{"version":9,"randomSeed":"ed6fe9ce8b7d71460e7748d3a3ee94ef","graph":{"viewport":{"xmin":-4.2319788101545734,"ymin":-2.190711696669098,"xmax":10.53818466405447,"ymax":7.742505113085224}},"expressions":{"list":[{"type":"folder","id":"48","title":"functions"},{"type":"text","id":"50","folderId":"48","text":"function of which to find extremum"},{"type":"expression","id":"51","folderId":"48","color":"#c74440","latex":"f\\left(x\\right)=\\sin x"},{"type":"text","id":"60","folderId":"48","text":"see https://www.desmos.com/calculator/y4is13zxxf"},{"type":"text","id":"55","folderId":"48","text":"linear function with points (a, f(a)) and (b, f(b))"},{"type":"expression","id":"56","folderId":"48","color":"#2d70b3","latex":"L\\left(a,b,x\\right)=\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}\\cdot\\left(x-a\\right)+f\\left(a\\right)"},{"type":"text","id":"67","folderId":"48","text":"root of linear function with points (a, f(a)) and (b, f(b))"},{"type":"expression","id":"65","folderId":"48","color":"#388c46","latex":"R_{o}\\left(a,b\\right)=-\\frac{b-a}{f\\left(b\\right)-f\\left(a\\right)}\\cdot f\\left(b\\right)+b"},{"type":"text","id":"63","folderId":"48","text":"secant approximation of derivative at (a + b) / 2"},{"type":"expression","id":"64","folderId":"48","color":"#2d70b3","latex":"D\\left(a,b\\right)=\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}"},{"type":"text","id":"69","folderId":"48","text":"root of linear approximation of derivative function"},{"type":"expression","id":"70","folderId":"48","color":"#c74440","latex":"R\\left(a,b,c\\right)=-\\frac{\\frac{b+c}{2}-\\frac{a+b}{2}}{D\\left(b,c\\right)-D\\left(a,b\\right)}\\cdot D\\left(b,c\\right)+\\frac{b+c}{2}"},{"type":"text","id":"79","folderId":"48","text":"see https://www.desmos.com/calculator/bqqy2kly3f"},{"type":"text","id":"82","folderId":"48","text":"three elements in increasing order from l (TEIIO) and p"},{"type":"expression","id":"80","folderId":"48","color":"#6042a6","latex":"S\\left(l,p\\right)=\\left\\{p>l\\left[2\\right]:\\left[l\\left[2\\right],p,l\\left[3\\right]\\right],\\left[l\\left[1\\right],p,l\\left[2\\right]\\right]\\right\\}"},{"type":"folder","id":"73","title":"step 0","collapsed":true},{"type":"expression","id":"84","folderId":"73","color":"#2d70b3","latex":"x_{0}=\\left[1,1.5,2\\right]"},{"type":"expression","id":"85","folderId":"73","color":"#388c46","latex":"d_{0}=R\\left(x_{0}\\left[1\\right],x_{0}\\left[2\\right],x_{0}\\left[3\\right]\\right)"},{"type":"expression","id":"86","folderId":"73","color":"#2d70b3","latex":"\\left(x_{0},f\\left(x_{0}\\right)\\right)"},{"type":"folder","id":"88","title":"step 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