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Alfred_F

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New attempt...

Alfred_F 325
September 04 2024
0 0

@Scot Gould 

I have tried to learn something from your suggestions. I have attached a new attempt. I would appreciate any critical comments.

p.s....

Alfred_F 325
September 01 2024
0 0

@Alfred_F Forgot attachmentAF_20240901.mw
 

Finden Sie den kleinsten Wert des Ausdrucks mit Logarithmen zur Basis a

sqrt[ 106 + log^2_a cos(a*x) + log_a cos^10(a*x) ] +
sqrt[ 58  + log^2_a sin(a*x) - log_a sin^6(a*x) ] +
sqrt[ 5 + log^2_a tan(a*x) + log_a tan^2(a*x) ]

und alle Paare (a,x), an denen das Minimum angenommen wird.

(Lösung: a=2, x=π/8+k*pi, Minimum=9*sqrt(5))

Lösung durch Substitution der Logarithmenterme und "tan=sin/cos" nach Anwendung binomischer Formel/quadratische Ergänzung.``

 

f(a, x) d √(81 + (LOG(COS(a·x), a) + 5)^2 ) + √(49 + (LOG(SIN(a·x), a) - 3)^2 ) + √(4 + (LOG(TAN(a·x), a) + 1)^2 )

 

         

 

g := sqrt(81+(u+5)^2)+sqrt(49+(v-3)^2)+sqrt(4+(v-u+1)^2)

(81+(u+5)^2)^(1/2)+(49+(v-3)^2)^(1/2)+(4+(v-u+1)^2)^(1/2)

 (1)
> 

"Berechnung der 1. Ableitungen nach u und v. Dies sind die Komponenten des Gradienten."

diff(g, u)

(1/2)*(2*u+10)/(81+(u+5)^2)^(1/2)+(1/2)*(-2*v+2*u-2)/(4+(v-u+1)^2)^(1/2)

 (2)

diff(g, v)

(1/2)*(2*v-6)/(49+(v-3)^2)^(1/2)+(1/2)*(2*v-2*u+2)/(4+(v-u+1)^2)^(1/2)

 (3)
> 

"Im lokalen Extremum sind die 1. Ableitungen =0. `Auflösung` des Systems nach u und v."

(2*u+10)/(2*sqrt(81+(u+5)^2))+(-2*v+2*u-2)/(2*sqrt(4+(v-u+1)^2)) = 0

(1/2)*(2*u+10)/(81+(u+5)^2)^(1/2)+(1/2)*(-2*v+2*u-2)/(4+(v-u+1)^2)^(1/2) = 0

 (4)

(2*v-6)/(2*sqrt(49+(v-3)^2))+(2*v-2*u+2)/(2*sqrt(4+(v-u+1)^2)) = 0

(1/2)*(2*v-6)/(49+(v-3)^2)^(1/2)+(1/2)*(2*v-2*u+2)/(4+(v-u+1)^2)^(1/2) = 0

 (5)

solve({(2*u+10)/(2*sqrt(81+(u+5)^2))+(-2*v+2*u-2)/(2*sqrt(4+(v-u+1)^2)) = 0, (2*v-6)/(2*sqrt(49+(v-3)^2))+(2*v-2*u+2)/(2*sqrt(4+(v-u+1)^2)) = 0}, [u, v])

[[u = -1/2, v = -1/2]]

 (6)

eval(g, [u = -1/2, v = -1/2]) = (1/4)*405^(1/2)*4^(1/2)+(1/4)*245^(1/2)*4^(1/2)+5^(1/2)NULL

simplify((1/4)*sqrt(405)*sqrt(4)+(1/4)*sqrt(245)*sqrt(4)+sqrt(5))

9*5^(1/2)

 (7)

NULL

Zur Übung die Ableitungen gemäß Hessian und Gradient:

 

with(Student[VectorCalculus]); Hessian(Student[VectorCalculus]:-`+`(Student[VectorCalculus]:-`+`(sqrt(Student[VectorCalculus]:-`+`(81, Student[VectorCalculus]:-`+`(u, 5)^2)), sqrt(Student[VectorCalculus]:-`+`(49, Student[VectorCalculus]:-`+`(v, -3)^2))), sqrt(Student[VectorCalculus]:-`+`(4, Student[VectorCalculus]:-`+`(Student[VectorCalculus]:-`+`(v, Student[VectorCalculus]:-`-`(u)), 1)^2))), [u, v], determinant); Gradient(Student[VectorCalculus]:-`+`(Student[VectorCalculus]:-`+`(sqrt(Student[VectorCalculus]:-`+`(81, Student[VectorCalculus]:-`+`(u, 5)^2)), sqrt(Student[VectorCalculus]:-`+`(49, Student[VectorCalculus]:-`+`(v, -3)^2))), sqrt(Student[VectorCalculus]:-`+`(4, Student[VectorCalculus]:-`+`(Student[VectorCalculus]:-`+`(v, Student[VectorCalculus]:-`-`(u)), 1)^2))), [u, v])

Vector(2, {(1) = (1/2)*(2*u+10)/sqrt(81+(u+5)^2)+(1/2)*(-2*v+2*u-2)/sqrt(4+(v-u+1)^2), (2) = (1/2)*(2*v-6)/sqrt(49+(v-3)^2)+(1/2)*(2*v-2*u+2)/sqrt(4+(v-u+1)^2)})

 (8)

simplify((196*(u^2+10*u+106)^(3/2)+3969*(u^2-2*u*v+v^2-2*u+2*v+5)^(3/2)+324*(v^2-6*v+58)^(3/2))/((u^2-2*u*v+v^2-2*u+2*v+5)^(3/2)*(u^2+10*u+106)^(3/2)*(v^2-6*v+58)^(3/2)))

196*((81/2)*(5/2+(1/2)*u^2+(-1-v)*u+(1/2)*v^2+v)*(u^2+(-2*v-2)*u+v^2+2*v+5)^(1/2)+(u^2+10*u+106)^(3/2)+(81/49)*(v^2-6*v+58)^(3/2))/((u^2+(-2*v-2)*u+v^2+2*v+5)^(3/2)*(v^2-6*v+58)^(3/2)*(u^2+10*u+106)^(3/2))

 (9)

rationalize((196*(u^2+10*u+106)^(3/2)+3969*(u^2-2*u*v+v^2-2*u+2*v+5)^(3/2)+324*(v^2-6*v+58)^(3/2))/((u^2-2*u*v+v^2-2*u+2*v+5)^(3/2)*(u^2+10*u+106)^(3/2)*(v^2-6*v+58)^(3/2)))

(196*(u^2+10*u+106)^(3/2)+3969*(u^2-2*u*v+v^2-2*u+2*v+5)^(3/2)+324*(v^2-6*v+58)^(3/2))/((u^2-2*u*v+v^2-2*u+2*v+5)^(3/2)*(u^2+10*u+106)^(3/2)*(v^2-6*v+58)^(3/2))

 (10)

with(LinearAlgebra)

[`&x`, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix, BidiagonalForm, BilinearForm, CARE, CharacteristicMatrix, CharacteristicPolynomial, Column, ColumnDimension, ColumnOperation, ColumnSpace, CompanionMatrix, CompressedSparseForm, ConditionNumber, ConstantMatrix, ConstantVector, Copy, CreatePermutation, CrossProduct, DARE, DeleteColumn, DeleteRow, Determinant, Diagonal, DiagonalMatrix, Dimension, Dimensions, DotProduct, EigenConditionNumbers, Eigenvalues, Eigenvectors, Equal, ForwardSubstitute, FrobeniusForm, FromCompressedSparseForm, FromSplitForm, GaussianElimination, GenerateEquations, GenerateMatrix, Generic, GetResultDataType, GetResultShape, GivensRotationMatrix, GramSchmidt, HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm, HilbertMatrix, HouseholderMatrix, IdentityMatrix, IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, JordanBlockMatrix, JordanForm, KroneckerProduct, LA_Main, LUDecomposition, LeastSquares, LinearSolve, LyapunovSolve, Map, Map2, MatrixAdd, MatrixExponential, MatrixFunction, MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower, MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, Minor, Modular, Multiply, NoUserValue, Norm, Normalize, NullSpace, OuterProductMatrix, Permanent, Pivot, PopovForm, ProjectionMatrix, QRDecomposition, RandomMatrix, RandomVector, Rank, RationalCanonicalForm, ReducedRowEchelonForm, Row, RowDimension, RowOperation, RowSpace, ScalarMatrix, ScalarMultiply, ScalarVector, SchurForm, SingularValues, SmithForm, SplitForm, StronglyConnectedBlocks, SubMatrix, SubVector, SumBasis, SylvesterMatrix, SylvesterSolve, ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd, VectorAngle, VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, ZeroMatrix, ZeroVector, Zip]

 (11)

IsDefinite([`?`])

Error, mscrolltable is not a command in the Typesetting package

  

NULL


 

Download AF_20240901.mw

 

Thank You!...

Alfred_F 325
September 01 2024
0 0

@Scot Gould 

I already know these instructions and couldn't do much with them. My goal is to transform tasks that were previously solved with the help of pen and paper and other programs into the much more powerful Maple. I'm currently trying to do this as a practice exercise with an old exam question that was set for the Abitur in a distant country ;-). Since I don't know a structural overview of Maple, I find it difficult to understand why "packages" are often required before entering a command.

I'm currently trying to check the positive definiteness of the matrix generated using "Hessian". The "IsDefinite" command produces an incomprehensible error message. I'm asking for advice.

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