2c:null 2d:["$","title",null,{"children":"Second derivative measures the curvature of a function | Mattias Villani | Observable"}] 32:[] 30:{"id":"4394f409c6a21a5e","slug":"second-derivative-measures-the-curvature-of-a-function","title":"Second derivative measures the curvature of a function","thumbnail":"3f0598fe57ed8a848fb21e7ad6b3d83c3152b4eda5a2639054757ff9fa2cbf92","access_level":"public","create_time":"$D2023-08-19T01:50:05.524Z","update_time":"$D2025-03-15T20:33:44.778Z","view_time":null,"first_public_time":"$D2023-08-19T02:10:52.562Z","fork":{"id":"a1846f134305662d","slug":"the-derivative","title":"The Derivative","owner":{"id":"66151b62b45d9fca","login":"credmond","name":"Charles Redmond"}},"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":248,"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"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"636adfa34cc7a030","slug":"sda3","title":"SDA3","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Second derivative measures the curvature of a function","cells":[{"id":0,"value":"# Second derivative measures the curvature of a function","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":156,"value":"The first derivative ${tex`f'(x)`} measures the slope of the function at ${tex`x`}. \n\nThe second derivative ${tex`f''(x)`} is the derivative of the first derivative.
\nA function with a large ${tex`\\vert f''(x) \\vert`} at ${tex`x`} has a slope that changes a lot around that point. \n\nAssume that ${tex`f(x)`} is concave and therefore has a unique maximum at some ${tex`x_{\\mathrm{max}}`}. The second derivative is negative at the maximal point.\n- If ${tex`\\vert f''(x_{\\mathrm{max}}) \\vert`} is large, then ${tex`f(x)`} is highly peaked around ${tex`x_{\\mathrm{max}}`}.
\n- If ${tex`\\vert f''(x_{\\mathrm{max}}) \\vert`} is small, then ${tex`f(x)`} is rather flat around ${tex`x_{\\mathrm{max}}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":182,"value":"disttext = md`

\nExample. The log density of a normal distribution ${tex`\\mathrm{N}(\\mu,\\sigma^2)`} is\n\n${tex`f(x) = -\\frac{1}{2}\\log(2 \\pi) - \\frac{1}{2\\sigma^2}(x-\\mu)^2`}\n\nwith first derivative\n\n${tex`f'(x) = -\\frac{1}{\\sigma^2}(x-\\mu)`}\n\nwhere of course ${tex`x_{\\max} = \\mu`}. The second derivative is\n\n${tex`f''(x) = -\\frac{1}{\\sigma^2}`}\n\nThe smaller the variance, the more peaked is the normal density function.\n` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":212,"value":"anchor = md`### Interactive illustration`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":161,"value":"The figure below illustrates how the second derivative measures how peaked a function is around its maximum. Change the point ${tex`x`} at which derivative is computed and see the how much more quickly the derivative (orange line) changes around the maximal point in the top figure with a highly peaked function, compared to the bottom figure where the function is rather flat.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":27,"value":"viewof x0 = Inputs.range([-2,2],{step:.01,label:\"Point where derivative is evaluated, x\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":11,"value":"plt1 = Plot.plot({\n x:{domain:[-5,5],label:\"x\"},\n y:{domain:[0,1.2],label:\"f(x)\"},\n marks:[\n Plot.ruleX([-5]),\n Plot.ruleY([0]),\n Plot.line(x,{x:x,y:y1, stroke: \"steelblue\", strokeWidth: 3}),\n Plot.line(ends1,{stroke:\"orange\", strokeWidth: 3}),\n Plot.dot(points1,{fill:\"black\",r:4})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":123,"value":"plt2 = Plot.plot({\n x:{domain:[-5,5],label:\"x\"},\n y:{domain:[0,1.5],label:\"f(x)\"},\n marks:[\n Plot.ruleX([-5]),\n Plot.ruleY([0]),\n Plot.line(x,{x:x, y:y2, stroke: \"steelblue\", strokeWidth: 3}),\n Plot.line(ends2,{stroke:\"orange\", strokeWidth: 3}),\n Plot.dot(points2,{fill:\"black\",r:4})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":209,"value":"
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\n\n---\n#### 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":32,"value":"h = 0.01","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":5,"value":"x = d3.range(-5, 5, 0.01)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"f1 = function(x){return Math.exp(-1/(2*0.3**2)*(x-0)**2)}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":47,"value":"line1 = function(x){\n const slope = (f1(x0+h)-f1(x0))/h\n const b = f1(x0)-slope*x0\n return slope*x+b\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":8,"value":"y1 = x.map(d => f1(d))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":36,"value":"points1 = [\n [x0,f1(x0)]\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":50,"value":"ends1 = [\n [-5,line1(-5)],\n [5,line1(5)]\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":94,"value":"f2 = function(x){return Math.exp(-1/(2*2**2)*(x-0)**2)}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":111,"value":"line2 = function(x){\n const slope = (f2(x0+h)-f2(x0))/h\n const b = f2(x0)-slope*x0\n return slope*x+b\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":114,"value":"y2 = x.map(d => f2(d))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":117,"value":"points2 = [\n [x0,f2(x0)]\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":120,"value":"ends2 = [\n [-5,line2(-5)],\n [5,line2(5)]\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 31:"$30:body"