----------------------------------------------------------------------------- -- Copyright 2013, Open Universiteit Nederland. This file is distributed -- under the terms of the GNU General Public License. For more information, -- see the file "LICENSE.txt", which is included in the distribution. ----------------------------------------------------------------------------- -- | -- Maintainer : alex.gerdes@ou.nl -- Stability : provisional -- Portability : portable (depends on ghc) -- -- Example exercises from the Digital Mathematics Environment (DWO), -- see: http://www.fi.uu.nl/dwo/gr/frameset.html. -- ----------------------------------------------------------------------------- module Domain.Math.Polynomial.RationalExamples ( brokenEquations, normBroken, normBroken2, normBrokenCon, deelUit ) where import Domain.Math.Data.Relation import Domain.Math.Expr import Ideas.Common.Rewriting import Prelude hiding ((^)) ---------------------------------------------------------- -- VWO B applets -- Hoofdstuk 1, gebroken vergelijkingen brokenEquations :: [[Equation Expr]] brokenEquations = -- Bereken exact de oplossingen let x = Var "x" in [ [ (2*x^2-10) / (x^2+3) :==: 0 , (7*x^2-21) / (2*x^2-5) :==: 0 , (3*x^2-6) / (4*x^2+1) :==: 0 , (4*x^2-24) / (6*x^2-2) :==: 0 , x^2 / (x+4) :==: (3*x+4) / (x+4) , (x^2+2) / (x-2) :==: (x+8) / (x-2) , (x^2+6*x-6)/(x^2-1) :==: (4*x+9)/(x^2-1) , (x^2+6)/(x^2-2) :==: (7*x)/(x^2-2) ] , [ (x^2+6*x)/(x^2-1) :==: (3*x+4)/(x^2-1) , (x^2+6)/(x-3) :==: (5*x)/(x-3) , (x^2+4*x)/(x^2-4) :==: (3*x + 6)/(x^2-4) , (x^2+2*x-4)/(x-5) :==: (4*x+11)/(x-5) , (5*x+2)/(2*x-1) :==: (5*x+2)/(3*x+5) , (x^2-9)/(4*x-1) :==: (x^2-9)/(2*x+7) , (3*x-2)/(2*x^2) :==: (3*x-2)/(x^2+4) , (2*x+1)/(x^2+3*x) :==: (2*x+1)/(5*x+8) ] , [ (x^2-1)/(2*x+2) :==: (x^2-1)/(x+8) , (x^2-4)/(3*x-6) :==: (x^2-4)/(2*x+1) , (x^2+5*x)/(2*x^2) :==: (x^2+5*x)/(x^2+4) , (x^2-3*x)/(2*x-6) :==: (x^2-3*x)/(4*x+2) , x/(x+1) :==: 1 + 3/4 , (x+2)/(3*x) :==: 1 + 1/3 , (2*x+3)/(x-1) :==: 3 + 1/2 , (x-3)/(1-x) :==: 1 + 2/5 ] , [ (x+4)/(x+3) :==: (x+1)/(x+2) , (2*x+3)/(x-1) :==: (2*x-1) / (x-2) , (3*x+6)/(3*x-1) :==: (x+4)/(x+1) , (x+2)/(2*x+5) :==: (x+4)/(2*x-3) , (x+5)/(2*x) + 2 :==: 5 , (3*x+4)/(x+2) - 3 :==: 2 , (x^2)/(5*x+6) + 4 :==: 5 , (x^2)/(2*x-3) + 3 :==: 7 ] , [ (x-2)/(x-3) :==: x/2 , (x+9)/(x-5) :==: 2/x , (x+2)/(x+4) :==: 2/(x+1) , (-3)/(x-5) :==: (x+3)/(x+1) , (x+1)/(x+2) :==: (7*x+1)/(2*x-4) , (2*x-7)/(5-x) :==: (x+1)/(3*x-7) , (x+1)/(x-1) :==: (3*x-7)/(x-2) , (3*x-7)/(x-2) :==: (7-x)/(3*x-3) ] ] -- Hoofdstuk 4, gebroken vorm herleiden (1 en 1a) normBroken :: [[Expr]] normBroken = -- Herleid let x = Var "x" in let y = Var "y" in let a = Var "a" in let b = Var "b" in [ [ 7/(2*x) + 3/(5*x), 3/(2*x) + 2/(3*x), 4/(5*x)-2/(3*x) , 2/(7*x) - 1/(4*x), 5/(6*a)+3/(7*a), 3/(8*a)+5/(3*a) , 7/(2*a)-2/(3*a), 9/(5*a)-1/(2*a) ] , [ 1/x+1/y, 2/(3*x)+1/(2*y), 3/(x^2*y) - 5/(2*x*y), 2/(x*y)-7/(5*y) , 2/a - 3/b, 4/(3*a)-2/(5*b), 2/(a*b)+4/(3*a), 7/(4*a)+3/(4*b) ] , [ 3+1/(2*x), 2*x+(3/(5*x)), 5/(2*x)-3, 3-5/(7*x), 5/(3*a)+1 , 4*a+3/(2*a), 2*a-1/(3*a), 7/(5*a)-2 ] , [ 5/(x+2)+4/(x+3), 3/(x-1)+2/(x+3), 4/(x+5)+2/(x-3), 3/(x-2)+2/(x-3) , 4/(x+3)-6/(x+2), 1/(x+5)-3/(x-4), 7/(x-3)-2/(x+1), 6/(x-1)-3/(x-2) ] , [ (x+1)/(x+2)+(x+2)/(x-3), (x-2)/(x+3)+(x-1)/(x+2), (x+3)/(x-1)+(x+2)/(x-4) , (x-4)/(x+5)+(x-2)/(x-3), (x-1)/(x+1)-(x+2)/(x-2), (x+5)/(x+3)-(x+3)/(x+5) , (x-1)/(x+2)-(x+4)/(x+1), (x-3)/(x-1)-(x+2)/(x+4) ] , [ (2*x)/(x-1)+x/(x+2), (3*x)/(x-4)+(5*x)/(x-2) , (4*x)/(x+2)-(2*x)/(x+1), x/(x+5)-(4*x)/(x+6) ] ] -- Hoofdstuk 4, gebroken vorm herleiden (2 en 2a) normBroken2 :: [[Expr]] normBroken2 = -- Herleid let x = Var "x" in let a = Var "a" in let p = Var "p" in [ [ (x^2+4*x-5)/(x^2+5*x-6), (x^2+2*x-8)/(x^2+10*x+24) , (x^2-7*x+12)/(x^2+x-20), (x^2+7*x+12)/(x^2+5*x+6) , (a^2-a-2)/(a^2+4*a-12), (a^2-3*a-10)/(a^2-a-20) , (a^2-2*a-15)/(a^2-3*a-18), (a^2+a-2)/(a^2+3*a+2) ] , [ (x^2-16)/(x^2+x-12), (x^2-2*x+1)/(x^2-1), (x^2-9)/(x^2+6*x+9) , (x^2-7*x+6)/(x^2-1), (2*p^2+8*p)/(p^2-16), (-(p^2)+5*p)/(p^2-10*p+25) , (p^2-4)/(4*p^2+8*p), (p^2-12*p+36)/(p^2-6*p) ] , [ (x^3+3*x^2+2*x)/(x^2+4*x+4), (x^3+10*x^2+24*x)/(x^2+7*x+6) , (x^2+5*x+6)/(x^3-x^2-6*x), (x^2+3*x-4)/(x^3-6*x^2+5*x) , (a^3+7*a^2+12*a)/(a^2+6*a+9), (a^3+7*a^2+10*a)/(a^2-a-6) , (a^2-9)/(a^3-4*a^2+3*a), (a^2-2*a-15)/(a^3-3*a^2-10*a) ] ] deelUit :: [[Expr]] deelUit = let x = Var "x" in let a = Var "a" in let p = Var "p" in let t = Var "t" in [ -- laatste sommen van gebroken vorm herleiden (2), niveau 5 [ (-6*a^2-1)/a, -2*p^2+3/(7*p), (7*t^2+4)/(-4*t), (9*x^2+8)/(8*x) ] , -- sommen (2a) [ (-7*a^2-4*a-6)/(-6*a), (3*p^2+6*p-8)/p, (2*t^2-9*t-8)/(-2*t) , (x^2+5*x+5)/(2*x), (5*a^3-4*a+2)/(9*a), (5*p^3-7*p^2+9)/(2*p) , (-3*t^3+6*t-4)/(3*t), (4*x^3-3*x^2+4)/(7*x) ] ] -- Vervolg hoofdstuk 4, gebroken vorm herleiden (2 en 2a), vanaf niveau 4 normBrokenCon :: [[Equation Expr]] normBrokenCon = -- Herleid let a = Var "a" in let p = Var "p" in let t = Var "t" in let ca = symbol (newSymbol "A") in let ct = symbol (newSymbol "T") in let cn = symbol (newSymbol "N") in [ [ ca :==: (p^2+2*p)/(p^2-4), ca :==: (6*p^2-18*p)/(p^2-9) , ca :==: (p^2-1)/(-2*p^2+2*p), ca :==: (p^2-16)/(4*p^2+16*p) , ct :==: (t^3-2*t^2)/(t^2-4), ct :==: (t^3+4*t^2)/(t^2-16) , ct :==: (t^2-1)/(t^3+t^2), ct :==: (t^2-25)/(t^3-5*t^2) ] , [ cn :==: (a^4+4*a^2-5)/(a^4-1), cn :==: (a^4+5*a^2+6)/(a^4+4*a^2+3) , cn :==: (a^4-5*a^2+6)/(a^4-7*a^2+10), cn :==: (a^4-8*a^2+16)/(a^4-5*a^2+4) ] ]