Gráficos e Aproximações Polinomiais

Vista previa del material en texto

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restart: with(plots); 
animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d,
conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot,
display, dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d,
inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d,
listdensityplot, listplot, listplot3d, loglogplot, logplot, matrixplot, multiple, odeplot, pareto,
plotcompare, pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d,
polyhedra_supported, polyhedraplot, rootlocus, semilogplot, setcolors, setoptions,
setoptions3d, spacecurve, sparsematrixplot, surfdata, textplot, textplot3d, tubeplot
f := x -> sqrt(1-x^2)+sin(3*x);
f := x/ 1K x2 C sin 3 x
plot(f(x),x=-1..1);
x
K1 K0.5 0 0.5 1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
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n:=7;
n := 7
AproxPot[n]:= taylor(f(x),x=0,n);
AproxPot7 := 1C 3 xK
1
2
 x2 K
9
2
 x3 K
1
8
 x4 C
81
40
 x5 K
1
16
 x6 CO x7
PlotFunc[n]:=convert(AproxPot[n],polynom);
PlotFunc7 := 1C 3 xK
1
2
 x2 K
9
2
 x3 K
1
8
 x4 C
81
40
 x5 K
1
16
 x6
plot([f(x),PlotFunc[n]],x=-1..1);
x
K1 K0.5 0 0.5 1
0.5
1
1.5
Int(x^l*x^m,x=-1..1)=int(x^l*x^m,x=-1..1);
K1
1
xl xm dx =
1C K1 lC m
lCmC 1
restart: with(plots): with(orthopoly);
G, H, L, P, T, U
P(0,x);P(1,x);P(2,x);P(3,x);P(4,x);P(5,x);
1
x
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K
1
2
C
3
2
 x2
5
2
 x3 K
3
2
 x
3
8
C
35
8
 x4 K
15
4
 x2
63
8
 x5 K
35
4
 x3 C
15
8
 x
int(P(n,x)*P(m,x),x=-1..1);
K1
1
P n, x P m, x dx
int(P(3,x)*P(4,x),x=-1..1);
0
int(P(3,x)*P(3,x),x=-1..1);
2
7
f := x -> sqrt(1-x^2)+sin(3*x);
f := x/ 1K x2 C sin 3 x
Int(f(x)*P(0,x),x=-1..1)/Int(P(0,x)*P(0,x),x=-1..1)=int(f(x)*P(0,
x),x=-1..1)/int(P(0,x)*P(0,x),x=-1..1);
K1
1
1K x2 C sin 3 x dx
K1
1
1 dx
=
1
4
 p
Int(f(x)*P(1,x),x=-1..1)/Int(P(1,x)*P(1,x),x=-1..1)=int(f(x)*P(1,
x),x=-1..1)/int(P(1,x)*P(1,x),x=-1..1);
K1
1
1K x2 C sin 3 x x dx
K1
1
x2 dx
= Kcos 3 C
1
3
 sin 3
Int(f(x)*P(2,x),x=-1..1)/Int(P(2,x)*P(2,x),x=-1..1)=int(f(x)*P(2,
x),x=-1..1)/int(P(2,x)*P(2,x),x=-1..1);
K1
1
1K x2 C sin 3 x K
1
2
C
3
2
 x2 dx
K1
1
K
1
2
C
3
2
 x2
2
dx
= K
5
32
 p
Int(f(x)*P(3,x),x=-1..1)/Int(P(3,x)*P(3,x),x=-1..1)=int(f(x)*P(3,
x),x=-1..1)/int(P(3,x)*P(3,x),x=-1..1);
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(8)(8)
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(16)(16)K1
1
1K x2 C sin 3 x 
5
2
 x3 K
3
2
 x dx
K1
1
5
2
 x3 K
3
2
 x
2
dx
=
14
9
 cos 3 C
91
27
 sin 3
Int(f(x)*P(4,x),x=-1..1)/Int(P(4,x)*P(4,x),x=-1..1)=int(f(x)*P(4,
x),x=-1..1)/int(P(4,x)*P(4,x),x=-1..1);
K1
1
1K x2 C sin 3 x 
3
8
C
35
8
 x4 K
15
4
 x2 dx
K1
1
3
8
C
35
8
 x4 K
15
4
 x2
2
dx
= K
9
256
 p
CoefLeg[k]:= int(f(x)*P(k,x),x=-1..1)/int(P(k,x)*P(k,x),x=-1..1);
CoefLegk :=
K1
1
1K x2 C sin 3 x P k, x dx
K1
1
P k, x 2 dx
n:=7;AproxLeg[n]:= sum(CoefLeg[k]*P(k,x),k=0..n);
n := 7
AproxLeg7 :=
1
4
 pC
3
2
 K
2
3
 cos 3 C
2
9
 sin 3 xK
5
32
 p K
1
2
C
3
2
 x2
C
7
2
 
4
9
 cos 3 C
26
27
 sin 3 
5
2
 x3 K
3
2
 x K
9
256
 p 
3
8
C
35
8
 x4 K
15
4
 x2
C
11
2
 K
2
3
 cos 3 K
40
9
 sin 3 
63
8
 x5 K
35
4
 x3 C
15
8
 x K
65
4096
 p K
5
16
C
231
16
 x6 K
315
16
 x4 C
105
16
 x2 C
15
2
 
674
81
 cos 3 C
14182
243
 sin 3 
429
16
 x7
K
693
16
 x5 C
315
16
 x3 K
35
16
 x
simplify(AproxLeg[n]);
20965
65536
 pK
131131
48
 cos 3 x5 K
2757755
144
 sin 3 x5 K
15015
65536
 p x6 C
240955
144
 cos 3 x7
C
5070065
432
 sin 3 x7 K
63385
432
 x cos 3 K
1306445
1296
 x sin 3 K
13545
65536
 p x2
C
182105
144
 cos 3 x3 C
3818815
432
 sin 3 x3 C
10395
65536
 p x4
plot([f(x),AproxLeg[n]],x=-1..1);
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x
K1 K0.5 0 0.5 1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8