definition": "random(0 .. 5#0.5)", "group": "Ungrouped variables", "name": "h", "description": "", "templateType": "randrange"}, "d": {"definition": "random(2..5)", "group": "Ungrouped variables", "name": "d", "description": "", "templateType": "anything"}, "maximum": {"definition": "f<0", "group": "Ungrouped variables", "name": "maximum", "description": "

Is the stationary point a maximum?

", "templateType": "anything"}}, "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "unitTests": [], "prompt": "

Find the gradient of the curve $y$ at the point $x=\\var{d}$, giving your answer to $2$ decimal places if necessary.

\\[ y = \\simplify{ {a}*x^2 + {b}x + {c}} \\]

Firstly, differentiate.

$\\displaystyle \\frac{dy}{dx}=$ [[1]]

Gradient at $x=\\var{d}\\;$ is [[0]]

", "scripts": {}, "customName": "", "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "2*a*d+b", "maxValue": "2*a*d+b", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "

You have not given your answer to the correct precision.

Find the coordinates of the turning point of the function below and state whether it is a maximum or a minimum point. Give your answers to $2$ decimal places where necessary.

$y=\\simplify {{f}x^2+{g}x+{h}}$

Firstly, find the first and second derivatives $y$.

$\\displaystyle \\frac{dy}{dx}=$ [[2]]

$\\displaystyle \\frac{d^2y}{dx^2}=$ [[3]]

Secondly, find $x$ such that $\\displaystyle \\frac{dy}{dx}=0$.

$x$-coordinate of the turning point $=$ [[0]]

$y$-coordinate of the turning point $=$ [[1]]

The turning point is a [[4]]

", "scripts": {}, "customName": "", "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "-g/(2*f)", "maxValue": "-g/(2*f)", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "

You have not given your answer to the correct precision.

", "type": "numberentry", "precision": "2"}, {"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "g^2/(4*f)-g^2/(2*f)+h", "maxValue": "g^2/(4*f)-g^2/(2*f)+h", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "

You have not given your answer to the correct precision.

", "type": "numberentry", "precision": "2"}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "2*{f}*x+{g}", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [{"value": "", "name": "x"}], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "2*{f}", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "displayColumns": 0, "showFeedbackIcon": true, "shuffleChoices": false, "variableReplacements": [], "unitTests": [], "maxMarks": "0", "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "showCellAnswerState": true, "distractors": ["", ""], "scripts": {}, "choices": ["

maximum

", "

minimum

"], "customName": "", "minMarks": 0, "customMarkingAlgorithm": "", "displayType": "radiogroup", "type": "1_n_2", "showCorrectAnswer": true}], "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "unitTests": [], "prompt": "

An unpowered missile is launched vertically from the ground.

At a time $t$ seconds after the instant of projection, its height, $y$ metres, above the ground is given by the formula

\\[ y=\\var{z}t-\\var{w}t^2. \\]

Calculate the maximum height reached by the missile.

Firstly, differentiate.

$\\displaystyle \\frac{dy}{dt}=$ [[0]]

Now use this result and your knowledge of differentiation to find the maximum height of the missile, rounding your answer to $2$ decimal places.

$y=$ [[1]]

", "scripts": {}, "customName": "", "gaps": [{"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "{z}-2*{w}*t", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [{"value": "", "name": "t"}], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "z^2/(4w)", "maxValue": "z^2/(4w)", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "type": "numberentry", "precision": "2"}], "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true}], "type": "question"}]}], "name": "Max min problems", "type": "exam", "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}], "extensions": [], "custom_part_types": [], "resources": []}