// Numbas version: exam_results_page_options
{"name": "Anna's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "statement": "<p>Consider the following equation.</p>\n<p>\\[\\simplify[std]{e^({m}x)+{b}x-{a}=0}\\]</p>\n<p>Find the approximate solution in the range $0 \\le x \\le 1$ by using the Newton-Raphson method.</p>\n<p>The following diagram demonstrates the method.</p>\n<p>$x_0$ is the starting value, you can slide it along the x-axis to see the effect of changing it.</p>\n<p></p>\n<p>{test(m,b,a,maxy)}</p>\n<p></p>\n<p></p>\n<p></p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "name": "Anna's copy of Maria's copy of Mario's copy of Interactive Newton-Raphson method", "preamble": {"js": "", "css": ""}, "extensions": ["jsxgraph"], "functions": {"fun": {"definition": "((m*c-1)*e^(m*c)+a)/(m*e^(m*c)+b)", "language": "jme", "type": "number", "parameters": [["m", "number"], ["b", "number"], ["a", "number"], ["c", "number"]]}, "test": {"definition": "var div = Numbas.extensions.jsxgraph.makeBoard('600px','400px', {boundingbox:[0,maxy,3,-30], axis:false});\n        var brd=div.board;\n        // Initial function term\n        var term = function(x) { return Math.exp(m*x)+b*x-a; };\n        var graph = function(x) { return term(x); };\n        // Recursion depth\n        var steps = 4;\n        // Start value\n        var s = 2;\n         \n        //for (i = 0; i < steps; i++) {\n         //document.write('<tr><td><nobr>x<sub>' + i + '</sub> = </nobr></td><td><font id=\"xv' + i + '\"></font></td></tr>');\n        //}\n        \n        var i;\n        var ax = brd.create('axis', [[0,0], [1,0]], {strokeColor: 'black'});\n        var ay = brd.create('axis', [[0,0], [0,1]], {strokeColor: 'black'});\n         \n        var g = brd.create('functiongraph', [function(x){return graph(x);}],{strokeWidth: 2, dash:0});\n       // var x = brd.create('glider',[s,0,ax], {name: 'x_{0}', strokeColor: 'magenta', fillColor: 'yellow'});\n         \n       // newton(x, steps, brd);\t\n         \n        //function xval() {\n       //for (i = 0; i < steps; i++)\n      //document.getElementById('xv' + i).innerHTML = (brd.select('x_{' + i + '}').X()).toFixed(14);\n        //}\n         \n        //brd.addHook(xval);\n         \n        function newton(p, i, board) {\t\n        board.suspendUpdate();\t\n        if(i>0) {\n        var f = board.create('glider',[function(){return p.X();}, function(){return graph(p.X())},g], {name: '', style: 3, strokeColor: 'green', fillColor: 'yellow'});\n        var l = board.create('line', [p,f],{strokeWidth: 0.5, dash: 1, straightFirst: false, straightLast: false, strokeColor: 'black'});\n       // var t = board.create('tangent',[f],{strokeWidth: 0.5, strokeColor: '#0080c0', dash: 0});\n        //var x = board.create('intersection',[ax,t,0],{name: 'x_{'+(steps-i+1) + '}', style: 4, strokeColor: 'magenta', fillColor: 'yellow'});\n      //  newton(x,--i, board);\n        }\n        board.unsuspendUpdate(); \n      \n      \n        }\t\n        return div;", "language": "javascript", "type": "html", "parameters": [["m", "number"], ["b", "number"], ["a", "number"], ["maxy", "number"]]}, "funfornr": {"definition": "var div = Numbas.extensions.jsxgraph.makeBoard('600px','200px',{boundingBox:[0,10,1,-8],grid:false});\n        var board = div.board;\n        var m = Numbas.jme.unwrapValue(scope.variables.m);\n        var be = Numbas.jme.unwrapValue(scope.variables.b);\n        var al = Numbas.jme.unwrapValue(scope.variables.a);\n        var ans = Numbas.jme.unwrapValue(scope.variables.ans1)\n        var a = board.create('point',[ans,0],'$A$');\n        \n        var graph = board.create('functiongraph',function(x){return Math.exp(m*x)+be*x-al});\n        \n        return div;\n        ", "language": "javascript", "type": "html", "parameters": []}, "nr": {"definition": "if(n=5,l+[fun(m,b,a,c)]+[fun(m,b,a,fun(m,b,a,c))],nr(m,b,a,fun(m,b,a,c),n-1,l+[c]))", "language": "jme", "type": "list", "parameters": [["m", "number"], ["b", "number"], ["a", "number"], ["c", "number"], ["n", "number"], ["l", "list"]]}}, "advice": "<p>a)</p>\n<p>Recall that the Newton-Raphson method is defined by:<br/>\\[x_{n+1}=x_n-\\frac{g(x_n)}{g'(x_n)}\\]<br/>where we would like to find the root of the equation $g(x)=0$</p>\n<p>In this question we have:<br/>\\[\\simplify[std]{g(x) = Exp({m} * x) + {b} * x + { -a}} \\Rightarrow \\simplify[std]{g'(x) = {m}*Exp({m} * x) + {b}}\\]<br/>Substituting these expressions into the formula we have:<br/>\\[x_{n+1} =\\simplify[std]{ x_n -((Exp({m} * x_n) + {b} * x_n + { -a}) / ({m} * Exp({m} * x_n) + {b}))}\\]</p>\n<p>which can be rearranged to give:<br/>\\[x_{n + 1} = \\simplify[std]{(({m} * x_n -1) * Exp({m} * x_n) + {a}) / ({m} * Exp({m} * x_n) + {b})}\\]</p>\n<p>(In your answers you would input $x$ rather than $x_n$.)</p>\n<p>In the following let $\\displaystyle f(x)=\\simplify[std]{ (({m} * x -1) * Exp({m} * x ) + {a}) / ({m} * Exp({m} * x ) + {b})}$</p>\n<p>b)</p>\n<p>If $x_0=2$ then $x_1$ is simply given by:<br/>\\[\\simplify[std]{x_1 = (({2*m} -1) * Exp({2*m}) + {a}) / ({m} * Exp({2*m}) + {b})}\\]</p>\n<p>which to 4 decimal places is: $\\;\\;x_1= \\var{ans}$</p>\n<p>We find on running the iteration that the first six values are:</p>\n<p>\\[\\begin{align}x_1&amp;=f(x_0)=f(2)&amp;=\\var{results[1]}\\\\x_2&amp;=f(x_1)=f(\\var{results[1]})&amp;=\\var{results[2]}\\\\x_3&amp;=f(x_2)=f(\\var{results[2]})&amp;=\\var{results[3]}\\\\x_4&amp;=f(x_3)=f(\\var{results[3]})&amp;=\\var{results[4]}\\\\x_5&amp;=f(x_4)=f(\\var{results[4]})&amp;=\\var{results[5]}\\\\x_6&amp;=f(x_5)=f(\\var{results[5]})&amp;=\\var{results[6]}\\end{align}\\]</p>\n<p>So the solution to the equation to four decimal places for $0 \\le x \\le 1$ is $x=\\var{precround(ans1,4)}$</p>\n<p>Here we see the graph of $\\simplify{e^({m}*x)+{b}*x-{a}}$ and the first four successive approximations to the root:</p>\n<p>{test(m,b,a,maxy)}</p>\n<p></p>\n<p>Note that you can slide the first approximation $x_0$ along the x-axis to see the effect of changing the starting value.</p>", "metadata": {"description": "<p>Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.</p>\n    <p>Included in the Advice of this question are:</p>\n    <p>6 iterations of the method.</p>\n    <p>Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "maxy", "ans1", "m", "results", "b", "tol", "ans", "tans", ""], "variables": {"": {"templateType": "anything", "name": "", "description": "", "group": "Ungrouped variables", "definition": ""}, "m": {"templateType": "anything", "name": "m", "description": "", "group": "Ungrouped variables", "definition": "random(1.5..2#0.1)"}, "b": {"templateType": "anything", "name": "b", "description": "", "group": "Ungrouped variables", "definition": "random((a+1)..9)"}, "tans": {"templateType": "anything", "name": "tans", "description": "", "group": "Ungrouped variables", "definition": "((2*m-1)*exp(2*m)+a)/(m*exp(2*m)+b)"}, "results": {"templateType": "anything", "name": "results", "description": "", "group": "Ungrouped variables", "definition": "nr(m,b,a,2,10,[])"}, "ans1": {"templateType": "anything", "name": "ans1", "description": "", "group": "Ungrouped variables", "definition": "precround(results[6],4)"}, "maxy": {"templateType": "anything", "name": "maxy", "description": "", "group": "Ungrouped variables", "definition": "ceil(e^(2*m)+2*b-a)+5"}, "a": {"templateType": "anything", "name": "a", "description": "", "group": "Ungrouped variables", "definition": "random(8..15)"}, "ans": {"templateType": "anything", "name": "ans", "description": "", "group": "Ungrouped variables", "definition": "precround(tans,4)"}, "tol": {"templateType": "anything", "name": "tol", "description": "", "group": "Ungrouped variables", "definition": "0"}}, "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "variable_groups": [], "parts": [{"scripts": {}, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "stepsPenalty": 0, "sortAnswers": false, "useCustomName": false, "prompt": "\n        <p>This equation has a root in the range $0 \\lt x \\lt 1$.</p>\n      <p>Using the Newton-Raphson formula, if $x_n$ is the $n$th estimate for this root, show that the next estimate can be written in the form \\[x_{n+1}= \\frac{p(x_n)}{g'(x_n)}\\]<br/>Enter $p(x_n)$ and $g'(x_n)$ in the boxes below.</p>\n      <p>Please note that if you enter a function of the form $xe^{ax}$, then you must input it as $x*e^{ax}$.</p>\n      <p>$p(x_n)=\\;\\;$[[0]]&nbsp;In your answer use $x$ instead of $x_n$.</p>\n      <p>$g'(x_n)=\\;\\;$[[1]]&nbsp;In your answer use $x$ instead of $x_n$.</p>\n      <p>If you have forgotten the Newton-Raphson formula you can click on Steps to see it. You will not lose any marks in doing so.</p>\n        ", "gaps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.0001, "showPreview": true, "failureRate": 1, "checkingType": "absdiff", "vsetRange": [1, 1.5], "unitTests": [], "answer": "(((({m} * x) -1) * Exp(({m} * x))) + {a})", "variableReplacements": [], "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "std", "showCorrectAnswer": true, "type": "jme", "scripts": {}, "marks": 1, "showFeedbackIcon": true, "useCustomName": false, "checkVariableNames": false, "customName": ""}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "showPreview": true, "failureRate": 1, "checkingType": "absdiff", "vsetRange": [0, 1], "unitTests": [], "answer": "(({m} * Exp(({m} * x))) + {b})", "variableReplacements": [], "valuegenerators": [{"name": "x", "value": ""}], "answerSimplification": "std", "showCorrectAnswer": true, "type": "jme", "scripts": {}, "marks": 1, "showFeedbackIcon": true, "useCustomName": false, "checkVariableNames": false, "customName": ""}], "unitTests": [], "marks": 0, "variableReplacements": [], "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "showPreview": true, "failureRate": 1, "checkingType": "absdiff", "vsetRange": [0, 1], "unitTests": [], "answer": "", "variableReplacements": [], "valuegenerators": [], "checkVariableNames": false, "type": "jme", "scripts": {}, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "useCustomName": false, "prompt": "<p>Recall that the Newton-Raphson method is defined by:<br/>\\[x_{n+1}=x_n-\\frac{g(x_n)}{g'(x_n)}\\]<br/>where we would like to find the root of the equation $g(x)=0$</p>", "customName": ""}], "customName": "", "showCorrectAnswer": true, "type": "gapfill"}, {"scripts": {}, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "gaps": [{"scripts": {}, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "minValue": "ans-tol", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "useCustomName": false, "correctAnswerFraction": false, "allowFractions": false, "maxValue": "ans+tol", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "marks": 1, "variableReplacements": [], "mustBeReducedPC": 0, "customName": "", "showCorrectAnswer": true, "type": "numberentry", "mustBeReduced": false, "showFractionHint": true}], "sortAnswers": false, "useCustomName": false, "prompt": "<p>If $x_0=2\\;\\;\\;$what is $x_1$ correct to $4$ decimal places?</p>\n        <p>$x_1=\\;\\;$ [[0]]</p>\n        <p>Enter your answer to 4 decimal places.</p>", "unitTests": [], "marks": 0, "variableReplacements": [], "customName": "", "showCorrectAnswer": true, "type": "gapfill"}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showCellAnswerState": true, "unitTests": [], "matrix": [0, 0, 0], "variableReplacements": [], "minMarks": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "scripts": {}, "distractors": ["", "", ""], "marks": 0, "choices": ["Choice 1<br/>{test(m,b,a,maxy)}", "Choice 2<br/>{test(-m,b,a,maxy)}", "Choice 3<br/>{test(m,-b,a,maxy)}"], "showFeedbackIcon": true, "maxMarks": 0, "useCustomName": false, "prompt": "<p>Which graph represent the following equation:</p>\n<p>\\[\\simplify[std]{e^({m}x)+{b}x-{a}=0}\\]</p>", "customName": "", "shuffleChoices": false, "displayColumns": 0}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showCellAnswerState": true, "unitTests": [], "matrix": [0, 0, 0], "variableReplacements": [], "minMarks": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "scripts": {}, "distractors": ["", "", ""], "marks": 0, "choices": ["Choice 1<br/>{test(m,b,a,maxy)}", "Choice 2<br/>{test(-m,b,a,maxy)}", "Choice 3<br/>{test(m,-b,a,maxy)}"], "showFeedbackIcon": true, "maxMarks": 0, "useCustomName": false, "prompt": "<p>Which graph represent the following equation:</p>\n<p>\\[\\simplify[std]{e^({m}x)+{b}x-{a}=0}\\]</p>", "customName": "", "shuffleChoices": false, "displayColumns": 0}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Anna Strzelecka", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2945/"}, {"name": "Mario Stevanovski", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2979/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Anna Strzelecka", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2945/"}, {"name": "Mario Stevanovski", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2979/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}