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"definition": "y7-y6"}}, "tags": [], "metadata": {"description": "<p>Use&nbsp;a series of difference tables to determine the coefficients of a degree three polynomial function from a set of data values, equally spaced over the x-axis.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "name": "Difference tables 2", "parts": [{"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{a}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "{a}", "mustBeReduced": false, "mustBeReducedPC": 0, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}], "prompt": "<p>Use a difference table to show it was generated by a degree 3 polynomial \\(f(x)=ax^3+bx^2+cx+d\\) &nbsp;and hence calculate the leading coefficient, &nbsp;\\(a\\).</p>\n<p style=\"padding-left: 30px;\">\\(a=\\) [[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{b}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "{b}", "mustBeReduced": false, "mustBeReducedPC": 0, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}], "prompt": "<p>Use a modified difference table to calculate the coefficient of \\(x^2\\).</p>\n<p style=\"padding-left: 30px;\">\\(b=\\) [[0]]&nbsp;</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{c}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "{c}", "mustBeReduced": false, "mustBeReducedPC": 0, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}], "prompt": "<p>Use a modified difference table to calculate the coefficient of \\(x\\).</p>\n<p style=\"padding-left: 30px;\">\\(c=\\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{d}", "showCorrectAnswer": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "{d}", "mustBeReduced": false, "mustBeReducedPC": 0, "allowFractions": false, "scripts": {}, "marks": 1, "correctAnswerStyle": "plain"}], "prompt": "<p>Finally, calculate the constant value in the polynomial.</p>\n<p style=\"padding-left: 30px;\">\\(d=\\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "rulesets": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "variable_groups": [], "statement": "<p>The following set of data was generated by a polynomial.</p>\n<p style=\"padding-left: 90px;\">\\(\\begin{matrix} {x}&amp;{f(x)} \\\\ \\var{x0}&amp;\\var{y0}\\\\\\var{x1}&amp;\\var{y1}\\\\\\var{x2}&amp;\\var{y2}\\\\\\var{x3}&amp;\\var{y3}\\\\\\var{x4}&amp;\\var{y4}\\\\\\var{x5}&amp;\\var{y5}\\\\\\var{x6}&amp;\\var{y6}\\\\\\var{x7}&amp;\\var{y7}\\\\\\end{matrix}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["x0", "stp", "x1", "x2", "x3", "x4", "x5", "x6", "x7", "a", "b", "c", "d", "y0", "y1", "y2", "y3", "y4", "y5", "y6", "y7", "d10", "d11", "d12", "d13", "d14", "d15", "d16", "d20", "d21", "d22", "d23", "d24", "d25", "d30", "d31", "d32", "y10", "y11", "y12", "y13", "y14", "y15", "y16", "y17", "e10", "e11", "e12", "e13", "e14", "e15", "e16", "e20", "e21", "e22", "e23", "e24", "e25", "y20", "y21", "y22", "y23", "y24", "y25", "y26", "y27", "f1"], "advice": "<p>The difference table should take the form:</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{matrix} {x}&amp;{f(x)}&amp;\\delta_1&amp;\\delta_2&amp;\\delta_3 \\\\ \\var{x0}&amp;\\var{y0}&amp;\\var{d10}&amp;\\var{d20}&amp;\\var{d30}\\\\\\var{x1}&amp;\\var{y1}&amp;\\var{d11}&amp;\\var{d21}&amp;\\var{d31}\\\\\\var{x2}&amp;\\var{y2}&amp;\\var{d12}&amp;\\var{d22}&amp;\\var{d32}\\\\\\var{x3}&amp;\\var{y3}&amp;\\var{d13}&amp;\\var{d23}&amp;\\var{d32}\\\\\\var{x4}&amp;\\var{y4}&amp;\\var{d14}&amp;\\var{d24}&amp;\\var{d32}\\\\\\var{x5}&amp;\\var{y5}&amp;\\var{d15}&amp;\\var{d25}\\\\\\var{x6}&amp;\\var{y6}&amp;\\var{d16}\\\\\\var{x7}&amp;\\var{y7}\\\\\\end{matrix}\\)</p>\n<p>All the 3rd differences equal the constant \\(\\var{d32}\\)</p>\n<p>The coefficient of \\(x^3\\) is given by the formula: &nbsp; \\(a=\\frac{constant}{3!*(step-size)^3}\\)</p>\n<p style=\"padding-left: 30px;\">\\(a=\\frac{\\var{d32}}{3!*(\\var{stp})^3}=\\var{a}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>The next difference table should relate the \\(x\\) values to the \\(f(x)-\\var{a}x^3\\) values.</p>\n<p>Repeat the process:</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{matrix} {x}&amp;{f(x)-\\var{a}x^3}&amp;\\delta_1&amp;\\delta_2 \\\\ \\var{x0}&amp;\\var{y10}&amp;\\var{e10}&amp;\\var{e20}\\\\\\var{x1}&amp;\\var{y11}&amp;\\var{e11}&amp;\\var{e21}\\\\\\var{x2}&amp;\\var{y12}&amp;\\var{e12}&amp;\\var{e22}\\\\\\var{x3}&amp;\\var{y13}&amp;\\var{e13}&amp;\\var{e23}\\\\\\var{x4}&amp;\\var{y14}&amp;\\var{e14}&amp;\\var{e24}\\\\\\var{x5}&amp;\\var{y15}&amp;\\var{e15}&amp;\\var{e25}\\\\\\var{x6}&amp;\\var{y16}&amp;\\var{e16}\\\\\\var{x7}&amp;\\var{y17}\\\\\\end{matrix}\\)</p>\n<p>All the 2nd differences equal the constant \\(\\var{e25}\\)</p>\n<p>The coefficient of \\(x^2\\) is given by: &nbsp; \\(b=\\frac{\\var{e25}}{2!*(\\var{stp})^2}=\\var{b}\\)</p>\n<p></p>\n<p>The next difference table should relate the \\(x\\) values to the \\(f(x)-\\var{a}x^3-(\\var{b}x^2)\\) values.</p>\n<p>Repeat the process:</p>\n<p style=\"padding-left: 60px;\">\\(\\begin{matrix} {x}&amp;{f(x)-\\var{a}x^3}-(\\var{b}x^2)&amp;\\delta_1 \\\\ \\var{x0}&amp;\\var{y20}&amp;\\var{f1}&amp;\\\\\\var{x1}&amp;\\var{y21}&amp;\\var{f1}\\\\\\var{x2}&amp;\\var{y22}&amp;\\var{f1}\\\\\\var{x3}&amp;\\var{y23}&amp;\\var{f1}\\\\\\var{x4}&amp;\\var{y24}&amp;\\var{f1}\\\\\\var{x5}&amp;\\var{y25}&amp;\\var{f1}\\\\\\var{x6}&amp;\\var{y26}&amp;\\var{f1}\\\\\\var{x7}&amp;\\var{y27}\\\\\\end{matrix}\\)</p>\n<p>All the 1st differences equal the constant \\(\\var{f1}\\)</p>\n<p>The coefficient of \\(x\\) is given by: &nbsp; \\(c=\\frac{\\var{f1}}{1!*(\\var{stp})}=\\var{c}\\)</p>\n<p></p>\n<p>\\(f(x)=\\simplify{{a}x^3+{b}x^2}+\\var{c}x+{d}\\)</p>\n<p>From the initial set of data we see that \\((\\var{x0},\\var{y0})\\) satisfies the polynomial function.</p>\n<p>\\(\\var{y0}=\\var{a}(\\var{x0})^3+(\\var{b})(\\var{x0}^2)+\\var{c}(\\var{x0})+{d}\\)</p>\n<p>\\({d}=\\var{d}\\)</p>\n<p style=\"padding-left: 120px;\">Therefore \\(f(x)=\\simplify{{a}x^3+{b}x^2}+\\var{c}x+\\var{d}\\)</p>", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}