Matematika 1 - 8. vaja - University of...
Transcript of Matematika 1 - 8. vaja - University of...
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Matematika 18. vaja
B. Jurcic Zlobec1
1Univerza v Ljubljani,Fakulteta za Elektrotehniko
1000 Ljubljana, Trzaska 25, Slovenija
Matematika FE, Ljubljana, Slovenija 3. januar 2013
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Parametricna oblikax = x(t), y = y(t), t ∈ [t1, t2] ⊂ R.
1. Funkciji x = x(t) in y = y(t) sta zvezni in odvedljividefinirani na [t1, t2].
2. Smerni koeficient tangentedydx
=y(t)x(t)
.
3. Ploscina zanke S =12
∫ t2
t1(x(t)y(t)− y(t)x(t))dt .
4. Dolzina loka s =
∫ t2
t1
√x(t)2 + y(t)2dt .
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Eksplicitna oblika y = f (x), x ∈ [x1, x2] ⊂ Df ⊂ R.
I Kot poseben primer parametricne oblike x = x , y = f (x).
I Smerni koeficient tangentedydx
= f ′(x).
I Ploscina zanke, ki jo doloca krivulja in os x
S =
∫ x2
x1
f (x)dx .
I Dolzina loka s =
∫ x2
x1
√1 + f ′(x)2dx .
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Polarna oblika r = r(ϕ), ϕ ∈ [ϕ1, ϕ2] ⊂ R
.I Kot poseben primer parametricne oblike
x = r(ϕ) cosϕ, y = r(ϕ) sinϕ.I Smerni koeficient tangente
dydx
=r(ϕ) cosϕ+ r ′(ϕ) sinϕ−r(ϕ) sinϕ+ r ′(ϕ) cosϕ
.
I Ploscina obmocja, ki jo doloca krivulja in kota ϕ1, ϕ2 je
S =12
∫ ϕ2
ϕ1
r(ϕ)2 dϕ.
I Dolzina loka med kotoma ϕ1 in ϕ2 je
s =
∫ ϕ2
ϕ1
√r(ϕ)2 + r ′(ϕ)2dϕ.
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Implicitna oblika f (x , y) = 0.
I Smerni koeficient tangente. Odvajamo implicitno.ddx
f (x , y(x)) = 0 in izrazimo y ′(x).
I Primer x2 + xy + y2
x = 0,→I 2x + y + xy ′ + 2yy ′x + y2 = 0,→I y ′ = −2x + y + y2
x + 2xyI Vecinoma bomo imeli opravka s primeri, ko z uvedbo
polarnih koordinat, implicitno obliko prevedemo na polarno.
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Graf srcnice (x2 + y2 − x)2 = (x2 + y2)
Vpeljemo polarne koordinate: r2 = x2 + y2,x = r cosϕ in dobimo r = 1 + cosϕ.PolarPlot[1+Cos[t],{t,0,2Pi},PlotStyle->Thick]
0.5 1.0 1.5 2.0
-1.0
-0.5
0.5
1.0
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Graf lemniskate (x2 + y2)2 = (x2 − y2)
Vpeljemo polarne koordinate in dobimor =
√cos(2ϕ), −π
4 ≤ ϕ ≤π4 ,
PolarPlot[Sqrt[Cos[2t]],{t,-Pi/4,Pi/4},PlotStyle->Thick]
-1.0 -0.5 0.5 1.0
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Graf cikloide x(t) = t − sin t , y(t) = 1− cos t .
ParametricPlot[t-Sin[t],1-Cos[t],{t,0,2Pi},PlotStyle->Thick]
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0.5
1.0
1.5
2.0
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Graf asteroide x(t) = cost , y(t) = sin3 t .
ParametricPlot[Cos[t]ˆ3,Sin[t]ˆ3,{t,0,2Pi},PlotStyle->Thick]
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Narisi graf krivulje y2 = x(1− x)2.
I Krivulja je sestavljena iz dveh delov.I y = ±
√x |1− x |, x > 0.
Plot[Sqrt[x]*Abs[1-x],-Sqrt[x]*Abs[1-x],x,0,1.2];
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Graf
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Narisi graf krivulje y2 =x
2− x(1− x)2.
I Krivulja je sestavljena iz dveh delov.
I y = ±√
x2−x |1− x |, x > 0.
Plot[Sqrt[x/(2-x)]*Abs[1-x],-Sqrt[x/(2-x)]*Abs[1-x],x,0,2];
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Graf
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-1.0
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0.5
1.0
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Izracunaj ploscino zanke y2 = x(1− x)2.
I S = 2∫ 1
0
√x(1− x)dx →.
I 22
15x3/2(−5 + 3x)
∣∣∣∣10.
I S =4
15.
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Izracunaj ploscino pod lokom cikloidex = t − sin t , y = 1− cos t .
I S =
∫ 2π
0y(t)x(t)dt ,→
I
∫ 2π
0(1− cos t)2dt ,→
I3t2− 2 sin t +
14
sin(2t)∣∣∣∣2π0
.
I S = 3π.
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Izracunaj ploscino srcnice r(ϕ) = 1 + cosϕ.
I S =12
∫ 2π
0(1 + cosϕ)2dϕ,→
I
∫ 2π
0(1 + 2 cosϕ+ cos2 ϕ)dϕ,→
I3t2
+ 2 sin t +14
sin(2t)∣∣∣∣2π0
.
I S = 3π.
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Izracunaj ploscino lemniskate (x2 + y2)2 = x2 − y2.
I Polarna oblika r(ϕ) =√
cos(2ϕ).
I 2∫ π/4
−π/4cos(2ϕ)dϕ.
I S = 2.
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Izracunaj ploscino asteroide x2/3 − y2/3 = 1.
I Parametricna oblika x(t) = cos3 t , y(t) = sin3 t .
I S =12
∫ 2π
0(x(t)y(t)− y(t)x(t))dt →
I 3∫ 2π
0sin2 t cos2 t dt → 3
4
∫ 2π
0sin2(2t)dt →
38
∫ 2π
0(1− cos(4x)dx
I S = 3t8− 1
32sin(4t)
∣∣∣∣2π0
, S =3π4
.
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Izracunaj dolzino enega loka cikloidex(t) = t − sin t , y(t) = 1− cos t , t ∈ [0,2π].
I s =
∫ 2π
0
√x2(t) + y2(t)dt ,→
I
∫ 2π
0
√2− 2 cos tdt ,→
I
∫ 2π
02| sin(t/2)|dt = 4 cos(t/2)
∣∣∣2π0
.
I s = 8.
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Izracunaj dolzino krivulje r(ϕ) = 1cosϕ , ϕ ∈ [0, π4 ].
I x(ϕ) = r(ϕ) cosϕ = 1,I y(ϕ) = r(ϕ) sinϕ = tanϕ.
I S =
∫ π/4
0(1 + tan2 ϕ)dϕ = tanϕ
∣∣∣π/4
0→ S = 1.
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Prostornina in povrsina vrtenine, parametricna oblika.
Vrtenina je dobljena z vrtenjem krivulje y = f (x) okoli osi x zax1 ≤ x ≤ x2.
I Prostornina
V = π
∫ x2
x1
f (x)2dx
I Povrsina
P = 2π∫ x2
x1
f (x)√
1 + f ′(x)2 dx
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Prostornina in povrsina vrtenine, parametricna oblika.
Vrtenina je dobljena z vrtenjem krivulje x = x(t), y = y(t) okoliosi x za t1 ≤ t ≤ t2.
I Prostornina
V = π
∫ t2
t1y(t)2x(t)dt
I Povrsina
P = 2π∫ t2
t1y(t)
√x(t)2 + y(t)2 dt