{polar_curves(f1)}
", "shuffleChoices": true, "showFeedbackIcon": true, "displayType": "radiogroup", "choices": ["$ r = 3 $
", "$ r = 0.8 \\theta $
", "$ r = 1 $
", "$ r = 2 \\theta $
"], "variableReplacements": [], "matrix": "[mod(0+f1,2),mod(1+f1,2),0,0]", "marks": 0, "maxMarks": 0}, {"showCorrectAnswer": true, "minMarks": 0, "type": "1_n_2", "scripts": {}, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "prompt": "{polar_curves(f2)}
", "shuffleChoices": true, "showFeedbackIcon": true, "displayType": "radiogroup", "choices": ["$ r = 3 \\sqrt{ 1+\\cos^2(\\theta) }$
", "$ r = 3 + \\cos(6\\theta) $
", "$ r = 3 \\sqrt{ 1+\\cos(\\theta) }$
", "$ r = 3 + \\cos(\\theta) $
"], "variableReplacements": [], "matrix": "[mod(0+f2,2),mod(1+f2,2),0,0]", "marks": 0, "maxMarks": 0}, {"showCorrectAnswer": true, "minMarks": 0, "type": "1_n_2", "scripts": {}, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "prompt": "{polar_curves(f3)}
", "shuffleChoices": true, "showFeedbackIcon": true, "displayType": "radiogroup", "choices": ["$r = 3 \\sqrt{ 1+\\tan^2(\\theta)} $
", "$ r = \\frac{ \\sin(\\theta) \\sqrt{\\left|\\cos(\\theta)\\right|} }{\\sin(\\theta)+1.4} - 2 \\sin(\\theta)+2 $
", "$r = 3 \\sqrt{ 1+\\sin^2(\\theta)} $
", "$ r = \\frac{ \\sin(\\theta) \\sqrt{\\left|\\cos(\\theta)\\right|} }{\\sin(\\theta)+1.4}$
"], "variableReplacements": [], "matrix": "[mod(0+f3,2),mod(1+f3,2),0,0]", "marks": 0, "maxMarks": 0}, {"showCorrectAnswer": true, "minMarks": 0, "type": "1_n_2", "scripts": {}, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "prompt": "{polar_curves(f4)}
", "shuffleChoices": true, "showFeedbackIcon": true, "displayType": "radiogroup", "choices": ["$ r = 4 \\cos(2\\theta) $
", "$ r = 4\\cos^2(\\theta) $
", "$ r = 4 \\cos(4\\theta) $
", "$ r = 4\\cos^2(2\\theta) $
"], "variableReplacements": [], "matrix": "[mod(0+f4,2),mod(1+f4,2),0,0]", "marks": 0, "maxMarks": 0}], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "functions": {"polar_curves": {"type": "html", "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('300px','300px',\n {boundingBox: [-5,6,6,-5], axis: true, showNavigation: true, grid: true});\n \nvar board = div.board;\nvar a = board.create('slider', [[1,5],[4,5],[0,1,2*Math.PI]], {style:6, name:'theta'});\n\nvar r = function(t) {\n switch (fcn_number) {\n case 1:\n return 3;\n break;\n case 2:\n return 0.8*t;\n break;\n case 3:\n return 3*Math.pow((1+Math.cos(t)*Math.cos(t)),1/2);\n break;\n case 4:\n return 3+Math.cos(6*t);\n break;\n case 5:\n return 3*Math.pow((1+Math.tan(t)*Math.tan(t)),1/2);\n break; \n case 6:\n return (Math.sin(t)*Math.sqrt(Math.abs(Math.cos(t))))/(Math.sin(t)+7/5)-2*Math.sin(t)+2;\n break; \n case 7:\n return 4*Math.cos(2*t);\n break;\n case 8:\n return 4*Math.cos(t)*Math.cos(t);\n break; \n }\n}\n\nvar graph = board.create('curve', [r, [0,0], 0, function(){return a.Value();}],\n {curveType:'polar', strokeColor:'red', strokeWidth:5});\n\nreturn div;", "parameters": [["fcn_number", "number"]], "language": "javascript"}}, "statement": "You are shown four different curves below. For each of them, select the correct formula for the radius $r$.
\nThe phase $\\theta$ is always in the interval $[0,2\\pi)$. Use the slider in the upper right of the canvas to display more of the curve, and to see how $r$ depends on $\\theta$.
\n(Hint: Some of the given expressions are quite complicated -- it is not necessary to understand them completely. Try to think of ways to eliminate the wrong formulas.)
", "type": "question", "contributors": [{"name": "Pascal Philipp", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1569/"}]}]}], "contributors": [{"name": "Pascal Philipp", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1569/"}]}