Nekonecne rady s programem Maple

Vypocet limity posloupnosti funkci, animovani posloupnosti funkci

Karel Srot, xsrot@math.muni.cz

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Zapisnik ilustruje vypocet limity posloupnosti funkci v programu Maple.

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Priklad 1: Znazornete prvních n  clenu posloupnosti funkcí { }, x=0..1 a urcete jeji limitu.

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restart:

>	f:=(n,x)->x^n;

>	plots[animate](f(n,x),x=0..1, n=0..19, frames=20, thickness=2);

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vypocet limity bez omezeni na interval

>	limit(f(n,x),n=infinity);

omezime se na interval [0,1]

>	assume(x::RealRange(0,1));

>	limit(f(n,x),n=infinity);

odpoved presto neni spravna, protoze pro x=1  je limita rovna jedne.

>	limit(f(n,1),n=infinity);

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Priklad 2: Znazornete prvních n  clenu posloupnosti funkcí {arctan( nx )} a urcete jeji limitu.

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restart:

>	f:=(n,x)->arctan(n*x);

>	plots[animate](f(n,x),x=-5..5, n=0..39, frames=20, thickness=2, scaling=constrained, numpoints=200);

>	limit(f(n,x),n=infinity);

Odpoved Maple je tentokrat spravna, funkce csgn  je funkce signum komplexni prommene. Upresnime-li, ze promenna x  je realna, ziskame odpoved obsahujici funkci signum .

>	assume(x::real);

>	limit(f(n,x),n=infinity);

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Priklad 3:

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restart:

>	f:=(n,x)->exp(-n*x^2);

>	plots[animate](f(n,x),x=-2..2, n=0..19, frames=20, thickness=2, numpoints=200, scaling=constrained);

>	assume(x::real);

>	limit(f(n,x), n=infinity);

>	limit(f(n,0), n=infinity);

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Priklad 4:  Vysetrete konvergenci funkce  na mnozine M=[-1,1]x[-1,1] .

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restart;

>	f:=(n,x,y)->abs(x)^n+abs(y)^n;

>	plots[animate](plot3d,[f(n,x,y),x=-1..1,y=-1..1], n=0..29,frames=30,scaling=constrained,axes=framed,view=0..2);

Z grafu je patrne, ze pro body ve vnitrku mnoziny je limita rovna 0, pro body na hranici rovna 1 s vyjimkou ctyr "rohu", kde je limita rovna dvema.

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Vypocet limity pomoci procedury FindSequenceLimit

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restart:

>	currentdir("c:/devel/maple/phd/export"):

>	read "posl_a_rady_fci_proc.mpl":

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>	infolevel[FindSequenceLimit]:=2:

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limita posloupnosti {x^n} pro x z R

>	FindSequenceLimit(x^n,n,x);

FindSequenceLimit:   Could not find the limit function

totez s omezenim na interval [0,1]

>	FindSequenceLimit(x^n,n,x=RealRange(0,1));

FindSequenceLimit:   Limit function found by Maple   0

FindSequenceLimit:   Derivative of the function    x^(n-1)*n

FindSequenceLimit:   We will explore points   x   =   0, 1

totez s omezenim na interval [0,1)

>	FindSequenceLimit(x^n,n,x=RealRange(0,Open(1)));

FindSequenceLimit:   Limit function found by Maple   0

FindSequenceLimit:   Derivative of the function    x^(n-1)*n

FindSequenceLimit:   We will explore points   x   =   0

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Dalsi priklady:

>	FindSequenceLimit(exp(-n*x^2),n,x);

FindSequenceLimit:   Limit function found by Maple   0

FindSequenceLimit:   Derivative of the function    -2*n*x*exp(-n*x^2)

FindSequenceLimit:   We will explore points   x   =   -infinity, 0, infinity

>	FindSequenceLimit(piecewise(x<0,1/n,1/(2*n)),n,x);

FindSequenceLimit:   Limit function found by Maple   0

FindSequenceLimit:   Derivative of the function    0

FindSequenceLimit:   We will explore points   x   =   -infinity, 0, infinity

>	FindSequenceLimit(arctan(n*x),n,x);

FindSequenceLimit:   Limit function found by Maple   1/2*signum(x)*Pi

FindSequenceLimit:   Derivative of the function    n/(1+n^2*x^2)

FindSequenceLimit:   We will explore points   x   =   -infinity, infinity

>	FindSequenceLimit(piecewise(x<0,1/n,x<=1, x^n, x<2, x, 1/(2*n)),n,x);

FindSequenceLimit:   Limit function found by Maple   piecewise(x < 1,0,x < 2,x,2 <= x,0)

FindSequenceLimit:   Derivative of the function    piecewise(x < 0,0,x < 1,x^(n-1)*n,x < 2,1,2 <= x,0)

FindSequenceLimit:   We will explore points   x   =   -infinity, 0, 1, 2, infinity

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>	FindSequenceLimit(x^n-x^(2*n),n,x);

FindSequenceLimit:   Could not find the limit function

>	FindSequenceLimit(x^n-x^(2*n),n,x=RealRange(0,1));

FindSequenceLimit:   Limit function found by Maple   0

FindSequenceLimit:   Derivative of the function    x^(n-1)*n-2*x^(2*n-1)*n

FindSequenceLimit:   We will explore points   x   =   0, 1

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