2a:null 2b:["$","title",null,{"children":"The Taylor approximation | Mattias Villani | Observable"}] 31:T632,The *n*th order **Taylor approximation** of the function ${tex`f(x)`} around ${tex`x=x_0`} is ${tex.block`\hat{f}_n(x) := f(x_0) + \sum_{k=1}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k,`} where the hat symbol (^) is used to denote that it is an approximation, and the subscript *n* to denote the choice of approximation order. The approximation is based on the derivatives of the function at ${tex`x=x_0`}, with the notation ${tex.block`f^{(k)}(x_0) = \frac{\mathrm{d}^k}{\mathrm{d} x^k}f(x)\vert_{x=x_0}`} for the *k*th derivative of ${tex`f(x)`} evaluated at ${tex`x=x_0`}. The function ${tex`f(x)`} must have *n* continuous derivatives. The approximation is local around ${tex`x=x_0`} and may not be at all accurate far from that point. The approximation is named after the mathematician Brook Taylor, but a more descriptive, and a little playful, name would be **tailored polynomial approximation**. This is because the approximation is based on tailoring a polynomial ${tex`b_0 + \sum_{k=1}^n b_k (x-x_0)^k`} of degree *n* to the function ${tex`f(x)`} by matching its derivatives at ${tex`x=x_0`}. Specifically, the Taylor approximation ${tex`\hat{f}_n(x)`} satisfies ${tex.block`\hat{f}^{(k)}(x_0) = f^{(k)}(x_0)\quad\text{ for } k = 0,1,\ldots,n.`} Note that repeated application of the power rule for differentiation gives that ${tex`\frac{\mathrm{d}^k}{\mathrm{d} x^k} x^k = k!`} , which accounts for the appearance of the ${tex`k!`} in the Taylor approximation. Experiment below with different functions ${tex`f(x)`}, approximation orders *n*, and expansions points ${tex`x_0`}.32:Tb83,f = { if (funcname == "exp"){ return [function(x){return Math.exp(x)}, function(x){return Math.exp(x)}, function(x){return Math.exp(x)}, function(x){return Math.exp(x)}, function(x){return Math.exp(x)}, function(x){return Math.exp(x)} ] }else if(funcname == "square"){ return [function(x){return x**2}, function(x){return 2*x}, function(x){return 2}, function(x){return 0}, function(x){return 0}, function(x){return 0} ] }else if(funcname == "cube"){ return [function(x){return x**3}, function(x){return 3*x**2}, function(x){return 6*x}, function(x){return 6}, function(x){return 0}, function(x){return 0} ] } else if(funcname == "log"){ return [function(x){return Math.log(x)}, function(x){return 1/x}, function(x){return -1/(x**2)}, function(x){return 2/(x**3)}, function(x){return -6/(x**4)}, function(x){return 24/(x**5)} ] } else if(funcname == "sqrt"){ return [function(x){return Math.sqrt(x)}, function(x){return (1/2)/x**(1/2)}, function(x){return -(1/4)/x**(3/2)}, function(x){return (3/8)/x**(5/2)}, function(x){return -(15/16)/x**(7/2)}, function(x){return (105/32)/x**(9/2)} ] } else if(funcname == "sin"){ return [function(x){return Math.sin(x)}, function(x){return Math.cos(x)}, function(x){return -Math.sin(x)}, function(x){return -Math.cos(x)}, function(x){return Math.sin(x)}, function(x){return Math.cos(x)} ] } else if(funcname == "cos"){ return [function(x){return Math.cos(x)}, function(x){return -Math.sin(x)}, function(x){return -Math.cos(x)}, function(x){return Math.sin(x)}, function(x){return Math.cos(x)}, function(x){return -Math.sin(x)} ] } else if(funcname == "gaussian"){ return [function(x){return Math.cos(x)}, function(x){return -Math.sin(x)}, function(x){return -Math.cos(x)}, function(x){return Math.sin(x)}, function(x){return Math.cos(x)}, function(x){return -Math.sin(x)} ] } else if(funcname == "cauchy kernel"){ return [function(x){return 1/(1 + x**2)}, function(x){return -2*x/((1 + x**2)**2)}, function(x){return (8*x**2)/(1 + x**2)**3 - 2/(1 + x**2)**2}, function(x){return (24*x)/(1 + x**2)**3 - (48*x**3)/(1 + x**2)**4}, function(x){return (24*(1 - 10*x**2 + 5*x**4))/(1 + x**2)**5}, function(x){return (-240*x*(-3 + x**2)*(-1 + 3*x**2))/(1 + x**2)**6} ] } }33:[] 2e:{"id":"4a49ee2a3be496f7","slug":"taylor-approximation","title":"The Taylor approximation","thumbnail":"96286c7c8d561def9568cf4aadf8cf52e34a4ac967612c24a239084b87b68b6e","access_level":"public","create_time":"$D2025-03-12T20:39:32.334Z","update_time":"$D2025-09-20T09:24:34.318Z","view_time":null,"first_public_time":"$D2025-03-12T22:16:29.747Z","fork":{"id":"32fd1a513378ecaa","slug":"some-functions-and-their-derivatives","title":"A function and its derivatives","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":495,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":{"id":"186f48bbffc13b97","slug":"stm","title":"STM","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"The Taylor approximation","cells":[{"id":0,"value":"# The Taylor approximation","pinned":false,"hidden":false,"mode":"md"},{"id":185,"value":"> *This notebook shows how a smooth function ${tex`f(x)`} can be approximated locally by polynomials tailored to ${tex`f(x)`}.*","pinned":false,"hidden":false,"mode":"md"},{"id":360,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":297,"value":"viewof funcname = Inputs.select([\"exp\", \"log\", \"square\", \"cube\", \"sqrt\", \"sin\", \"cos\", \"cauchy kernel\"], {value: \"exp\", label: \"Function f(x)\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"viewof inputs = Inputs.form({\n order: Inputs.range([1, 5], {label: tex`\\text{Taylor order, }n\n`, value: 1, step: 1}),\n x0: Inputs.range(xdomain, {label: tex`\\text{Expansion point, }x_0`, value: x0init, step: 0.1})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":143,"value":"Plot.plot({\n title: latexfunc,\n style: \"font-size: 12px;\",\n color: {\n legend: true,\n domain: plotlabels.labels,\n range: [colors[0],colors[1]]\n },\n x: {\n label: \"x\",\n axis: true,\n },\n y: {\n label: \"f(x)\",\n axis: true,\n domain: ydomain\n },\n grid: false,\n marks: [\n Plot.ruleY([ydomain[0]]),\n //Plot.ruleX([0]),\n Plot.line([{x: inputs.x0, y: ydomain[0]}, {x: inputs.x0, y: f[0](inputs.x0)}], \n {x: \"x\", y: \"y\", stroke: \"gray\", strokeDasharray: \"4,4\", label: \"x\"}),\n Plot.text([{x: inputs.x0, y: ydomain[0]}], \n {x: \"x\", y: \"y\", text: [\"x₀\"], dx: 0, dy: 20, fill: \"gray\", fontSize: 18}),\n Plot.lineY(funcdata, {x: \"x\", y: \"f\", stroke: colors[0], strokeWidth: 3}),\n Plot.lineY(funcdata, {x: \"x\", y: \"fapprox\", stroke: colors[1], strokeWidth: 3}),\n Plot.dot([{x: inputs.x0, y: f[0](inputs.x0)}], {x: \"x\", y: \"y\", stroke: colors[1], fill: colors[1]})\n \n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":31,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":28,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":96,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":166,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":16,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":19,"value":"latexfunc = {\n if (funcname == \"linear\"){\n return tex`f(x) = x`\n }else if(funcname == \"exp\"){\n return tex`f(x) = \\exp(x)`\n }else if(funcname == \"log\"){\n return tex`f(x) = \\log(x)`\n }else if(funcname == \"square\"){\n return tex`f(x) = x^2`\n }else if(funcname == \"cube\"){\n return tex`f(x) = x^3`\n }else if(funcname == \"sqrt\"){\n return tex`f(x) = \\sqrt{x}`\n }else if(funcname == \"cos\"){\n return tex`f(x) = \\cos(x)`\n }else if(funcname == \"tan\"){\n return tex`f(x) = \\tan(x)`\n }else if(funcname == \"gaussian\"){\n return tex`f(x) = \\frac{1}{\\sqrt{2\\pi}}\\exp(-\\frac{1}{2}x^2)`\n }else if(funcname == \"cauchy kernel\"){\n return tex`f(x) = \\frac{1}{1 + x^2}`\n }else{\n return tex`f(x) = \\sin(x)`\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":21,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":36,"value":"xdomain = {\n if (funcname == \"log\"){\n return [0.1, 5]\n }\n if (funcname == \"exp\"){\n return [-3,3]\n }\n if (funcname == \"square\"){\n return [-2,2]\n }\n if (funcname == \"cube\"){\n return [-2,2]\n }\n if (funcname == \"sqrt\"){\n return [0.01,5]\n }\n if (funcname == \"cos\"){\n return [-2*Math.PI,2*Math.PI]\n }\n if (funcname == \"sin\"){\n return [-2*Math.PI,2*Math.PI]\n }\n if (funcname == \"cauchy kernel\"){\n return [-1,3]\n }\n else{\n return [-5,5]\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":385,"value":"x0init = {\n if (funcname == \"log\"){\n return 0.5\n }\n if (funcname == \"exp\"){\n return 0\n }\n if (funcname == \"square\"){\n return 0.5\n }\n if (funcname == \"cube\"){\n return 0.5\n }\n if (funcname == \"sqrt\"){\n return 0.5\n }\n if (funcname == \"cos\"){\n return 0.5\n }\n if (funcname == \"sin\"){\n return 0.5\n }\n if (funcname == \"cauchy kernel\"){\n return 0\n }\n else{\n return 0\n }\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":145,"value":"funcdata = {\n const ngrid = 500;\n let fval = [];\n const xgrid = d3.range(xdomain[0], xdomain[1], (xdomain[1] - xdomain[0])/ngrid)\n for (let i = 0; i < ngrid; ++i) {\n let fapprox = 0;\n for (let j = 0; j < inputs.order + 1; j++){\n fapprox += (f[j](inputs.x0)/mathjs.factorial(j))*(xgrid[i] - inputs.x0)**j\n }\n fval.push({x: xgrid[i], f: f[0](xgrid[i]), fapprox: fapprox})\n }\n return fval\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":173,"value":"plotlabels = {\n let labels = [`f(x)`, `Taylor approximation of order ${inputs.order}`];\n let labelColors = [colors[0],colors[inputs.order + 1]]\n return {labels: labels, colors: labelColors}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":202,"value":"ydomain = d3.extent(funcdata.map(d => d.f))","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"