----------------------------------------------------------------------------- -- 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.Power.Equation.Examples ( powerEquations, expEquations, logEquations, higherPowerEquations , rootEquations, rootEquations2, rootSubstEquations, expEquations2 ) where import Domain.Math.Data.Relation import Domain.Math.Expr import Prelude hiding ((^)) ---------------------------------------------------------- -- HAVO B applets -- Hoofdstuk 7, vergelijkingen met machten algebraisch (6) powerEquations :: [[Equation Expr]] powerEquations = -- los vergelijkingen algebraisch op let x = Var "x" in [ [ x^14 :==: 25 , x^(-7) :==: 110 , 2*x^3.5 :==: 70 , 8*x^(-9.2) :==: 1000 ] , [ root x 5 :==: 2.9 , 5 * root x 3 :==: 7 , root (x^3) 4 :==: 720 , root (x^2) 5 :==: 5.5 ] , [ 4*x^(-12) :==: 28 , 7*x^5.1 + 16 :==: 100 , 8*x^(-1.9) - 5 :==: 2 , 0.8 * x^0.7 + 7 :==: 12.5 ] , [ 4*root x 7 + 7 :==: 11.8 , 9*x^3.2+17 :==: 37 , 6*x^(-3.1)-9 :==: 12 , 0.7 * x^(-1.1) + 17 :==: 40 ] ] -- Hoofdstuk 7, exponentiele vergelijkingen algebraisch (7) expEquations :: [[Equation Expr]] expEquations = -- los exponentiele vergelijkingen algebraisch op let x = Var "x" in [ [ 2^x :==: 16 * sqrt 2 , 2^(x+2) :==: 1/4 , 3^(x-1) :==: 81 , 3^(x+5) :==: 243/sqrt 3 ] , [ 5^(2-x) :==: 0.04 , 3^(2*x) :==: 1/9 , 3^(1-3*x) :==: 81 , 3^(3*x-2) :==: 3*sqrt 3 ] , [ 5*2^(x-1) :==: 20*sqrt 2 , 6*5^(2-x) :==: 150 , 2*7^(4*x-1) :==: 98 , 8*3^(5-2*x) :==: 72*sqrt 3 ] , [ 2^x-7 :==: 9 , 4^(3*x)+5 :==: 69 , 7*3^(2*x+1) :==: 189 , 5*2^(1-4*x)+11 :==: 51 ] , [ 5^(x-4) :==: (1/5)^(2*x+1) , 7^(1-2*x) :==: 1 , 4^(2*x-3) :==: 2*sqrt 2 , 2*9^(1-2*x) :==: 6*sqrt 3 ] ] -- Hoofdstuk 7, logaritmische vergelijkingen algebraisch (8) logEquations :: [[Equation Expr]] logEquations = -- los algebraisch op let x = Var "x" in [ [ logBase 2 x :==: 7 , logBase 3 (x-2) :==: 2 , logBase 4 (x-3) :==: 1+(1/2) , logBase 5 ((1/10)*x-3) :==: -1 , logBase x 7 :==: 1 , logBase x 4 :==: -1 , logBase 2 (x^2-1) :==: 3 , logBase (1/3) (1-5*x) :==: -1 ] ] ---------------------------------------------------------- -- VWO A/C applets -- Hoofdstuk 5, hogeremachtswortels (1) higherPowerEquations :: [[Equation Expr]] higherPowerEquations = -- bereken exacte oplossing let x = Var "x" in [ [ 2*x^3+9 :==: 19 , 4*x^5-17 :==: 27 , 3*x^7+8 :==: 62 , 5*x^3-1 :==: 9 , 6-5*x^3 :==: 76 , 11-7*x^5 :==: 53 , 4-(1/5)*x^7 :==: 9 , 18-11*x^7 :==: 62 ] , [ (1/2)*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 ] , [ 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 , (1/5)*x^3+27 :==: 52 , 32*x^3+18 :==: 22 , 4*x^3-8 :==: 100 ] , [ 14-2*x^3 :==: 700 , 4-3*x^5 :==: 100 , 14-11*x^7 :==: 25 , 1-3*x^5 :==: 97 ] -- Geef in twee decimalen nauwkeurig , [ 3*x^5+7 :==: 15 , 0.7 * x^4 - 1.3 :==: 2 , (1/3)*x^7 :==: 720 ] ] -- Hoofdstuk 5, hogeremachtswortels (2) rootEquations :: [[Equation Expr]] rootEquations = -- Bereken exacte oplossing let x = Var "x" in let y = Var "y" in [ [ x^4 :==: 6 , root x 4 :==: 6 , sqrt x :==: 10 , root x 5 :==: 2 ] , [ 3*x^5-1 :==: 20 , 3*root (x-1) 5 - 1 :==: 20 , (1/10)*sqrt x + 2 :==: 12 , (1/5)*x^7+8 :==: 26 ] , [ 3*root x 4+2 :==: 14 , (1/2)*x^8-2 :==: 18 , 5-2*root x 3 :==: 3 ] -- Maak x vrij , [ y :==: x^5 , y :==: 2*x^5+4 , y :==: (1/10)*x^3-6 , y :==: root x 7 , y :==: 2*root x 3+8 , y :==: (1/10)*root x 5-6 ] , [ y :==: 3*root x 7-6 , y :==: (1/4)*x^9-6 , y :==: 8+(1/2)*root x 3 ] ] ---------------------------------------------------------- -- VWO B applets -- Hoofdstuk 1, wortelvergelijkingen rootEquations2 :: [[Equation Expr]] rootEquations2 = let x = Var "x" in -- los algebraisch op [ [ 5-2*sqrt x :==: 1 , 7-3*sqrt x :==: 5 , 4-2*sqrt x :==: -3 , 6-3*sqrt x :==: 2 ] , [ 2*sqrt x :==: x , 2*sqrt x :==: 3*x , x-3*sqrt x :==: 0 , 3*x-5*sqrt x :==: 0 ] , [ x :==: sqrt (2*x+3) , x :==: sqrt (3*x+10) , x :==: sqrt (4*x+21) , x :==: sqrt (3*x+4) ] , [ 5*x :==: sqrt (50*x+75) , 2*x :==: sqrt (24*x+28) , 3*x :==: sqrt (27*x-18) , 2*x :==: sqrt (28*x-40) , 3*x :==: sqrt (3*x+42) , 5*x :==: sqrt (49*x+2) , 3*x :==: sqrt (10*x-1) , 5*x :==: sqrt (30*x-5) ] , [ x-sqrt x :==: 6 , x-4*sqrt x :==: 12 , x-sqrt x :==: 12 , x-sqrt x :==: 2 , 2*x+sqrt x :==: 3 , 3*x+4*sqrt x :==: 20 , 2*x+sqrt x :==: 15 , 2*x-3*sqrt x :==: 27 ] ] -- Hoofdstuk 1, wortelvergelijkingen rootSubstEquations :: [[Equation Expr]] rootSubstEquations = let x = Var "x" in -- los algebraisch op [ [ 8*x^3+1 :==: 9*x*sqrt x , 27*x^3 :==: 28*x*sqrt x-1 , x^3+3 :==: 4*x*sqrt x , x^3 :==: 10*x*sqrt x-16 ] , [ x^3 :==: 6*x*sqrt x+16 , x^3-24*x*sqrt x :==: 81 , x^3+x*sqrt x :==: 20 , x^3-15 :==: 2*x*sqrt x ] , [ x^5+32 :==: 33*x^2*sqrt x , 243*x^5-244*x^2*sqrt x+1 :==: 0 , 32*x^5+31*x^2*sqrt x :==: 1 , x^5 :==: 242*x^2*sqrt x+243 ] , [ x^5+8 :==: 6*x^2*sqrt x , x^5 :==: 9*x^2*sqrt x-18 , x^5 :==: 5*x^2*sqrt x+24 , x^5+4*x^2*sqrt x :==:12 ] ] -- Hoofdstuk 5, exponentiele vergelijkingen exact oplossen (1, 2, 2a) expEquations2 :: [[Equation Expr]] expEquations2 = let x = Var "x" in -- los algebraisch op -- 1 [ [ 2^(2*x-1) :==: 1/16 , 3^(1-x) :==: 81 , 5^(1-2*x) :==: 1/5 , (1/2)^(4*x-3) :==: 1/4 , (1/3)^(5*x+2) :==: 1/3 , 6^(3*x-2) :==: 1/216 ] , [ 2^(3*x+2) :==: 2*sqrt 2 , 3^(2*x+1) :==: 9*sqrt 3 , 5^(4*x+3) :==: 625*sqrt 5 , (1/2)^(x+1) :==: 4 , (1/3)^(x-3) :==: 3 , 4^(x+2) :==: 64*root 4 3 ] , [ 2^(x+3) :==: (1/2)*root 2 3 , 3^(4*x+1) :==: 27 , 5^(-x+2) :==: 1/25 , (1/2)^(1-x) :==: sqrt 2 , (1/3)^(x+1) :==: (1/9)*sqrt 3 , 2^(1-3*x) :==: (1/8)*sqrt 2 ] , [ 3*2^x+1 :==: 25 , 4*3^x-9 :==: 27 , 2*5^x+4 :==: 14 , 5*(1/2)^x+11 :==: 51 , 8*(1/3)^x+27 :==: 99 , 3*(1/5)^x-35 :==: 40 ] , [ 2^(4*x+3) :==: 1 , (1/2)^(2*x-1) :==: 1 , 3^(2*x+4) :==: 1 , (1/3)^(x-3) :==: 1 , 4^(4*x-7) :==: 1 , 5^(3*x-6) :==: 1 ] -- 2 , [ 2^(2*x+1) :==: (1/2)^(x+2) , 4^(2*x-1) :==: 2^(3*x+2) , 2^(5*x-4) :==: 8^(x-3) , (1/4)^(2*x+1) :==: 2^(6-2*x) , (1/3)^(2*x-3) :==: 3^(4*x-3) , 3^(3*x-2) :==: 9^(2-x) , 27^(2*x+1) :==: 3^(2*x-5) , 3^(5*x-1) :==: (1/9)^(2*x-1) ] , [ 6^(7*x-3) :==: 36^(2*x+3) , (1/7)^(2*x-1) :==: 7^(2*x-7) , 5^(5-2*x) :==: (1/5)^(x+2) , 25^(4*x+1) :==: 5^(5*x-4) , 3^(x^2) :==: (1/3)^(2*x) , (1/2)^(x^2) :==: 2^(2*x) , 5^(x^2) :==: 25^(3*x) , 2^(x^2) :==: (1/8)^(-x) ] , [ (1/2)^(2-2*x) :==: 4^(3*x+5) , 8^(x+1) :==: (1/2)^(x+7) , (1/4)^(x+2) :==: 8^(2*x-1) , 8^(2*x-3) :==: 16^(2*x+3) , (1/3)^(x-2) :==: 9^(x+4) , 9^(2*x-1) :==: 27^(2*x-1) , (1/9)^(x+3) :==: 27^(2*x+2) , 27^(3-2*x) :==: (1/3)^(4*x+3) ] , [ 4*2^x :==: 2^(3*x-2) , 2^(5*x-9) :==: (1/8)*2^x , 3^(4*x+6) :==: 27*3^x , (1/9)*3^x :==: 3^(2-3*x) , 3*3^x :==: (1/3)^(2*x+5) , 4^(x+1) :==: 8*2^x , (1/2)*2^x :==: (1/2)^x , 9^(x+2) :==: (1/3)*3^x ] , [ (1/5)*5^(3*x-2) :==: 25^(x+1) , 9*3^(2*x+1) :==: (1/3)^(4*x-3) , 4^(3*x-5) :==: 8*2^(x+2) , (1/2)^(3-2*x) :==: (1/4)*2^(3*x-4) , 2^(x+2)+2^x :==: 40 , 2^(x+4) :==: 3/4+2^(x+2) , 2^(x-2)+2^(x+1) :==: 9 , 2^(x+5)-2^(x+4) :==: 16 ] -- 2a , [ 3^(x+2) :==: 72+3^x , 3^(x-1)+3^(x+1) :==: 10 , 3^(x+3)+3^(x+2) :==: 12 , 3^x-3^(x-1) :==: 54 ] , [ 5^(x+1)+5^x :==: 150 , 5^(x+1) :==: 100+5^x , 5^(x+2)+5^x :==:1+1/25 , 5^(x+1)+5^(x+2) :==: 30 ] , [ 2^(x+4)-2^(x-2) :==: 63*sqrt 2 , 3^(x-1)+3^x :==: 12*sqrt 3 , 5^x-5^(x-1) :==: 4*sqrt 5 , 2^(x+2)+2^(x-3) :==: 66*sqrt 2 ] ]