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{"name": "Fourth order Taylor approximation of cos(2x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Fourth order Taylor approximation of cos(2x)", "tags": [], "metadata": {"description": "<p>Taylor Approximation ofr $\\cos(2x)$</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Fin the fourth order Taylor Approximation of $\\cos(2x)$</p>", "advice": "<p><strong>a)</strong></p>\n<p>We need the derivatives of $\\cos(2x)$ up to fourth order. They are&nbsp;</p>\n<p>\\begin{align}<br/>\\cos^{(0)}(2x) &amp;= \\cos(2x) \\quad&nbsp; &nbsp;&amp;\\cos^{(1)}(2x) &amp;= -2\\sin(2x)\\\\<br/>\\cos^{(2)}(2x) &amp;= -4\\cos(2x) \\quad&nbsp; &amp;\\cos^{(3)}(2x) &amp;= 8\\sin(2x)\\\\<br/>\\cos^{(4)}(2x) &amp;= 16\\cos(2x) \\quad&nbsp;&nbsp;<br/>\\end{align}</p>\n<p>Then the fourth order Taylor Approximation to the given function is&nbsp;</p>\n<p>\\[\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right) - <br/>2\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right) - <br/>\\frac{4\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{2}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^2 <br/>+\\frac{8\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{3!}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^3 + <br/>\\frac{16\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{4!}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^4 = <br/>\\\\[3mm]<br/>\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right) - <br/>2\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right) - <br/>2\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^2 + <br/>\\frac{4\\sin\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{3}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^3 + <br/>\\frac{2\\cos\\left(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi}\\right)}{3}\\left(\\simplify[zeroTerm, fractionNumbers]{x-{r}*pi}\\right)^4<br/>\\]</p>\n<p><strong>(Note that in the answer above, the algorithm approximates sine and cosine values that appear in the polynomial we gave here)</strong></p>\n<p><strong></strong></p>\n<p><strong>b)</strong></p>\n<p>We need the fifth derivative for the error term. That is $\\cos^{(5)}(2x) = -32\\sin(2x)$. Then by Taylor's Theorem the error can computed by&nbsp;</p>\n<p>\\[R_5(\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}) = \\frac{(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi - {r}*pi})}{5!}\\cos^{(5)}(2\\eta) = <br/>\\frac{(\\simplify[fractionNumbers, zeroFactor, unitFactor]{{r2}*pi - {r}*pi})}{5!}(-32(\\sin(2\\eta)) \\quad \\\\[1mm] \\mbox{for some } \\eta\\in \\left[\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r}*pi}, \\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}\\right] \\quad (\\mbox{or }\\; <br/>\\eta\\in \\left[\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}, \\simplify[fractionNumbers, unitFactor, zeroFactor]{{r}*pi}\\right])\\]</p>\n<p></p>\n<p>Note that findin what $\\eta$ should is not possible at this stage. However, one can still bound the error. For that you need to find the $\\eta \\in \\left[\\simplify[fractionNumbers, unitFactor, zeroFactor]{{r}*pi}, \\simplify[fractionNumbers, unitFactor, zeroFactor]{{r2}*pi}\\right]$ such that $\\sin(2\\eta)$ is maximal.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"r": {"name": "r", "group": "Ungrouped variables", "definition": "random(0, -1/4, 1/3, 1/6, 1/4, 1/2)", "description": "", "templateType": "anything", "can_override": false}, "a0": {"name": "a0", "group": "Ungrouped variables", "definition": "cos(2*(r*pi))", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "-2*sin(2*(r*pi))", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "-2*cos(2*(r*pi))", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "4*sin(2*(r*pi))/3", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "2*cos(2*(r*pi))/3", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "random([0, -1/4, 1/3, 1/6, 1/4, 1/2] except {r}) ", "description": "", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "2*r", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["r", "a0", "a1", "a2", "a3", "a4", "r1", "r2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the $4$-th order Taylor Approximation to $\\cos x$ around $x=\\simplify[zeroFactor, fractionNumbers]{{r}*pi}$.</p>\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a0} + {a1}*(x-{r}*pi) + {a2}*(x-{r}*pi)^2 + {a3}*(x-{r}*pi)^3 + {a4}*(x-{r}*pi)^4", "answerSimplification": "basic, trig, zeroFactor, zeroTerm, collectNumbers, simplifyFractions", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": "2", "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "information", "useCustomName": true, "customName": "b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find an upper bound for the error between the approximation you found above and $\\cos(2x)$ at $x=\\simplify[zeroFactor, unitFactor, fractionNumbers]{{r2}*pi}$.</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}