// Numbas version: exam_results_page_options {"name": "Stationary Points 02", "extensions": [], "custom_part_types": [], "resources": [["question-resources/cubic1.PNG", "/srv/numbas/media/question-resources/cubic1.PNG"], ["question-resources/ball1.png", "/srv/numbas/media/question-resources/ball1.png"], ["question-resources/ball2.png", "/srv/numbas/media/question-resources/ball2.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Stationary Points 02", "tags": [], "metadata": {"description": "Instructional \"drill\" exercise to emphasize the method.", "licence": "All rights reserved"}, "statement": "

Points on the curve $y = f(x)$ at which $\\frac{dy}{dx}=0$ are called stationary points of the function.

If $x_0$ locates a stationary point of $f(x)$, so that $\\frac{dy}{dx} \\large \\vert_{x_0}=0$ then the stationary point is:

a local minimum if $\\frac{d^2y}{dx^2} \\Large \\vert_{x_0}>0$

a local maximum if $\\frac{d^2y}{dx^2} \\Large \\vert_{x_0}<0$

inconclusive if $\\frac{d^2y}{dx^2} \\Large \\vert_{x_0}=0$

", "advice": "

We are asked to find the greatest height reached by a thrown ball whose height can be modelled by the function $h=\\simplify{ {C}+{CF2}t-{CF1}t^{2} }$

You don't need to produce a graph to do this but right now, it may help us to visualise what it is that we are trying to do. Here is a graph of this function with the highest point indicated:

It is easy to see that the highest point is where the graph \"turns\" and the gradient changes from positive to negative. That is the point that we are looking for.

At this point the gradient will be zero. The gradient is given by the derivative of the function $\\frac{dh}{dt}$.

Differentiate the function:

$\\frac{dh}{dt}=\\simplify{{CF2}-{CF1}*2*t}$

The stationary points will be where this is zero, so:

$\\simplify{{CF2}-{CF1}*2*t}=0$

If we now solve this equation it will give us the time to reach that height.

$\\simplify{{CF2}-{CF1}*2*t}=0$

$\\var{CF2}=\\simplify{{CF1}*2*t}$

$\\frac{\\var{CF2}}{\\simplify{{CF1}*2}}=t$

$t=\\var{t}$ seconds

Now that we know the time it takes for the ball to reach its highest point, we are ask to calculate how high it reaches.

We simply need to use the original equation that models the height of the ball and substitute $t=\\var{t}$

$h=\\simplify{ {C}+{CF2}t-{CF1}t^{2} }$

becomes:

$h= \\var{C}+\\var{CF2}(\\var{t})-\\var{CF1}(\\var{t})^{2} $

$h=\\var{h}$ m

In this example it seems logically obvious that the stationary point is a maximum. After all, balls fly up and then fall down, they don't fly down and then float up again.

However, it is always good practice to check and it will help to stop you making silly mistakes - and it is quick and easy to do.

To find out, we need the second derivative $\\frac{d^2h}{dt^2}$.

Here is the original function:

$h=\\simplify{ {C}+{CF2}t-{CF1}t^{2} }$

and its first derivative, that we worked out earlier:

$\\frac{dh}{dt}=\\simplify{{CF2}-{CF1}*2*t}$

Differentiate a second time to give:

$\\frac{d^2h}{dt^2}=\\simplify{-{CF1}*2}$

Which is negative, confirming that this point is a local maximum.

", "rulesets": {}, "extensions": [], "variables": {"CF1": {"name": "CF1", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "

x^2 co-efficient

", "templateType": "anything"}, "CF2": {"name": "CF2", "group": "Ungrouped variables", "definition": "random(11..19)", "description": "

x co-efficient

", "templateType": "anything"}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "

constant

", "templateType": "anything"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "precround(({CF2})/(2*{CF1}),2)", "description": "

pre worked out answer for my convenience

", "templateType": "anything"}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "precround(3+{CF2}*{t}-{CF1}*{t}^2,2)", "description": "

pre worked out answer for my convenience

", "templateType": "anything"}}, "variablesTest": {"condition": "-2*{CF1}<0", "maxRuns": "181"}, "ungrouped_variables": ["CF1", "CF2", "C", "t", "h"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

A ball is thrown in the air. Its height (in metres) at any time $t$ is given by:

$h=\\simplify{ {C}+{CF2}t-{CF1}t^{2} }$

What is its maximum height?

The ball will reach its highest at the stationary (turning) point.

First differentiate the function:

$\\large \\frac{dh}{dt}=$ [[0]]

At the stationary point, this gradient will be zero.

Enter the equation that shows this:

[[1]]

Solve this equation to find the time when this happens:

$t=$ [[2]] seconds

Substitute this values, into the original function to find the height reached at that time:

$h=$[[3]] m

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{CF2}-{CF1}*2*t", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{CF2}-{CF1}*2*t=0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{t}", "maxValue": "{t}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": "0", "precisionMessage": "You have not given your answer to the correct decimal places.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{h}", "maxValue": "{h}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct decimal places.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Check to ensure that this is a maximum.

Whilst, in this case, it is not strictly necessary, it is good practice to check the nature of the stationary points that you find.

To determine the nature of the stationary point we need the second derivative $\\frac{d^2h}{dt^2}$

$\\frac{d^2h}{dt^2}=$ [[0]]

The value of this second derivative is [[1]]

Therefore, this is [[2]]