diff ln Error invalid input diff epects or more arguments but received? diff diff ln Om man vill sätta nytt värde på fariabeln f så använd = (inte = ) f d sin sqrt f = sin f d diff f cos sin ep cos 7 ln abs tan sqrt 8 5$5 7 diff f cos f = e cos 7 8 5 ln tan sin e cos 7 8 5 ln tan () () () (4) (5) K abs tan 8 5 tan 8 5 K K K 8 5 tan 8 5 4 K K K 8 7 6 K e cos 7 ln tan 8 5 sin e K7 cos 6 sin K4 abs tan 8 5 tan 8 5 tan 8 5 8 5 6 ln plot f Warning computation interrupted sin f d ep cos 7 ln abs tan sqrt 8 5$ 5 7 Df d diff f plot f cos Df = e cos 7 ln tan 8 5 K e cos 7 ln tan 8 5 8 5 sin e K7 cos 6 sin FUNKTIONEN - ^4 g d / 4 g f d 4 Df d diff f Dg d diff g Dg d D g Dg 5 Dg sin 8 f = 4 g = / 4 6 Df = 4 (6) (7)
f d / $sin Df d D f Df Dg = 4 Dg = /4 5 4 sin 8 f = / sin (8).4... plot fk.... plot Df =K.... Df = / sin sin Kcos Kcos (9) K K 6 K K. K. K. K.4 6
K K 6 K.5 K.5 6 K I 9 K K I 9 Notera här att = tilldelar ett nytt värde till variabeln eq medan = ger en ekvation. eq d K$ 7$ K8= eq = K 7 K8= fsolve eq solve eq.46 5 858 / 7 K 5 858 / K 5 858 / 6 7 6 5 858 / I 5 858 / 7 5 858 / K 5 858 / 7 6 6 5 858 / K I 5 858 / 7 5 858 / () (4) (5) plot K$ 7$ K8 K Gränsvärden f limit f = Error (in f) numeric eception division by zero limit =? limit limit / =left limit / = right undefined KN N fsolve för numerisk lösning och solve för att lösa "symboliskt" fsolve $ 7= solve $ 7= () () () f K I 9 K K I 9
5 K K5 5 K5 RandomMatri RandomVector Rank RationalanonicalForm ReducedRowEchelonForm Row RowDimension RowOperation RowSpace ScalarMatri ScalarMultiply ScalarVector SchurForm SingularValues SmithForm StronglyonnectedBlocks SubMatri SubVector SumBasis SylvesterMatri SylvesterSolve ToeplitzMatri Trace Transpose TridiagonalForm UnitVector VandermondeMatri VectorAdd VectorAngle VectorMatriMultiply VectorNorm VectorScalarMultiply ZeroMatri ZeroVector Zip? ReducedRowEchelonForm f d v/ f 6 cos v Ksin v sin v cos v K (7) K m m m m 4 m 5 m 6 m 7 m 8 m 9 m m m m m 4 m 5 m 6 m 7 m 8 m 9 m m m m m 4 m 5 m 6 m 7 m 8 m 9 m M d m 4 m 4 m 4 m 4 4 m 4 5 m 4 6 m 4 7 m 4 8 m 4 9 m 4 m 5 m 5 m 5 m 5 4 m 5 5 m 5 6 m 5 7 m 5 8 m 5 9 m 5 m 6 m 6 m 6 m 6 4 m 6 5 m 6 6 m 6 7 m 6 8 m 6 9 m 6 Linjär algebra with LinearAlgebra & Add Adjoint BackwardSubstitute BandMatri Basis BezoutMatri BidiagonalForm BilinearForm ARE haracteristicmatri haracteristicpolynomial olumn olumndimension olumnoperation olumnspace ompanionmatri onditionnumber onstantmatri onstantvector opy reatepermutation rossproduct DARE Deleteolumn DeleteRow Determinant Diagonal DiagonalMatri Dimension Dimensions DotProduct EigenonditionNumbers Eigenvalues Eigenvectors Equal ForwardSubstitute FrobeniusForm GaussianElimination GenerateEquations GenerateMatri Generic GetResultDataType GetResultShape GivensRotationMatri GramSchmidt HankelMatri HermiteForm HermitianTranspose HessenbergForm HilbertMatri HouseholderMatri IdentityMatri IntersectionBasis IsDefinite IsOrthogonal IsSimilar IsUnitary JordanBlockMatri JordanForm KroneckerProduct LA_Main LUDecomposition LeastSquares LinearSolve LyapunovSolve Map Map MatriAdd MatriEponential MatriFunction MatriInverse MatriMatriMultiply MatriNorm MatriPower MatriScalarMultiply MatriVectorMultiply MinimalPolynomial Minor Modular Multiply NoUserValue Norm Normalize NullSpace OuterProductMatri Permanent vot PopovForm QRDecomposition (6) m 7 m 7 m 7 m 7 4 m 7 5 m 7 6 m 7 7 m 7 8 m 7 9 m 7 A B D v d E F G H I J Mv d M.v Mv = m A m B m m 4 Dm 5 E m 6 F m 7 Gm 8 H (8)
A d I m 9 m J m A m B m m 4 Dm 5 E m 6 F m 7 Gm 8 HI m 9 m J m A m B m m 4 Dm 5 E m 6 F m 7 Gm 8 HI m 9 m J m 4 A m 4 B m 4 m 4 4 Dm 4 5 E m 4 6 F m 4 7 Gm 4 8 HI m 4 9 m 4 J m 5 A m 5 B m 5 m 5 4 Dm 5 5 E m 5 6 F m 5 7 Gm 5 8 HI m 5 9 m 5 J m 6 A m 6 B m 6 m 6 4 Dm 6 5 E m 6 6 F m 6 7 Gm 6 8 HI m 6 9 m 6 J m 7 A m 7 B m 7 m 7 4 Dm 7 5 E m 7 6 F m 7 7 Gm 7 8 HI m 7 9 m 7 J Matrismultiplikation a a a a 4 M d AM d A.M MA d M.A a a a a 4 a a a a 4 m m m m m m m m m m 4 m 4 m 4 A = M = a a a a 4 a a a a 4 a a a a 4 m m m m m m m m m m 4 m 4 m 4 AM = a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 a m a m a m a 4 m 4 MA = a m a m a m m a m a m a m a m a m a m a 4 m a 4 m a 4 m a m a m a a m a m a m m a m a m a m a 4 m a 4 m a 4 m a m a m a m a m a m a a m a m a m m a 4 m a 4 m a 4 m 4 a m 4 a m 4 a m 4 a m 4 a m 4 a m 4 a m 4 a m 4 a a 4 m 4 a 4 m 4 a 4 m 4 Kan v skrivas som linjärkombination av kolumnerna i M? Dvs finns lösning till M =v? 8 K 99 65 K8 K9 K98 K74 9 K K6 6 86 4 K5 K77 K4 88 5 K K95 5 K 57 7 K8 K K68 K K6 94 45 7 8 M d K7 8 K67 K5 K48 K8 K9 69 4 7 5 77 K K8 K76 99 9 4 4 76 9 5 K8 K7 9 9 8 6 K44 87 K 44 7 K59 9 4 K5 K6 K 9 45 v d K4 6 K5 9 GaussianElimination M v 8 K 99 65 K8 K9 K98 K74 (9) ()
5 K97 99 8 9 6 K89 6 8 K49 4 54 4 57 8 675 K 8 44 K 9859 K 8875 5775 K 56 66 K747 66 K 798565 856 K4 659 856 4 4 44 75 7 K76675 48 9664 48 K 988 K 9758 648 648 67777 648 K46489986 648 K595476 648 797 648 8467897 854988 84889889 854988 56679674 47994 97595459 548985 554747 47994 K 56478586474 7864 K5564464 7864 K487947 8467897 K 444474 8467897 K 6854854796 576497 K7657575987 594857594 K8967994 759598797 8885594568 5677875449 6796667665 5677875449 GaussianElimination M v 'method'='fractionfree' 8 K 99 65 K8 K9 K98 K74 58 K588 99 5645 496 K89 K498 86 57 48 K696 K5946 K7848 75 468 K7464 K676 K99865 K57665 78 87875 4585 K76675 9664 854988 59687765 K9795555 8499 867894 K548955 8467897 84889889 659486 4777496868 948666474 K594857594 K45686487 K487947 K444474 646895776 7657575987 65787459884 7895465857 496797469 ReducedRowEchelonForm M v ()? LinearSolve LinearSolve M v 5455457788589 76577848556 4579964548 7895465857 K 5688479987579 7895465857 8455784894 65946789756 47867459 76577848556 K 66969567959 455546975 988877875745 8885594568 957658645 7895465857 6796667665 8885594568 5455457788589 76577848556 4579964548 7895465857 K 5688479987579 7895465857 8455784894 65946789756 47867459 76577848556 K 66969567959 455546975 988877875745 8885594568 957658645 7895465857 () () 6796667665 8885594568 LinearSolve klarar även fria variabler
M d K6 95 K6 5 K8 K K44 K8 6 K K9 9 K6 6 9 K78 K44 K K6 K 88 K6 K4 K8 6 v d K88 99 K59 ReducedRowEchelonForm M LinearSolve M v M = K6 95 K6 5 K8 K K44 K8 6 K K9 9 K6 6 9 K78 K44 K K6 K 88 K6 K4 K8 6 v = 797466 865 K 57687 47665 K 756 47665 488 865 K88 99 K59 K 96 865 5595 865 748 865 K 744 865 4657 865 8894 47665 9997 47665 495556 47665 5768865 755 K 4988 755 _t 54875 755 _t K 67 755 _t 6 _t _t K 54779 546 _t 6458 755 _t 566 755 K 75 546 _t 6 9997 755 _t K 8894 755 _t K 47 546 96 755 _t 6 _t 6 756 755 _t K 57687 755 _t K 85 546 48858 755 _t 6 En lösning med fria variabler i R^ kan även tolkas geometriskt som visat på lektion och för fallet två fria variabler kan man göra samma t e på följande sätt i Maple. Vi kan börja med kommandot restart för att få mmapåle att glömma alla tidigare räkningar (används här enbart för att få samam namn på fria variablerna vid varje ny körning). Sedan lösning av ekvationen M=v restart with LinearAlgebra M d 4 6 6 9 v d 7 4 d LinearSolve M v = 7 K _t K _t _t _t (4) (5) (4) Välj en punkt p på planet och två vektorer som det är parallellt med p d subs _t =_t = v d subs _t =_t = Kp v d subs _t =_t = Kp with Student LinearAlgebra PlanePlot v v p
7 K9 K7 5 K6 p = M d K55 5 K94 K69 7 K6 K97 69 v = v = K K K5 K86 K8 K5 MatriInverse M 44859 775 M = K9 K7 5 K6 K55 5 K94 K69 7 K6 K97 69 K5 K86 K8 K5 85 555 74 687555 K 499 775 K 5786 4587 657 775 667 85674 K 79 985 K 647 4587 K 467 555 K 5864 687555 K 498 775 (6) K 758 985 K 96 775 K 547 985 474 687555 Graph of a plane and related vectors. Included on the graph the plane (leafgreen) a normal to the plane (purple) two basis vectors for the plane (navy) a vector (burgundy) from the origin to a specified point on the plane