// Numbas version: exam_results_page_options {"name": "Oracle function root estimation", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": [], "name": "Oracle function root estimation", "tags": [], "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "type": "question", "advice": "

Interval bisection means, simply, taking an interval and cutting it in half.

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Let us define $I_0 = (\\var{intervalBisected[0]},\\var{intervalBisected[2]})$.

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In this instance, the idea is that we generate two subintervals of our initial interval, $I_0$, and by the intermediate value theorem, we can guarantee that one of them will contain our root. Here are the first steps.

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Firstly, we check the values of the function at the endpoints of $I_0$: $f(\\var{intervalBisected[0]}) = \\var{functionValues[0]}$ and $f(\\var{intervalBisected[2]}) = \\var{functionValues[2]}$. Hence, by the intermediate value theorem, there is at least one $c \\in I_0$ such that $f(c) = 0$.

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Now, we cut the interval $I_0$ in half, producing the subintervals $(\\var{intervalBisected[0]},\\var{intervalBisected[1]})$ and $(\\var{intervalBisected[1]},\\var{intervalBisected[2]})$. Now, $f(\\var{intervalBisected[1]}) = \\var{functionValues[1]}$, and so we choose the subinterval where the sign of $f$ differs at the endpoints, which is to say, we take $I_1 = (\\var{intervalChoice[0]},\\var{intervalChoice[1]})$. By the intermediate value theorem, we guarantee that there is a $c \\in I_1$ such that $f(c) = 0$.

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We repeat this procedure on our new, smaller interval, $I_1$, choosing a smaller interval still, until eventually we have an interval so small that we can guarantee the answer to two decimal places, which is $c = \\var{dpformat(roots[pos],2)}$.

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I will also tell you that there is a root in the open interval $(\\var{intervalBisected[0]},\\var{intervalBisected[2]})$. Using interval bisection and the intermediate value theorem, identify the location of this root to two decimal places.

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The root is located at $x=\\;$[[0]]

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I am thinking of a function, but I will not tell you what it is! However:

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1. I will tell you that it is continuous everywhere.
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3. I will tell you the value of the function anywhere you ask, to three decimal places. You can use the box below to inquire of me. (You may need to wait a little for it to load.)
4. \n
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{geogebra_applet('ufgKrAzb',defs)}

\n", "variable_groups": [{"variables": ["a", "b", "c", "d"], "name": "Coefficients"}, {"variables": ["rootsInteger", "rootsDecimal", "roots"], "name": "Roots"}, {"variables": ["k", "testRoots", "testRootsExplicit", "testRootsInteger", "testRootsDecimal"], "name": "Testing"}, {"variables": ["pos", "extend", "minExtent"], "name": "Question definition"}, {"variables": ["defs"], "name": "Geogebra"}, {"variables": ["intervalBisected", "functionValues", "intervalChoice"], "name": "Worked solution"}], "variablesTest": {"maxRuns": "1", "condition": "roots[pos] < intervalBisected[2]"}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(-3..3 except 0)", "templateType": "anything", "group": "Coefficients", "name": "a", "description": ""}, "defs": {"definition": "[\n ['a',a],\n ['b',b],\n ['c',c],\n ['d',d]\n]", "templateType": "anything", "group": "Geogebra", "name": "defs", "description": ""}, "c": {"definition": "-d*sum(map(1/x,x,roots))", "templateType": "anything", "group": "Coefficients", "name": "c", "description": ""}, "b": {"definition": "-a*sum(roots)", "templateType": "anything", "group": "Coefficients", "name": "b", "description": ""}, "functionValues": {"definition": "map(precround(a*x^3+b*x^2+c*x+d,3),x,intervalBisected)", "templateType": "anything", "group": "Worked solution", "name": "functionValues", "description": ""}, "d": {"definition": "-a*roots[0]*roots[1]*roots[2]", "templateType": "anything", "group": "Coefficients", "name": "d", "description": ""}, "extend": {"definition": "repeat(precround(random(50..99)*minExtent/100,2),2)", "templateType": "anything", "group": "Question definition", "name": "extend", "description": ""}, "testRootsInteger": {"definition": "rootsInteger[k]", "templateType": "anything", "group": "Testing", "name": "testRootsInteger", "description": ""}, "k": {"definition": "1", "templateType": "anything", "group": "Testing", "name": "k", "description": ""}, "pos": {"definition": "random(0..2)", "templateType": "anything", "group": "Question definition", "name": "pos", "description": ""}, "testRoots": {"definition": "roots[k]", "templateType": "anything", "group": "Testing", "name": "testRoots", "description": ""}, "intervalChoice": {"definition": "if(functionValues[0]*functionValues[1] < 0,intervalBisected[0..2],intervalBisected[1..3])", "templateType": "anything", "group": "Worked solution", "name": "intervalChoice", "description": ""}, "testRootsExplicit": {"definition": "rootsInteger[k]+rootsDecimal[k]", "templateType": "anything", "group": "Testing", "name": "testRootsExplicit", "description": ""}, "testRootsDecimal": {"definition": "rootsDecimal[k]", "templateType": "anything", "group": "Testing", "name": "testRootsDecimal", "description": ""}, "minExtent": {"definition": "precround(min([abs(roots[2]-roots[1]),abs(roots[2]-roots[0]),abs(roots[1]-roots[0])]),2)", "templateType": "anything", "group": "Question definition", "name": "minExtent", "description": ""}, "rootsInteger": {"definition": "sort(map(x-5,x,deal(10)[0..3]))", "templateType": "anything", "group": "Roots", "name": "rootsInteger", "description": ""}, "intervalBisected": {"definition": "[\n precround(roots[pos]-extend[0],2),\n precround((roots[pos]+extend[1]+roots[pos]-extend[0])/2,2),\n precround(roots[pos]+extend[1],2)\n]", "templateType": "anything", "group": "Worked solution", "name": "intervalBisected", "description": ""}, "rootsDecimal": {"definition": "repeat(random(1..9999)/10000,3)", "templateType": "anything", "group": "Roots", "name": "rootsDecimal", "description": ""}, "roots": {"definition": "map(sum(x),x,zip(rootsInteger,rootsDecimal))", "templateType": "anything", "group": "Roots", "name": "roots", "description": ""}}, "showQuestionGroupNames": false, "metadata": {"description": "

Given an oracle function that will output its value given an input, and an interval within which a root exists, find the root to a given precision.

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Makes use of a Geogebra applet.

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