----------------------------------------------------------------------------- -- 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 : bastiaan.heeren@ou.nl -- Stability : provisional -- Portability : portable (depends on ghc) -- -- Example exercises from the Digital Mathematics Environment (DWO) -- ----------------------------------------------------------------------------- module Domain.Math.Polynomial.Examples ( linearExamples, quadraticExamples, higherDegreeExamples , factorizeExamples, expandExamples , ineqLin1, ineqQuad1, ineqQuad2, extraIneqQuad, ineqHigh ) where import Domain.Math.Data.Relation import Domain.Math.Expr import Ideas.Common.Exercise import Ideas.Common.Rewriting import Prelude hiding ((^)) x :: Expr x = variable "x" linearExamples :: Examples (Equation Expr) linearExamples = level Easy -- applet level 1 [ 5*x + 3 :==: 18 , 11*x - 12 :==: 21 , 19 - 3*x :==: -5 , -12 + 5*x :==: 33 , 15 - 9*x :==: 6 , 4*x + 18 :==: 0 , 11*x - 12 :==: -34 , -2*x - 3 :==: -4 , 6*x - 12 :==: 2 , -4*x - 13 :==: -11 ] ++ level Easy -- applet level 2 [ 6*x-2 :==: 2*x+14 , 3+6*x :==: 3*x+24 , 5*x+7 :==: 2*x - 10 , 2*x-8 :==: 18 - x , 4*x - 6 :==: 7*x - 14 , -1 -5*x :==: 3*x - 20 , 4*x - 7 :==: -5*x - 24 , 4*x - 18 :==: 14 + 11*x , 17 :==: 4 - 10*x , -5*x + 6 :==: 2 - 3*x ] ++ level Medium -- applet level 3 [ 4*(x-1) :==: 11*x - 12 , 4*(x-4) :==: 5*(2*x+1) , 2*(5-3*x) :==: 6-x , 4*x - (x-2) :==: 12 + 5*(x-1) , -3*(x-2) :==: 3*(x+4) - 7 , 3*(4*x-1) + 3 :==: 7*x - 14 , 4*(4*x - 1) - 2 :==: -3*x + 3*(2*x -5) , 2*x - (3*x + 5) :==: 10 + 5*(x-1) , -5*(x+1) :==: 9*(x+4)-5 , 18 - 2*(4*x + 2) :==: 7*x - 4*(4*x -2) ] ++ level Medium -- applet level 4 [ (1/2)*x - 4 :==: 2*x + mixed 2 1 2 , (1/4)*x + (1/2) :==: (5/2)*x + 2 , (1/4)*x - (3/4) :==: 2*x + (1/2) , -(1/2)*x + (3/4) :==: (5/2)*x + 3 , -(1/2)*x + mixed 1 1 2 :==: 2*x - 5 , -(1/3)*x + (3/4) :==: (1/4)*x + (1/6) , (3/4)*x - (1/3) :==: (2/3)*x - (3/4) , (2/5)*x - (1/4) :==: (1/2)*x + (3/4) , (2/3)*x - 2 :==: (1/5)*x - (3/5) , (-mixed 1 2 5)*x + mixed 3 1 2 :==: (3/5)*x + (9/10) ] ++ level Medium -- applet level 4 [ (1/4)*(x-3) :==: (1/2)*x - 4 , (x+3)/2 :==: 5*((1/2)*x + mixed 1 1 2) , (1/2)*(7-(2/3)*x) :==: 2 + (1/9)*x , (3/4)*x - (x-1) :==: 3 + mixed 2 1 2*(x-1) , -(5/4)*(x-7) :==: (3/4)*(x+2) - mixed 4 1 2 , 3*((1/5)*x - 1) + 5 :==: 7*x - 14 , ((5*x - 1) / 6) - 2 :==: -4*x + (3*x - 6)/2 , 2*x - ((2*x+2)/5) :==: 12 + (x-1)/6 , (-3*(x+2))/6 :==: 9*((2/3)*x + (1/3)) - (5/3) , 1 - ((4*x + 2)/3) :==: 3*x - ((5*x - 1) / 4) ] quadraticExamples :: Examples (Equation Expr) quadraticExamples = level Easy -- applet level 1 [ x^2 :==: 2 , x^2+3 :==: 52 , x^2-7 :==: 0 , 9*x^2 - 6 :==: 75 , 32 - 2*x^2 :==: 14 , 2*(x^2 - 3) :==: 12 , (1/4)*x^2 + 12 :==: 16 , (x-1)^2 :==: 100 , 14 - 2*x^2 :==: 6 , (1/4)*(17-x^2) :==: 2 ] ++ level Medium -- applet level 2 [ (x-7)^2 + 3 :==: 11 , (6-2*x)^2 :==: 81 , (1/2)*(x+9)^2 :==: 4 , (3-x^2)/10 :==: 2 , 5*x^2 + 3*x :==: 3*x + 2 , 11 - (2*x + 1)^2 :==: 5 , (6*x - 3)^2 + 6 :==: 12 , (7+2*x)^2 :==: 7 , 4 - (x^2 / 10) :==: 6 , 12 - (2*x + 3)^2 :==: 6 ] ++ level Medium -- applet level 3 [ x^2 :==: 5*x , x^2 - 6*x :==: 0 , 6*x + x^2 :==: 0 , x*(x+4) :==: 0 , x*(2*x-4) :==: 0 , 3*x^2 :==: 6*x , 3*x :==: 2*x^2 , x*(1-6*x) :==: 0 , (x+5)*(x-8) :==: 0 , (3*x-1)*(x+3) :==: 0 ] ++ level Medium -- applet level 4 [ x^2-2*x :==: 3 , x^2+12*x+20 :==: 0 , x^2-x :==: 30 , x*(x+2) :==: 8 , x*(x-3) :==: 4 , 2*x+15 :==: x^2 , 4*x :==: 12 - x^2 , x^2 :==: 15 - 8*x , x^2-9*x+18 :==: 0 , x^2+14*x+24 :==: 0 ] ++ level Difficult -- applet level 5 [ (3*x+5)^2+(x-5)^2 :==: 40 , 4*(10-x^2) :==: -2*x*(2*x + 10) , x*(x+12) :==: 2*x^2 , 3*(x-2)*(x+6) :==: 12*x , 8*x^2+4*x-24 :==: (x+3)*(x-8) , 3*x^2 - 11 :==: (3+2*x)^2 , 2*x*(x-3)-3 :==: (x+2)*(x+6) , 12*(x^2-3*x)+8 :==: 56 , 4*x^2-6*x :==: x^2+9 , (x+1)*(x-5) :==: (x-2)*(x-3) ] ++ level Difficult -- applet level 6 [ x^2+4*x-4 :==: 0 , x^2-6*x :==: 4 , x^2-12*x+34 :==: 0 , 2*x^2+4*x-8 :==: 0 , (x-4)*(x-1) :==: 11 , (x-(7/2))^2 :==: 2*(x+4) , x^2-3*x :==: 3*(x-2) , (4-x)*(1-x) :==: 3*x , 2*x^2 :==: x*(x+2)+7 , (1-x)^2 :==: x+2 ] factorizeExamples :: Examples Expr factorizeExamples = level Easy [ -- (buiten haakjes brengen) 4*x^2 -4*x , 36*x^2+30*x , -6*x^2-18*x , 14*x^2-10*x --(product-som methode) , x^2+11*x+24 , x^2-8*x+15 , x^2-x-2 , x^2-11*x+28 ] expandExamples :: Examples Expr expandExamples = level Easy [ 5*(x+1), -3*(x-3), (x-1)*7 , 4*(3-2*x), (x+1)*(x-3), (x+1)*(1-x) , x*(x-1), 3*(x-2)*2*x , (x-1)^2, (x+1)^2, (x-1)^2*(x+1) , (x+1)^3, (x-1)^3*x, (x-1)*(x+3)*(x-5) , x/2, (x+1)/2, (x+1)/2 + (x+2)/3, ((x+1)/2) * ((x+2)/3) ] -------------------------------------------------------------------- -- Algemene applet higherDegreeExamples :: Examples (Equation Expr) higherDegreeExamples = -- Havo B hoofdstuk 3, Hogeregraadsvgl. level Easy [ -- level 1 (1/3)*x^3 :==: 9 , x^5 - 12 :==: 20 , 1 - 8*x^3 :==: -124 , 16 - 32*x^5 :==: - 227 -- level 2 , 3*x^4 :==: 48 , (1/9)*x^6 + 12 :==: 93 , 39 - 8*x^2 :==: 21 , (1/2)*x^4 - 13 :==: 27.5 -- level 3 , 3*(2*x-1)^3 + 11 :==: 659 , 0.5*(3*x-4)^5 + 7 :==: 23 , 2*(0.5*x+3)^7 - 11 :==: -9 , 5*(1-4*x)^3 + 4 :==: -621 -- level 4 , 3*(2*x + 5)^2 + 9 :==: 21 , 2*(3*x-6)^6 - 24 :==: -22 , -2*(4*x-5)^4 + 192 :==: -8000 , (3-2*x)^4 + 23 :==: 279 ] ++ level Easy [ -- level 1 2*x^3 + 9 :==: 19 , 4*x^5 - 17 :==: 27 , 3*x^7 + 9 :==: 62 , 5*x^3 - 1 :==: 9 , 6 - 5*x^3 :==: 76 , 11 - 7*x^5 :==: 53 , 4 - 0.2*x^7 :==: 9 , 18 - 11*x^7 :==: 62 -- level 2 , 0.5*x^4 + 5 :==: 12 , 5*x^6 - 37 :==: 68 , 4*x^8 - 19 :==: 9 , 5*x^6 + 7 :==: 97 , 18 - 7*x^4 :==: -38 , 3 + (1/3)*x^6 :==: 7 , 1 - (1/9)*x^8 :==: -4 , 47 + 15*x^8 :==: 77 -- level 3 , 18*x^8 - 11 :==: 7 , (1/4)*x^6 + 14 :==: 30 , 5*x^4 + 67 :==: 472 , 5*x^4 - 1 :==: 4 , (1/8)*x^7 + 24 :==: 40 , 0.2*x^3 + 27 :==: 52 , 32*x^3 + 18 :==: 22 , 4*x^3 - 8 :==: 100 -- level 4 , 14 -2*x^3 :==: 700 , 4-3*x^5 :==: 100 , 14 - 11*x^7 :==: 25 , 1 - 3*x^5 :==: 97 , 3*(x-2)^4 + 7 :==: 37 , 6 - (2*x-1)^3 :==: 1 , (1/3)*(x+5)^6 - 4 :==: 3 , 6 - 0.5*(x-1)^5 :==: 10 -- level 5 , (1/2)*(3*x-1)^4 :==: 8 , 100-(1/3)*(4*x-3)^5 :==: 19 , 4*(0.5*x+2)^6 + 5 :==: 9 , 3*(2*x + 7)^3 + 11 :==: 35 ] ++ level Medium -- (Ontbinden applet) [ -- level 1 x^3 - 5*x^2 + 4*x :==: 0 , x^3 :==: 3*x^2 + 10*x , 14*x :==: x^3 + 5*x^2 , (1/2)*x^3 + 3*x^2 + 4*x :==: 0 , x^3 + 6*x^2 + 9*x :==: 0 , 5*x^2 :==: x^3 + 6*x , x^3 - 5*x^2 :==: 6*x , x^3 :==: 4*x^2 + 12*x -- level 2 , x^4 + 36 :==: 13*x^2 , x^4 - 9*x^2 + 20 :==: 0 , x^4 :==: 2*x^2 + 3 , x^4 + 2*x^2 :==: 24 , 7*x^2 + 18 :==: x^4 , x^4 :==: x^2 + 12 , 29*x^2 :==: x^4 + 100 , 2*x^4 + 2*x^2 :==: 12 -- (abc-form applet) -- level 1 , 4*x^4 + 4 :==: 17*x^2 , 16*x^4 + 225 :==: 136*x^2 , 2*x^4 - 15*x^2 + 25 :==: 0 , 9*x^4 - 28*x^2 + 3 :==: 0 , 3*x^4 - 14*x^2 - 5 :==: 0 , 2*x^4 :==: x^2 + 3 , 9*x^4 + 14*x^2 :==: 8 , 4*x^4 - 29*x^2 - 24 :==: 0 -- level 2 , 8*x^6 - 9*x^3 + 1 :==: 0 , 27*x^6 + 8 :==: 217*x^3 , 2*x^6 + x^3 - 1 :==: 0 , 8*x^6 + 31*x^3 :==: 4 , 3*x^6 - 80*x^3 - 27 :==: 0 , 5*x^6 :==: 39*x^3 + 8 , 7*x^6 + 8*x^3 + 1 :==: 0 , 4*x^6 + 2 :==: -9*x^3 ] ++ level Difficult [ x^3 + x^2 :==: 0 , x^3 - 5*x :==: 0 , x^3 - 11*x^2 + 18*x :==: 0 , x^3 + 36*x :==: 13*x^2 , x^3 + 2*x^2 :==: 24*x , 7*x^3 :==: 8*x^2 , x^4 :==: 9*x^2 , 64*x^7 :==: x^5 , x^3 - 4*x^2 - 9*x :==: 0 , (x-1)*(x^3 - 6*x) :==: 3*x^3 - 3*x^2 ] -------------------------------------------------------------------- -- Havo applets -- Havo B Voorkennis: lineaire ongelijkheden ineqLin1 :: [[Inequality Expr]] ineqLin1 = let a = Var "a" in [ [ 7*x - 12 :<: 5*x + 3 , 4*(x-3) :>: 3*(x-4) , 6*(a+1) :<: 3*(a-2)+4 , 5 - 2*(a-3) :>: 5*(3-a) ] , [ 4*x+5 :<: 5*x - 3 , (1/3)*x+10 :>: (1/2)*x , 3*x+1 :<: 7*x + 5 , x+6 :>: 2 - (3/4)*x ] , [ 5*(x-1) :<: 7*x - 1 , -3*(4*x-1) :>: 2-(x-1) , 2*(3*x-1) :<: 5-(2-9*x) , 2*(x-1)-3*(x-2) :>: 6 ] ] -- Havo B Voorkennis: kwadratische ongelijkheden -- (door eerst gelijkheid op te lossen) -- (level 2 uit Hoofdstuk 3) ineqQuad1 :: [[Inequality Expr]] ineqQuad1 = [ [ x^2 +3*x-4 :<: 0 , x^2-4*x-12 :>: 0 , -x^2 - 4*x + 5 :<: 0 , -x^2 + 3*x + 18 :>: 0 , (1/2)*x^2 - 3*x - 8 :<: 0 , -2*x^2 + 10*x :>: 0 ] , [ x^2 + 9*x :<: 3*x - 5 , x^2 - x :>: 12 , x^2 - 4.5*x :<: 7-3*x , 2*x^2 - 10*x :>: x^2 - 3*x , 4*x^2 + 6*x :<: x^2 + 3*x + 18 , 2*x^2 + 6*x - 10 :>: x^2 + 2*x - 5 ] ] -- Havo B hoofdstuk 3, hogeregraadsongelijkheid exact -- (door eerst gelijkheid op te lossen) ineqHigh :: [Inequality Expr] ineqHigh = [ 2*x^3 :>: 54 , -0.5*x^4 :<: -40.5 , 1 - 2*x^5 :<: -485 , (2*x-3)^4 :>: 1 , -(0.5*x+2)^3 :<: -1 , 0.25*(0.5*x-2)^4 :<: 4 ] -------------------------------------------------------------------- -- VWO A/C applets -- hoofdstuk 2 ineqQuad2 :: [Inequality Expr] ineqQuad2 = [ x^2 + 9*x :<: 3*x - 5 , x^2 - x :>: 12 , x^2 - 4.5*x :<: 7 - 3*x , 2*x^2 - 10*x :>: x^2 - 3*x , 4*x^2 + 6*x :<: x^2 + 3*x + 18 , 2*x^2 + 6*x - 10 :>: x^2 +2*x - 5 ] -------------------------------------------------------------------- -- Extra test cases extraIneqQuad :: [Inequality Expr] extraIneqQuad = [ x^2-x-7 :>: -100, x^2-x-7 :<: -100, x^2 :<: x^2, x :>=: x , x^2 :>=: 0, x^2 :>: 0, x^2 :<: 0, x^2 :<=: 0 ]